.

Advunces in ATOMIC A N D MOLECULAR PHYSICS

VOLUME 25

Alexander Dalgarno

ADVANCES I N

ATOMIC AND MOLECULAR PHYSICS

Edited by

Sir David Bates

DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE QUEENS UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

VOLUME 25

ACADEMIC PRESS, INC.

Harcourt Brace Jovanovich, Publishers

Boston San Diego New York Berkeley London Sydney Tokyo Toronto

Copyright &) 1988 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

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United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. 24 28 Oval Road, London NWI 7DX

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 65-18423

ISBN 0- 12-003825-0

PRINTED IN THE UNITED STATES OF AMERICA 8 9 9 0 9 1 9 2 9 8 7 6 5 4 3 2 1

Contents

CONTRIBUTORS

Alexander Dalgarno: Life and Personality

Daz~id R . Bates and George A . Victor

Alexander Dalgarno: Contributions to Atomic and Molecular Physics

Neal Lane

I . Introduction IT. Atomic and Molecular Structure

111. Atomic and Molecular Interactions IV. Atomic and Molecular Collisions V. Summary

References

Alexander Dalgarno: Contributions to Aeronomy

Michael 3. McElroy

I. Introduction

References 11. Selected Contributions

Alexander Dalgarno: Contributions to Astrophysics

David A . Williams

I. Introduction 11. Selected Areas of Contribution

V

... Xlll

1

7

7 8

12 14 18 19

23

23 23 27

29

29 29

vi CONTENTS

111. Dalgarno’s Wider Contributions to Astrophysics References

Electric Dipole Polarizability Measurements

Thomas M . Miller and Benjamin Bederson

I. Introduction 11. Bulk Measurements

111. Atomic Beam Measurements IV. Conclusions

Acknowledgment References

Flow Tube Studies of Ion-Molecule Reactions

Eldon Ferguson

I. Introduction 11. Ion-Molecule Reactions at Thermal Energies

111. Negative Ion Kinetics IV. Vibrational Energy Transfer in Ion-Neutral Collisions V. The 0; + CH, 3 H,COOH+ + H Reaction:

VI. Conclusions A Detailed Mechanistic Study


Differential Scattering in He-He and He+-He Collisions at KeV Energies

R . F . Stebbings

I. Introduction 11. He-He Collisions at Small Angles

111. He+ + He Collisions at Small Angles IV. He-He Scattering at Large Angles V. Conclusion


33 34

37

37 40 42 56 58 58

61

61 63 69 71

76 78 79 79

83

83 84 91 95 98 98 98

CONTENTS vii

Atomic Excitation in Dense Plasmas

Jon C . Weisheit

I . Introduction 11. Characteristics of Dense Plasmas

111. Excitation Models for Dense Plasmas IV. Conclusion

Acknowledgments References

Pressure Broadening and Laser-Induced Spectral Line Shapes

Kenneth M . Sando and Shih-I Chu

I . Atomic Line Shape Theory in the Weak Field Limit 11. Spectral Line Shapes in Strong Fields


Model-Potential Methods

G . Laughlin and G . A . Victor

I . Introduction 11. Development of Model Potentials

111. Applications of Model Potentials IV. Molecular Model Potentials

References

Z-Expansion Methods

M . Cohen

1. Introduction 11. Z - '-Expansion of Schrodinger's Equation

111. The Screening Approximation IV. The Hartree-Fock Approximation V. Some Representative Results V1. Summary and Conclusions

References

101

101 102 115 127 129 129

133

133 146 160 160

163

163 164 173 186 190

195

195 197 204 206 210 219 219

CONTENTS ...

V l l l

Schwinger Variational Methods

Deborah K a y Watson

I. Introduction 11. Early Development

111. Studies by Nuclear Physicists IV. The Schwinger Variational Method in Atomic

and Molecular Physics V. Summary

References

Fine-Structure Transitions in Proton-Ion Collisions

R . H . G . Reid

I . Introduction 11. Semiclassical Calculations

111. Close-Coupled Quanta1 Calculations IV. Summary

References

Electron Impact Excitation

R . J. W. Henry and A . E. Kingston

I. Introduction 11. The Close-Coupling Approximation

111. Convergence of the Close-Coupling Expansion 1V. The Effect of Resonances on Electron Excitation Rates V. Inner Shell Excitation Autoionization

VI. Resonances in Cu References

Recent Advances in the Numerical Calculation of Ionization Amplitudes

Christopher Bot tcher

1. Introduction 11. Formal Solutions of the Stationary Schrodinger Equation IIT. The Boundary Function Method IV. Path Integral and Semiclassical Methods

22 1

22 1 222 228

230 247 247

25 1

25 1 255 26 1 265 265

267

267 268 272 282 289 298 300

303

303 305 308 31 1

CONTENTS ix

V. Calculations on a Two-Dimensional Model VI. Calculations in Three Dimensions


The Numerical Solution of the Equations of Molecular Scattering

A . C. Allison

I. Introduction 11. Numerical Methods

111. Close-Coupled Equations IV. Solution Following Methods V. Potential Following Methods

VI. Adiabatic and Diabatic Representations VII. Propagators

VIII. Summary References

31 5 320 32 1 321

323

323 3 24 327 33 1 334 336 337 338 339

High Energy Charge Transfer 343

I. Introduction 343 11. Transition Amplitudes 345

111. Distorted Wave Series 348 IV. First Order Models 349

B. H . Bransden and D . P . Dewangan

V. The Continuum Distorted Wave, Vainshtein Presnyakov and Sobelman, Glauber and Symmetrical Eikonal Models 357

VI. Second Order Theories 363 VII. Relativistic Electron Capture 369

Acknowledgment 37 1 References 37 1

Relativistic Random-Phase Approximation

W R . Johnson

I . Introduction 11. Derivation of the RRPA Equations

375

375 376

X CONTENTS

111. Reduction to Radial Equations IV. Basis Set Expansion of the Radial RRPA Equations


Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics

G. W F. Drake and S . P . Goldman

I. Introduction 11. Variational Representation of the Dirac Equation

111. Relativistic Sturmian Basis Sets IV. Test Calculations with Relativistic Sturmian Basis Sets and

V. Variational Dirac-Hartree-Fock Calculations VI. Suggestions for Future Work

Comparisons with B-Spline Methods


Dissociation Dynamics of Polyatomic Molecules

T. Uzer

I. Introduction 11. Unimolecular Reaction Rate Theories

111. Semiclassical and Quasiclassical Trajectory Methods IV. Unimolecular Dissociation through State Selection

VI. Case Study: Overtone-Induced Dissociation of

VII. Dissociation through Electronically Excited

V. Overtone-Excited Processes

Hydrogen Peroxide-Experiment and Theory

States-Interface Between Photodissociation and IVR

Molecular Dissociation IX Concluding Remarks


VIII. Unimolecular Quantum Dynamics and

379 386 390 390

393

393 396 402

404 410 414

414

417

417 418 422 424 425

426

428

43 1 43 1 432 433

CONTENTS xi

Photodissociation Processes in Diatomic Molecules of Astrophysical Interest

K a t e P . Kirby and Ewine F. van Dishoeck

I. Introduction 11. Direct Photodissociation

111. Spontaneous Radiative Dissociation IV. Predissociation V, Coupled States Photodissociation

VI. Near-Threshold Photodissociation VII. Concluding Remarks


The Abundances and Excitation of Interstellar Molecules

John H . Black

I. Introduction 11. Molecular Hydrogen

111. Ion-Molecule Chemistry IV. Chemistry of Shock-Heated Gas

VI. The Excitation of Interstellar CN V. The CH' Problem

VII. Models of Interstellar Clouds VIII. Summary


Index Contents of Previous Volumes

437

437 442 453 456 464 469 473 473 473

477

477 479 48 3 495 497 50 1 503 505 505 506

513 56 1

This Page Intentionally Left Blank

Contributors

Numbers in parentheses refer to the pages on which the authors contributions begin.

A. C. Allison (323), Department of Computing Science, University of Glasgow, Glasgow GI2 8QQ, Scotland, United Kingdom

David R. Bates (l), Department of Applied Mathematics and Theoretical Physics, Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland, United Kingdom

Benjamin Bederson (37) Department of Physics, New York University, Wash- ington Square, New York, New York 10003.

John H. Black (477), Steward Observatory, University of Arizona, ncson,

Christopher Bottcher* (303), Physics Division, Argonne National Laboratory,

B. H . Bransden (343), Department of Physics, University of Durham, South

Shih-I Chu (1 33), Department of Chemistry, University of Kansas, Lawrence,

M. Cohen (195), Department of Physical Chemistry, The Hebrew University,

D. P. Dewangent (343), Department of Physics, University of Durham, South

G. W. F. Drake (393), Department of Physics, University of Windsor, Windsor

Arizona 85721

Argonne, Illinois 60439-4843

Road, Durham DHI 3LE, England, United Kingdom

Kansas 66045

Jerusalem 91904, Israel

Road, Durham DHI 3LE, England, United Kingdom

N9B 3P4, Canada

* Permanent address: Physics Division, Oak Ridge National Laboratory, Oak Ridge,

f Permanent address: Physical Research Laboratory, Naurangpura, Ahmedabad 380 009, Tennessee 37831-6373

India ...

X l l l

xiv CONTRIBUTORS

Eldon Ferguson (61), UniversitC de Paris-Sud, Bdtiment 350, Centre d’Orsay,

S . P. Goldman (393), Department of Physics, University of Western Ontario,

R. J. W . Henry (267), Department of Physics and Astronomy, Louisiana State

W . R. Johnson (379, Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556

A. E. Kingston (267), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Bevast BT7 INN, Northern Ireland, United Kingdom

Kate P. Kirby (437), Hurvard-Smithsonian Center for Astrophysics, Cam- bridge, Massachusetts 02138

F-91405 Orsay Cedex, France

London N6A 3K7, Canada

University, Baton Rouge, Lousiana 70803

Neal Lane (7), Ofice of the Chancellor, Rice University, Houston, Texas

C. Laughlin (163), Mathematics Department, University of Nottingham, Nottingham NC7 2RD, England, United Kingdom

Michael B. McElroy (23), Department of Earth and Planetary Sciences and Division of Applied Sciences, Harvard University, Cambridge, Massachu- setts 02138

77251-1892

Thomas M. Miller (37), Department of Physics and Astronomy, University of Oklahoma, 440 West Brooks, Room 131, Norman, Oklahoma 73019

R. H. G. Reid (251), Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland, United Kingdom

Kenneth M. Sando (133), Department of Chemistry, University of Iowa, Iowa City, Iowa 52242

R. F. Stebbings (83), Department of Space Physics and Astronomy and The Rice University Quantum Institute, Rice University, Houston, Texas 77251- 1892

T. Uzer (417), School of Physics, Georgia Institute of Technology, Atlanta,

Ewine F. Van Dishoeck (437), Princeton University Observatory, Princeton,

Georgia 30332-0430

New Jersey 08544

CONTRIBUTORS xv

George A. Victor ( 1 , 163), Center for Astrophysics, Harvard University, 60

Deborah Kay Watson (221), Department of Physics and Astronomy, Univer-

Jon C . Weisheit (101), Physics Department, Lawrence Livermore National

David A. Williams (29), Mathematics Department, UMIST, PO Box 88,

Garden Street, Cambridge, Massachusetts 02138

sity of Oklahoma, Norman, Oklahoma 73019

Laboratory, Livermore, California 94550

Manchester M60 100, England, United Kingdom


ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 25

l l ALEXANDER DALGARNO LIFE AND PERSONALITY D A V I D R. BATES Depurtment of Applied Morhemuiic.s & Theorvtiwl Physics Queen’.p Uniiw-sity of‘ Belfast Belfusi. United Kingdom

GEORGE A . V I C T O R Hurvard-Smithsonian Center for Astrophysics Harvard UniversiQ Cambridge, Massachusetts

His name is more Scottish than the finest malt whisky. Walter Scott saw fit to make a Lord Dalgarno one of the principal Scottish characters in T h e Fortunes of Nigel. The focus of the Dalgarnos is Aberdeen, which has over 150 Dalgarnos to grace its telephone directory. Alex’s roots lie in that city. His paternal grandfather (after whom he was named) owned a mill in Aberdeen and his maternal grandfather was a blacksmith there. As far as is known, the only scholar in his background was his paternal granduncle, Alexander Low, who was a Professor of Anatomy at Aberdeen University and a perceptive phrenologist. After he examined Alex and his twin sister Pamela as babies, he pronounced, “The male will be brilliant.”

Alex’s fat her, William, and mother, Margaret (Murray), were born and married in Aberdeen. William Dalgarno worked for an insurance company. He was transferred to the capital about 1920 and bought a house in Wood Green, a suburb in North London, where Alex and Pamela were born on January 5, 1928.

The twins were the youngest of five children. They have a brother Murray and a sister Margaret. Another brother died before they were born. Alex’s siblings have provided much information about him. The boy and youth they remember will be recognized by all who know the man. He was outstanding at competitive sports-good enough at football to be invited to join, as an amateur, the roster of players from which the famous London club Totten- ham Hotspur made up their teams. He inherited from his father a fine sense of humour and a dry wit. He was attractive to girls. He had amazing powers of concentration and was superbly intelligent. Because of his love of reading he was known as “the Professor” from an early age. He used his bedroom as a study and would often not emerge for meals with the family. Instead, his mother would take a tray up to him. She would also look in on her way to bed and might find him fast asleep with open books all around him.

1

Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

l C U N & L 1 7 . M I P ? C n

2 David R. Bates and George A. Victor

Already comfortably middle class, the Dalgarnos prospered and in 1935-36 moved to a rather more affluent North London suburb, Winchmore Hill. Alex attended a primary school in the adjacent suburb of Palmers Green. It was a poor school with no library and no books except those required for a rigid curriculum. Alex attained the top class at the age of only nine. Since the state Qualifying Examination for entry to grammar schools was intended for pupils who had reached the age of eleven Alex had the frustration of being compelled to repeat the same course three times. Those responsible for his education were completely unresponsive to the plight of the prodigy in their charge.

In September 1939 Alex was visiting his grandparents and on the outbreak of war was enrolled at Robert Gordon’s College in Aberdeen. He was not happy there. After one term he returned to his parents’ home and attended Southgate Grammar School in Palmers Green. This school had some excellent teachers. Despite many hours spent in underground shelters during air raids, Alex enjoyed himself. He found the work undemanding but got satisfaction in playing football and cricket for his school.

William Dalgarno thought that it would be a waste of time and money for his son to proceed to a university, but was persuaded by his wife to allow Alex to go to University College, London (which he could attend without incurring the expense of residence away from home). He enrolled in the Mathematics Department in 1945 with no particular career in mind although he supposed that he would follow one in which he could use his numerical skills. Stimulated by the challenge of mathematics at university level, he began working in earnest and has never stopped since. In 1947 he was rewarded with a First Class Honours degree.

During the following year he took and won Distinction in a course called Advanced Subjects. Interviewing him, the Head of Department, Harrie Massey, asked what he intended to do when he left. When Alex replied that he did not know, financial support was offered to enable him to work for a Ph.D. degree in atomic physics. He accepted although he still had no intention of making research his career (the decision to do so not being reached until 5 or 6 years later). With Richard Buckingham as his supervisor he completed a Ph.D. thesis on Metastable Helium in 1951.

David Bates, who was then also at University College, had recently been appointed to the Chair of Applied Mathematics at the Queen’s University of Belfast. Having an Assistant Lectureship to fill, he sought the advice of Harrie Massey, who told him that Alex was the best research student in an exceptionally good year. Alex took the post that was immediately offered to him. A succession of promotions followed quickly: 1952 Lecturer; 1956 Reader; 1961 Professor.

As well as continuing his investigations on atomic physics, Alex Dalgarno

ALEXANDER DALGARNO-LIFE A N D PERSONALITY 3

developed an interest in aeronomy. In 1955 he and Brian Armstrong helped organize an international conference in Belfast on The Airgtow and the Aurorae and edited the proceedings for publication. He played an important part in negotations that took place with the U.S. Office of Naval Research for a contract that included providing the university with its first computer. He was Director of the Computing Laboratory from 1960 until 1965. He went to MIT to learn to program Whirlwind and to develop methods for evaluating molecular integrals. He commonly visited the United States during the summers to work at NASA, the Air Force Cambridge Research Laboratories, or the Geophysics Corporation of America (of which he was Chief Scientist in

Yet he made time to play tennis, squash, five-a-side indoor football and bridge. Illustrative of the width of the interests of his intimates, the poet Philip Larkin was best man at Alex’s 1957 wedding to Barbara Kane, by whom he had four children, Penelope, Rebecca, Piers and Fergus.

He worked with many good research students at Queen’s University and has remained in touch with most of them to this day. One of them, Michael McElroy, has written, “I met Alex first in 1958. I was a young undergraduate in Belfast. He was my teacher in a class on Classical Mechanics. I came to appreciate over the next several years the power of his intellect. More importantly, I came to value the fact that his door was ever open. He was always available to discuss a problem, whether he had a personal interest in its solution or not. He had an inexhaustible store of interesting research topics and was generous always in sharing his ideas. He was an inspiration, constantly challenging those he interacted with to strive for excellence. It was he who first inspired my interest in the atmosphere. He was invariably patient as I sought to find my feet-always supportive and constructive. I came to know him as a friend, and the friendship has endured a lifetime. I continue to seek his advice and know that these sentiments are shared by many”.

Alex moved to Cambridge, Massachusetts, in 1967 as Professor of Astron- omy at Harvard University and as a Physicist at the Smithsonian Astrophysi- cal Observatory. He was Chairman of the Department of Astronomy 1971-76, Acting Director of Harvard College Observatory 1971-73 and Associate Director of the Center for Astrophysics 1973-80. In 1977 he became Philips Professor of Astronomy. While he was Chairman he insti- tuted reforms to reduce the time needed for graduate students to complete their degrees.

On arriving in Cambridge, Alex extended his own interests in astrophysics proper. Because of his renown, he attracted good research students from the departments of astronomy, physics, and chemistry at Harvard, and from other American and European universities. He proved himself adept at identifying important problems and bringing together teams to solve them.

1962-63).

4 David R . Bates and George A. Victor

He has made the Center for Astrophysics a research Mecca to which scientists return year after year.

Since January 1974 Alex has been the capable and firm Editor of Astrophysical Journal Letters. As senior figure in the astrophysics community and as a founder of the subject of molecular astrophysics, he is often asked to summarize the discussions at scientific meetings. David Williams has written: “At the IAU Symposium 120 on Astrochemistry, Dalgarno (1987) reviewed the present state of the subject and of its future prospects. He reminded the audience that the subject comprised chemistry of the early Universe, of the interstellar medium, of circumstellar and stellar envelopes of comets and meteorites and of planetary atmospheres, and that the unification of these studies presents a challenge which is stimulating research in many branches of physics, chemistry, geology and astronomy. However, the rewards are correspondingly great and will teach us much about the Universe and its evolution. That so much has been achieved and that future propsects are so good is in large measure due to the achievements of Alexander Dalgarno, his leadership in his subject, and the inspiration given so many other astrophysi- cists.”

Having deep concern for the future well-being of atomic physics in the United States, Alex has worked long and hard professing the vitality of the subject in academia and to funding agencies. He has been Chairman of the Division of Atomic, Molecular and Optical Physics of the American Physical Society and has served effectively on the Committee on Atomic and Molecu- lar Sciences of the National Academy of Sciences. He has served on many committees of the American Physical Society and on numerous other scientific advisory committees. He has given generously of his time in writing reports and in travelling far to attend committee meetings. He is much sought after as a lecturer and has been selected to give special lectures at Notre Dame University, Brandeis University, the University of Oklahoma, the University of Georgia, Rice University, and the Indian Association for the Cultivation of Science, Calcutta. But he always gives his research students priority for his time.

In spite of the pressure on him, Alex still plays tennis and squash regularly. He listens to music. He enjoys social activity.

Some of the characteristics of this multisided man have already been indicated in connection with the recollections of his siblings. Before turning to special attributes, his great kindness and consideration at a personal level and his protectiveness towards the weak must be put on record. He is a true friend to many.

Alex is adroit at mental arithmetic and has a phenomenal memory-he can recall a complete citation including the page number, and any of the important results down to the last important decimal place. After he has

ALEXANDER DALGARNO- LIFE AND PERSONALITY 5

reflected thoroughly on the subject matter of a projected scientific paper, he writes the paper so quickly that it would seem that he must be copying from a version completed and stored in his mind. His style is simple and clear with no superfluous words. As his prowess at sports suggests, he has a very fast reaction time. The old DEUCE computer at Queen’s University had a game in which the players’ success depended on their reaction times. Alex would win so easily that it appeared he knew something about the game that nobody else knew.

Naturally, many honours have come Alex’s way. He was elected Fellow of the American Academy of Arts and Sciences 1968, Fellow of the American Geophysical Union 1972, Fellow of the Royal Society 1972, Corresponding Member of the International Academy of Astronautics 1972, Member 1985 Fellow of the American Physical Society 1980. He was made a Fellow of University College, London, 1976 and an Honorary Doctor of Science of Queen’s University of Belfast 1980. He has been awarded the Prize of the International Academy of Quantum Molecular Science 1969, the Hodkins Medal of the Smithsonian Institution 1977, the Davidson-Germer Prize of the American Physical Society 1980, the Meggers Award of the Optical Society of America 1986, and the Gold Medal of the Royal Astronomical Society 1987. There is no greater figure than Alex in the history of atomic physics and its applications.

REFERENCE

Dalgarno, A. (1987). In Astrochemistry (M. S. Vardya and S. P. Tarafda, eds.) D. Reidel, Dordrecht, p 577.


ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 25

l l ALEXANDER DALGARNO CONTRIBUTIONS TO ATOMIC AND MOLECULAR PHYSICS NEAL L A N E Depariment of Physics

Rice University Houston, Texas

and Rice Quanrum Instiiute

I. Introduction . . . . . . . . . . . . . . . 11. Atomic and Molecular Structure . . . . . . . .

A. Perturbation, Variational and Expansion Methods B. Coupled, Time-Dependent, Hartree-Fock Theory . C. Autoionization and Electron Scattering . . . . D. Relativistic Quanta1 Treatments . . . . . . . E. Molecular Properties . . . . . . . . . . .

111. Atomic and Molecular Interactions . . . . . . . A. Long-Range Forces . . . . . . . . . . . B. Model Potentials and Pseudo-Potentials . . . .

IV. Atomic and Molecular Collisions . . . . . . . . A. Near-Resonance Electronic Transitions . . . . B. Excitation and Charge Transfer . . . . . . . C. Radiative Collisions . . . . . . . . . . . D. Rotational Excitation of Molecules . . . . . .

V. Summary . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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I. Introduction

The contributions Alex Dalgarno has made to the field of atomic and molecular physics over a period of more than thirty-five years defy any brief description that this author is capable of rendering. It is correct but inadequate to say that he has advanced in a major way our understanding of basic atomic and molecular phenomena and their applications to astronomy, atmospheric sciences, the physics and chemistry of ionized gases, and other complex systems in nature. He has contributed through his own research, through the research of the many students and postdoctoral researchers whose work he has directed, and through his personal high standards and his enthusiasm for the science of atoms and molecules.

7


8 Neal Lane

If one includes under the classification of atomic and molecular physics the internal structure of atoms, molecules, and ions, and their responses to external fields and to collisions, including the scattering of electrons and photons, then it is difficult to identify an area in which Alex Dalgarno has not made significant contributions. Any effort to select out the most important of these contributions is likely to fail, since each worker in the field probably has a slightly different list. However, most atomic and molecular researchers would agree, perhaps, that all such lists should include the following: atomic collisions, particularly low-energy fine-structure excitation, spin-changing, and charge-transfer processes; rotational excitation of molecules in collisions with electrons, atoms and ions; photoionization and photodissociation, including multistate theory; coupled and time-dependent Hartree-Fock (TDHF) theory; relativistic random phase approximation (RRPA); the double perturbation theory; the Dalgarno-Lewis sum rule and the interchange theorem; the theory of long-range forces; and model potential methods.

In the few pages available here, an effort will be made to briefly describe selected accomplishments of Alex Dalgarno in these areas. The selections, to some extent, reflect the author’s own biases, as well as the impossibility of referring to more than a small fraction of the voluminous published works of Dalgarno and his collaborators.

11. Atomic and Molecular Structure

Alex Dalgarno’s contributions to the understanding of the quantum mechanical structure of atomic and molecular systems tend to cluster around the development of new perturbation and variational methods and new and efficient techniques for applying them to complex systems (Dalgarno and Stewart, 1956a, 1958; Dalgarno, 1959).

A. PERTURBATION, VARIATIONAL AND EXPANSION METHODS

Double perturbation theory, enhanced by the powerful interchange theorem (Dalgarno and Stewart, 1956a, 1958), was shown to be an effective method for including first-order electron correlation effects in calculations of frequency-dependent (dynamic) polarizabilities, long-range forces, and other atomic properties.

Dalgarno and Lewis (1955) developed a powerful sum-rule technique that eliminates the need to carry out actual summations of terms of the type that arise in perturbation theory and that, in principle, permits exact calculation

CONTRIBUTIONS TO ATOMIC AND MOLECULAR PHYSICS 9

of the coefficients of each term in the expansion (see also Dalgarno and Lynn, 1957a; Dalgarno, 1963).

Dalgarno and Lewis (1956b) demonstrated the equivalence of variational and perturbation treatments of small disturbances. Cohen and Dalgarno (1961a, 1961 b) proved that for any operator that can be expressed as the sum of one-electron operators, the expectation value with respect to a Hartree- Fock wavefunction vanishes.

The early development by Dalgarno (1960) of a theory which permits the exact evaluation of the series expansion of Hartree-Fock energies in powers of 1/Z provided the foundation for a powerful expansion method for calculating atomic properties (Chisolm and Dalgarno, 1966). The method was applied to correlation energies, transition probabilities, and other properties of helium, lithium, and other sequences. (See, for example, Onello, Ford, and Dalgarno, 1974, and Dalgarno and Parkinson, 1967.) Many studies of correlation and relativistic effects in atoms, based on increasingly sophisticated methods, followed these earlier investigations.

Chan and Dalgarno (196%) applied a powerful variational method for evaluating infinite summations (Dalgarno 1961a, 1963) to the first calculation of frequency-dependent properties of atomic systems; this method was a precursor to subsequent pseudo-state methods.

B. COUPLED, TIME-DEPENDENT, HARTREE-FOCK THEORY

The coupled, time-dependent Hartree-Fock theory was developed by Dalgarno and Victor (1966b), who showed its equivalence to the random phase approximation (RPA); it is generally referred to simply as the time- dependent Hartree-Fock (TDHF) method. Dalgarno and Victor (1966b) used the TDHF method to calculate accurate values for several atomic properties, including interatomic long-range interactions (see also Dalgarno and Victor, 1967). Applications of TDHF theory to molecules were facilitated by the development of variational techniques (Stewart, Watson, and Dalgarno, 1975, 1976; Watson, Stewart, and Dalgarno, 1976a), and the further simplification obtained by employing pseudopotentials (Watson, Stewart and Dalgarno, 1976b).

c. AUTOIONIZATION A N D ELECTRON SCATTERING

Bransden and Dalgarno (1952, 1953b, 1953c, 1956) developed a time- independent variational approach to the calculation of the energies and lifetimes of autoionization states, and showed, in the earliest calculation using

10 Neal Lane

accurate wavefunctions, that for several doubly-excited states of He and H -, both the time-dependent and time-independent theories give the same results. Later work by Drake and Dalgarno (1970,1971; Dalgarno and Drake, 1971) included the application of a high-order Z-expansion perturbation procedure and an energy maximization method.

Bransden and Dalgarno (1953a) first generalized the variational methods of Hulthen and Kohn to treat electron-ion collisions, and then applied the method to the calculation of the phase shifts and differential cross section for electron-He' scattering. Variational functions of this form were used by Bransden, Dalgarno, and King (1953) in the earliest distorted-waves treatment of 1s -, 2s excitation in electron-He' collisions; thus the inadequacy of the Born-Oppenheimer approximation was shown. Bransden, Dalgarno, John, and Seaton (1958) showed, in an application of the variational approach to electron scattering from neutral hydrogen, that the long-range polarization interaction must be taken into consideration.

D. RELATIVISTIC QUANTAL TREATMENTS

Dalgarno and his collaborators (see, for example, Shorer and Dalgarno, 1980, 1977) have made significant contributions to the theory of relativistic atomic structure through the development (Johnson, Lin, and Dalgarno, 1976; Lin, Johnson, and Dalgarno, 1977) and application of the relativistic random phase approximation (RRPA) and through early accurate variational calculations of energy levels and transition rates. An example of the latter is the study of helium-like ions by Drake and Dalgarno (1969), who showed, for example, that the (spin-forbidden) intercombination rate exceeds rates for allowed transitions for helium-like ions beyond C4+. Goldman and Dalgarno (1 986) have developed a finite-basis-set approach to the Dirac- Hartree-Fock equations.

Drake and Dalgarno (1968) pointed out that in two-photon decay of the 2 3 S , state of helium-like ions,.the proper mechanism is the spin-orbit mixing of singlet and triplet character in the intermediate P state. Accurate rates were calculated by Drake, Victor, and Dalgarno (1969). (The decay of triplet metastable helium is dominated by an M1 transition (see, for example, Drake, 197 l).)

E. MOLECULAR PROPERTIES

In an early series of papers, beginning with Dalgarno and Poots (1954), the one-electron H l system was used as a test bed for evaluating the accuracy of variational methods applied to the determination of molecular orbitals,


which in the one-electron case could be determined exactly (Bates, Ledsham, and Stewart, 1953). It was shown, for example, that the united-atom construction of the excited MO was superior to the separated-atom construction for this system as well as for HeH2+, even for relatively large internuclear separations (Dalgarno, Moiseiwitsch, and Stewart, 1957). Accurate calculations of properties of these systems were also performed using perturbation methods (Dalgarno and Stewart, 1956b, 1957,1960) and exact wavefunctions (Dalgarno, Lynn, and Williams, 1956; Dalgarno and McCarroll, 1957b; Dalgarno, Patterson, and Somerville, 1960).

After the early study of photoionization of methane by Dalgarno (1952), there followed a continuous stream of significant contributions on the optical properties of molecules, including photoionization (PI), photoabsorption (PA), photodissociation (PD), oscillator strengths (OS), radiative lifetimes (RL), dynamic polarizabilities (DP) (which are related to the refractive index), Verdet constants, Rayleigh scattering cross sections, and Rayleigh and Raman depolarization factors. The first calculation of the Rayleigh scattering cross section for H2 over a full range of wavelengths was a semi-empirical study by Dalgarno and Williams (1962); a variational treatment was given by Dalgarno, Ford, and Browne (1971).

The molecular studies include further investigations of H, (Dalgarno and Williams, 1965 (OS, DP); Victor, Browne, and Dalgarno, 1967 (DP); Victor and Dalgarno, 1969 (DP); Allison and Dalgarno 1969b (PD), 1970 (0s); Dalgarno and Stephens, 1970 (PD, PA, RL); Ford, Docken, and Dalgarno, 1975a,b (PD, PI); Dalgarno, Ford, and Browne, 1971 (DP); Kirby, Guberman, and Dalgarno, 1979 (PD, PI); Kwok, Dalgarno, and Posen, 1985 (RL); Kwok, Guberman, Dalgarno, and Posen, 1986 (RL); Kirby, Uzer, Allison, and Dalgarno, 1981 (PI, PD)); Li, (Uzer, Watson, and Dalgarno, 1978 (RL); Uzer and Dalgarno, 1980 (PD)); N, (Dalgarno, Degges, and Williams, 1967 (DP)); 0, (Guberman and Dalgarno, 1979 (PA); Roche, Kirby, Guberman, and Dalgarno, 1981 (PI); Allison, Guberman, and Dalgarno 1982 (PA)); 0; (Wetmore, Fox, and Dalgarno, 1984 (OS, RL)); Lif and Naf (Kirby-Docken, Cerjan, and Dalgarno, 1976 (OS, PD); Uzer and Dalgarno, 1979a (PD), 1979c (PD)); CH' (Uzer and Dalgarno, 1979b (PD)); LiF (Asaro and Dalgarno, 1985); HC1 (van Dishoeck, van Hemert, and Dalgarno, 1982 (PD)); OH (van Dishoeck and Dalgarno, 1983 (PD)); OH (van Dishoeck, Langhoff, and Dalgarno, 1983; van Dishoeck, van Hemert, Allison, and Dalgarno, 1984 (PD)).

The photodissociation studies are particularly interesting from a fundamental perspective. Allison and Dalgarno (197 l), considering an analogy with photoexcitation and photoionization, derived a continuity relationship for molecules that relates discrete absorption oscillator strengths to photodissociation cross sections. Uzer and Dalgarno (1979b), in their study of

12 Neal Lane

CH+ , calculated the explicit contributions of predissociating shape resonances associated with the upper electronic state. Kirby, Guberman, and Dalgarno (1979) showed by means of an ab initio calculation that the process of resonant dissociative photoionization, in which a photon is absorbed causing a transition into a repulsive autoionizing final state, can explain the appearance of fast protons in photoabsorption of energetic photons by H,.

Uzer and Dalgarno (1 980) have quantitatively examined the significance of “accidental” predissociation in Li, , that is, via avoided crossings, and have formulated a general theory of the process in terms of Feshbach projection operators. Asaro and Dalgarno (1983), in their study of photodissociation of LiH and H,, showed that the Stieltjes imaging method can yield accurate values for dissociation cross sections, even using modest basis sets; they also applied the method to the photodissociation of H:.

In a major theoretical study of photodissociation of OH by van Dishoeck and Dalgarno (1983) and van Dishoeck, van Hemert, Allison, and Dalgarno (1984), the traditional view was shown to be incorrect and the dominant mechanism was identified. For the first time, fully quanta1 calculations were carried out with coupled final electronic states; vibrational states of diabatic curves were shown to generate resonances in the photodissociation cross sections.

111. Atomic and Molecular Interactions

At large separations, the interactions between atomic and molecular systems are dominated by polarization and dispersion forces that arise from the mutual perturbations of transient or permanent multipole moments of the electronic charge distributions of the interacting systems. The short-range interactions of such systems, on the other hand, are strongly influenced by electron exchange effects.

A. LONG RANGE FORCES

It is an analytic convenience to describe long-range interactions by means of an expansion of the potential energy in inverse powers of the internuclear separation, R. Dalgarno and Lewis (1956a) showed that the formal divergence of such a power-series representation at all values of R is fundamental to the nature of the expansion, but that appropriate truncation can lead to accurate results.

CONTRIBUTIONS TO ATOMIC A N D MOLECULAR PHYSICS 13

The long-range adiabatic interactions of complex atomic systems can be viewed as an average of all possible permanent and transient multipole interactions. For example, the second-order interaction potential energy- the leading term for neutral systems is the Van der Waals contribution-can be expressed in terms of the multipole dynamic polarizabilities of the two systems (see, for example, Dalgarno and Davison, 1966).

Chan and Dalgarno (1965a), using a variational treatment that eliminates the infinite summations, calculated the second-order interaction energy of two ground-state hydrogen atoms, obtaining impressive accuracy. A similar approach was found to be successful for other systems: He-He (Chan and Dalgarno, 1965a; Dalgarno and Victor, 1966a); He-H (Chan and Dalgarno 1965b); and Li-Li (Stacey and Dalgarno, 1968). Similarly, the leading non- additive contribution to the three-body interactions among He and H atoms was calculated by Chan and Dalgarno 1965b) and Dalgarno and Victor (1 966a).

Semi-empirical representations of the dynamic polarizabilities, based on measured oscillator strengths, were shown to be useful in determining the Van der Waals coefficients, particularly for complex systems (Dalgarno and Victor, 1968; Victor and Dalgarno, 1970).

Sum rules can be called on when the data on oscillator strengths are inadequate. Dalgarno and Lynn (1 957b) found that by modifying theoretical oscillator strengths of helium so that they satisfy four sum rules, reasonably accurate values of physical parameters, including the Van der Waals coefficient, could be calculated (see also Dalgarno and Kingston, 1959, 1961). It was shown to be possible to bypass the oscillator strengths and use the measured frequency-dependent refractive index, which is proportional to the polarizability (Dalgarno, Morrison, and Pengelly, 1967).

Expressions for the leading non-adiabatic correction to the static interaction between a charged or neutral system and a spherically symmetric atom were shown by Dalgarno, Drake, and Victor (1968) to be expressible in terms of oscillator strengths in a generalization of the earlier study of hydrogen by Dalgarno and MeCarroll (1956, 1957a, 1957b).

B. MODEL POTENTIALS AND PSEUDO-POTENTIALS

In many important atomic processes only a few electrons are truly “active.” The “inactive” electrons along with the nucleus can often be accurately represented by a pseudo-potential, thus reducing the many-electron problem to that of one electron in a potential field.

Dalgarno, Bottcher and Victor (1970), in an application of pseudopotential theory to the molecular ion Li:, showed that the ground-state g

14 Neal Lane

and u internuclear potentials could be well represented by this method, including the Van der Waals term. The approach, which essentiaIly describes a single electron in the field of two polarizable cores, was shown to be useful in describing excited states as well (Bottcher and Dalgarno, 1975). For the system N a l , the method was successfully used to determine the potential curves (Bottcher, Allison, and Dalgarno, 1971) and various one-electron properties (Cerjan, Docken, and Dalgarno, 1976).

Caves and Dalgarno (1972) adopted a powerful semi-empirical, model- potential scheme in which the potential energy, which includes core-polarization effects, was adjusted to yield accurate atomic energies, and then successfully used to calculate dynamic polarizabilities, discrete oscillator strengths, photoionization cross sections, and radiative recombination coefficients. Weisheit and Dalgarno (1971a, 1971b) have shown that excellent results for the oscillator strengths of sodium and potassium as well as spin- orbit effects in the photoionization of potassium can be obtained using model potentials.

Bottcher and Dalgarno (1974) developed a model potential method that is more systematic than the ad hoc procedures used earlier and that applies to many-electron single-center and two-center systems.

IV. Atomic and Molecular Collisions

Cross sections for inelastic processes in slow atomic and molecular collisions are generally very small, except for near-resonance cases, where the excitation energy is small, or systems where avoided curve crossings provide near-resonance conditions at some point during the collision.

A. NEAR-RESONANCE ELECTRONIC TRANSITIONS

Spin change in collisions between H atoms was studied by Dalgarno (1961b), who gave a semi-classical description and pointed out the analogy to accidental resonant charge transfer, and by Dalgarno and Henry (1964), who employed the full quanta1 description, necessary at low energies.

More accurate cross sections for spin change in collisions between H atoms were calculated by Allison and Dalgarno (1969a), and found to be in agreement with earlier studies except at thermal energies, where the more accurate results are smaller by an order of magnitude. Semi-classical calculations for the alkali atoms were carried out by Dalgarno and Rudge (1965) and found to be in good agreement with existing measurements. Wofsy, Reid,


and Dalgarno (1971) further improved the theory in application to spin change of Cf and 0 atoms in collisions with H. Fine-structure excitation of Cf by H, impact was studied by Chu and Dalgarno (1975a) and that of C by H impact, by Yau and Dalgarno (1976). Recently, elastic scattering and fine- structure excitation in low-energy collisions between oxygen atoms were studied by Yee and Dalgarno (1985, 1987).

For ion-ion collisions, the Coulomb barrier suppresses inelastic cross sections in the low-energy limit. On the other hand, the relevant couplings can be long range, inducing transitions at large separations. Theoretical studies of fine-structure excitation of ions by proton impact have been carried out for the systems 03+ (Heil, Green, and Dalgarno, 1982); Fe13+ (Heil, Kirby, and Dalgarno, 1983); and several hydrogenic ions by electron as well as proton impact (Zygelman and Dalgarno, 1987).

Penning and associative-ionization processes, where an excited atom, usually metastable helium, gives up its energy to ionization of the collision partner, are cases of exact, electronic-energy resonance, since the initial electronic state of the system He* + atom (or molecule) lies in the continuum of the system He + ion + e. Model studies of Penning ionization of several neutral systems by metastable helium were carried out by Bell, Dalgarno, and Kingston (1968), where for the first time, accurate Van der Waals interactions were used. Dalgarno and Browne (1967), Browne and Dalgarno (1969), and Bieniek and Dalgarno (1 979) investigated the equivalent process involving negative ions, i.e. electron detachment, for collisions of H- with H.

B. EXCITATION AND CHARGE TRANSFER

In the first ah initio investigation of excitation transfer in collisions of metastable and ground-state helium atoms, Buckingham and Dalgarno (1952a, 1952b) obtained an unexpected hump in the potential curve that suppressed the excitation-transfer cross section at low energies.

Bates and Dalgarno (1952) carried out the first correct Born treatment of resonant excitation transfer in collisions between protons and hydrogen atoms, correcting earlier theory; and Dalgarno and Yadav (1953) performed the first perturbed-stationary-states (PSS) calculation for this system, obtaining agreement with the Born results at the higher energies. Bates and Dalgarno (1 953) confirmed the small cross sections for non-resonant charge transfer into excited states of hydrogen. Boyd and Dalgarno (1958) showed, using the PSS method, that the cross section for resonant charge transfer in proton collisions with excited H atoms is much greater than that for the ground-state. Dalgarno and McDowell (1 956) provided an early theoretical

16 N e d Lane

description of charge transfer in collisions of H- with H, which is in agreement with recent measurements.

Dalgarno (1 954) performed one of the earliest Stueckelberg-Landau- Zener studies of avoided crossings in charge-transfer collisions between Al’ +, Liz + and A12 + ions and atomic hydrogen.

Dalgarno and his collaborators have continued to refine the theory of charge-transfer for low and intermediate-energy collisions, both close- coupling treatments (Heil and Dalgarno, 1979) and distorted-waves extensions to higher energies (Bienstock, Heil, and Dalgarno, 1984). They have applied these techniques to a variety of systems of fundamental and practical interest.

A particularly illuminating series of studies includes the systems O2 + on He (Butler, Heil, and Dalgarno, 1984); 03+ on H (Dalgarno, Heil, and Butler, 1981; Bienstock, Heil, and Dalgarno, 1983); N2+ and C3+ on H (Bienstock, Heil, Bottcher, and Dalgarno, 1982; Heil, Butler, and Dalgarno, 1981); and N3+ on H (Bienstock, Dalgarno, and Heil, 1984). For all these systems, it was shown that the charge transfer occurs primarily at avoided crossings, so that an accurate determination of the molecular potential curves is necessary.

Mutual neutralization of colliding positive and negative alkali ions and the inverse chemi-ionization process were described by Cooper, Bienstock, and Dalgarno (1 987), who provided an accurate treatment of the localized avoided crossings between the A + + B- ionic and the A + B covalent curves, which dominate the collision dynamics.

C. RADIATIVE COLLISIONS

Except for the special cases of electronic energy resonance, the cross sections for excitation (or de-excitation) or charge transfer tend to fall off rapidly with decreasing relative velocity. However, photoemission processes are enhanced by long collision times. Thus, radiative deactivation and radiative charge-transfer tend to dominate at low energies for many systems.

The first theoretical descriptions of radiative deactivation of metastable helium were performed by Allison and Dalgarno (1963) for proton collisions and Allison, Browne, and Dalgarno (1966) for He collisions. The rate for the former was shown to be much greater because of the long-range dipole interaction induced by the proton charge.

Dalgarno and Sando (1973) performed the first quanta1 calculation of spectral line broadening as a dynamic process; they pointed out the connection between the occurrence of an extremum in the difference between the relevant initial and final state interatomic potentials and satellite bands in


the spectrum. The direct application of collision theory to a radiative process was carried out by Sando and Dalgarno (1971).

Bottcher, Dalgarno, and Wright (1973) applied model potential methods to calculate collision-induced absorption coefficients for thermal collisions between alkali and inert-gas atoms, showing that the effect should be experimentally observable and that both quanta1 and classical-path methods give equivalent results.

Allison and Dalgarno (1965) significantly improved the accuracy of the theoretical cross section for radiative charge transfer in a collision of He2 + on H. The radiative process was shown to dominate charge transfer at low energies. The induced dipole moment for this system was shown to fall off much more rapidly with R than that for the proton-metastable He system, leading to a smaller cross section for the former.

The radiative mechanism was shown by Butler, Guberman, and Dalgarno (1977) to dominate charge transfer for thermal collisions of C2+, C3+, and N2+ with H, since favorable avoided crossings are not present. Radiative charge transfer and radiative association have been studied for collisions of He+ ions with neon atoms by Cooper, Kirby, and Dalgarno (1984), and in comparison with the direct process (Zygelman and Dalgarno, 1986) are found to dominate for collision energies less than 25 eV.

Electrons elastically scattered from atoms, molecules, or ions emit and absorb continuous radiation in a so-called free-free process. Using an approximation based on modified-effective-range theory, Lane and Dalgarno (1966) calculated free-free absorption coefficients for a number of atoms and molecules.

D. ROTATIONAL EXCITATION OF MOLECULES

Arthurs and Dalgarno (1960) formulated the theory for the scattering of an electron from a rigid rotator in terms of an expansion in total angular momentum states of the system; this formalism provided the foundation for essentially all subsequent studies of rotational excitation by electron, atom, or ion impact. Dalgarno and Henry (1965) performed the first application of distorted-waves theory to electron-H, collisions.

Dalgarno and Moffett (1963) showed, by employing the first Born approximation to rotational excitation of nonpolar molecules, that the nonspherical polarization interaction can significantly increase or decrease the cross section at low energies, depending on the sign of quadrupole moment of the molecule.

Crawford, Dalgarno, and Hays (1967) questioned the physical relevance of a so-called “critical dipole moment,” and argued that the first Born approxi-

18 Neal Lane

mation to the momentum-transfer cross section for polar molecules is superior to the exact fixed-dipole calculation because the former does not show the mathematical anomaly when the dipole moment exceeds the critical value. Dalgarno, Crawford, and Allison (1968) went on to show that a close- coupling treatment of the problem, properly accounting for molecular rotation, did not show any anomalous behavior. Chu and Dalgarno (1974) applied the Coulomb-Born approximation to rotational excitation in electron-CH + scattering.

The initial formulation of the theory of atomic collisions with rotating molecules was given by Bernstein, Dalgarno, Massey, and Percival (1963); a distorted-waves treatment of rotational excitation of H, and D, by H impact was carried out by Dalgarno, Henry, and Roberts (1966), followed by the first close-coupling treatment of the problem by Allison and Dalgarno (1967). The close-coupling treatment was shown to correct the breakdown of the distorted-waves approximation at higher energies, where the coupling is stronger. Reid and Dalgarno (1969) pointed out the resemblance of molecular rotational excitation to atomic fine-structure excitation.

Rotational excitation of molecules by atom or ion impact (Chu and Dalgarno, 1975b) is found to be sensitive both to the range and anisotropy (with respect to orientation of the molecular axis) of the interaction. Jamieson, Kalaghan, and Dalgarno (1975) carried out theoretical studies of rotational excitation of CN by proton impact.

V. Summary

Alex Dalgarno has had a major impact on essentially all areas of atomic and molecular physics through the discovery of the basic principles that underlie atomic and molecular phenomena, the development of powerful theoretical and computational methods, and the explication of a vast array of atomic and molecular processes through applications. Moreover, his contributions continue to be substantial in number, in variety, and in depth of content.

ACKNOWLEDGMENTS

The author wishes to acknowledge the helpful suggestions of several colleagues at various stages of the writing as well as the invaluable assistance of Laura Montagne, particularly for her work with the references. The author also asknowledges the US. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences and the Robert A. Welch Foundation for their support.


REFERENCES

Allison, A. C., and Dalgarno, A. (1967). Proc. Phys. Soc. London 90, 609-614. Allison, A. C., and Dalgarno, A. (1969a). Astrophys. J . 158, 423-425. Allison, A. C., and Dalgarno, A. (1969b). At. Data 1, 91-102. Allison, A. C., and Dalgarno, A. (1970). At. Data 1, 289--304. Allison, A. C., and Dalgarno, A. (1971). J. Chem. Phys. 55, 4342-4344. Allison. A. C., Guherman, S. L., and Dalgarno, A. (1982). J. Geophys. Res. 87, 923-925. Allison, D. C., Browne, J. C., and Dalgarno, A. (1966). Proc. Phys. Soc. London 89,41. Allison, D. C. S., and Dalgarno, A. (1963). Proc. Phys. Soc. London 81, 23-27. Allison, D. C. S., and Dalgarno, A. (1965). Proc. Phys. Soc. London 85, 845. Arthurs, A. M., and Dalgarno, A. (1960). Proc. R. Soc. London, A 256, 540-551. Asaro, C., and Dalgarno, A. (1983). J. Chem. Phys., 78, 200-205. Asdro, C., and Dalgarno, A. (1985). Chem. Phys. Lett, 118, 64-66. Bates, D. R., and Dalgarno, A. (1952). Proc. Phys. Soc. London, A 65, 919-925. Bates, D. R., and Dalgarno, A. (1953). Proc. Ph.vs. Soc. London, A 66, 972-976. Bates, D. R., Ledsham, K., and Stewart, A. L. (1953). Phil. Trans. R. Soc. London, A 246, 215. Bell, K. L., Dalgarno, A,, and Kingston, A. E. (1968). J . Phys. B 1, 18. Bernstein, R. B., Dalgarno, A,, Sir Harrie Massey, and Percival, I. C. (1963). Proc. R. Soc.

Bieniek, R. J., and Dalgarno, A. (1979). Astrophys. J. 228. 635-639. Bienstock, S., Dalgarno, A,, and Heil. T. G. (1984). Phys. Rev. A 29, 2239-2241. Bienstock, S., Heil, T. G., Bottcher, C., and Dalgarno, A. (1982). Phys. Rev. A 25, 2850-2852. Bienstock, S., Heil, T. G., and Dalgarno, A. (1983). Php. Rev. A 27, 2741-2743. Bienstock, S., Heil, T. G., and Dalgarno, A. (1984). Phys. Rev. A 29, 503-508. Bottcher, C., and Dalgarno, A. (1974). Proc. R. Soc. London, A 340, 187-198. Bottcher, C., and Dalgarno, A. (1975). Chem. Phys. Lett. 36, 137-144. Bottcher, C., Allison, A. C., and Dalgarno, A. (1971). Chem. Phys. Lett. 11, 307-309. Bottcher, C., Dalgarno, A,, and Wright, E. L. (1973). Phys. Rev. A 7, 1606-1609. Boyd, T. J. M., and Dalgarno, A. (1958). Proc. Phys. Soc. London 72, 694-700. Bransden, B. H., and Dalgarno, A. (1952). Phys. Rev. 88, 148. Bransden, B. H., and Dalgarno, A. (1953a). Proc. Phys. Soc. London, Sect. A 66, 268. Bransden, B. H., and Dalgarno, A. (1953b). Proc. Phys. Soc. London. Seer. A 66, 904. Bransden, B. H., and Dalgarno, A. (1953~). Proc. Phys. Soc. London, Sect. A 66,911-920. Bransden, B. H., and Dalgarno, A. (1956). Proc. Phys. Soc. London, Sect. A 69, 65-69. Bransden, B. H., Dalgarno, A,, and King, N. M. (1953). Proc. Phys. Soc. London, A 66, 1097. Bransden, B. H., Dalgarno, A,, John, T. L., and Seaton, M. J. (1958). Proc. Phys. Soc. London, A

Browne, J. C., and Dalgarno, A. (1969). J. Phys. B 2, 885-889. Buckingham, R. A,, and Dalgarno, A. (1952a). Proc. R. Soc. London, A 213, 327-349. Buckingham, R. A,, and Dalgarno, A. (1952b). Proc. R. Soc. London, A 213, 506-519. Butler, S. E., Guherman, S. L., and Dalgarno, A. (1977). Phys. Rev. A 16, 500-507. Butler, S. E., Heil, T. G., and Dalgarno, A. (1984). J . Chem. Phys. 80, 4986-4988. Caves, T., and Dalgarno, A. (1972). J . Quant. Spectrosc. Radial. Transfer 12, 1539-1552. Cerjan, C. J., Docken, K. K., and Dalgarno, A. ( 1 976). Chem. Phys. Lett, 38,401 -404. Chan, Y. M., and Dalgarno, A. (1965a). Mol. Phys. 9, 525-528. Chan, Y. M., and Dalgarno, A. (1965b). Proc. Phys. Soc. London 86, 777-782. Chan, Y. M., and Dalgarno, A. (1965~). Proc. Phys. Soc. London 85, 227-230. Chisolm, C. D. H., and Dalgarno, A. (1966). Proc. R. Soc. London, A 290, 264.

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Chu, S.-I., and Dalgarno, A. (1974). Phys. Rev. A 10, 788-792. Chu, S.-I., and Dalgarno, A. (1975a). J. Chem. Phys. 62,4009-4015. Chu, S.-I., and Dalgarno, A. (1975b). J. Chem. Phys. 63, 2115-2118. Cohen, M., and Dalgarno, A. (1961a). Proc. Phys. Soe. London 77, 748-750. Cohen, M., and Dalgarno, A. (1961b). Proc. R. Sac. London, A 261, 565-576. Cooper, D. L., Kirby, K., and Dalgarno, A. (1984). Can. J. Phys. 62, 1622-1628. Cooper, D. L., Bienstock, S., and Dalgarno, A. (1987). J. Chem. Phys. 86, 3845-3851. Crawford, 0. H., Dalgarno, A,, and Hays, P. B. (1967). Mol. Phys. 13, 181-192. Dalgarno, A. (1952). Proc. Phys. Soc. London, Sect A 65, 663. Dalgarno, A. (1954). Proc. Phys. Soc. London, Sect. A 67, 1010-1017. Dalgarno, A. (1959). Proc. R. Soc. London, A 251, 282-290. Dalgarno, A. (1960). Proc. Phys. Soc., London 75,439-441. Dalgarno, A. (1961a). In Quantum Theory (D. R. Bates, ed.), Vol. I, pp. 171-209. Academic Press,

Dalgarno, A. (1961 b). Proc. R. Soe. London, A 262, 132- 135. Dalgarno, A, (1963). Rev. Mod. Phys. 35, 522. Dalgarno, A,, and Browne, J. C. (1967). Astrophys. J. 149, 231-232. Dalgarno, A,, and Davison, W. D. (1966). Adv. At. Mol. Phys. 2, pp . 1-32. Academic Press,

Dalgarno, A., and Drake, G. W. F. (1971). Chem. Phys. Left. 11, 509-511. Dalgarno, A,, and Henry, R. J. W. (1964). Proc. Phys. Soc. London 83, 157-158. Dalgarno, A., and Henry, R. J. W. (1965). Proc. Phys. Soc. London 85,679. Dalgarno, A,, and Kingston, A. E. (1959). Proc. Phys. Soc. London 73, 455. Dalgarno, A,, and Kingston, A. E. (1961). Proc. Phys. Soc. London 78, 607-609. Dalgarno, A,, and Lane, N. L. (1966). Astrophys. J . 145, 623. Dalgarno, A,, and Lewis, J. T. (1955). Proc. R. Soc. London, A 233, 70-74. Dalgarno, A,, and Lewis, J. T. (1956a). Proc. Phys. Soc. London, Serf. A 69, 57-64. Dalgarno, A,, and Lewis, J. T. (1956b). Proc. Phys. Soc. London, Sect. A 69, 628-630. Dalgarno, A., and Lynn, N. (1957a). Proc. Phys. Soc. London, Sect. A 70, 223-225. Dalgarno, A., and Lynn, N. (1957b). Proc. Phys. Soc. London, Sect. A 70, 802-808. Dalgarno, A,, and McCarroll, R. (1956). Proc. R. Soc. London, A 237, 383-394. Dalgarno, A,, and McCarroll, R. (1957a). Proc. R. Soc. London, A 239,413-419. Dalgarno, A., and McCarroll, R. (1957b). Proc. Phys. Soc. London, Sect. A 70, 501-506. Dalgarno, A,, and McDowell, M. R. C. (1956). Proc. Phys. SOC. London, Sect. A 69, 615-623. Dalgarno, A., and Moffet, R. J. (1963). Proc. Nut. Acud. Sci., India, Sect. A 33, 511. Dalgarno, A,, and Parkinson, E. M. (1967). Proc. R. Soc. 301, 253-260. Dalgarno, A,, and Poots, 0. (1954). Proc. Phys. Soc. A67, 343-350. Dalgarno, A., and Rudge, M. R. H. (1965). Proc. R. Soc. London, A 286, 519-524. Dalgarno, A., and Sando, K. M. (1973). Comments At. Mol. Phys. 4, 29-33. Dalgarno, A,, and Stephens, T. L. (1970). Astrophys. J. 160, L107-LlO9. Dalgarno, A,, and Stewart, A. L. (1956a). Proc. R. Soc. London, A 238,269-275. Dalgarno, A,, and Stewart, A. L. (1956b). Proc. R. Soc. London, A 238, 276-285. Dalgarno, A., and Stewart, A. L. (1957). Proc. R. Soc. London, A 240, 274-283. Dalgarno, A,, and Stewart, A. L. (1958). Proc. R. Soc. London, A 247,245-259. Dalgarno, A,, and Stewart, A. L. (1960). Proc. R. Soc. London, A 254, 570-574. Dalgarno, A,, and Victor, G. A. (1966a). Mol. Phys. 10, 333-337. Dalgarno, A,, and Victor, G. A. (1966b). Proc. R. Soc. London, A 291, 291-295. Dalgarno, A,, and Victor, G. A. (1967). Proc. Phys. Soc., London 90, 605. Dalgarno, A., and Victor, G. A. (1968). J. Chem. Phys. 49, 1982-1983. Dalgarno, A,, and Williams, D. A. (1962). Astrophys. J . 136, 690-692.

New York, New York.

Orlando, Florida.


Dalgarno, A., and Williams, D. A. (1965). Proc. Phys. Soc. London 85, 685-689. Dalgarno, A., and Yadav, H. N. (1953). Proc. Phys. Sue. London, Sect. A 66, 173- 177. Dalgarno. A., Bottcher, C., and Victor, G. A. (1970). Chem. Phys. k t i . 7, 265-267. Dalgarno, A., Crawford, 0. H., and Allison, A. C. (1968). Chem. Phys. Lett. 2, 381-382. Dalgarno, A., Drake, G. W. F., and Victor, G. A. (1968). Phys. Rev. 176, 194-197. Dalgarno, A., Degges, T., and Williams, D. A. (1967). Proc. Phys. SOC. London 92,291-295. Dalgarno, A., Ford, A. L., and Browne, J. C. (1971). Phys. Rev. Lett. 27, 1033-1036. Dalgarno, A., Heil, T. G., and Butler, S. E. (198 1 ) . Astrophys. J. 245, 793-797. Dalgarno, A., Henry, R. J. W., and Roberts, C. S . (1966). Proc. Phys. Soc. London 88,611-615. Dalgarno, A., Lynn, N., and Williams, E. J. A. (1956). Proc. Phys. SOC. London, Sect. A 69,

Dalgarno, A., Moiseiwitsch, B. L., and Stewart, A. L. (1957). J. Chem. Phys. 26, 965-966. Dalgarno, A,, Morrison, I. H., and Pengelly, R. M. (1967). Int. J. Quantum Chem. 1, 161-168. Dalgarno, A., Patterson, T. N. L., and Somerville, W. B. (1960). Proc. R. Soc. London, A 259,

Drake, G. W. F. (1971). Phys. Rev. A 3, 908. Drake, G. W. F., and Dalgarno, A. (1968). Astrophys. J. 152, L121LL123. Drake, G. W. F., and Dalgrano, A. (1969). Astrophys. J. 157,459-462. Drake, G. W. F., and Dalgarno, A. (1970). Phys. Rev. A 1, 1325-1329. Drake, G. W. F., and Dalgdrno, A. (1971). Proc. R. SOC. London, A. 320, 549-560. Drake, G. W. F., Victor, G. A., and Dalgarno, A. (1969). Phys. Reu. 180, 25-32. Ford, A. L., Docken, K. K., and Dalgarno, A. (1975a). Asirophys. J. 195, 819-824. Ford, A. L., Docken, K. K., and Dalgarno, A. (1975b). Astrophys. J. 200, 788-789. Goldman, S. P., and Dalgarno, A. (1986). Phys. Rev. Lett. 57, 408-411. Guberman, S. L., and Dalgarno, A. (1979). J. Geuphys. Res. 84,4437-4440. Heil, T. G., and Dalgarno, A. (1979). J. Phys. B 12, L557-L560. Heil, T. G., Butler, S. E., and Dalgarno, A. (1981). Phys. Rev. A 23, 1100-1109. Heil, T. G., Green, S. . and Dalgarno. A. (1982). Phys. Rev. A 26, 3293-3298. Heil, T. G., Kirby, K., and Dalgarno, A. (1983). Phys. Rev. A 27, 2826-2830. Jamieson, M. J., Kalaghan, P. M., and Dalgarno, A. (1975). J. Phys. B 8,2140-2148. Johnson, W. R., Lin, C. D., and Dalgarno. A. (1976). J. Phys. B9, L303-L306. Kirby, K., Guberman, S., and Dalgarno, A. (1979). J. Chem. Phys. 70,4635-4639. Kirby, K., Uzer, T., Allison, A. C., and Dalgarno, A. (1981). J. Chem. Phys. 75,2820-2825. Kirby-Docken, K.. Cejan, C. J., and Dalgarno, A. (1976). Chem. Phys. Lerf. 40,2055209. Kwok, T. L., Dalgarno, A., and Posen A. (1985). Phys. Rev. A 32, 646-649. Kwok, T. L., Cuberman, S., Dalgarno, A,, and Posen, A. (1986). Phys. Rev. A 34, 1962-1965. Lin, C. D., Johnson, W. R.. and Dalgarno, A. (1977). Phys. Rev. A 15, 154-161. Onello, J. S., Ford, L., and Dalgarno, A. (1974). Phys. Rev. A 10, 9. Reid, R. H. C., and Dalgarno, A. (1969). Phys. Rev. Lett. 22, 1029-1030. Roche, A.-L., Kirby, K., Guberman, S. L., and Dalgarno, A. (1981). J. Electron Spectrosc. Relut.

Sando, K. M., and Dalgarno, A. (1971). Mol. Phys. 20, 103-112. Shorer, P., and Dalgarno, A. (1977). Phys. Rev. A 16, 1502-1506. Shorer, P., and Dalgarno, A. (1980). Phys. Scr. 21, 432-435. Stacey, G. M., and Dalgarno, A. (1968). J . Chem. Phys. 48, 2515-2518. Stewart, R. F., Watson, D. K., and Dalgarno, A. (1975). J. Chenz. Phys. 63, 3222-3227. Stewart, R. F., Watson, D. K., and Dalgarno, A. (1976). J. Chem. Phys. 65, 2104-2111. Uzer, T., and Dalgarno, A. (1979a). Chem. Phys. Lett. 61, 213-215. Uzer. T., and Dalgarno, A. (1979b). Chem. Phys. Lett. 63, 22-24. Uzer, T., and Dalgarno, A. (1979~). Chem. Phys. Lett. 65, 1-3.

6 10.- 6 14.

100-109.

Phenom. 22, 223-235.

22 Neal Lane

Uzer, T., and Dalgarno, A. (1980). Chem. Phys. 51, 271-277. Uzer, T., Watson, D. K., and Dalgarno, A. (1978). Chem. Phys. Lett. 55, 6-8. van Dishoeck, E. F., and Dalgarno, A. (1983). J. Chem. Phys. 79, 873-888. van Dishoeck, E. F., Langhoff, S. R., and Dalgarno, A. (1983). J. Chem. Phys. 78, 4552-4561. van Dishoeck, E. F., van Hernert M. C., and Dalgarno, A. (1982). J. Chem. Phys. 77,3693-3702. van Dishoeck, E. F., van Hernert, M. C., Allison, A. C., and Dalgarno, A. (1984). J. Chem. Phys.

Victor, G. A., and Dalgarno, A. (1969). J. Chem. Phys. 50, 2535-2539. Victor, G. A., and Dalgarno, A. (1970). J. Chem. Phvs. 53, 1316-1317. Victor, G. A., Browne, J. C., and Dalgarno, A. (1967). Proc. Phys. Soc. London 92,42-49. Watson, D. K., Stewart, R. F., and Dalgarno, A. (1976a). J. Chem. Phys. 64,4995-4999. Watson, D. K., Stewart, R. F., and Dalgarno, A. (1976b). Mul. Phys. 32, 1661-1670. Weisheit, J. C., and Dalgarno, A. (1971a). Chem. Phys. L e f f . 9, 517-520. Weisheit, J. C., and Dalgdrno, A. (1971b). Phys. Rec. Lett. 27, 701-703. Wetmore, R. W., Fox, J. L., and Dalgarno, A. (1984). Planel. Space Sci. 32, 11 11-1 113. Wofsy, S., Reid, R. H. G., and Dalgarno, A. (1971). Astrophys. J. 168, 161-167. Yau, A. W., and Dalgarno, A. (1976). Astrophys. J. 206, 652-657. Yee, J.-H., and Dalgarno, A. (1985). Planet. Space Sci. 33, 825-830. Yee, J.-H., and Dalgarno, A. (1987). Planet. Space Sci. 35, 399-404. Zygelman, B., and Dalgarno, A. (1986). Phys. Reu. A 33, 3853-3858. Zygelman, B., and Dalgarno, A. (1987). Phys. Rev. A 35,4085-4100.

81, 5709-5724.

ADVANCES I N ATOMIC AND MOLECULAR PHYSICS, VOL. 25

1 1 ALEXANDER DALGARNO CONTRIBUTIONS TO AERONOMY MICHAEL B. MCELROY Depurirnenr of’ Eurlh and Plunecary Sciences

Hurvurd Unit1ersily Cumbridge, Mussuchusetts

and Diuision of Applied Sciences

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 23 11. Selected Contributions . . . . . . . . . . . . . . . . . . . . 23

References . . . . . . . . . . . . . . . . . . . . . . . . 27

I. Introduction

It is impossible in a few brief pages to summarize the breadth and depth of Alexander Dalgarno’s contributions to aeronomy. In a career extending over 35 years, Dalgarno has authored or coauthored almost 500 papers, an average of more than 14 per year. Approximately 70 of his papers are devoted specifically to aeronomical topics. At least an equal number involve investigations of atomic and molecular processes directly applicable to aeronomy. He has been a dominant figure in the field, and continues to play a major role, with contributions not only to studies of the Earth’s upper atmosphere but also to investigations of the stratosphere and to studies of the atmospheres of Mercury, Venus, Mars, Jupiter and comets.

We shall attempt in what follows to offer a guided tour to some of the highlights of Alexander Dalgarno’s contributions to the field. The selection, of necessity, reflects the prejudice of the author: omission of specific contributions should not be interpreted as a slight, merely choice imposed by the limitations of space.

11. Selected Contributions

Studies of the upper atmosphere prior to 1950 were based on inferences drawn from remote sensing, primarily radio probing of the ionosphere and observations of the aurora and airglow. It was a subject suited ideally to the

23


24 Michael B. McElroy

talents of an atomic and molecular physicist. The goal was to define the nature of the processes at work in the atmosphere and, from the limited data available, to draw inferences concerning the structure and composition of the atmosphere. Atomic and molecular physicists were remarkably successful in this endeavor. It is difficult now to appreciate the challenge they faced. Basic data on oscillator strengths were unavailable. Approximate schemes had to be devised to calculate the essential properties of gases in the atmosphere. The very nature of the reactions at work in the atmosphere were unknown. The importance of dissociative recombination and ion-atom interchange in determining the structure of the ionosphere had been recognized but a few years earlier by David R. Bates and Harrie Massey.

The processes responsible for the airglow and aurora were topics of vigorous research. The atmosphere represented a natural laboratory. One could hope to learn about the atmosphere by applying the insights of atomic and molecular physics. And it was a two-way street: observations of the atmosphere could be used to enhance our understanding of specific atomic and molecular processes. Such was the nature of the subject when Dalgarno made his first direct contributions almost 35 years ago.

He wrote two papers in 1953 and 1954, both with David Bates. One concerned the altitude of the layers responsible for the airglow (Bates and Dalgarno, 1953). The second anticipated what was to become a major theme of his research in the 1960s and 197Os, the nature of the dayglow (Bates and Dalgarno, 1954). Dalgarno had an encyclopedic knowledge of the literature. He traveled widely and was exceptionally well informed on what was being measured and what could be measured. Experimentalists sought him out. They recognized an invaluable resource. He sensed immediately what was important and feasible. He was quick to capitalize on new information, extracting more than was obvious. Though not directly involved in measurement, he had an immense influence on what was measured.

The first reliable measurements of the extreme ultraviolet spectrum of the sun were published by Hans Hinteregger in 1961 (Hinteregger, 1961). Dalgarno appreciated immediately the significance of these data. It was possible now for the first time to develop a quantitative theory of the dayglow. It required a detailed model to describe the path by which energy at extreme ultraviolet wavelengths was degraded to heat. He was aware also of measurements indicating that the electron temperature in the ionosphere was elevated relative to the temperature of the neutral gas (Boggess et af., 1959, Spencer et al., 1962). Hanson and Johnson (1961) had pointed out that this could arise as a consequence of the efficiency of electron-electron collisions in competing for energy stored initially in photoelectrons. What was needed was a model to describe the production and energy degradation of photoelectrons.

CONTRIBUTIONS TO AERONOMY 25

There was nobody better equipped to meet this challenge. A few years earlier, Dalgarno had published a comprehensive review of cross sections for collisions of electrons with atmospheric gases (Dalgarno, 1961). The initial discussion of the photoelectron problem appeared in 1963 (Dalgarno et al., 1963). I t provided the stimulus for a series of follow-up studies, involving both improvements in data for the cross sections of a number of the key proceses, in addition to applications of increasing breadth and complexity.

Cross sections for photoionization of atomic oxygen, identifying the nature of the ionic products, were evaluated and published in 1964 (Dalgarno et al., 1964). Cross sections for rotational excitation of molecular nitrogen were presented by Dalgarno and Moffett in 1963, while analogous data for molecular oxygen were evaluated with Lane in 1969 (Dalgarno and Lane, 1969). Dalgarno and Degges drew attention in 1968 to the importance of electron excitation of the fine structure levels of atomic oxygen as an energy loss mechanism for hot electrons in the upper ionosphere. A series of papers (Dalgarno and McElroy, 1965a; Dalgarno and Henry, 1965; Dalgarno et al., 1968) explored the implications of electron-electron collisions for electron temperatures. Ion temperatures were discussed in a paper with Walker in 1967 (Dalgarno and Walker, 1967). Excitation of the dayglow was discussed by Dalgarno and Walker (1964) and Walker (1964) and by Dalgarno et al. (1969). A paper by Dalgarno and McElroy (1965b) raised the possibility that observations of the O f doublet at 7330 A could be used to monitor the flux of extreme ultraviolet solar radiation from the ground. It was a remarkable decade of sustained and diverse accomplishment.

Dalgarno played a leading role in the Atmosphere Explorer Satellite Project in the 1970s (Dalgarno et al., 1973). The Atmosphere Explorer was designed to provide simultaneous measurements of a wide range of atmospheric parameters, including neutral composition, ion composition, electron temperature, and the electron energy distribution function, in addition to data defining the flux of ultraviolet solar radiation. It was a mission, or series of missions, made to order for an atomic physicist. For the first time it was possible to check the reliability of models for ionospheric chemistry, to explore the role of reactions involving metastable species, and at the same time to check the accuracy of claims made for the precision of particular measurements. It was an intense, interactive experience, documented in a remarkable series of papers (Victor et al., 1976; Oppenheimer et al., 1976a; Oppenheimer et al., 1976b; Oppenheimer et al., 1977a; Oppenheimer et al., 1977b). Atmosphere Explorer changed the state of upper atmospheric aeronomy in five brief years, providing a wealth of new information and detail, leaving little scope for speculation. An entire area of research came of age and Alexander Dalgarno played a major role in its transformation.


There was a major shift in emphasis in aeronomy in the 197Os, to problems concerned with assessment of the effects of industrial activity on stratospheric ozone. Eighteen years ago (Crutzen, 1970; Johnston, 1971), drew attention to the role of oxides of nitrogen as catalysts for removal of stratospheric ozone. They pointed out that exhaust gases of high-flying supersonic aircraft could add to the burden of stratospheric NO, with a potentially significant increase in the transmission of ultraviolet solar radiation to the surface. It was realized a few years later that small concentrations of chlorine radicals could have an even more potent impact. An ethereal concern changed to reality with the paper by Molina and Rowland, 1974, pointing out that decomposition of industrial chlorofluorocarbons could provide a major source of stratospheric chlorine. Stratospheric research has been intense and vigorous ever since. The need to assess the impact of industrial gases placed unprecedented new burdens on aeronomical research. What passed previously as acceptable error was no longer tolerable.

It was desirable after 1974 that we strive to describe ozone to at least the precision with which it could be measured. This was no small task. The initial step in production of stratospheric ozone involves photodissociation of molecular oxygen in the Herzberg continuum. Dalgarno had already anticipated the need for more precise data on the transmission of ultraviolet radiation through oxygen. His paper with Allison and Passachoff (Allison et al., 1971), examining absorption in the Schumann-Runge continuum, and the subsequent paper with Fang and Wofsy (Fang et al., 1974), presenting opacity distribution functions for absorption in the Schumann-Runge bands, filled an urgent need and were widely adopted in atmospheric models. More recently, Dalgarno responded again to the need for data, computing cross sections for photodissociation of metastable O,( 'D) (Dalgarno and McElroy, 1986), reacting rapidly to a suggestion that this process could represent a significant additional source of stratospheric odd oxygen over and above that from dissociation of ground state 0, (atoms released by photolysis of 0, react rapidly with 0, to form 0,).

Dalgarno has been an active participant also in attempts to interpret data from the fly-bys and orbiters of Mars and Venus (Fox et a!., 1977; Fox and Dalgarno, 1979a; Fox and Dalgarno, 1979b; Fox and Dalgarno, 1980; Fox and Dalgarno, 1983), and in analysis of data from the Voyager fly-bys of Jupiter (Broadfoot et al., 1977; Broadfoot et al., 1979; Sandel et al., 1979). His work with Fox on the escape of nitrogen from Mars is particularly notable (Fox and Dalgarno, 1980; Fox and Dalgarno, 1983). The atmosphere of Mars is enriched with the heavy isotope "N relative to 14N, indicating that substantial quantities of nitrogen have escaped over the age of the planet. A careful analysis of the processes responsible for production of fast atoms in the planetary exosphere is key to interpretation of the isotopic results and can

CONTRIBUTIONS TO AERONOMY 21

provide potentially unique clues as to the history of volatiles on Mars. The analysis with Fox provides the most comprehensive discussion of this problem in the literature to date.

Alexander Dalgarno has had a profound and lasting influence on the development of aeronomy over the past 35 years.

REFERENCES

Allison, A. C., McElroy, M. B., and Passachoff, N. W. (1971). Planer. Space Sci. 19, 1463. Bates, D. R., and Dalgarno, A. (1953). J. Armos. Terr. Phys. 4, 112. Bates, D. R., and Dalgarno, A. (1954). J . Atmos. Terr. Phys. 5, 329. Boggess. R. L., Bruce, L. H., and Spencer, N. W. (1959). J. Geophys. Res. 64, 1627. Broadfoot, A. L., Sandel, B. R., Shemansky, D. E., Atreya, S. K., Donahue, T. M., Moos, H. W.,

Bertaux, J. L., Blamont, J. E., Ajello, J. M., Strobel, D. F., McConnell, J. C., Dalgarno, A,, Goody, R., McElroy, M. B., and Yung, Y. L. (1977). Space Science Reviews 21, 183-205.

Broadfoot, A. L., Belton, M. J. S., Takacs, P. Z., Sandel, B. R., Shemansky, D. E., Holberg, J. B., Ajello, J. M., Atreya, S. K., Donahue, T. M., Moos, H. W., Bertaux, J. L., Blamont, J. E., Strobel, D. F., McConnell, J. C., Dalgarno, A,, Goody, R.. and McElroy, M. B. (1979). Science 204, 979-982.

Crutzen, P. J., (1971). Quart. J. Roy. Meteorol. Soc., 96, 320. Dalgarno, A. (1961). Anna1e.r de GPophysique 17, 16. Dalgarno, A., and Degges, T. C. (1968). Planer. Space Sci. 16, 125. Dalgdrno, A,, and Henry, R. J. W. (1965). Proc. Roy. Soc. A 288, 521. Dalgarno, A,, and Lane, N. F. (1969). J. Geophys. Res. 74, 301 1 . Dalgarno, A,, and McElroy, M. B. (1965a). Planer. Space Sci. 13, 143. Dalgarno, A., and McElroy, M. B. (1965b). Planer. Space Sci. 13, 947. Dalgarno, A,, and McElroy, M. B. (1986). Geophys. Rex Letts. 13, 660-663. Dalgdrno, A., McElroy, M. B., and Moffett, R. J . (1963). Planet. Space Sci. 11, 463. Dalgarno, A. and Moffett, R. J. (1963). Proc. Nut. Acad. Sci. India A. 33, 511. Dalgarno, A., and Walker, J. C. G . (1967). Planer. Space Sci. 15, 200. Dalgarno, A., and Walker, J. C. G. (1964). J . Armos. Sci. 21, 463. Dalgarno, A,, Henry, R. J. W., and Stewart, A. L. (1964). Plmet. Space Sci. 12, 235. Dalgarno, A., McElroy, M. B., Rees, M. H., and Walker, J. C. G. (1968). Planet. Space Sci. 16,

Dalgarno, A., McElroy, M. B., and Stewart, A. 1. (1969). J. Atmos. Sci. 26, 253. Dalgarno, A., Hanson, W. B.. Spencer, N. W., and Schmerling, E. R. (1973). Radio Science 8,263. Fang, T.-M., Wofsy, S. C., and Dalgarno, A. (1974). Planer. Space Sci. 22, 413. Fox, J. L., and Dalgarno, A. (1979). Planer. Space Sci. 27, 491-502. Fox, J. L. and Dalgarno, A. (1980). Planet. Space Sci. 28, 41-46. Fox, J. L., and Dalgdrno, A. (1983). J . Geophys. Res. 88,9027-9032. Fox, J. L., Dalgarno. A., Constantinides, E. R., and Victor, G. A. (1977). J. Ge0phy.r. Res. 82,

Hanson, W. B., and Johnson, F. S. (1961). Mem. Soc. Sci. Liege, Series 5,4, 390. Hinteregger, H. (1961). J . Geophys. Rex 66, 2367. Johnston, H. S. (1971). Science, 173, 517.

1371.

161 5- 1616.


Molina, M. J., and Rowland, F. S. (1974). Nature, 249, 810. Oppenheimer, M., Dalgarno. A,, and Brinton, H. C. (1976a). J. Geophys. Res. 81, 3762-3766. Oppenheimer, M., Dalgarno, A,, and Brinton, H. C. (1976b). J . Geophys. Res. 81, 4678-4684. Oppenheimer, M., Constantinides, E. R., Kirby-Docken, K. Victor, G. A., and Dalgarno, A.

(1977a). J. Geophys. Res. 82, 5485-5492. Oppenheimer, M., Dalgarno, A., Trebino, F. P., Brace, L. H., Brinton, H. C., and Hoffman, J. H.

(1977b). J. Geophys. Res. 82, 191-194. Sandel, B. R., Shemansky, D. E., Broadfoot, A. L., Bertaux, J. L., Blamont, J. E., Belton, M. J. S.,

Ajello, J. M., Holberg, J. B., Atreya, S. K., Donahue, T. M., Moos, H. W., Strobel, D. F., McConnell, J. C., Dalgarno, A., Goody, R., McElroy, M. B., and Takacs, P. 2. (1979). Science 206, 962-966.

Spencer, N. W., Brace, L. H., and Cariynan, G. R. (1962). J. Geophys. Res. 67, 157. Victor, G. A., Kirby-Docken, K., and Dalgarno, A. (1976). Planet. Space Sci. 24, 679-681.


l l ALEXANDER DALGARNO CONTRIBUTIONS TO ASTROPHYSICS DA VID A. WILLIAMS Mathematics Depurtmeni UMIST Munchester United Kingdom

I. Introduction . . . . . . . . . . . . .

A. Molecular Hydrogen . . . . . . . . . B. Interstellar Chemistry. . . . . . . . . C. Interstellar Shocks. . . . . . . . . . D. Structure of the Interstellar Medium . . .

111. Dalgarno’s Wider Contributions to Astrophysics References. . . . . . . . . . . . . .

11. Selected Areas of Contribution. . . . . . . . . . . . . . . . . 29 . . . . . . . . . . 29 . . . . . . . . . . 29 . . . . . . . . . . 31 . . . . . . . . . . 32 . . . . . . . . . . 33 . . . . . . . . . . 33 . . . . . . . . . . 34

I. Introduction

Dalgarno has made such a variety of major contributions to astrophysics that in a short article one can only highlight a few of his outstanding achievements. Many of his contributions have been in the area of molecular astrophysics. His detailed knowledge of atomic and molecular physics and his ability to calculate accurate data have enabled him to make fundamental advances when applying the data to astrophysical situations. He and his collaborators (of whom there are many, to whom he is an inspiration, and from whom he inspires great achievements) can claim to have laid the foundations-and built much of the structure-of molecular astrophysics.

11. Selected Areas of Contribution

A. MOLECULAR HYDROGEN

It is now generally accepted that much of the interstellar gas is H, and that the behavior of H, controls to a large extent the neutral component of the interstellar medium. Interstellar H, is observed in absorption by cold

29


~

30 David A . Williams

interstellar gas in the Lyman bands near 110 nm, and in emission in various vibrational and rotational infrared lines in interstellar regions of high excitation. Much of the data required for the interpretation of such observations has been calculated by Dalgarno and his collaborators in an extensive series of papers published over many years.

For more than two decades it has been accepted that H,, formed at the surface of dust grains, is predominantly destroyed in the interstellar medium following absorption in the Lyman bands:

The molecule is left in an excited vibrational level of the ground electronic state or in its vibrational continuum. Both the initial absorption and the subsequent emission are important processes in interstellar clouds. Allison and Dalgarno (1969) computed accurate transition probabilities for the B-X band system, and these calculations enabled an accurate estimate of the H, excitation and photodissociation processes to be made by Dalgarno and Stephens (1970). This work was brilliantly confirmed by the laboratory detection of the broad band continuous emission from the various B, v' levels into the vibrational continuum of the ground state, X (Dalgarno, Herzberg, and Stephens, 1970). This emission peaks around 160nm, and may be detectable in the interstellar medium.

Absorption in the Lyman and Werner bands and the subsequent cascade into the rotational-vibrational ladder of the ground state is a complex network which Dalgarno has explored in several papers, beginning with a study in 1969 (Dalgarno et al., 1969) and culminating in the paper with Black (1976) which follows the cascade in a detailed way. These calculations have become of great significance in infrared astronomy. The extensive regions of H, vibrational emission that are observed may be either shock excited or radiatively pumped. Use of Dalgarno's data suggests that both types of regions are detected.

Since H, is the dominant form of interstellar neutral matter, many processes involving H, may be significant. Dalgarno has examined, for example, the photoionization of H, by hard radiation (A < 91 nm) (Ford et al., 1975a), the input of kinetic energy into the gas consequent upon the photodissociation of H, (Stephens and Dalgarno, 1973), and absorption of radiation by vibrationally excited H, (Ford et al., 1975b). This last process is now recognized as important in regions of intense irradiation where the relaxation time of excited vibrational levels is comparable to the mean absorption time. Such situations may occur in star forming regions where young hot stars are still closely associated with dense molecular clouds.

CONTRIBUTIONS TO ASTROPHYSICS 31

B. INTERSTELLAR CHEMISTRY

In 1967 four types of interstellar molecules had been identified; in 1977 the figure was 75 and at the time of this writing, about 100 types of molecules (including isotopic varieties) are known to be present in the interstellar medium. The subject of interstellar chemistry is obviously one of growing complexity and its importance for astronomy is now well recognized. Molecules such as CO, NH,, H,CO, and C,H act as convenient tracers of the molecular gas, since these molecules are readily detected by their rotational emission spectra. Each molecular transition is favored in particular density and temperature regimes. A full understanding of the chemistry that gives rise to the molecules, however, brings with it a wealth of information about the conditions where the molecules are found. Not only are density and temperature indicated in such studies, but such parameters as radiation field, level of ionization, cosmic ray flux, relative abundances of the elements, and so forth are also suggested by these chemical models. Such models have been developed by many authors and are now quite detailed. The number of chemical species may be large (possibly several hundred) and many involve a large number of reactions (possibly several thousand). In addition, the astronomical model itself may be quite complex, involving either steady-state or time-dependent situations. Dalgarno and his collaborators have made such major contributions to the subject of interstellar chemistry that they have largely dictated the development of this exciting and rapidly developing subject.

Dalgarno and collaborators (Black and Dalgarno, 1973; Black, Dalgarno, and Oppenheimer, 1975) discussed the chemistry of CH and CH' and proposed a route in carbon chemistry in diffuse clouds involving the radiative association of C + with H,: C+(H2, hv) CH:, where CH; then takes part in further reactions. This proposal was an elegant solution to the problem of initiating the chemistry of diffuse interstellar clouds. The rate coefficient is not well known, but the value required to establish amounts of CH consistent with those observed is modest, and in harmony with various estimates. The observed high relative abundance of CH' along some lines of sight, however, requires a more efficient mechanism, and this may occur in interstellar shocks (see Section 1l.C).

Dalgarno and Black (1977) constructed a model of the chemistry occurring in the clouds of gas towards the star Oph. This was a model of exceptional sophistication for the time, and it became a benchmark against which all chemical models were compared. It included an extensive chemistry, involving known and predicted molecules, driven by the ambient radiation field and by cosmic ray ionization. The radiation field also controls the H, rotational

32 David A. Williams

distribution and so H, excitation was included. The model was successful in predicting chemical species in abundances largely similar to those observed, in proposing other chemical species to be abundant (later confirmed by observation), and in suggesting a plausible description of the astrophysical situation. Although much of the detail has changed in the succeeding decade, many of the robust conclusions are still valid, and the model is still a useful source and valuable inspiration to other workers in the field. The methods of this model have been applied to the molecular cloud towards Per (Black, Hartquist, and Dalgarno, 1978).

C. INTERSTELLAR SHOCKS

It is clear from observed interstellar line widths that superthermal motions are common in the interstellar medium. While the origin of such motions is unclear, their effect will be quite dramatic on the local scale through the action of shocks. If magnetic fields do not seriously modify the motions, then hydrodynamic shocks cause abrupt increases in density and temperature, with compression accompanying the subsequent cooling. Chemistry in such environments was studied by Dalgarno (Hartquist, Oppenheimer, and Dal- garno, 1980) with particular reference to sulfur-bearing molecules. At the high temperatures of interstellar shocks (several thousand K), the endothermic reaction of S with H, can be driven, forming SH. SH then enters a network of reactions. Similarly, the reaction of 0 and H, leads to abundant OH and H,O. Reactions of Si with OH form the very stable molecule SiO. Thus, Dalgarno and collaborators predicted high abundances of molecules such as H,S, SO, SiO in shock regions relative to cold cloud abundances. High abundances of sulfur molecules, in particular, are now regarded as indicative of shocks.

Magnetic fields may substantially modify the dynamical structure of shocks, since the ion- and neutral-fluids behave differently with respect to the field. Ion-neutral flows will occur. The discrete nature of the hydrodynamic shock may become a smoothly continuous magnetohydrodynamic shock wave, so that the chemical effects in such a shock are subtly different. The shock structure and its chemistry were described in a major and comprehensive study by Dalgarno and collaborators (Draine, Roberge, and Dalgarno, 1983). They showed that the effect of the magnetic field is to deposit the energy over a wide region. Thus, the elevated shock temperatures are generally lower in a magnetohydrodynamic shock but persist for a longer period than for a corresponding simple hydrodynamic shock. In addition, the ambipolar diffusion (where the ion-neutral relative velocity is a substantial


fraction of the shock speed) may drive chemical reactions which would otherwise be suppressed.

As a result of this study by Dalgarno and collaborators, observational criteria can be established to distinguish between the magnetic and nonmagnetic shock cases-a remarkable example of the power of chemical modeling. The ion-neutral flow also affects those dust grains which are charged. They suffer impact with neutral atoms and molecules at considerable speeds. Dalgarno and his collaborators have shown that this may be an important eroding effect on the ice mantles expected to accumulate on grains in dense clouds.

Dalgarno has also considered the effect of shocks in diffuse clouds (Pineau des For&ts et al., 1986) with particular reference to the problem of the interstellar CH + abundance. Hydrodynamic shocks were found to predict amounts of CH + inadequate to explain the observations, whereas magnetohydrodynamic models, in which the ambipolar diffusion drives the endothermic reaction C + ( H,, H)CH +, were more successful. Nonthermal internal energy in reacting species can affect the overall rate of their reaction. Graff and Dalgarno (1987) have examined oxygen chemistry of shocked gas and have shown that these effects substantially modify results of earlier calculations and bring the predicted OH abundances for diffuse clouds more into harmony with observations.

D. STRUCTURE OF THE INTERSTELLAR MEDIUM

A variety of components (hot, warm, and cool gas) can be identified in the interstellar medium. I t is important to understand how such a structure of density and temperature is maintained. Dalgarno and collaborators have described the heating and cooling processes characteristic of the various diffuse components (Dalgarno and McCray, 1972) and denser clouds (Dal- garno et al., 1974; Oppenheimer and Dalgarno, 1975). Jura and Dalgarno (1972) and Dalgarno and McCray (1972) have developed time dependent models of the cooling of interstellar gas, and describe in particular statistical models in which occasional supernova explosions heat the ambient gas. Such models are still the current view.

111. Dalgarno’s Wider Contributions to Astrophysics

In this brief article I have necessarily selected a very few of the astrophysical topics to which Dalgarno has contributed. These, and many other contributions not referenced here, have at their origin Dalgarno’s deep

34 David A . Williams

understanding of the diverse range of processes in atomic and molecular physics, and his unerring intuition where formal approaches are inadequate. He and his collaborators have produced much atomic and molecular data for use in an astrophysical context. These include an extensive series of results on charge transfer with applications to gaseous nebulae (e.g. Dalgarno, Heil, and Butler, 1981; Neufeld and Dalgarno, 1987; Preston and Dalgarno, 1987), photodissociation processes (e.g. as in OH) with applications to comets and to interstellar clouds (van Dishoeck and Dalgarno, 1984), and collisionally induced dissociation of CO and H, for use in interstellar shock models (Roberge and Dalgarno, 1982). His review (Dalgarno, 1983) of electron-ion and proton-ion collisions in astrophysics is definitive.

Alexander Dalgarno’s contributions to the study of planetary atmospheres and space physics require a separate review to be properly treated. As just one example, his work on nitrogen escape from the Martian atmosphere (Fox and Dalgarno, 1983) illustrates his imaginative use of accurate basic molecular data. His use of deuterium bearing molecules as probes of molecular clouds (Dalgarno and Lepp, 1984; Croswell and Dalgarno, 1985) testify to his imaginative and forceful application of atomic and molecular physics. His work characteristically anticipates developments in the subject, avoids very large scale computations, and presents clear conclusions based on reliable data and sound physical insight.

REFERENCES

Allison, A. C., and Dalgarno, A. (1969). J. Quant. Spectrosc. Radiat. Transfer 9, 1543. Black, J. H., and Dalgarno, A. (1973). Asrrophys. Lett. 15, 79. Black, J. H., and Dalgarno, A. (1976). Astrophys. J. 203, 132. Black, J. H., and Dalgarno, A. (1977). Astrophys. J. Suppl. 34, 405. Black, J. H., Dalgarno, A., and Oppenheimer, M. (1975). Asrrophys. J. 199, 633. Black, J. H., Hartquist, T. W., and Dalgarno, A. (1978). Astrophys. J. 224, 448. Croswell, K., and Dalgarno, A. (1985). Asrrophys. J . 289, 618. Dalgarno, A. (1983). In Physics of Ion-ion and Electron-ion Collisions. Plenum, New York, New

Dalgarno, A., Allison, A. C., and Browne, J. C. (1969). J. Amlos. Sc. 26, 946. Dalgarno, A., Herzberg, G., and Stephens, T. L. (1970). Astrophys. J. 162, L49. Dalgarno, A., de Jong, T., Oppenheimer, M., and Black, J. H. (1974). Astrophys. J. 192, L37. Dalgarno, A., Heil, T. G., and Butler, S. E. (1981). Astrophys. J. 245, 793. Dalgarno, A., and Lepp, S. (1984). Astrophys. J. 287, L47. Dalgarno, A., and McCray, R. A. (1972). Ann. Rev. Astron. Astrophys. 10, 375. Dalgarno, A., and Stephens, T. L. (1970). Astrophys. J. 160, L107. Draine, B. T., Roberge, W. G., and Dalgarno, A. (1983). Astrophys. J . 264, 485. Ford, A. L., Docken, K. K., and Dalgarno, A. (1975a). Astrophys. J. 195, 819.

York, p. 1.


Ford, A. L., Docken, K. K., and Dalgarno, A. (1975b). Astrophys. J. 200, 788. Fox, J . L., and Dalgarno, A. (1983). J. Geophys. Res. 88, A1 1 , 9027. Graf, M . M., and Dalgarno, A. (1987). Astrophys. J . 317, 432. Hartquist, T. W., Oppenheimer, M., and Dalgarno, A. (1980). Astrophys. J. 236, 182. Jura, M., and Dalgarno, A. (1972). Astrophys. J. 174, 365. Neufeld, D. A,, and Dalgarno, A. (1987). Phys. Rev. A., 35, 3142-3144. Oppenheimer, M., and Dalgarno, A. (1975). Astrophys. J. 200,419. Pineau des Forets, G., Flower, D. R., Hartquist, T. W., and Dalgarno, A. (1986). Mon. Not. Roy.

Preston, S., and Dalgarno, A. (1987). Chem. Phys. Lett., 138, 157-161. Roberge, W., and Dalgarno, A. (1982). Astrophys. J . 255, 176. Stephens, T. L., and Dalgarno, A. (1973). Astrophys. J . 186, 165. van Dishoeck, E., and Dalgarno, A. (1984). Astrophys. J. 277, 576.

Astr. Soc 220, 801.



l l ELECTRIC DIPOLE POLARIZABILITY MEASUREMENTS THOMAS M. MILLER Department of Physics and Astronomy University of‘ Oklahoma Norman. Oklahoma

BENJAMIN BEDERSON Department of Physics New York University New York. New York

I. Introduction . . . . . . 11. Bulk Measurements . . .

111. Atomic Beam Methods . . A. Indium and Thallium. . B. The Alkali Metal Dimers C. The Alkali Halide Dimers

IV. Conclusions . . . . . . Acknowledgments References . . . . . . .

. . . . . . . . . . . . . . . . . 37

. . . . . . . . . . . . . . . . . 40

. . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . . . 46

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. . . . . . . . . . . . . . . . . 58

. . . . . . . . . . . . . . . . . 58

I. Introduction

In 1962 Alex Dalgarno published an article in Advances in Physics entitled, “Atomic Polarizabilities and Shielding Factors.” It contained a comprehensive review of the state of calculations of atomic polarizabilities and shielding factors as of that time. It also happens that this article is a “Citation Classic.” According to Current Contents (see Dalgarno, 1978), the article was cited in the published literature 302 times in the period 1962-1976. Basically, it is a discussion of perturbation and nonperturbation methods of calculating these important atomic parameters (including higher order dc polarizabilities). In view of the nature of the present volume of Advances in Atomic and Molecular Physics, it is of interest to quote from Dalgarno’s explanation of the reasons for such a citation record (Dalgarno, 1978).

“This article was written in response to an invitation from Dr. B. H. Flowers (now Sir Brian Flowers), who had assumed responsibility for the

31


ISBN 1-12-003825-0

38 Thomas M . Miller and Benjamin Bederson

editorship of Advances in Physics and who was, I believe, attempting to extend the range of subject matter published in the journal beyond its earlier devotion to solid state physics. The subject was not specified, but I understood that he was seeking a review of some aspect of atomic and molecular physics. I had worked for several years on atomic perturbation theory and I had been impressed by the confusion which attended the development of an accurate description of the response of an atomic or molecular system to the application of a static electric field. The subject was a simple one, and the confusion lay in the propagation of the inevitable inaccuracies in the description of the unperturbed system, into the effects of the perturbation. There were two obvious ways of proceeding: in one, the unperturbed and perturbed systems could be treated simultaneously at the same level of approximation, and in the other, the error in the description of the unperturbed system could be ascribed to an additional perturbation and double perturbation theory used to identify the sources of uncertainty in the calculation of the response. It was not difficult to organize the two viewpoints into a unified presentation and I thought that a review with this end would clarify my understanding and perhaps be more generally useful. Perturbation theory can also be expressed usefully in variational terms, and a review of some aspect of perturbation theory would allow me to emphasize the close relationship between the two apparently disparate approaches. The history of atomic polarizabilities had been a long one in quantum mechanics and the theory had retained its earlier lack of mathematical sophistication. The introduction of more recent angular momentum techniques scarcely merited an original paper but could be conveniently incorporated into a review.

‘‘. . . The article was useful also in that its essential theme, a self-consistent theory of atomic and molecular perturbation, was later to be readily generalized to the description of frequency-dependent response functions, to the calculations of long range intermolecular interactions and to multiphoton processesses. It is not without interest to note that the article was a review which had no original content.”

Shielding factors are of considerable specialized interest because of the role they play in the determination of nuclear properties obtained from atomic spectroscopy experiments. Atomic and molecular polarizabilities, however, are of general interest because they appear as parameters in a large variety of atomic, molecular, and condensed matter properties. We refer the reader to earlier review articles by these authors and others for general discussions of polarizabilities and their methods of calculation and measurement (Bederson and Robinson, 1966; Buckingham, 1967; Bogaard and Orr, 1975; Miller and Bederson, 1977). An extensive tabulation of experimental atomic and molecular polarizabilities may be found in the Handbook ofchemistry and Physics beginning with the 66th edition and updated annually (Miller, 1987).

ELECTRIC DIPOLE POLARIZABILITY MEASUREMENTS 39

We would like to note one matter of historic interest concerning the polarizabilities of the alkali metals. The alkalis, as is well-known, have served as marvelous testing grounds for every variety of atomic structure, collision theory, and spectroscopy since the very beginnings of modern atomic physics. With regard to polarizabilities, the alkalis were the first atomic systems whose polarizabilities were studied using modern atomic beam methods (as opposed to measurements of dielectric constants in bulk). We will only mention the so-called “oscillator strength” formula, first employed by Dalgarno and Kingston (1959) to obtain surprisingly reliable values of the ground-state polarizabilities of the alkalis. Virtually no quantitative information is required on atomic wavefunctions or expectation values of electron correlation to obtain these values, other than the knowledge that the oscillator strength of the “resonance” transition from the ground state, nS -+ nP, is essentially unity for all the alkalis except lithium. The “oscillator strength” formula is

with CI the electric dipole polarizability (assumed scalar here); Ek, the energy of excited and ground states, respectively; and fo, the oscillator strength. The sum is taken over all allowed states including continuum states. For a one- electron system, the oscillator strength sum rule, Eq. (l), can be expressed as a bound rather than an equality,

m

where E, refers to the first electronic excited state, [although, in fact, for the alkalis the equals sign obtains, to a very good approximation, as first noted by Dalgarno and Kingston]. It was this observation that helped resolve discrepancies in experiment at the time. Table 1 shows the results of the

TABLE 1 ALKALI METAL ATOM POLARIZABlLITlES IN UNITS OF CM3

Li Na K Rb cs

Dalgarno and Kingston 24.4 24.6 41.6 43.8 53.7 (1959)-0scillator Strength Sum Rule k2.4 k2.5 f2.1 k4.2 k5.4

Molof et al. (1974a)-Experiment 24.3 23.6 43.4 47.3 59.6 - f0.5 f0.5 k0.9 k0.9 k1.2

Reinsch and Meyer, and Werner and 24.38 24.45 42.63 - -

Meyer (1976) -Theory


oscillator strength formula, and the most recent reliable results from both computation and experiment.

It is not the purpose of this short article to present a complete summary of recent advances in polarizability theory and experiment. We will instead concentrate on our own recent work, connecting it to theory where possible.

11. Bulk Measurements

The most accurate polarizabilities available today are determined from bulk measurements of a dielectric constant (or refractive index) for a gas of atoms or molecules. The determination of a dielectric constant is accomplished by measuring a capacitance change with and without the gas present. This procedure can be done quite accurately, particularly using low frequency ac fields in the GHz range (Newell and Baird, 1965). The polarizability is then given by M = ( K - 1)4xN, in the low density approximation, where N is the gas number density and K is the dielectric constant of the gas. (The experimental data are normally adjusted so that N may be taken to be the density of 1 atmosphere at 0°C.) These experiments are usually carried out at varying pressures up to many atmospheres in order to obtain virial coefficients, and the polarizability is obtained from the molar polarization in an extrapolation to zero pressure. The polarizability is 3/47cN, times the molar polarization, where N o is Avogadro’s number. Many such measurements were performed at Brown University in the late 1960s and early 1970s (Orcutt and Coie, 1967; Sutter and Cole, 1970; Bose and Cole, 1970, 1971; Nelson and Cole, 1971; Bose et al., 1972; Kirouac and Bose, 1973, 1976). For example, the polarizability of argon was measured to be 1.641 1 x cm3, accurate to 0.05% (Orcutt and Cole, 1967; also, Newell and Baird, 1965). Many molecular gases were studied by the Brown group and by Newell and Baird. It is regretable that even more measurements were not done, particularly since the method allows one to study the temperature dependence of the molecular polarizability. A criticism of the data interpretation for polar molecules has been given by Barnes et al. (197 l), who pursued similar work and pointed out the need to separate the effects of the induced and permanent moments.

Akin to these measurements are those of the refractive index of a gas. One obtains the dynamic polarizability corresponding to the frequency of the radiation by replacing the molar polarization in the formula above with the molar refraction.

These techniques necessarily give the average polarizability since the gas sample covers all orientations of the atoms or molecules. In some cases, such


as argon, where the polarizability is a scalar, there is no further information to be obtained. When the polarizability is anisotropic, Bridge and Buck- ingham (1966) and Alms et al. (1975) have shown in some elegant and difficult experiments that the polarizability anisotropy y = all - a l , can be deduced from measurements of optical depolarization in gases. Taking both sets of data together, Alms et al. extrapolated to zero frequency to find static values of the polarizability anistropy. The depolarization measurements yielded the ratio y/a. The authors then gave values of y using existing literature data on a. Persons using these results today should be aware that much better data for a are generally available and should redetermine y using the newer data. As an example, consider the important case of CO,. The extrapolated (static) value of y/a is 0.783 k 0.023, and the best average polarizability for CO, is (2.91 1 f 0.005) x cm3 (Orcutt and Cole, 1967); so we find y = (2.28 k 0.07) x cm3. This static value is 7 % smaller than the anisotropy at 458 nm.

Cai et al. (1 987) used a molecular-beam laser-Stark-spectroscopy technique to measure y for CO,. They found y = (2.215 f 0.007) x cm3, for the 00'1 state of CO,, in excellent agreement with the value deduced above from older measurements. Cai et a!. also measured y = (2.244 f 0.007) x

These bulk methods are unsurpassed in accuracy for atomic and molecular species that are unreactive at room temperature. It is tempting to say that for all other species one must resort to atomic beam techniques. In 1959, however, Alpher and White showed at atoms such as oxygen and nitrogen may be studied in a bulk manner using a shock tube to dissociate the molecular gas. They determined atomic polarizabilities for 0 and N from the refractive index at three different wavelengths. A few years later, Marlow and Bershader (1964) used the same method to measure the polarizability of atomic hydrogen. The shock tube results represent the only experimental data for 0 and N polarizabilities. Although the results are for visible frequencies, it is usually assumed that the static polarizabilities are within a few percent of the optical values, since the atomic excitation frequencies lie still higher. In principle, one could measure the average and nonspherical polarizabilities of any atom using the shock tube method, but the technique has not been used since the mid 1960s. The reason is that the greatest interest in polarizabilities is among physicists and chemists, but as a rule shock tubes are built and operated in aeronautical engineering groups. It is also important to note, however, that such methods deal very roughly with bulk samples, so that the presence of excited states, ions, and radiation can produce unknown systematic errors.

Thus we arrive at our primary topic, atomic and molecular beam measurements of polarizabilities. The beam method may never match the precision of

cm3 for the 02'0 state of CO,.


the bulk measurements; the low density of very narrow, well-collimated atomic beams is antithetical to good signal-to-noise. But the advantages of the atomic beam method are (1) the ability to study labile species, e.g., metastable-excited xenon, and (2) the possibility of state selecting the atoms or molecules in order to determine the tensor components of the polarizability. In our earlier review (Miller and Bederson, 1977), we discussed a number of modern beam measurements of polarizabilities: Salop et al. (1961), Pollack et al. (1964), Robinson et al. (1966), Hall (1968), Levine et al. (1968), Player and Sandars (1969), Nelissen et al. (1969), Ramsey and Petrasso (1969), Johnson (1970), English and MacAdam (1970), MacAdam and Ramsey (1972), Hall and Zorn (1974), Molof et al. (1974a, 1974b), English and Kagann (1974), Schwartz et al. (1974), Gould (1976), and Crosby and Zorn (1977). A few other experiments will be described below. Atomic beam methods have also been used to determine excited state polarizabilities (very accurately) by optical means (see Marrus et al., 1966; Khadjavi et al., 1968; Marrus and Yellin, 1969; v. Oppen and Piosczy, 1969; v. Oppen, 1969, 1970; Schmieder et al., 1971; Kaul and Latshaw, 1972; Kreutztrager and v. Oppen, 1973; Bhaskar and Lurio, 1974; Kreutztrager et al., 1974; Hohervorst and Svanberg, 1974,1975; Fabre and Haroche, 1975; Harvery et al., 1975; Sandle et al., 1975; Baravian et al., 1976; Hawkins et al., 1977; Gallagher et al., 1977; and Tanner and Wieman, 1988).

111. Atomic Beam Methods

A. INDIUM AND THALLIUM

In our own beam experiments, we employ several related though distinct techniques to extract polarizabilities from deflection (or null) data, using essentially a single apparatus. One class of such techniques, which is the one most generally used in virtually all beam deflection experiments, simply relies on analysis of the deflection profile of a beam passing parallel to the electrodes of an inhomogeneous electric field. The Stark energy of an induced dipole in an electric field is

where a is the (tensor) electric dipole polarizability. Accordingly, the force is


which, for scalar u, reduces to

F, = i a V ( E 2 ) . (5)

Eq. ( 5 ) generally appears in molecular beam literature as

where it is assumed that the electric field is strictly transverse to the beam axis, which is parallel to the pole faces. The operational expression that is used generally is

d E d z

F = F J = ~ ~ ~ 2 6 . (7)

although, of course, this can only be an approximation if the field is to be inhomogeneous.

Errors due to variations of E and of dE/dx are generally ignored or a t least minimized by use of the so-called “Rabi” condition, z = 1.2a, where z is the position of the beam as measured from the center of the convex equivalent “two-wire’’ field configuration (Ramsey, 1956), and a is the radius of the convex pole piece. In fact, if one wishes to employ deflection data to obtain polarizabilities with accuracies at the several percent level, it is necessary to calculate two-dimensional analyses of the full beam deflection profile, adjust- ing the polarizability as a parameter until a best fit is obtained to the data. It is, understood, of course, that the velocity distribution of the beam is known, either by assuming an ideal Maxwellian distribution of the source of a low- density beam, with source temperature accurately and reliably known, or by direct measurement. Alternatively, and more reliably, beam parameters and, to some extent, detailed knowledge of the velocity distribution, can be determined by use of polarizability “standards,” of which there are several now whose polarizabilities are known to an accuracy of a few percent, e.g., Li and Na.

A simpler, though somewhat less reliable deflection method may be called the “slope” method. A plot of the positions of the peaks of a family of deflection curves at constant voltage, as a function of the voltage squared, yields the polarizability directly, as the slope is proportional to the polarizability. This method is accurate only in the limit of large deflections, however, compared to the undeflected beam width. In most cases, this criterion is not well satisfied. A typical beam deflection experiment, without normalization, could yield results with errors of perhaps + 10%. As always, comparative results, i.e., taking the ratio of the unknown to a “standard” slope will be more reliable, with errors at perhaps the +4-5 % level. A detailed analysis of the slope method, which includes the systematic uncertainties, is contained in


the Molof PhD thesis (1974). A discussion of the effect of the finite beam size is given by Molof et al. (1974b) in their Appendix.

The second technique that we have used is based on the “E-H Gradient Balance” device developed by Bederson et al. (1960) for atomic beam state selection. The E-H gradient balance device depends for its operation on the effective magnetic moment and electric moment of an atom passing through it; since the magnetic moments of the atoms are known through the Breit- Rabi formula, one can deduce the electric moment tensor element of the state- selected atom. A major advantage experimentally in using this device is that the beam strength on axis is independent of the velocity distribution in the beam, in sharp contrast to the simple electric deflection process. The resolution of the apparatus is thereby increased without having to use a velocity selector, and the sensitivity is far greater.

In the E-H gradient balance apparatus, the electrodes of the applied inhomogeneous electric field serve also as the pole pieces for an applied magnetic field. The inhomogeneous magnetic field exerts a “Stern-Gerlach” force, acting on the effective magnetic moment p(mj) of the atomic magnetic substate Inj. For states of negative magnetic moment, this force is opposite to F,, which for atomic ground states is always dielectric, i.e., towards strong field. When the electric and magnetic fields are adjusted so that these two forces are equal and opposite, the balance condition

F , = F , (8)

(9)

and those atoms in the magnetic sublevel mj pass through the pole pieces undeflected, while other atoms are moved off the beam v i s . If the electric and magnetic fields are congruent, their gradient-to-field ratios, C, are the same. The electric field strength is determined experimentally from the applied potential by E(x, z) = K(x, z)V, where K(x , z ) is a geometric parameter assumed constant for a narrow beam at the “Rabi” position 1.2a (Ramsey, 1956). Substituting these quantities, the balance condition becomes

holds, and so

E dEldz = p(mj) dHldz

K 2 V 2 = p(mj)H (10)

from which the polarizability may be obtained if V and H are measured. In practice, in all of the beam measurements we have made in the past 15 years, we have normalized our results to a known polarizability, e.g., that for metastable-excited helium, whether using the balance method or the electric deflection method, because of the intrinsic unreliability of determining accurately the geometric constants. A sketch of our apparatus for measuring polarizabilities is shown in Fig. 1.


OVEN INTERACTION

(u R E G I O N

SIDE

V I E W T l f 'I

DETECTOR

0.

L

LD

0

' -0 r

FIG. I . A diagram of the polarizability apparatus at New York University. At the bottom of the figure is a detail cross section of the field electrodes, based on the Rabi "two-wire" design (Ramsey, 1956). Quartz spacers are marked with a Q. The design parameters are a = 0.159 cm, h = 0.1 72 crn, and c = 0.006 cm.

We discussed our work on the alkali atoms and the metastable noble gases in a review article earlier in the Advances series (Miller and Bederson, 1977). We have made only two measurements using the E-H gradient balance method since then, on atomic indium (Guella et al., 1984) and thallium (Stockdale et al., 1976). The indium case illustrates the limitations of the E-H gradient balance method. If the nuclear spin quantum number is large (9/2 for indium-1 I5,96 natural abundance), the number of magnetic sublevels is correspondingly large (20 for the ground 2P1,2 state of indium). Thus, the intensity of any one of the state selected beams is low compared to the full


beam intensity. Furthermore, if the polarizability is small, higher electric field strengths are needed to balance the magnetic force. Electric breakdown problems with the electrodes then limit the applied electric field strengths, resulting in lower resolution than optimal for indium. In Fig. 2 we compare E-H gradient balance peaks for indium and metastable-excited krypton. The polarizability of 2P2 krypton is nearly seven times that of indium, and the krypton peak intensities are strengthened by the fact that 88 % of the krypton atoms have no nuclear spin. The remainder contribute to the background.

For these reasons, the E-H gradient balance results for indium were combined with straight electric deflection data on indium to obtain a final result of (10.2 i 1.2) x cm3 (Guella e f al., 1984). At the temperatures we operated, the indium beam consisted mostly of the ground 2P1,2 state, but with an admixture of about 9 % 'P,/,. Our result compares quite well with the only modern theoretical calculations for indium. Liberman and Zangwill (1984) used a fully relativistic density-functional approach to calculate 9.66 x cm3 for the average polarizability of In(2P,,2), and 11.19 x

Atomic thallium has a lower nuclear spin (1/2 for the naturally occurring isotopes thallium-203 and -205), but since the dominant isotope only accounts for 70% of the beam intensity, the remainder adds to the background level. For 2P,i2 thallium, we found (7.6 * 0.8) x cm3 for the average polarizability (Stockdale et al., 1976). Two calculations give results close to this value: 7.74 x cm3 from Liberman and Zangwill (1984) and 7.1 1 x cm3 from a summation of theoretical oscillator strengths by Flambaum and Sushkov (1978).

In both of these cases, the polarizability is a scalar since J = 1/2 represents a spherically symmetric state.

cm3 for 1n(2P,,2).

B. THE ALKALI METAL DIMERS

The alkali metal dimers, e.g., Na,, cannot be studied with the E-H gradient balance method since they do not possess significant magnetic moments, is., of the order of Bohr magnetons. Nuclear magnetic moments are too small to be balanced against a large polarizability with our apparatus. The combination of electric and magnetic fields in the apparatus proved useful, however, in measuring the polarizabilities of the alkali metal dimers. The effusive alkali oven gave us beams that were mostly atomic, but with approximately 1 % diatomic molecules. The inhomogeneous magnetic field was used to rid the beam of the atoms; the molecules passed through virtually undisturbed. The inhomogeneous electric field was used to perform electric deflection measurements on the molecular component of the beam. For deflections of the beam

ELECTRIC DIPOLE POLARIZABILITY MEASUREMENTS

1 0 -

c. U ) .

k z 3

> - - a < F 8 - O

m L < - t w 4 - <

t z

a

47

1 I I I 1 1 1 1 1 1 + - 8 2 .. -

i KRYPTON

5 -

: - - . . - S -

(m,=t> 4 A - , -

s

h

v)

z 3 z a K

LD

t

a

t a a

k

Y

z v) z W I- z

INDIUM

F.4

mFz- 1

111111111 0 2 4 6 8 0 2 4 6 8

VOLTAGE (kV) (b)

FIG. 2. A comparison of E-H gradient-balance spectra for (a) metastable-excited krypton and (b) indium. The data in both cases were obtained with the beam detector on-axis and with fixed magnetic field strength, scanning the potential applied across the electrodes. For 'P, krypton, the magnetic field strength was 313 G, and for indium, 150 G.


which are large compared to the beam width, the peak position of the deflected beam should be proportional to V 2 , the square of the potential applied to the electrodes.

An example of the result of about 50 such measurements is given in Fig. 3. Measurements were made on both the atomic and molecular components of the beam at a particular detector position, Z , so that the molecular polarizability may be normalized to the atomic polarizability, which is known within 2 % from E-H gradient balance measurements. Molof et al. (1974b) has shown that the curvature in the plot of V 2 versus Z for small Z is due to the finite width of the beam.

These measurements were made almost 15 years ago (Molof et al., 1974b) and were discussed in greater detail in our earlier Advances article (Miller and Bederson, 1977). We bring up the subject in the present article because of renewed interest in the alkali dimers. At the time these measurements were made, theorists did not seem able to cope with the large internuclear spacing of the dimer nuclei and the large polarizabilities. Calculations on electron interactions with these dimers (primarily through the long range polarization potential) require incredibly large angular momentum quantum numbers.

n L J l A - L J 0 ’L - 0 0.2 0.4 0.6 0 0.2 0.4 0.6

BEAM DEFLECTION (mm)

FIG. 3. Typical data for the “slope” method of determining polarizabilities. The position of the peak of the deflection spectrum is plotted against the square of the potential applied to the field electrodes, here for K and K, at a source temperature of T = 579 K. The slope, for deflections large compared to the beam width, is proportional to T/a.

BEAM DEFLECTION (mm)


Since that time, however, theorists have been able to calculate electron scattering cross sections (Padial, 1985; Michels et al., 1985) and neutral and ionic spectroscopic constants and polarizabilities (Muller and Meyer, 1986 and references therein).

Table 2 gives a comparison of our alkali-metal dimer polarizabilities to those calculated by Muller and Meyer (1986). Muller and Meyer used an all- electron SCF plus valence CI method they developed, in which intershell correlation effects are accounted for by effective core polarization potentials. Their method is efficient enough to permit them to use a minicomputer for this work. They have previously found excellent agreement with experiment for spectroscopic constants, dipole and transition moments, coupling constants, and lifetimes (Muller et al., 1984). Their calculations of polarizabilities show similarly excellent agreement with our measured value for Li,, but the calculated results fall outside of the experimental uncertainty limits for Na, and K,. Muller and Meyer estimate the theoretical uncertainty at 2%. It is worth noting that their calculations for the alkali metal atoms (Muller et al., 1984) agree very well with our measurements (Molof et al., 1974a). For the purposes of comparison with experiment, Muller and Meyer calculated polarizabilities averaged over a thermal distribution of vibrational and rotational states. The resulting polarizabilities a t elevated temperatures are 5-7 higher than those corresponding to re . (A number of earlier calculations of the polarizabilities of Li, and Na, are listed by Muller and Meyer (1986))

The discrepancy between experiment and theory for Na, and K, are troubling because the experimental normalization between the atomic alkali

TABLE 2 A COMPARISON OF EXPERIMENTAL AND THEORETICAL AVERAGE POLARIZABILITIES FOR THE ALKALI METAL

DIMERS, AT ELEVATED TEMPERATURES ~~ ~

Molecule Temperature Theory" Experimentb

Li, 990 K 33.3 34 & 3 Na, 1 3 6 39.9 30 + 3

K, 569 72.2 61 & 5 Rb2 534 68 k I

515 ~ 9 1 k 7 cs2

37'

~

The polarizabilities are given in units of " Miiller and Meyer (1986). ' Molof, Miller, Schwartz, Bederson, and Park

(1974b). except as noted. ' Greene and Milne (1968).

cm3.


and the molecular alkali seems foolproof the two beams are necessarily congruent as they enter the electric field region, and are detected with the same detector. There are two differences: the atomic beam is 100 times as intense as the molecular beam, and the molecular measurement is made in the presence of a magnetic field on the order of 1 kG. Neither difference should affect the results. Nevertheless, we intend to repeat these measurements in an attempt to clear up the discrepancy. The electric deflection measurement of Greene and Milne (1968) is unpublished. Since only one other measurement was made with their apparatus (for the NaCl dimer), it is difficult for us to assess the accuracy of that work.

One other measurement of the polarizability for Na, and K, exists, from a remarkable experiment by Knight et al., (1985) in which the polarizabilities of alkali metal clusters were determined for cluster sizes from 2 to 40 atoms for Na, and n = 2, 5, 7, 8, 9, 1 1 , and 20 for K , . Their data are shown in Fig. 4. (The polarizability of Na,, is given as 600 x 10-24cm3.) Germane to our topic, Knight et al. measured polarizabilities of Na, and K,, normalized to the respective atomic polarizabilities of Molof et al. (1974a). The Knight et al. results correspond to very low internal temperatures since a high pressure nozzle source was used. Therefore, it seems proper to compare the results of Knight et al. with the polarizabilities calculated by Miiller and Meyer for I,,

which correspond to a temperature close to 0 K. This comparison is shown in Table 3, where we also give the parallel and perpendicular components of the

FIG. 4. Average polarizability per atom for clusters of sodium, relative to the polarizability of the sodium atom.

ELECTRIC DIPOLE POLARIZABILITY MEASUREMENTS 5 1

TABLE 3 A COMPARISON OF THEORETICAL AND

EXPERIMENTAL ALKALI METAL DIMER

TEMPERATURES)

POLARlZABlLlTIES AT re (VERY LOW

Molecule Theory“ Experimentb all a, U

Liz 44.3 24.3 31.0 -

KZ 102.5 51.6 68.6 12 * 5 Na, 55.6 29.2 38.0 38 2

-

The polarizabilities are given in units of

* Miiller and Meyer (1986). cm3.

Knight et a/., (1985).

polarizability calculated by Muller and Meyer (1986). Only the average polarizabilities have been measured thus far. It is interesting to note the large size of the ratio of the parallel to perpendicular polarizabilities: 1.82 (Li,), 1.90 (Na,), and 1.99 (K,), representative of the large internuclear spacing in these molecules.

The deflection results of Knight et al. (1985) presume that the alkali clusters do not possess permanent dipole moments. A permanent electric dipole moment would tend to cancel some of the deflection of the molecular beam that one would observe for an induced moment alone (see below), leading one to underestimate the induced polarizability. While the alkali atoms and dimers clearly do not possess permanent dipole moments, Ray et al. (1985) have calculated the dipole moment of Li, to be 0.375 D (for the most likely geometry, isosceles), and they found 0.045 D for the most stable geometry for Li, (parallelpiped). Ray et al. conclude, “Since most stable Li, systems should have non-zero dipole moment, experimental determination of dipole moments can help determine the cluster geometries.” Permanent dipole moments as small as those calculated by Ray et al. should have a very small effect on electric deflection experiments for the alkali clusters.

In our 1977 Advances review, we noted that as yet no one had used a supersonic beam in a polarizability measurement. Knight et al. (1985) have accomplished this. The supersonic beam has the advantages that the beam intensity is stronger, so that even a pulsed beam may be used, and the narrow velocity distribution of the supersonic beam yields sharper deflection profiles. One must know, of course, what the velocity distribution is, because the deflection in the inhomogeneous electric field is velocity dependent. Knight et al. used a time-of-flight measurement to determine the velocity distributions.


In 1977 we also lamented the lack of a good “universal” detector that would detect any atom or molecule, unlike the surface ionizers used at that time. Electron bombardment ionizers only ionize about 1 in every lo4 or lo5 of the atoms or molecules in the beam, and we had only moderate success with one we had used for H, (Schwartz, 1970). Resonance induced fluorescence provides a good detector candidate because each atom or molecule could give off a hundred photons in the detector region. Since 1977 this method has fluorished, so that there have already been four international conferences held on the subject since that time. Such a “universal” detection scheme, however, has not been used yet in beam polarizability experiments, to our knowledge. Knight et al. (1985) have used an ultraviolet lamp to ionize their alkali cluster beams. The resulting ions were then mass analyzed and detected. (Knight et al. (1986) used this system, in fact, to determine ionization potentials of the alkali clusters.)

Finally, we should note the deflection and molecular beam resonance experiments performed by Dagdigian et al. (1971), Graff et al. (1972), and Dagdigian and Wharton (1972), who measured the average polarizability [(40 f 5 ) x 10-24cm3] and the polarizability anisotropy [(24 k 2) x

cm3] for NaLi. To our knowledge there are no corresponding theoretical data.

C . THE ALKALI HALIDE DIMERS

The alkali halide dimers are interesting because they are composed of highly polar molecules bound together; the opposing dipoles cancel, and the dimer thus has no permanent dipole moment. The structure of the alkali halide dimer is planar rhombic. The dipolar bond strength is 2-3 eV. Our interest in the polarizabilities of the alkali halide systems began with low- energy electron scattering, where the polarization potential dominates. (There are fundamental questions regarding electron interactions with polar molecules, and practical applications in energy conversion devices and, by analogy, in rare-gas halide excimer laser media.) The alkali halide dimers are readily produced in an oven loaded with the appropriate salt; indeed, they are difficult to avoid. At our operating temperatures of about 1OOOK (vapor pressures of to torr), the dimers constitute 10-40% of the vapor.

The average polarizabilities of all 20 of the accessible alkali halide dimers-from (LiF), through (CsI), and omitting, obviously, francium and astatine compounds-have been measured in the New York University laboratory (Kremens et al., 1984; Guella et al., 1988). The data contain information on the polarizabilities of the monomers as well, but as yet we have not perfected the data analysis to the point of deducing monomer polarizabilities in the face of the huge permanent dipole moment deflections.


An interesting feature of this experiment is that no attempt is made to separate the monomer and dimer beams, as we did with the alkali atoms and dimers prior to detection of the deflected beams, because, (fortuitously) the net monomer and dimer forces and deflections are in opposing directions (see below).

The deflection of the alkali halide dimer component of the molecular beam in an inhomogeneous electric field is the same as described above for other atoms or molecules. The deflection, z , of an infinitesimal beam is proportional to the component of the polarizability a, along the applied field direction, and the square of the applied potential, V :

where A is a geometric factor, A = CK2L,(L, + 2L2)/2, C is the field gradient to field ratio characteristic of the electrodes, K relates the field to the applied potential I/, L , is the length of the electrodes, L, is the field-free distance from the end of the electrode pair to the detector, rn is the molecular mass, and u is its velocity. The deflected beam is broadened by the distribution of velocities in the beam, but the net result is a single peak of detected intensity toward the strong side of the applied electric field.

The deflection of a molecule possessing a permanent dipole moment is quite different. The amount and direction of the deflection of a polar molecule in the inhomogeneous electric field depends on the size and orientation of the dipole in the applied field. The displacement of the molecular beam in the plane of the detector is an old problem (Ramsey, 1956). For a rigid linear rotator in a uniform electric field, the first nonvanishing Stark energy, E, is

J ( J + 1)(2J - 1)(2J + 3)

where p is the permanent dipole moment, E is the electric field strength, B is the rotational constant in cm-I, and J and M are the rotational quantum number and its projection along the electric field direction. (We estimate that the next higher order Stark energy contributes only about 0.01% for the alkali halides.) We will neglect the effect of the polarizability term for now; it amounts to at most a few percent.

For the experiments described here, the average J value is 50-150, so we may use a large-J approximation, and the Stark energy reduces to the essentially classical result

where cos 0 = MIJ.


The deflection of the polar molecule by an inhomogeneous field, as measured in the detector plane is

(14) mu2 4J2Bhc

for fixed u, J , and 0 (or M ) . The deflected beam profile Z(x) is the undeflected beam profile Zo(x) shifted by a distance z :

A V ~ p2(3 C O S ~ e - 1) z = - .

Z(x) = Io ( z - x) (15)

for fixed u, J , and 8. In our experiments, there is a thermal distribution F , ( J ) of J values, a

uniform distribution F2(@ of M / J values, and a modified Maxwellian velocity distribution F3(u) appropriate for an effusive oven and a flux-type detector. The actual beam intensity along the detector path is given by the average of Z(x) over these distribution functions:

Z(X) = Z ~ ( Z - x ) F , ( J ) F ~ ( B ) F , ( u ) ~ J dB d ~ . (16) s The normalized distribution functions are

Bhc(2J + 1) k T k T F , ( J ) =

and

There is a dramatic difference in the deflections of the monomeric and dimeric components of the alkali halide molecular beam. The nonpolar dimers are deflected toward the strong field side of inhomogeneous electric field. The polar monomers may be deflected in either direction-or possibly not at all-depending on whether M is positive, negative, or zero. The maximum in the monomeric beam deflection profile occurs for negative effective moments, i.e., toward the weak. field side of the inhomogeneous electric field, because the population of molecules possessing negative M exceeds that for positive M (they subtend a larger solid angle). This extreme broadening of the monomeric deflection profile allows one to resolve the sharper beam profile of the deflected dimers even though the dimer may be a


FIG. 5. Deflection of an alkali halide beam composed mostly of RbCl, with about 5 % (RbCI),. The beam detector position was scanned about the apparatus axis, with potentials applied to the field electrodes: (a) zero, (b) 2.5 kV, and (c) 3.0 kV.

small fraction of the full beam intensity. An example is given in Fig. 5, where the undeflected beam profile is also shown.

The polarizability of the dimer can be determined in two ways. First, if data such as shown in Fig. 5 are obtained for a range of applied potentials, the position of the maximum in the dimeric deflection profile should be proportional to V 2 and to the polarizability. In this case, a comparison must be made to a species of known polarizability (normalization). Alternatively, the deflected beam profile may be fit using the deflection equations above, with variable parameters being the dimer polarizability, the background intensity, and the fraction of dimers in the beam. Here, the normalization may be to the known dipole moment of the monomer, or to a separate data run for a species of known polarizability. The greatest uncertainty in the normalization to a species of known polarizability is in reloading the beam oven and setting it exactly along the path taken by the alkali halide beams. In a separate publication (Guella et al., 1988) we will give the details of the data analysis; results of the two methods agree within about 10%.


TABLE 4 POLARIZABILITIES OF ALKALI HALIDE DIMERS IN UNITS OF

CM3

Dimer Polarizabilit y

Since the permanent dipole moment of the monomers is known, we plan eventually to fit the monomeric deflection profile to obtain the induced electric moment of the monomer. Very accurate data will be required. In Table 4 we give the dimer polarizabilities published thus far (Kremens et al., 1984). Generally speaking, the polarizabilities increase with molecular weight, presumably reflecting the increase in the ionic polarizabilities with atomic number and the increase in the bond lengths as the sizes of the atomic ions increase.

It is possible to model the alkali halide monomers and dimers assuming simple electrostatic bonds (Brumer and Karplus, 1973; Berkowitz, 1958a, 1958b, 1980; O’Konski and Higuche, 1955). The net polarizability of the dimer may be taken as that due to the alkali positive ions, halide negative ions, and a bond polarizability which depends on the strength of the ionic bonds. The bond polarizability reflects the change in energy as the bond is stretched. Using measured and estimated frequencies for the normal modes of oscillation in the alkali halide dimers, and the “effective” ionic polarizabilities of Brumer and Karplus (1973), we find that net polarizabilities calculated for the alkali halide dimers agree with the measured values within 25 % at worst (Guella et al., 1988). A precise comparison is clouded by the quality of the vibrational frequency estimates for many of the molecules and by the inherent uncertainties in our polarizability data.

IV. Conclusions

In our 1977 Adoances review, we noted that experimentalists were lagging behind theorists in dealing with polarizabilities, especially atomic polarizabilities. This situation was not the fault of the experimentalists, but was due


‘1 [UV, k:, , , . . , , , , , , , , , , , , , , , , , . , , , , I I . , , , , , ,

largely to Alex Dalgarno’s leaps ahead of everyone else, as we noted in the Introduction to this article. We see encouraging signs for the experimentalists now that Dalgarno is more preoccupied with astrophysics. In 1977, accurate atomic polarizabilities existed primarily for the two ends of the periodic table-the alkali metals and the noble gas atoms-and some inroads have been made since, notably with indium and thallium. More molecular polarizabilities have been measured with beam techniques recently; we have discussed the cases of the alkali dimers, the alkali halide dimers, and alkali clusters, and we applaud the introduction (by Knight et al., 1985) of supersonic beams and “universal” beam detection into this small corner of physics. An efficient universal beam detector will create vast opportunities for atomic and molecular beam measurements of electric moments.

Fortunately, theorists have been equally active in this area, if not more so. The greatest advances have been in techniques for handling electron correlation. It is appropriate to quote Miiller and Meyer (1984) in this context: “Bottcher and Dalgarno [1974] were the first to give a systematic approach to the treatment of atomic interactions with inclusion of molecular polarization terms.” We have discussed some of the theoretical results available

,

POLARIZABILITY

FIG. 6. Calculated polarizabilities of the elements, in units of cm3, plotted against atomic number. The dots represent experimental values. Except for the alkaline earths, the experimental uncertainties are within the size of the dot.


today, for example, Miiller and Meyer’s (1986) work on the alkali dimers (using a minicomputer!). In 1977, calculated polarizabilities for most of the open shell atoms were very crude; most of the “recommended” polarizabilities that we gave (Miller and Bederson, 1977) had 50% uncertainty bounds. Doolen and Liberman (1984) have calculated polarizabilities for all atoms using a fully relativistic density-functional method described by Zangwill and Soven (1980). These results are plotted in Fig. 6. One surprise (for us) in these results is that the polarizability of francium is less than that of cesium by Is%, presumably due to a severe relativistic tightening of the valence orbital in francium. These caiculated results are a bit low compared to accurate experimental values for the alkalis, and a bit high for the noble gas atoms, but seem perfect for indium and thallium, within the accuracy of the experimental data. An attempt to scale the calculations to accurate experimental data has been made (Miller, 1987). The results for the open shell atoms are estimated good within 25 %.

ACKNOWLEDGMENTS

The experimental work at New York University described in this article was supported by grants from the National Science Foundation and the Department of Energy.

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1 .

60 Thomas M. Miller and Benjamin Bederson

Molof, R. W. (1974). PhD Thesis, New York University, New York, New York. Molof, R. W., Schwartz, H. L., Miller, T. M., and Bederson, B. (1974a). Phys. Rev. A 10, 1131. Molof, R. W., Miller, T. M., Schwartz, H. L., Bederson, B., and Park, J. T. (1974b). J . Chem. Phys.

61, 1816. In Eq. (A9) of this paper, replace II by II’ and w by w4. Muller, W. and Meyer, W. (1986). J . Chem. Phys. 85,953. Muller, W., Flesch, J., and Meyer, W. (1984). J . Chem. Phys. 80, 3297. Nelissen, L., Reuss, J., and Dymanus, A. (1969). Physica (Utrecht) 42, 619. In Table I of this

0.02)A3 (J. paper, the average polarizability of H, is misprinted and should read (0.75 Reuss, private communication).

Nelson, R. D. and Cole, R. H. (1971). J . Chem. Phys. 54,4033. Newell, A. C. and Baird, R. C. (1965). J . Appl. Phys. 36, 3751. OKonski, C. T. and Higuche, W. I . (1955). J . Chem. Phys. 23, 1175. Orcutt, R. H. and Cole, R. H. (1967). J . Chem. Phys. 46, 697. Padial, N. T. (1985). Phys. Rev. A 32, 1359. Player, M. A. and Sandars, P. G. H. (1969). Phys. Lett. A 30,475. Pollack, E., Robinson, E. J., and Bederson, B. (1964). Phys. Rev. 134, A 1210. Ramsey, N. F. (1956). Molecular Beams. Oxford Univ. Press, London and New York, Reprinted,

Ramsey, N. F. and Petrasso, R. (1969). Phys. Rev. Lett. 23, 1478. Ray, A. K., Fry, J. L., and Myles, C. W. (1985). J . Phys. B 18, 381. Reinsch, E.-A. and Meyer, W. (1976). Phys. Rev. A 14, 915. Robinson, E. J., Levine, J., and Bederson, B. (1966). Phys. Rev. 146, 95. Salop, A., Pollack, E., and Bederson, B. (1961). Phys. Rev. 124, 1431. Sandle, W. J., Standage, M. C., and Warrington, D. M. (1975). J . Phys. B 8, 1203. Schmieder, R. W., Lurio, A., and Happer, W. (1971). Phys. Rev. A 3, 1209. Schwartz, H. L. (1970). PhD Thesis, New York University, New York, New York. Schwartz, H. L., Miller, T. M., and Bederson, B. (1974). Phys. Rev. A 10, 1924. Stockdale, J. A. D., Efremov, I., Rubin, K., and Bederson, B. (1976). Int. Con/: At. Phys., Abstr. p.

Sutter, H. and Cole, R. H. (1970). J . Chem. Phys. 52, 132. Tanner, C. E. and Wieman, C. (1988). Phys. Rev. A, in press. v. Oppen, G. (1969). Z . Phys. 227, 207; (1970). 2. Phys. 232,473. v. Oppen, G . and Piosczyk, B. (1969). Z . Phys. 229, 163. Werner, H . J . and Meyer, W. (1976). Phys. Rev. A 13, 13. Zangwill, A. and Soven, P. (1980). Phys. Rev. A 21, 1561.

Oxford, 1985.

408.


1 1 FLOW TUBE STUDIES OF ION-MOLECULE REACTIONS ELDON FERGUSON Uniuersitk de Paris-Sud Centre dOrsuj, Orsu-v, Ccde.u, Fruncci

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . 11. Ion-Molecule Reactions at Thermal Energies, . . . . . . . . . . .

111. Negative Ion Kinetics . . . . . . . . . . . . . . . . . . . . IV. Vibrational Energy Transfer in Ion-Neutral Collisions . . . . . . . . V. The 0; + CH, --t H, COOH’ + H Reaction: A Detailed Mechanistic Study.

VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

61 63 69 71 76 78 79 79

I. Introduction

Twenty-five years ago, a flow tube technique was introduced for the measurement of ion-molecule reaction rate constants: the so-called “flowing afterglow” (FA), developed in the Commerce Department Laboratories, Upper Atmosphere and Space Physics Division of the National Bureau of Standards, Boulder, Colorado. This laboratory subsequently became the Aeronomy Laboratory of the National Oceanic and Atmospheric Adminis- tration. The first FA publication (Ferguson et al., 1964) reported measurement of the reactions that control the loss of helium ions in the earth’s ionosphere. At that time only a few dozen ion-molecule reaction rate constants had been measured and none of the reaction rate constants controlling the ion chemistry of the earth’s atmosphere had been reliably determined. After these first measurements were obtained, Hef + 0, and He’ + N,, i t was learned that Sayers and Smith (1964) had reported measurements of these reactions at a Faraday Society meeting shortly before, obtained using a stationary afterglow (SA) technique. The close agreement of the measurements provided the first indication that the FA was indeed a suitable method for quantitative rate constant measurements. In the SA technique, the evolution of the ion composition of a weakly ionized plasma was followed as a function of time after cessation of the discharge. This was the first gas discharge physics approach to ion-molecule reaction studies and

61


NRN n.i?.nnia?c.n

62 Eldon Ferguson

thus was the predecessor to the FA. Several investigators, in addition to Sayers and Smith, pursued this technique, notably Fite, Hasted, Lineberger and Puckett, and several successful studies were carried out. Serious problems with uncontrolled reactant states, however, as well as severe limitations in chemical versatility, restricted the use of SA’s to a minor role in ion chemistry. By 1977, when Albritton (1978) tabulated the ion-molecule reaction rate constants measured to that date in flow reactors, over 1600 reaction rate constants had been published and seven laboratories were actively engaged in the application of the technique to ion-molecule interaction studies. A detailed description of the technique was presented in Volume 5 of this series (Ferguson et al., 1969a) and by Twiddy (1974) and McDaniel et al. (1970). Since that time major advances in the technology have occurred and over 20 laboratories are now using ion flow tube techniques. Continued contributions to our understanding of the ion chemistry of planetary atmospheres and the interstellar medium, as well as to our understanding of basic aspects of ion-molecule interactions at the molecular level are being made.

A superb presentation of the history and development of the flow tube technology, with a detailed discussion of the experimental aspects has been given by Graul and Squires (1988). This valuable review includes 34 figures displaying various experimental configurations and details and contains over 400 references. Smith and Adams, of the University of Birmingham, have described in detail the major refinement of this technique, the Selected Ion Flow Tube (SIFT) (Smith and Adams, 1987). Smith and Adams have been leaders in the development of the flow tube technology, providing many important innovations, including the SIFT. They have been prolific contributors to ion-molecule reaction knowledge, most particularly to the field of molecular astrophysics, but to atmospheric physics and basic mechanistic knowledge as well.

The present review is modest in scope and has a different orientation. An attempt is made here to distil some of the understanding that has been acquired about ion-molecule interaction processes from the past 25 years of measurements. We will consider a few aspects of charge-transfer processes, some chemical reactions, and the recent vibrational energy transfer studies that are leading to useful generalities and insights into ion neutral interaction mechanisms.

The contribution that flow tube studies have made to basic molecular physics and ion chemistry is less well recognized than their contributions to ionospheric physics and astrophysics. This is a consequence, in part, of the fact that the motivating interests of the originators of these studies leaned heavily in the direction of atmospheric physics problems and later in the direction of astrophysical problems. A large fraction of the results, particularly the results of the first decade or so, have been published in the atmospheric

FLOW TUBE STUDIES OF ION-MOLECULE REACTIONS 63

and astrophysical literature, rather than in the chemical literature, and the data have been analyzed in much more detail from the point of view of their applications in these areas than from the point of view of their significance for reaction mechanisms.

Furthermore, flow tubes do not play the dominant role in the area of reaction mechanistic studies that they enjoy in their aeronomical and astrophysical applications by virtue of their chemical versatility. Other very powerful techniques, including mass spectrometry and ion cyclotron resonance spectrometry (whose temporal development has coincided with that of flow tubes almost precisely), have been applied primarily to basic studies and are in far wider use. For the most part, they have been developed and applied by chemists with stronger orientations toward mechanistic studies.

It might even be questioned whether flow techniques can be very useful at the current level of basic interaction studies, considering the diverse and sophisticated technology now available. Lasers and molecular beams permit individual reaction studies in kinematic detail, leading closer and closer to the holy grail of chemistry, state-to-state chemistry. It is the point of the present review that flow tube studies can indeed contribute to basic understanding of reaction mechanistic details, and in many cases in quite a unique manner. This potential of the technology has scarcely been tapped yet.

11. Ion-Molecule Reactions at Thermal Energies

Thermal energy ion-molecule reactions have a strong propensity to proceed efficiently along the lowest energy pathways between reactants and products, without steric or Franck-Condon, or sometimes even spin conservation restraints. Very few exothermic ion-molecule reactions are slow, whether charge-transfer or ion-atom interchange reactions. This, of course, makes the slow reactions more interesting and it is these that offer the best prospects for revealing new insights into mechanistic behavior. One of the earliest discoveries from Flowing Afterglow (FA) studies was that exothermic thermal energy charge-transfer reactions involving molecular neutrals were generally efficient, occurring usually in more than 5 % of the collisions and often at near the collision rate. One aspect of the FA that allowed many charge-transfer measurements to be made easily was the physical separation of the ion production region from the position of neutral reactant addition, which allowed ions to be produced from neutrals of high ionization potential in the absence of the neutral reactant, necessarily of lower I.P. A similar advantage applies in the case of negative ion charge-transfer.

The generality of fast charge-transfer implies that there are not simultaneous constraints on energy resonance and Franck-Condon factors. This was

64 Eldon Ferguson

not at all evident 25 years ago from a theoretical viewpoint. As an example, the reaction

N: +O,+O: + N 2 + 3 . 5 e V (1)

has a near energy resonance for 0, (v = 35) for which the ionization FC factor for 0, (v = 0) -+ 0; (v = 35) is vanishingly small. [Reaction (1) has a relatively small rate constant, 5 x lo-" cm3 s-', which is -5% of the Langevin collision rate constant = 2 x e G . l Reaction (1) has a negative temperature dependence at 300 K which has been interpreted as implying that the reaction occurs via a long-lived intermediate complex (Ferguson et al., 1969b), the negative temperature dependence presumably reflecting the increased lifetime with lowered energy content of the complex. The negative temperature dependences of charge-transfer and thermal energy ion-atom interchange reactions, however, are invariably weaker than either the experimental temperature dependences of lifetimes deduced from three-body associatibn rate constants or theoretical lifetimes deduced from statistical theories (Bates, 1979; Herbst, 1980). The temperature dependence of reaction (l), for example, is approximately T- I , whereas the statistical theory lifetime is approximately T-'. This emphasizes the fact that, theoretically, thermal energy ion-molecule reactions have not been dealt with satisfactorily.

No cases have been found where thermal energy charge-transfer rate constants exceed the collision rate constant, therefore, there is no evidence from rate constants that electrons jump distances greater than the impact parameters implied by typical values of k, - 7-8 i% with high probability.

All of the above suggests that during the intimate reactant encounter, that is during the long-lived complex lifetimes, the ion-molecule systems can relax to low energy configurations, thus Franck-Condon barriers are largely missing. Recent Innsbruck measurements (Richter et al., 1987) provide an example of this phenomenon:

Kr' + SF, + S F i + F + Kr,

AE(SF:/SF,) < IP(Kr) = 14.00 eV.

(2)

leading to

(2a)

On the other hand, a number of photon and electron impact experiments have led to values of AE > 15 eV (Rosenstock et al., 1977; Levin and Lias, 1982), referring clearly to vertical AEs which, in this case, are substantially larger than the adiabatic AE. In addition, Babcock and Streit (1981) measured

SF: + NH, + NH: + SF,. (3)


When the results of reactions (2) and (3) are combined with the results of Kiang and Zare (1980) on the dissociation energy of SF,-F, one obtains AE(SF:/SF,) = 13.98 k 0.03 eV (Tichy et ul., 1987). Reactions (2) and (3) both occur at nearly the collision rate constant, although both have less than 0.1 eV exothermicities, in spite of the fact that there are substantial changes in geometry upon ionization of the neutral in each case. Recently, the reaction

(4) HCI’ + SF, --* SF: + H F + C1

was also found to occur at near the collision rate, although it is only 0.2 eV exothermic (Tichy et al., 1987). This “adiabatic” behavior of ion-molecule reactions permits the determination of ion energies by bracketing techniques in favorable cases, sometimes where no other values are available.

The reaction

HCl’ + CF, + CF; + H F + C1 ( 5 )

has been found to be almost thermoneutral (Tichy et al., 1987) which yields the result that AP(CF:/CF,) = 14.2 f 0.1 eV, again very much lower than the “vertical” AP’s reported in the literature. In the case of both reactions (4) and (5 ) , the positively charged H atom of HCI’ abstracts an F- ion to produce ClHF which then breaks the weakest bond (CI-H) to yield C1 + HF.

The forces involved in ion-neutral attraction (when chemical bonds are not involved) are the classical electrostatic forces : ion-induced dipole and dispersion forces for all molecules, ion-quadrupole and ion-dipole forces for molecules with significant quadrupole or dipole moments and a somewhat less well appreciated electron transfer contribution, depending on a relatively insensitive electron exchange integral but critically on the difference in ionization potentials between the two molecules involved. A consideration of this electron transfer term (Gislason and Ferguson, 1987) has rationalized what seemed to be several anomalous vibrational relaxation and three-body association rate constants. The complex lifetime and formation probability depend critically on the interaction well depth between the ion and neutral, so that three-body association rate constants that are proportional to the lifetime also depend on the well depth, as do vibrational relaxation and other energy transfer processes. In addition, the electron transfer interaction may play an important role in providing anisotropy of the potential, which is critical for vibrational relaxation.

The importance of electron exchange is most evident when AIP = 0, that is for clusters between ions and their parents. The bond energies of CO’ .CO, N:*N,, NO’-NO and 0:-0, are 1.1, 1.0,0.60 and 0.43 eV respectively, far in excess of the expected electrostatic energy, approximately 0.2 eV, estimated

66 Eldon Ferguson

from D(O: -N2) = 0.24 eV and D(NO+.N,) = 0.20 eV (Keesee and Castle- man, 1986). The magnitudes of the bond energies of the above four ion-parent clusters correlate fairly well with the ionization 0 + 0 Franck-Condon factors (0.95,0.90,0.16,0.19), illustrating the role that FC factors play in the electron transfer stabilization. There are cases in which true chemical bonding, either attractive or repulsive, outweighs the electrostatic and electron transfer attractive forces. One example is the charge-transfer reaction

(6) Oy4S) + NO(%) + NOf('X) + O(3P) + 4.35 eV,

for which k < lo-'' cm3 s - l at 300 K, increasing rapidly with KE to 2 x lo-'' cm3 s- ' at 4 eV relative kinetic energy (Ferguson, 1975). In this case, the correlation of the reactant states is with a 3A NO: state that lies slightly above the separated O'(4S) + NO('n), leading to an activation barrier behavior. Such situations are rare and, in general, one does not know the details of the intermediate ion state involved in ion-molecule reactions. The isoenergetic (with (6)) charge-transfer reaction

H+('S) + NO('II) + NO+('X) + H('S) + 4.35 eV (7)

occurs with a very large rate constant, k = 1.9 x lop9 cm3 s-', which is essentially the collision rate constant. In this case, the reactants and products correlate adiabatically through the HNO'('X) ground state. Reactions (6) and (7), being isoenergetic, therefore have the same Franck-Condon factors and energy defects which thus clearly do not control the reaction probabilities. If one knows the state correlation of reactants with the intermediate ion, it should be possible to predict whether a reaction will be fast or slow, as in the case of reactions (6) and (7). It is not sufficient, however, to know the states of reactants and products, even so far as spin selection rules are concerned. For example, the reaction

(8)

occurs on every collision (Fehsenfeld et al., 1966a), even though spin is not conserved. Presumably, the O'(4S) and CO,('X) states have access on the attractive potential surface to a quartet state of CO: and in this state, during the complex lifetime, a quartet + doublet conversion occurs. This conversion need not be so efficient, if the complex traverses the quartet-doublet potential surface seam repeatedly. Another case in which spin conversion must occur is the charge-transfer

(9)

for which k = 1.2 x cm3 s- ' from thermal energy to approximately 2 eV (Ferguson, 1983). This is essentially the collision rate constant, whereas

0+(4s) + C O , ( ~ X ) + 0:(2r1) + c o ( 1 c )

HzO'('B) + NO,('A) -+ H,O('A) + NO:('X) + 2.8 eV


the statistically weighted (singlet/singlet + triplet) rate constant would be only one fourth of this. Evidently, spin conversion of the three fourths of the collisions leading to a triplet state occurs, presumably in the very stable complex (well depth approximately 3.6 eV corresponding to protonated nitric acid). This would be analogous to the familiar neutral quenching reaction

o('D) + NJZ) -, o ( 3 ~ ) + N,(IZ) (10)

discussed by Tully (1977) and Zahr et al. (1975) in which multiple curve crossings lead to spin exchange. Spin conservation is usually not a consideration in ion-molecule reactions which normally involve a doublet ion and a singlet neutral, with similar products, so that spin selection rules have rarely been tested. A chance to do so on a large scale came from a recent series of measurements of hydrocarbon ion reactions with the high spin (quartet) nitrogen atom (Federer et al., 1986). The result was found to be a wholesale violation of spin conservation. For example:

C,Hl('A) + N(4S) --t HC,N+(,A) + H,('C) + 0.43 eV (11)

and

C,Hi('A) + N(4S) -+ H2C3N+('A) + H(,S) + 0.30 eV (12)

which both occur with large rate constants, 2.2 and 1.3 x lo-'' cm3 s- l , respectively. Spin conservation thus does not appear to be a useful predictive tool for ion-molecule reaction occurrence. If one can deduce, from spin or other criteria, however, that attractive curves into the "complex" domain do not exist, as in the case of reaction (6), that is another matter. Presumably, a long-lived complex must be accessed for spin conversion. Otherwise stated, we believe that spin conversion should occur in the exit channel and not in the entrance channel of collisions.

Several examples of rather remarkable molecular behavior in ion-neutral complexes will be discussed. The recent work of Adams and Smith (1987) relating to the synthesis of cyclic-C,H, in interstellar clouds, in addition to its prime importance in astrophysics, reveals some very interesting ion chemistry. The reaction of linear C3H+ ions with H, has two product channels

(1 3)

-,C,H+ + H, (14)

C3H+ + H, 2 C3H:

a three-body association (reaction (1 3)) and slightly endothermic, approximately 0.04eV, H atom abstraction from H, (reaction (14)). From the differences in reactivities of the linear and the more stable cyclic C3H: and C3H: ions, Adams and Smith have been able to determine that the linear

68 Eldon Ferguson

C3H+ ion, which is 2.3 eV more stable than the cyclic C3H+ ion, reacts in a binary thermoneutral reaction to produce cyclic C 3 H i , and in a ternary association to produce equal amounts of linear and cyclic C3H: at both 300 K and 80 K. The closing of the C, ring by addition of either H or H, is quite a remarkable demonstration of complicated concerted rearrangement occurring on the C,H:* potential surface. The potential surface in this case is almost entirely chemical since the low polarizability of H, leads to only very weak electrostatic attraction to ions.

Another example of remarkable concerted activity in a long-lived complex is the reaction (Fehsenfeld et al., 1971)

NO+(H,O), + H,O + H3O+(H,O), + HNO,, (15)

which is the major source of the proton hydrates that dominate the positive ion chemistry of the D-region of the earth's ionosphere, approximately 60 to 80 km altitude. Since theory shows that the oxygen atoms of the three water molecules clustered to the NO+ bond to the N atom (Pullman and Ranganathan, 1984), a substantial rearrangement is required in the collision with another water molecule to account for the concerted formation of HONO and H+(H,O),. The three oxygen atoms in this case are now equivalently bound to a central proton. This rearrangement must occur in a concerted fashion in order for the necessary exothermicity to exist. The potential barrier to the rearrangement, if any, cannot exceed 0.08 eV, in view of the magnitude of the rate constant, k = 8 x lo-" cm3 s-'.

A novel illustration of the lowering of usual neutral reaction barriers in ion complexes was demonstrated in a series of ion catalyzed reactions (Rowe et al., 1982) including

N a + * 0 3 + NO 4 NO, + 0, + Na+, k = 6.5 x lo-' ' cm3 s - ' (16)

and

Li+*N,O, + NO -+ Li+*N204 + NO,, k = 1.2 x lo-" cm3 s-'. (17)

For reaction (16), the rate constant for 0, + NO + NO, + 0, is approximately 5 x lo3 faster when the O3 is clustered to Na' (or Li+) than for the unclustered neutrals. In the case of reaction (17), the exothermic gas phase reaction of N,O, with NO has not been detected, k < lo-" cm3 s-', so that there is a greater than nine order of magnitude enhancement due to the ion clustering. In this case, the N,04 product stays clustered to the Li', thereby offsetting the bonding of N,O, to the Li', which would otherwise make the reaction endothermic. The mechanism involved has not been clarified, however the potential barrier lowering is dramatic.


111. Negative Ion Kinetics

The first laboratory measurements of associative-detachment reactions of negative ions were made in the FA system (Fehsenfeld et a/., 1966b) and the FA technique has dominated this field, in part because many of the associative detachment reactions involve reactions with chemically unstable atoms or neutrals. The chemical bonding potential of the radicals provides the electron detachment energy. Early examples measured included:

H - + H + H, + e, k = 1.8 x lop9 cm3 sC1

0- + 0 + 0, + e, k = 1.9 x lo-'' cm3 s- '

0- + N + N O + e, k = 2.2 x lo-'' cm3 s-'

F- + H + HF + e, k = 1.6 x cm3 s- '

(1 8)

(19)

(20)

(21)

(Schmeltekopf et a/., 1967; Fehsenfeld et al., 1966b, 1973). These reactions occur by autodetachment after the reactants cross the neutral association product potential curve where autodetachment becomes exothermic. Autodetachment is usually fast. Reaction (18) is of major astrophysical importance and has been calculated theoretically (Browne and Dalgarno, 1969). Reaction (19) is of interest because it implies that approximately one third of the 240; curves arising from O-(,P) + O(3P) are attractive, which was quite unexpected at the time but has subsequently been supported by the theoretical calculation of Michels (1975). Reaction (20) has the same implica- tion for N O - : namely, greater than one third of the O-(*P) + N(4S) potential curves must be attractive into the NO + e intersection. A calculation of the NO- potential curves has not yet been carried out. Reaction (21) leads to only one 'C potential curve and the fast reaction establishes that this curve is attractive. At the time of the measurement, four theoretical calculations had been made, two of which yielded attractive potential curves and two of which yielded repulsive potential curves (Fehsenfeld et a/., 1973). Associative detachment reactions thus have simpler physical interpretations than is usually the case for charge-transfer or ion-atom interchange reactions.

In the case of molecular rather than atomic neutral reactions, exothermic associative detachment may be either fast or slow. One of the first associative detachment reactions measured (Fehsenfeld et al., 1966b) was

0- + H, + H,O + e + 3.6 eV, k = 6.0 x lo-'' cm3 s-'. (22)

The question of mechanism arose, namely, how does the oxygen insert into the hydrogen bond. This occurs in two steps: H atom abstraction, followed by OH- associative detachment

0- + H, +(OH- - H)* -+ H,O + e (23)

70 Eldon Ferguson

(Ferguson, 1970). This was confirmed by observing a small OH- signal which increased with average reactant KE, since the time available for the OH- + H associative detachment to occur in the intermediate complex decreases with energy. The onset of OH- production occurred at lower energy than the onset of OD- from the 0- + D, reaction (McFarland et al., 1973), consistent with a faster H separation from OH- due to the lighter mass of H. It is necessary that the initial H atom abstraction step be exothermic for this mechanism to be effective, and it is exothermic by 0.16 eV. The ensuing associative detachment

cm3 s K 1 (24)

was independently measured to be fast (Howard et al., 1974). By contrast, the exothermic associative detachment insertion reactions

(25)

OH- + H + H,O + e + 3.2 eV, k = 1.4 x

S - + H, + H,S + e + 0.9 eV, k <

C- + H, + CH, + e + 2.0 eV, k <

cm3 s-l

and

cm3 s - l (26)

were not observed, and in each case the necessary H atom abstraction first step is endothermic (by 0.4 eV and 0.6 eV respectively) so that the overall exothermic process cannot occur.

In an analogous way, the reaction

0- + CzH4 --* H,O + CCH; + 1.5 eV, k = 7 x lo-'' cm3 s-l (27)

has been found to be rapid (Lindinger et al., 1975), whereas the more exothermic process,

CH; + C,H, --* CH, + CCH; + 1.7 eV (28)

does not occur (DePuy et al., 1987). This is presumably a consequence of the fact that H atom abstraction from ethylene by 0- to produce OH- is exothermic (by 0.2 eV) while CH; abstraction to produce CH; in (28) is endothermic (by 0.6 eV).

Reactions involving H, and H i transfer are well known in organic positive ion chemistry. For example,

(29) C2H2 + CzD4 -P C2H,f + CZD4H2, k - lo-'* cm3 s - '

and

C2Hf + C,D, + C,Df + C,H4D,, k - 10" cm3 s f l (30)

(Lias and Ausloos, 1975). Lias and Ausloos have shown that the process occurs by two successive atom transfers in a complex and that a necessary condition is that each step be exothermic. In general, it is found that the


double transfer, H, or H;, dominates the single transfer, H or H-, in the observed products.

A very important technological advance has recently been made (Van Doren et al., 1987) with the construction of a tandem FA-SIFT instrument with considerably enhanced sensitivity and resolution. With this instrument, it is possible to study reactions with the naturally abundant "0- isotope ion. Surprisingly, it has been found that isotope exchange, l 8 0 - + CO +

l60- + C"0, competes with associative detachment, 0- + CO + CO, + e, to the extent of approximately 12 %. This requires a modified view of this process in which it was believed that the two thirds efficiency of associative detachment reflected an attractive autodetaching 0- ('P) + CO(1X)211, (in linear configuration) potential curve and a repulsive nonautodetaching 5: (linear) curve and that the lifetime of the 'nu resonance was too short (approximately 10- l 5 s) to allow for isotope exchange. The new experimental result suggests that non-linear configurations are accessed in collisions that are not autodetaching, presumably the exchange occurs at angles where the CO, curve lies below CO, + e. This should provide an incentive for more detailed potential surface calculations. Isotope exchange is also found to occur in "0- collisions with SO,, NO, N,O, H,O, CO, and 0,. This new capability has great promise as a valuable probe into reaction dynamics and potential curve topologies. It is very interesting to note that the vinylidine anion produced in reaction (27) is unstable toward an isomerization- autodetachment,

(31)

This metastable ion persists for at least 0.01 s in the FA experiments and for lifetimes greater than 1 s in recent triple ICR experiments carried out at Orsay by Heninger, Mauclaire and Marx (private communication). By contrast, the neutral vinylidine isomerizes to acetylene on a time scale of 10- l 1 s or less (Oshamura et al., 1981) with a barrier of less than 0.2 eV.

CCH; + HCCH + e + 1.4 eV.

IV. Vibrational Energy Transfer in Ion-Neutral Collisions

In the past several years, the first quantitative measurements of vibrational quenching of small molecular ions in collisions with neutrals have been obtained using FA'S and FDT's. Experimentally, this involved the extension of the monitor ion technique, used previously for the study of reactions of electronically excited metastable ions, for example O$(a41T,), NO+(A3X), O'(,D) (Glosik et al., 1978). A neutral, with which the excited species reacts but with which the ground state ion does not, is introduced into the flowing

72 Eldon Ferguson

stream before the ion mass analysis. The monitor ion appearance is then a measure of the concentration of excited ions and the disappearance of this monitor ion signal with added neutral reactant then leads directly to a rate constant for the excited state loss, either by reaction or quenching. In the case of N O + ions, ground vibrational state NO+ ions do not react with CH,I, the reaction being endothermic, while vibrationally excited ions exothermically charge-transfer on every collision. Vibrational quenching of NO+(u) ions was carried out in Innsbruck (Dober et al., 1983; Federer et al., 1985) and measurements of 0; (0) quenching were carried out in Boulder (Bohringer et al., 1983a,b). In the case of O:(u), Xe served as a monitor for u > 0, SO, for u > 1 and H,O for u > 2. The concentration of ions dropped rapidly with u and only u = 1 and u = 2 states were measured. The Innsbruck and Boulder experiments were carried out in FDT's so that the kinetic energy dependence of the quenching could be measured. Morris et al. (1988) have greatly extended the NO+(u) quenching studies and have obtained the first temperature dependences of quenching rate constants.

In most cases studied, thermal energy vibrational quenching is very efficient, in contrast with neutral vibrational quenching occurring with rate constants greater than cm3 s-', or requiring less than lo3 collisions for relaxation. The rate constants generally decrease with KE or T in the 300 K temperature range. The large rate constants and their negative energy dependences clearly establish that ion vibrational relaxation is generally due to the long range attractive forces. By contrast, the vibrational relaxation of neutral diatomics such as N, and C O is very inefficient at 300 K, requiring approximately lo8 or more collisions and the efficiency increases with KE (or T), the quenching arising from the repulsive interaction potential (Landau and Teller, 1936). The ion vibrational relaxation situation is thus very much like the vibrational relaxation of neutral radicals with collision partners having substantial chemical bonds (Fernando and Smith, 1980, 1981) for which large quenching probabilities have been measured.

A simple model has been developed for this process (Ferguson, 1986)

AB+(v) + C $ [AB+(v).C]*2AB+(v' < v) + C. (32) ku

In cases where k,, 4 k , the quenching rate constant is given by

This is the usual case, but there are cases in which k,, > k , in which quenching occurs on every collision and k , = k , , examples being the quenching of O:(v) by H 2 0 and NO+(v) by HN,. Since k , is proportional to the complex lifetime, 2, it follows that there will be a strong correlation with the


attractive well depth, controlled by such parameters as the neutral polarizability, dipole moment and the difference in ionization potentials between the neutral and the parent of the ion. Using these parameters to estimate the interaction, a very strong correlation between k , and well depth has been found (Gislason and Ferguson, 1987). Since the rate constant for three-body association also is proportional to the complex lifetime, z, (indeed the product k,z) , k 3 = kck , z , where k, is the complex stabilization rate constant, there is a strong correlation between k , and k 3 . One can eliminate k c z from the expression for k 3 and k , to obtain an expression for kvp, the vibrational predissociation rate constant k,, = k,k, /k3, in terms of the experimental k , and k3 and the approximately calculable collisional stabilization rate constant k, . A number of values of k,, and several upper and lower limits so determined are given in Table I. The remarkable finding is the limited spread in values: For the most part, they lie between lo9 and 10'' s-'. This is a significant addition to the limited quantitative data available on such energy transfer rates. The situation in these collision complexes is markedly different from that in the "half-collisions'' exemplified by vibrational predissociation of weakly bound neutral van der Waals molecules, in which rates are found ranging from greater than 10l2 s-' to less than lo4 s-' (Celii and Janda, 1986). This is an interesting generality which as yet has not been theoretically explained.

If the complex-vibrational predissociation model is valid, this regularity should find its explanation in the extension of the theory of the vibrational predissociation of weakly bound van der Waals molecules (Beswick and Jortner, 1981); an extension from the bound-free half collision van der Waals vibrational predissociation to the free-free vibrational quenching collisions. It remains to be determined whether this extension will yield the relatively invariant values of k v p . The situation in ion vibrational quenching is similar to that in ion-molecule reactions generally, in that no energy resonance criteria are evidenced, that is there is no enhancement of rate constants when near resonant V -+ V transfer can occur, as compared to cases where only V + 7: R transfer is possible (Ferguson 1986).

NO+(u = 1) + N,(u = 0) -+ NO+(u = 0) + N,(u = 1) + 14 cm-'

In one particular case, the quenching

(34a)

(34b)

was determined to occur by I/ -+ T exchange (Reaction (34b)) rather than near resonant V + V exchange (Reaction (34a)) (Ferguson et al., 1984). This is consistent with the complex model in which translational energy is transferred into internal rotational energy in forming the complex, prior to V + T transfer. Thus, in the case of Reaction (34a), V + I/ transfer is

+ NO+(u = 0) + N,(u = 0) + 2344 cm-'

74 Eldon Ferguson

TABLE I VIBRATIONAL PREDISSOCIATION RATE CONSTANTS

[AB+(u = l).C]' + AB+(u = 0) + C k",

~

ABf(u) C k,10-12(cm3 s- ' ) k310-30(~m6 s - l ) k,,(109 s-I)

0: Ar 1 .O" 0.35b 2.1 Kr 11" 0.94b 7.9 H, 2.5" 0.053b 2.5 N, 1.9" 1 .o I .4 co, low 26' 2.7 so, 57" 550' 0.7 H,O 1200" 250 > 3

NO' Kr 0.2d 0.097b < 0.7 0, < 0.3d 0.09 < 0.7 N, 2Sd 0.3 6.3 co 4.9d 1.9 1.8 co, llOd 1.4 5.5 CH, 50d < 58 > 0.7 NH, 1500' 880 > 1.2 H,O 90" 150 0.4

co + N, 1301 80 0.9 0; CO, - 27g 47 - 0.35

" H. Bohringer, M. Durup-Ferguson, D. W. Fahey, F. C. Fehsenfeld, and E. E. Ferguson, (1983). J . Chem. Phys. 79,4201.

extrapolated from M = He, 80 K to M = N,, 300 K. extrapolated from M = He, 200 K to M = N,, 300 K by 3.8 (T/300) 1/2 + 1,

R. A. Morris, A. A. Viggiano, J. F., Paulson, and F. Dale, (1988). J. Chem.

W. Federer, W. Dobler, F. Howorka, W. Lindinger, M. Durup-Ferguson,

I' C. E. Hamilton, V. M. Bierbaum and S. R. Leone, (1985). J . Chem. Phys. 83,

1 = total number of rotational degrees of freedom.

Phys. in press.

and E. E. Ferguson, (1985). J . Chem. Phys. 83, 1032.

601. T. M. Miller and W. C. Lineberger, private communication.

rendered endothermic in the separation coordinate. [This determination was made by observing that the reverse reaction to reaction (34a), V -+ V transfer from N,(u = 1) to NO+(u = 0) does not occur, k < cm3 s - l , invoking detailed balance.] This result has some geophysical significance, implying that the highly vibrationally excited N, molecules in auroras will not pump the infrared emitter NO'.

When the attractive well depth is less than the collision energy, one expects the vibrational quenching to be dominated by the long range forces. The situation is exactly the same in principle for molecular ion vibrational


relaxation as for neutral molecule vibrational relaxation, in practice, the difference is that, in the usual situation for ion-molecule collisions, the interaction well depth is greater than kT. This is a more exceptional situation for small neutral-neutral collisions. In one case so far,

Ol(v = 1) + Kr -+ O:(U = 0) + Kr (35)

the well depth ( - 0.33 eV) is large enough to dominate the interaction at thermal energy ( - 0.026 eV) and lead to a large quenching rate constant, k , = 1.1 x lo - ' ' cm3 s- ' (Z = 69 collisions/quenching), and yet small enough so that this energy can be exceeded in a FDT and a minimum in k , was observed, broadly in the range of - 0.33 eV, below which k, decreased with KE and above which k , increased with KE due to the influence of the short range repulsive forces (Kriegel et al., 1986).

Both the vibrational quenching and the excitation of N: by He,

N:(u = 1) + He + N,'(u = 1) + He (36)

have been studied recently (the first ion-neutral in which both have been measured) as a function of KE (Kriegel et al., 1988). Detailed balance is found to apply, indicating that there are no significant effects due to rotation. If the vibrational excitation produced a very non-Maxwellian rotational distribution, and vibrational relaxation was very sensitive to the rotational distribution, then a detailed balance would not have been observed since the quenching was done on a thermalized 300 K rotational distribution and not on the (unknown) distribution resulting from vibrational excitation. This appears to be the first test of detailed balance for molecular-ion vibrational excitation-quenching. It is also the first fit of ion vibrational quenching to a Landau-Teller plot, -In k , - KE- ' I 3 . The vibrational quenching of N:(u =

1 ) by He appears to have a minimum near 300 K (the energy dependence of k , approaches zero there) implying an interaction well depth broadly in the range of 0.026 eV for N l - He.

The only case of a slow ( k , < cm3 s - l ) quenching for O:, NO+ or N i ions by a molecular neutral (except for H2) is the quenching

(37)

for which k , < 3 x cm3 s- ' (Morris et al., 1988). This is a result of the chemical interaction overriding the electrostatic interaction as a consequence of NO: being a stable ion whose ground state is a singlet which is not accessed on the triplet interaction curve of reaction (37). By contrast, O2(lAg) molecules quench NO+(u) very efficiently, k , = 3 x lo-'' cm3 s- ' (Dotan et al., 1985) by virtue of the attractive singlet chemical well interaction which they do access.

NOC('& u = 1) + 0 2 ( 3 Z ) + NO+(u = 0) + 0 2

76 Eldon Ferguson

V. The 0; + CH, + HzCOOH+ + H Reaction: A Detailed Mechanistic Study

Measurements on the reaction

0; + CH, + H,COOH+ + H (38)

have recently been carried out in five different laboratories using conventional SIFTDT techniques for the most part but also involving measurements down to 20 K in the enormous supersonic flow tube at Meudon (Rowe et al., 1984), the so-called CRESU (CinCtique de RCactions en Ecoulement Super- sonique Uniforme). The most extensive data set available for any ion- molecule reaction of this complexity ( > 5 atoms) was obtained; T dependence from 20 to 560 K, kinetic energy dependence from 0.01 to 2 eV, dependence on 0: vibrational state and pressure dependence (Bohringer and Arnold, 1986) and data on five isotopes of CH,. As a consequence of this, a detailed reaction mechanism has been developed, providing perhaps the most detailed mechanistic understanding available for any ion-molecule reaction of such complexity (Barlow et al., 1986).

The reaction involves a sequence of three successive steps, proceeding over a double minimum potential surface with a large intermediate barrier. The reaction is slow at 300 K and increases at both higher and lower temperature. The double potential minimum surface as an explanation for slow exothermic ion-molecule reactions has been established by Brauman and co-workers over the past several years (Olmstead and Brauman, 1977). The 0; + CH, reaction provides one of the most detailed and quantitative elucidations of the surface involved. The first step in the reaction is the formation of long- lived complex in a 0.4 eV potential well, essentially an electrostatic well depth with a substantial added electron transfer stabilization. The lifetime of the complex is approximately 10-9-10-'0 s at 300 K. Unimolecular decomposition back to reactants dominates over reaction (38) by 200:l at 300 K. Reaction to products is irreversibly initiated by H- transfer from CH, to O:, over an approximately 0.35 eV potential barrier, leading to the cation of CH,OOH. The chemical stability of this ion is the origin of the second well of 2.4 eV depth (measured from separated reactants). The ratio of H- transfer to unimolecular decomposition is governed by the relative densities of states of these two exit channels from the complex. A statistical model thus yields the rate constant and its temperature dependence, and somewhat less satisfactorily the isotopic ratio for the CH, and CD, rate constants. The measured ratio of approximately three for the CH,/CD, reaction rate constants is larger than the expected statistical ratio of two (the ratio of rotational constants) and may be due to H atom tunneling through the hydride ion transfer barrier.


After the irreversible H - transfer to produce a very energetic CH,OOH' ion, this ion ejects an H atom weakly bound to C (with a typical R+-H bond energy of 1.4 eV), leading to the final product H,COOH+ + H. The resulting reaction is the least exothermic of six exothermic reactions possible for 0: + CH,. It is, however, the only mechanistically simple reaction in the sense that no more than one concerted bond breaking-bond formation act is required in any single step. One of the most exothermic possibilities for 0: + CH, reaction would be to produce HC(0H); + H (protonated formic acid) which is 4.8 eV exothermic, as contrasted to 1 eV for reaction (38). This was actually proposed as the product of reaction (38) by three published measurements which misidentified the reaction products. The production of protonated formic acid, however, would require the concerted breaking (and making) of four bonds and this does not occur in spite of the large exothermicity.

The structure of the CH,O: product of reaction (38) was determined in a detailed study of its reactions with a large number of neutrals which characterized the structure unambiguously (Van Doren et al., 1986). The chemistry of H,COOH+ is quite different than that of HC(COH);, as expected, due largely to the 3.8 eV higher energy content of the HzCOOH' ion.

Certain chemical reactions (OH' transfer to CS,, for example) allowed a distinction between D atoms bonded to 0 atoms and those bonded to the C atom arising from the deuterated methane reactions with 0;. These data and the H and D distributions of the product ions themselves and the relative rate constants for the five CH,D, -" isomers yielded ten experimental isotopic parameters which were fit to within the _+ 5 % experimental uncertainty of measurements by the two isotopic factors occurring in the theory. There is one isotopic factor in the initial H - transfer step, H to D transfer rates occur there in the ratio of 3: l . Since this irreversible step controls the overall reaction rate, this factor leads to a nearly linear dependence of rate constant on number of H atoms; the overall factor between CH, and CD, rate constants is three. The second isotopic parameter involves the H(D) atom ejection from the highly energized H,COOH+ ion and is simply - $, the ratio of H-R+ and D-R+ reduced masses. This theoretical value provided a fit to the product distributions observed.

The rate constant for reaction (38) increases by a factor of 100 from 300 K to 20 K, an increase dominated by the temperature dependence of the rotational partition functions. At higher temperature, and especially at higher KE's, the endothermic production of CH: + HO, (by 0.24 eV) dominates the H,COOH + production. The same is true for vibrationally excited O:(v = 2) ions when CH: production becomes energetically allowed. This is consistent with the H- transfer being the rate controlling step. When the

78 Eldon Ferguson

resulting [H3COOH]+ ion can energetically separate into CH: + H,O, it does so.

The production of CH: + HO, at higher kinetic or vibrational energy of the reactants is a serious breakdown of statistical or phase space theory in that decomposition of [CH300H+]* into CH; + HO, is less exothermic than decomposition into C H 2 0 0 H + + H by 1.2 eV and yet dominates the decomposition when the 0.24 eV endothermicity to break the CO bond is provided. Evidently, when sufficient energy (0.24 eV) is provided reaction (3913

0; + CH, + CH: + HO, - 0.24 eV, (39) occurs by a direct process, rather than via a long-lived CH,OOH + complex. At still higher kinetic energy, or for O;(v = 3), the charge-transfer reaction

0; + CH, --t CH; + 0, - 0.6 eV (40) becomes energetically possible and dominates the reaction process.

When reactants do not have the necessary 0.24 eV for reaction (39) to occur, then a CH,OOH+ complex necessarily is formed upon H - transfer and the weakest bond breaks to produce products. The CH bond energy is 1.4 eV, the 0-0 bond energy is approximately 2.3 eV, the CO bond energy is 2.6 eV and the OH bond energy is approximately 3.9 eV. Only the CH bond and probably the 0-0 bond could be broken with the energy available in the 0; + CH, reaction at 300 K for O;(v = 0).

It would be gratifying to be able to report this detailed mechanistic study as an indicator of future capabilities. Unfortunately, this reaction may be almost unique in its possibility of being elucidated in such experimental detail. Aside from the fact that so many scientists in so many laboratories converged on this problem to give such an unprecedented wealth of detail, a circumstance not likely to become common, the reaction itself has unique aspects that permit the study that are not apparent at this time for any other ion molecule reaction. For example, 0; simply exothermically charge- transfers with all other hydrocarbons. Reaction (38) thus may remain as somewhat of a landmark and in this sense it deserves further continued effort, most particularly theoretical attempts to calculate the potential surface and elucidate the reaction mechanism, but also additional experiments. Troe (1988) has made a detailed theoretical analysis of reaction (38) particularly emphasizing the complex rotational effects involved in the reaction.

VI. Conclusions

In this brief discussion we have attempted to make the point that ion- molecule flow tube studies, in addition to contributing large numbers of reaction rate constants for which they are well recognized, also have the

FLOW TUBE STUDIES O F ION-MOLECULE REACTIONS 79

potential for making fundamental contributions to the understanding of reaction mechanisms and energy transfer processes. This is a potential that is only beginning to be exploited for the most part. This capability is now being extended to a number of laboratories around the world.

ACKNOWLEDGMENTS

This chapter is dedicated to Professor Alexander Dalgarno on the occasion of his 60th birthday. Professor Dalgarno provided valuable guidance on problems of aeronomy, astrophysics and molecular processes from the earliest days of development of the flowing afterglow technique in Boulder. He has been, and remains, a source of inspiration, epitomizing the highest ideals of the scientific endeavor.

The Author acknowledges with appreciation a short period in residence at the Joint Institute for Laboratory Astrophysics as an Adjoint Fellow and the valuable assistance of the JILA manuscript preparation group.

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Beswick, J. A,, and Jortner, J. (1981). In Intramolecular Dynamics of oan der Waals Molecules in Photoselectiiv Chemistry (A. Jortner, R. D. Levine and S. A. Rice, eds.), Wiley, New York.

Bohringer, H., and Arnold, F. (1986). J . Chem. Phys. 87, 2097. Bohringer, H., Durup-Ferguson, M., Ferguson, E. E., and Fahey, D. W. (1983a). Planet-Space

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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 25 1 1 DIFFERENTIAL SCATTERING IN He-He A N D He+-He COLLISIONS AT KeV ENERGIES R. F. STEBBINGS Department of Spare Physics and Astronomy

Rice Univrrsity Houston, Texus

und Thr Rice Quuntum Institute

I. Introduction . . . . . . . . . 11. He-He Collisions at Small Angles. .

A. Measured Quantities . . . . . B. Data Analysis . . . . . . . C. Results and Discussion . . . .

111. He' + He Collisions at Small Angles A. Results and Discussion . . . .

IV. He-He Scattering at Large Angles . V. Conclusion. . . . . . . . . .

Acknowledgments . . . . . . . References . . . . . . . . . .

. . . . . . . . . . . . . . 83

. . . . . . . . . . . . . . 84

. . . . . . . . . . . . . . 86

. . . . . . . . . . . . . . 87

. . . . . . . . . . . . . . 88

. . . . . . . . . . . . . . 91

. . . . . . . . . . . . . . 92

. . . . . . . . . . . . . . 95

. . . . . . . . . . . . . . 98

. . . . . . . . . . . . . . 98

. . . . . . . . . . . . . . 98

I. Introduction

Measurements of the scattering of beams of ions and atoms by gases have provided a valuable source of information on the forces between the colliding particles at close distances of approach. Early studies as exemplified by the work of Amdur and Jordan (1 966) measured partial total cross sections for scattering outside some minimum angle. Such measurements, however, do not lead unequivocally to the intermolecular potential (Mason and Vanders- lice, 1962); instead they provide the parameters for assumed forms of the potential. Consequently, such measurements have been in recent years, largely superceded by studies of the angular distribution of the scattered collision products, since these provide a much more stringent test of the theory. Numerous studies of differential scattering in ion-atom and atom- atom collisions have been reported. Notable among them is the work of Fedorenko et al. (1960), Lockwood et al. (1963), Lorents and Aberth (19659, Baudon et al. ( 1968) and Wijnaendts and Los (1 979).

83


ISBN 1-12-003825-0

84 R . F . Stebbings

Derivation of the interaction energy from measurements of differential scattering is most readily accomplished when the scattering is free from quanta1 interference patterns. In general, these are a consequence of effects that can be related semiclassically to the presence of two or more trajectories leading to scattering at the same angle and at the same final velocity. Such patterns are observed in both elastic and inelastic scattering and commonly arise from the crossing of two molecular electronic energy curves. The abundance of curve crossings in most atomic and molecular systems ensures that such effects will be observed in most differential cross section measurements. Structure also arises due to rainbow, glory, and diffraction effects, while in the case of symmetric systems, oscillations also appear due to both electronic and nuclear symmetry. Where oscillations of more than one origin are superimposed, the interpretation becomes quite complex but it is nonetheless generally possible to distinguish between the various effects and ascertain their separate contributions to the scattering.

Despite the substantial body of data on differential scattering, there are only very few absolute measurements. Lorents and Aberth (1965) reported absolute cross sections for elastic scattering in He+-He collisions while Smith et al. (1970) have published data in which relative differential charge transfer cross sections were placed on an absolute footing by integrating them and normalizing the resulting cross sections to previously measured partial total cross sections.

A program of measurements of absolute differential cross sections for scattering of neutral atoms and positive ions was begun at Rice University in 1983. This work was motivated both by the need for improved interaction potentials and also for cross section data for the interpretation of the atmospheric effects (Dalgarno, 1979) due, for example, to precipitation of ring current ions during times of geomagnetic disturbance. This paper presents some findings of this group.

11. He-He Collisions at Small Angles

Fig. 1 shows a schematic of the apparatus used for small angle scattering. He' ions generated by electron bombardment in a low voltage, medium pressure, magnetically confined arc plasma source are extracted and focused by a three element einzel lens before passing through a pair of 60" sector confocal bending magnets. The beam is then partially neutralized by helium in the charge transfer cell (CTC) before entering the target cell (TC) where a small percentage of the beam is scattered by helium target gas admitted through a variable leak valve. Typical pressures in the target cell of a few

DIFFERENTIAL SCATTERING 85

S C ATT ERE D PARTICLES

COMPUTER

D P 1 DP2

FIG. 1. Schematic of the apparatus used for small angle scattering.

millitorr are measured with an MKS Baratron Model 390 capacitance manometer. Background pressure in the main vacuum chamber is typically 2 x

The laser drilled apertures at the exit of the charge-transfer cell and at the entrance of the target cell at 20 pm and 30 pm in diameter, respectively, and are separated by 49 cm so that the neutral beam is collimated to less than 0.003" divergence. The TC is 0.35 cm in length and has an exit aperture 300 pm in diameter. Electrostatic deflection plates DP1 remove the residual ion beam, while plates DP2 remove charged collision products due to stripping, for example. Both the unscattered primary beam particles and the scattered collision products strike the front face of a position sensitive detector (PSD) whose properties have been discussed by Gao et al. (1984) and by Newman et al. (1985). The PSD is located 109 cm beyond the TC on the beam axis. This detector has a 2.5cm diameter active area, thereby limiting the maximum observable scattering angle to about 0.7". The PSD operates as a single-particle detector, and consists of two microchannel electron multiplier plates (MCPs) and a specially shaped resistive anode. When a particle strikes a channel wall in the first microchannel plate, secondary electrons are produced, and are accelerated down the channel by an applied voltage. When these electrons strike the walls of the channel, more secondary electrons are produced, leading to a small cascade of electrons. The small ( - lo3) cloud of electrons leaving the first MCP enters several channels on the second MCP, increasing the number of electrons to about lo6. These electrons are accelerated toward and stike the resistive anode. The charges leave the anode through four wires located at the corners of the anode, whose resistive characteristics are such that the measured amounts of charge leaving the anode on each of the wires may be utilized to determine the location of the electron cloud impact on the anode, thus determining the location of the impact of the incident particle. An LSI 11/2 microcomputer

torr under operating conditions.


latches and stores the output of the PSD position-encoding electronics. As each event occurs, the computer sorts the x and y impact coordinates into a 90 x 90 array and increments the appropriate array element.

A. MEASURED QUANTITIES

For thin target conditions, the differential cross section, do(& q)/dR, is related to the measured quantities by the expression

where S is the primary beam flux, AS(& q) is the flux scattered into the solid angle AR, and 7 is the “target thickness.”

In these experiments, it is not necessary that the absolute efficiency of the detector be known, because the primary and scattered particles are identical and are thus detected with equal efficiency. There is a small effect which decreases the detection efficiency in regions on the detector where the count rate is above a few hundred Hz, but it is possible to operate the detector so that detection efficiencies for regions of high count rate and low count rate are equal to within a few percent. This operating point is determined before each data run, since the condition of the microchannel plates changes as a consequence of the plate’s history of particle impacts.

Newman et al. (1985) and Schafer (1987) have shown that for the cell used in this experiment, 7 is accurately given by the product nL where L is the geometric cell length and n is the number density obtained from a measurement of the gas pressure is the TC at a location far from the exit aperture.

The accuracy with which At2 = sin BABAq is measured is strictly limited by the accuracy of distance measurements, the size of the primary beam and the detector’s position-finding uncertainty. The PSD spatial resolution is related to the size of the electron pulse impinging on the resistive anode. The positions of large pulses are determined with more precision than are those of small pulses, due to noise in the amplification and summing circuits. This phenomenon was studied with the use of a single-channel analyzer (SCA) to record the contributions to the electronic image of the primary beam from different portions of the pulse height spectrum. In general, it is found that the largest pulses provide good signal-to-noise ratio for the position encoding electronics and result in accurate position data (within about 60 pm). On the other hand, the smallest pulses (amounting to a few percent of the total counts) may be registered as much as 1OOOpm outside the geometrically- limited impact region. The details of the distribution depend on operating conditions: the problem is accentuated by high local count rates and by low


PSD operating voltage, both of which increase the relative number of small output pulses. Under conditions appropriate to the collection of data for 0.5 keV collisions, it is found that 95 % of the counts are recorded with an error less than 0.016", 3 % with an error between 0.016" and 0.032", and the remaining 2 are distributed out to 0.05". The relatively inaccurate position assignment for these particles results in a loss of angular resolution, particularly where the measured cross section shows sharp features.

The inaccuracy in position assignment also interferes with measurement of the scattered signal at the smallest angles (0 < O.OSO), where inaccurate position indexing of some primary beam counts increases the apparent diameter of the primary beam. These counts can be eliminated (and the angular resolution enhanced) by using the SCA to reject the small pulses; but in this case, absolute cross sections are not determinable since the pulse height distributions for primary and scattered particles are no longer the same. As a consequence, different detection efficiencies for primary and scattered particles result when a limited range of the pulse height spectrum is sampled. Therefore, the procedure for measuring cross sections below 0.05" is to obtain relative data using the SCA, and then to normalize these to the absolute data at angles greater than 0.05", where the effects of the spurious primary beam counts are negligible.

B. DATA ANALYSIS

Advantage is taken of cylindrical symmetry when analyzing the data, and the array elements are summed into annular rings co-axial with the primary beam. Data are accumulated until the statistical uncertainty in the signal at a given angle is small, typically less than 10 %. Two data sets, one with gas in the target cell and one without, are taken to permit discrimination between counts due to scattering from the target gas and counts arising from other sources, such as scattering from the background gas, scattering from edges of apertures, and dark counts. The "gas out" data provide a measure of the primary flux S, since the dark current of the detector is negligible compared to the primary beam flux. The 90 x 90 data arrays are organized into concentric rings whose widths are chosen subject to the competing demands of good angular resolution and an acceptable rate of data accumulation. Typically, the ring widths are somewhat larger than the detector's uncertainty. Although the summation into rings does not take full advantage of the information available from the detector, the technique is necessary to obtain adequate counting statistics. The ring center is determined by fitting a smoothly peaked function to the data; the peak of this function is taken to be the center of the rings. In many cases, where the cross section does not show


sharp features, the results of the analysis are not particularly sensitive to the choice of beam center, Care is taken, however, to ensure that the center is chosen optimally and both the “gas-in” and “gas-out” peaks are fitted to determine a composite “best” center for analysis. The angle B is determined to within f (0.03 8 + 0.002) degrees. This error reflects the uncertainties in PSD calibration, distance from target cell to detector, and location of beam center. One is not only interested in the value of 6, however, but also in the range 66 of physical scattering angles contributing to the signal at 8. The angular resolution 68 arises from the finite width of the primary beam, the discrete nature of the analysis rings, and electronic noise in the detector’s position encoding circuits. Counts registered in the iZh ring are assigned to the angle Oi which is the average of the angles corresponding to the inner and outer radii of the ring. The scattered flux, AS(O), is obtained by subtracting the gas-out data from the gas-in data and the angular range A6 is taken to be the ring width.

c. RESULTS AND DISCUSSION

Differential cross sections have been determined at laboratory collision energies of 0.25, 0.5, 1.5, and 5.0 keV (Nitz et al. 1987). Previous results for small angle He-He scattering were reported by Leonas and Sermyagin (1977) who measured relative cross sections at approximately an order of magnitude less resolution.

He-He scattering at reduced scattering angles below 1 keV-degree is not expected to involve excited states of the collision complex. The scattering, therefore, occurs from a single potential and the differential cross section is expressed quantum mechanically as

This expression (Massey and Smith, 1930) takes account of the fact that because the projectile and the target are identical, the signal of atoms at angle 9 comprises:

(1) Projectiles scattering at 9 with amplitude f(9) and (2) Target atoms recoiling at angle 9. This occurs when the projectiles are

scattered at ( n - 9) with amplitude f ( n - 9).

At these small values of 9, however, f ( n - 9) is negligible compared to f(9) and the cross section is accordingly well-represented by


where the scattering amplitude j ( 9 ) is given by the partial wave summation formula

where 9 is the scattering angle in the center of mass frame, k is the wave number, 6, is the phase shift of the lth partial wave, and P,(cos 9) is the l th Legendre polynomial. Conversion into laboratory coordinates is required for comparison with experimental results.

Cross sections have been calculated with this equation, using phase shifts derived from various proposed forms of the interaction potential. The phase shifts are obtained using the semiclassical JWKB approximation, except in the limit of large I , when the phase shifts become small and they are then determined using the eikonal, or JB approximation.

Information about the He-He interaction potential has been derived from a combination of scattering experiments, dilute gas transport experiments, and theory. Recent attention in the literature has focused on the lower repulsive wall at internuclear separations less than 1.8 A, where the potential rises above the 0.1 eV level. In this region, the results of high temperature transport experiments indicate the validity of a more steeply rising potential than do scattering measurements of integral cross sections. Of the many potentials that could be investigated, two analytic forms which provide a convenient characterization of the situation are the potentials proposed by Aziz ef al. (1 979) and by Ceperley and Partridge (1 986). The potential of Aziz et a / . has an attractive well consistent with a large body of thermal-energy data and a steep repulsive wall consistent with the high temperature (2500 K) measurements of thermal conductivity by Jody et al. (1977). Ceperley and Partridge have proposed a composite potential based on ab initio quantum Monte Carlo calculations and an extrapolation to larger r. This potential follows Aziz et al. for I > 1.828 A, but at smaller r it agrees more closely with results of Feltgen et al. (1982) and Foreman et al. (1974), who obtained potentials by inverting integral cross section data. Theoretical differential cross sections derived from the Aziz et al. and Ceperley and Partridge potentials are plotted in Fig. 2 along with the 0.5 keV experimental results. The 0.5 keV results were selected for this comparison, since cross sections at low projectile energies are particularly sensitive to the choice of potential. In general, the agreement to both shape and magnitude of the cross sections is excellent, and i t is worthwhile to mention that the calculations include no adjustable parameters. At angles less than 0. lo, the predictions are almost identical and lie within the experimental uncertainty. At angles greater than O.V, the steeper Aziz et al. potential yields undulations larger than are observed experimentally and also gives slightly lower values of the cross

90 R. F . Stebbings

v

I o6

lo5

lo4

0,OI 0 , I I lo3

LAB ANGLE (deg) FIG. 2. Differential cross sections for He-He scattering at 0.5 keV. The measured values are

shown together with the calculations based on the potentials of Aziz e ta / . (shown dotted) and of Ceperley and Partridge (shown as the full line).

section, while the prediction based on the Ceperley and Partridge potential lies within the uncertainty of the data throughout almost the entire angular range of the experiment. Calculations have also been carried out using the exponential potential of Foreman et al. (1974) extrapolated to larger r. The resulting cross sections exhibit a slightly weaker undulation than do the data, but are otherwise in excellent agreement with the experimental results. The experimental data are thus most consistent with the less steeply rising of the He, potentials.

The range of the potential probed by the 0.5 keV data can be estimated in several ways. Calculations of a classical deflection function from the 0.5 keV phase shifts indicates that the experimental scattering angles correspond to impact parameters in the range 1.2-2.0 A. In addition, the 0.5 keV partial- wave series essentially converges at 1 = 1000, which translates into an impact parameter of 2.04 A. Finally, empirical tests show that the cross section predictions are insensitive to the behavior of the potential for r > 2 A.

Partial-wave calculations based on the Ceperley and Partridge potential have also been performed for collision energies of 1.5 keV and 5.0 keV. The agreement between experiment and theory is generally very good except at


the largest angles in the 5.0 keV data where the observed cross section deviates from the single-channel elastic scattering calculation. The onset of this behavior at an energy-angle product Ed of 2 keV-deg is consistent with previous observations at lower energies and corresponds to the opening of inelastic channels at internuclear separations of approximately 0.5 8, (Mor- genstern et al., 1973, Brenot et al., 1975, Guayacq, 1976).

The behavior of the cross section below 0.2" is of particular interest since, whereas the classical differential cross section rises monotonically and diverges as 0 --* 0, the observed behavior exhibits an undulating structure superimposed on the classical cross section and a leveling-off which varies as exp[ - cd2] at small angles. This behavior has been predicted theoretically (Mason et al. 1964) and was observed in thermal energy alkali-mercury and alkali-rate gas collisions (Berry, 1969). The undulation is referred to as the forward diffraction peak and is understood as arising from interference over a broad range of impact parameters associated with weak deflections from the tail of the potential. This phenomenon is markedly distinct from rainbow and glory scattering, which are associated with a few particular impact parameters. When observed as a function of energy, a given undulation feature (the first minimum, for example) resembles optical diffraction from a disk, moving to smaller angles as the de Broglie wavelength decreases. Beier (1973) has utilized this analogy to relate the undulation characteristics to potential parameters in the case of a screened Coulomb interaction. Depending on the collision energy and the potentials involved, the diffraction peak can be characteristic of either the attractive or the repulsive part of the potential. Partial wave calculations indicate that the influence of the weak van der Waals attraction is negligible in the present experiment.

111. He' + He Collisions at Small Angles

Ion-neutral collisions have also been investigated using the apparatus depicted in Fig. 1. For these measurements, the CTC is evacuated and the primary ion beam passes directly to the target cell. Both charged and neutral collision products have been investigated. The neutral collision products are measured by deflecting away the ions emerging from the target cell while the ionic collision products are determined as follows. Two files are taken with gas in the target cell; one where all energetic particles emerging from the target cell are collected (ASToT), and one where only neutral charge transfer collision products are collected (AS,,). Two files are also taken without gas in the target cell; one of the primary ion beam (AS,*) and one of the background noise (AS,,). The AS,,, file includes four contributions: 1) attenuated


primary ion beam, 2) elastically scattered ions, 3) charge transfer neutrals, and 4) brackground noise.

The ion scattering signal is therefore obtained as follows:

AsIs = (ASTOT + AS,,) - ( A h + ASCT). ( 5 )

A. RESULTS AND DISCUSSION

Raw data for both charge transfer and elastic scattering at 1.5 keV are shown in Fig. 3 which depicts the contents of the 90 x 90 array. The vertical displacement indicates the number of counts at a given location on the detector.

I . Elastic scattering

Elastic scattering cross sections have been obtained for 1.5 keV He' projectiles elastically scattered from neutral He, over the laboratory angular range 0.04" - 1.0" and are shown in Fig. 4 together with theoretical cross sections obtained using the potentials found in Marchi and Smith (1965).

The calculations are now complicated by the fact that for He+-He collisions the Hamiltonian is symmetric and two electronic states result when a ground state helium atom and ion are brought together adiabatically, one of which is symmetric (9) and the other antisymmetric (u) in the nuclei. Scattering occurs from each of these potentials and the differential cross section for elastic scattering is given by

1 dR 4

%) (elastic) = - If,($) - f,(n - 9) + f&S) + f,<n - $ ) I 2 . (6)

Charge Transfer Elastic Scattering FIG. 3. Raw data for He+-He scattering at 1.5 keV.


h

c v) \ cu E 0 U

8 (degrees) FIG. 4. Elastic scattering in He+-He colisions at 1.5 keV. The experimental data are shown

together with the results ofa partial wave calculation using the potentials of Marchi and Smith (1 965).

Once again, because the scattering is confined to such small angles, the cross section is well-represented by the expression

which accounts for the superposition of the scattering from the gerade and the ungerade potentials. The partial wave method therefore leads to two scattering amplitudes

94 R. F . Stebbings

where Sf." is the phase shift of the l fh partial wave. The agreement between experiment and theory is quite satisfactory with the calculations showing rather more pronounced oscillations than do the experiments.

2. Charge Transfer

Measurements of charge transfer have been obtained at several energies. A difficulty arises when placing these data on an absolute footing because the primary and product particles are no longer identical and attention must be given to the possibility that they may be detected with different efficiencies. Nagy et al. (197 1) have previously recognized this problem and offered an

0101 0, I 1'0 8 (degrees)

FIG. 5. Charge transfer in Het-He collisions. The solid lines are obtained using Eq. (9) together with the potentials of Marchi and Smith (1965).


ingenious solution. They noted that whereas both the elastic-scattering cross section and charge transfer cross section, which is given in its approximate form by

individually exhibit oscillatory behavior because of the interference between the two scattering amplitudes, their sum is nevertheless free from such oscillations (because the interference terms cancel) and decreases monotonically with increasing angle at angles larger than the rainbow angle. It follows that, if the detection efficiencies of the ions and the neutral atoms are different, an oscillatory structure will be observed in the combined signal whose phase will be determined by whichever of the efficiencies is the greater. No such oscillatory behavior is discernible in the data, however; the detection efficiencies of helium atoms and ions are thus identical to within experimental uncertainty. Experimental cross sections derived on this basis are plotted in Fig. 5 and are seen to be in good agreement with the theoretical cross sections obtained from Eq. (9) using the Marchi and Smith (1965) He: potentials. Furthermore, integration of these differential cross sections yields partial total cross sections that are in excellent agreement with direct measurements of total cross sections. Such comparisons are valid, in that the differential cross sections are so strongly peaked in the forward direction that only a small contribution to the total cross section results from scattering outside the range of the measurements.

IV. He-He Scattering at Large Angles

The experimental techniques described above may be used, with minor modification, to investigate scattering to larger angles. Indeed, measurements have been carried out in this way to scattering angles up to approximately 10" in the laboratory system. The differential cross sections typically fall off so abruptly with increasing angle, however, that at even larger angles alternative techniques must be adopted to achieve satisfactory signal to noise. Accord- ingly, the apparatus shown in Fig. 6 was developed for the study of scattering in the angular range 26" < 6 c 45". This apparatus differs from that used for the study of small angle scattering by the use of a target cell of length 450 pm and exit diameter 250pm and the employment of two PSDs which permit detection of both the scattered and the recoil particles. For a scattering event to be recorded, the projectile and the recoil particle must be detected in coincidence. The sequence of events is as follows: detection of a particle on

96 R. F . Stebbings

TO COMPUTER MAGNET I T I T O TIMING

P S D l .. r CIRCUIT

5 R 'I

ION SOURCE 5 ' ' INCIDENT BEAM ? AXIS

CIRCUIT

FIG. 6. Schematic of the apparatus used for the study of large angle He-He scattering.

PSDl opens a gate enabling recognition of counts from PSD2 for 0.5 psec. If PSD2 detects a particle within this interval, both the time difference between the arrival of the two particles and the position coordinates of the particle impacts are recorded. After rejection of spurious events that fail to meet certain momentum conservation requirements, the remainder are stored in an array analogous to that described earlier. Corrections to the contents of the array elements are then made to account for the fact that 1) only a fraction of atoms scattered at a given angle strike the detector, 2) particles scattered at different angles have different energies and are therefore detected with different efficiencies, and 3) the effective collision volume varies slightly with the scattering angle.

The measured relative differential cross sections for 4He-4He at 3 keV are plotted in Fig. 7. Pronounced oscillations are observed which are also present in the case of 3He-3He collisions. Such oscillations were previously observed by Dhuicq et al. (1973) and by Lorents and Aberth (1965) and are explained as follows.

For 4He-4He collisions, the projectiles are indistinguishable bosons and the cross section is accordingly given by

At these large scattering anglesf(n - 9) is not negligible in comparison with f ( s ) and the observed oscillations are thus a consequence of the interference


125

L

CJ v

cn 5 0 n 0

0 30 33 36 39 4 2 45

LAB ANGLE (deg ) FIG. 7. Relative differential scattering cross sections for 4He-4He collisions at 3 keV.

Experimental data are shown together with calculations based on Eq. (10).

between the two scattering amplitudes at supplementary CM scattering angles.

For the case of 3He-3He collisions involving indistinguishable fermions, the situation is analogous but even further complicated by the fact that the particles now have nuclear spin and the cross section is given by

(1 1)

From Eq. (10) and Eq. (ll), the amplitudes clearly interfere in different ways for bosons and fermions. In the asymmetric case 4He-3He, there is no nuclear symmetry and the differential cross section is given by Eq. (3).

The scattering amplitudes, and thus the cross sections, can be calculated if the interaction potential between the colliding atoms is known. For a 3 keV collision between 4He and 4He resulting in scattering at 45", the impact parameter is approximately 1.92 x lo-'' cm and the classical distance of closest approach is approximately 4.64 x 10- l o cm. Since the Bohr radius of the ground state He atom is approximately 2.65 x lo-' cm, it is reasonable to assume that the scattering is largely determined by the Coulomb interaction between the two He nuclei:

(lf(@l2 + If(. - S)I2> + lf(W - f ( . - $ ) I 2 2 2

a(9) =

98 R. F. Stebbings

Cross sections calculated in this manner agree very well with the experimental data. For 3 keV collisions, the best agreement between experiment and theory is achieved when the effective nuclear charge is taken to be 2 and the results are shown as the solid line in Fig. 7. For 1.5 keV collisions, Zeff = 1.8 gives the best fit, suggesting that, at this lower energy, partial shielding of the nuclei occurs.

V. Conclusion

Position sensitive detectors are seen to be an excellent tool for the investigation of absolute differential scattering cross sections for heavy particles at keV energies. Theoretical cross sections for He+-He and for He-He obtained using the method of partial waves are in very good agreement with the measurements when the Marchi and Smith (1965) potentials are used for He: and the Ceperley and Partridge (1986) potentials for He,.

ACKNOWLEDGMENTS

The work at Rice University described in this chapter was carried out with support from the National Science Foundation, the National Aeronautics and Space Administration, and the Robert A. Welch Foundation.

REFERENCES

Amdur, I., and Jordan, J. E. (1966). In Advances in Chemical Physics Vol. X (John Ross, ed.)

Aziz, R. A., Nain, V. P. S., Carley, J. S., Taylor, W. L., and McConville, G. T. (1979). J. Chem.

Baudon, J., Barat, M., and Abignoli, M. (1968). J. Phys. B. 1, 1083. Beier, H. J. (1973). J. Phys. 8. 4 683. Berry, M. V. (1969). J. Phys. B. 2, 381. Brenot, J. C., Dhuicq, D., Gauyacq, J. P., Pommier, J., Sidis, V., Barat, M., and Pollack, E. (1975).

Ceperley, D. M., and Partridge, H. (1986). J. Chem. Phys. 84, 820. Dalgarno, A. (1979). In Advances in Afomic and Molecular Physics Volume 15 (D. Bates and

Dhuicq, D., Baudin, J., and Barat M. (1973). J. Phys. 8.6, L1. Fedorenko, N. V., Filippenko, L. G., and Flaks, N. P. (1960). Soviet Phys.-Techn. Phys. 5 4 5 . Feltgen, R., Kirst, H., Kohler, K. A., Pauly, H., and Torello, F. (1982). J. Chem. Phys. 76, 2360. Foreman, P. B., Rol, P. K., and Coffin, K. P. (1974). J. Chem. Phys. 61, 1658.

Interscience publishers, New York.

Phys. 70, 4330.

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B. Bederson, eds.), page 37.


Gao, R. S., Gibner. P. S., Newman, J. H., Smith, K. A,, and Stebbings, R. F. (1984). Rev. Sci. Insr.

Guayacq, J. P. (1976). J . Phys. B. 9, 2289. Jody, B. J., Saxena, S. C., Nain, V. P. S., and Aziz, R. A. (1977). Chem. Phys. 22, 53. Leonas, V. B., and Sermyagin, A. V. (1977). Khim. Vys. Energ. 11, 296. Lockwood, G. J., Helbig, H. F., and Everhart, E. (1963). Phys. Rev. A. 132, 2078. Lorents, D. C., and Aberth, W. (1965). Phys. Rev. A. 139, 1017. Marchi, R. P., and Smith, F. T. (1965). Phys. Rev. A . 139, 1025. Mason, E. A., and Vanderslice, J. T. (1962). In Atomic and Moiecutar Processes (D. R. Bates, ed.)

Mason, E. A,, Vdnderslice, J. T., and Raw, C. J. G. (1964). J. Chem. Phys. 40, 2153. Massey, H. S. W., and Smith, R. A. (1930). Proc. Roy. Soc. (London) A . 126, 259. Morgenstern, R., Barat, M., and Lorents, D. C. (1973). J. Phys. B. 6, L330. Nagy. S. W., Fernandez, S. M., and Pollack, E. (1971). Phys. Rev. A . 3,280. Newman, J. H.. Smith, K. A., Stebbings, R. F., and Chen, Y. S. (1985). J . Geophys. Res. 90, 11045. Nitz, D. E., Gao, R . S., Johnson, L. K., Smith, K. A,, and Stebbings, R. F. (1987). Phys. Reo. A . 35,

Schafer, D. A. (1987). Private communication. Smith, F. T., Fleischmann, H. H., and Young, R. A. (1970). Phys. Rev. A . 2, 379. Wijnaendts van Resandt, R. W., and Los, J. (1979). Proc XI 1.C.P.E.A.C.

55, 1756.

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4541.



1 1 ATOMIC EXCITATION IN DENSE PLASMAS JON C. WEISHEIT* Physics Depar men I Lawrence Livermore National Laborarory Livermore, Cutiforma

I . Introduction . . . . . . . . . 11. Characteristics ol Dense Plasmas . .

A. Plasma Parameters . . . . . B. Plasma Timescales. . . . . . C. Dynamic Properties . . . . . D. Static Properties . . . . . .

111. Excitation Models for Dense Plasmas A. Quantum States of the Target. . B. Weak Electron-Ion Interactions . C. Strong Electron-Ion Interactions. D. Ion-Ion Interactions . . . . .

IV. Conclusion. . . . . . . . . . Acknowledgments . . . . . . . References . . . . . . . . . .

. . . . . . . . . . . . . . 101

. . . . . . . . . . . . . . I02

. . . . . . . . . . . . . . 102

. . . . . . . . . . . . . . 104

. . . . . . . . . . . . . . 106

. . . . . . . . . . . . . . 113

. . . . . . . . . . . . . . 115

. . . . . . . . . . . . . . 116

. . . . . . . . . . . . . . 116

. . . . . . . . . . . . . . 122

. . . . . . . . . . . . . . 125

. . . . . . . . . . . . . . 127

. . . . . . . . . . . . . . 129

. . . . . . . . . . . . . . 129

I. Introduction

High temperature, high density plasmas have long been studied in connection with stellar interiors. Interest in the properties of such plasmas has increased markedly during the past decade or so because of the large international effort to achieve controlled thermonuclear fusion by means of inertial confinement (ICF). The development of short-wavelength (EUV and X-ray) lasers has provided further impetus to understand atomic phenomena in hot and dense plasmas (Sobel’man and Vinogradov, 1985).

Atomic physicists, for the most part, have paid attention to the adjective “hot,” and have continued to develop experimental and theoretical techniques for determining energy levels, oscillator strengths, and collision cross sections for highly stripped ions in uucuo. Progress in these areas has been notable (see, for example, the compilation of transition wavelengths by Kelly, 1982; and the compilation of electron-impact ionization studies by Tawara and Kato, 1987). By contrast, implications of the adjective “dense” have largely been ignored, and our knowledge of plasma effects on atomic

* Present address: Department of Space Physics and Astronomy, Rice University, Houston, Texas.

101

Copyright 6 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. _^_.. ” .- ̂^_^^_ -

102 Jon C. Weisheit

collisions is particularly limited (Weisheit, 1984; Pratt, 1987). A rigorous treatment of inelastic scattering in a dense plasma would lead to an intractable many-body calculation. This tactic will not be pursued here. Instead, we intend to discuss two much simpler-but still credible-descriptions of the environmental influence of a dense plasma on atomic collisions.

In the development of these models, it is necessary to identify the “strength” of collisions resulting in a transition between a particular pair of quantum states, y o and yl. Throughout this paper we use a conventional measure of a collision’s strength (Baranger, 1962; Smith et al., 1981) that -

involves its time-dependent interaction

I j-+; VIO(t) dt 1

One model forsakes the treatment of individual scattering events, and views a transition in any one ion as being due to the action of a stochastic electrodynamic perturbation arising from all other plasma particles; this model includes dynamic plasma response, but in applications is restricted to weak perturbations that can be treated in the Born approximation. In the other model, advantage is taken of the ordering of various timescales (such as collision duration and plasma response time) to obtain two-body potentials that mimic the average effect of the plasma environment on a given collisional interaction. Although dynamic plasma response is excluded, more elaborate scattering theories can be used in this model to treat strong collisions.

In Section 111 these two, somewhat complementary, approaches are developed, and an empirical connection is made between them. New calculations, as well as previously published results, are discussed. Prior to that, in Section 11, we summarize needed concepts and equations from plasma kinetic theory. For background material in this subject, the reader is referred to monographs by Ichimaru (1973), and Lifshitz and Pitaevskii (1981). Finally, in Section IV, we list some topics for future research. The literature cited generally includes work published by mid 1987.

11. Characteristics of Dense Plasmas

A. PLASMA PARAMETERS

We shall restrict our attention to homogeneous plasmas having a well- defined temperature 0 = k, T, and consisting of electrons (mean number density = N , ) and ions (charge = Z e and mean number density = N , =

ATOMIC EXCITATION IN DENSE PLASMAS 103

NJZ). Generalizations of the statistical description to more complicated situations-ionic mixtures, different ion and electron temperatures, and so on-can be made (see, for example, Deutsch, 1982; Boercker and More, 1986).

The key dimensionless parameters characterizing such two-component plasmas are the Coulomb coupling constants,

where Re and R, are, respectively, the electron- and ion-sphere radii, defined by

i (Q)nR,3Ne = 1 ,

(4)nRZN, = 1 , (3)

and where R = ( R e + Rz)/2. Numerically, one has Tee = (leV/O)(Ne/8.0 x 1019cm-3)1/3. When a particular Tab is much less than unity, the relative motion of a particle pair (a, b) is only weakly influenced by Coulomb forces; the opposite is true when Tab % 1. For either plasma constituent, Taa measures the ratio of R, to that constituent’s Debye length,

D, = ( 0 >”’. 4xZ,2e2 N,

It is readily found that

(2) =a.

(4)

( 5 )

When coupling constants are small, there are many particles in a Debye sphere, that is, (4/3)nD,3Na % 1, and the BBGKY hierarchical formalism can be used to investigate kinetics problems in plasmas (Ichimaru, 1973). O n the other hand, elaborate numerical simulations are generally required to investigate plasmas with large coupling constants. This survey will emphasize weakly coupled plasmas, which can be treated in the random phase approximation (RPA) developed,by Pines and Bohm (1952).

A second limitation on the plasmas to be considered here is that they be “classical,” and have temperatures well in excess of the Fermi energy, E,. This is tantamount to the requirement (4/3)nA:N, G 1, where A, =

( h 2 / 2 m , 0 ) ’ 1 2 is the thermal deBroglie wavelength for particles having mass ma. As shown in Fig. 1 for electrons, plasmas that satisfy both of these criteria include ICF conditions presently achievable, virtually all x-ray laser media now under laboratory investigation, and even the cores of all but the least massive main-sequence stars. Evidently, many important high-density plasmas are classical and only weakly coupled.

104 Jon C. Weisheit

4

s

5 m 2

a 3 F

Y

0 - 1

0 18 20 22 24 26

log (NJ1 crnw3) FIG. 1. Boundaries in density-temperature plane that identify regions of strong coupling

(Fee > 1) and classical behavior (Q > EF) for plasma electrons. Typical regimes for inertial confinement fusion (ICF), and EUV and X-ray laser experiments are noted. Also plotted is the track of central densities and temperatures of main-sequence stars, with the numbered points denoting masses in units of the solar mass.

Other practical constraints on the plasma density and temperature can be imposed: unless the temperature 0 < $Z21H N 5Z2 eV ( IH = 13.6 eV), the relative abundance of Z-times ionized atoms is very small (Post et al., 1977). And, unless the density is such that ZR, > lOa,, pressure ionization (cf. Stewart and Pyatt, 1966; More, 1985) eliminates all excited ionic states.

B. PLASMA TIMESCALES

According to Bogoliubov (1962), there exist three important timescales in plasma kinetics problems:

system size sound speed ’

z(hydrodynamic) - collision-free path length

particle speed z(re1axation) -

interaction range t(corre1ation) -


where ( ) indicates an average value for the plasma. Bogoliubov showed that great simplification arises when one takes advantage of the ordering of these timescales that is characteristic of weakly coupled plasmas, namely,

( 6 )

In particular, in an equilibrium many-body system satisfying the conditions of Eq. (6) , correlations among particles as well as fluctuations in local values of various physical quantities can be expressed in terms of single particle distribution functions Fa(r, p, t). The averaging process for a given species then reduces to (.. .), = N ; ’ dp Fa(p) [...I. Inasmuch as we are considering only homogeneous, equilibrium plasmas, each species has a Maxwellian distribution of momenta p = mav,

t( hydrodynamic) $ t(re1axation) $ z(corre1ation).

Here, and below, the superscript (0) will be used to distinguish results pertaining to the ideal gas distribution.

Of the three timescales in Eq. (6), we will need an expression only for r(corre1ation). It is conventional to put the interaction range between particles of type “a” equal to their Debye screening length D,, and to define the correlation time for that species as the time it takes a particle “a” with thermal speed to move this distance. This time turns out to be the reciprocal of that species’ plasma frequency, ma,

r,(correlation) = - =

For electrons, one has Besides the correlation time, which relates to elastic Coulomb interactions

among plasma particles, we will also need an estimate of the corresponding timescale set by inelastic collisions:

= (NJ7.26 x lozo ~ m - ~ ) ’ / ’ eV.

>. interaction length for excitation

particle speed t(duration) -

This z-value, which clearly depends upon the transition in question, is not so easily determined. Smith (1960) argued for a collision lifetime matrix that measures the time delay of the scattered wave packet. It is not obvious, however, how one should deal with the numerous scattering channels that may contribute to a given atomic transition. Cooper (1986) has published an alternative, semiclassical estimate of t(duration) for electron-ion collisions. His result depends upon the trajectory’s impact parameter, and, in this case, it is unclear how one should deal with the range of impact parameters that may

106 Jon C . Weisheit

contribute significantly to the total cross section. In light of these ambiguities, we decided to otherwise estimate r(duration).

Perturbation theory leads to the factor exp[ - i(AElo/h)t] in the time- dependent amplitude for a transition with energy defect AElo. This suggests a reasonable upper bound on the duration of the yo + y1 collision,

h z(duration, yo -+ y ) c __.

-AE,, (9)

A lower bound can also be identified, because the collisional interaction must exist for at least as long as it takes the incident particle Z,e to move a distance equal to its de Broglie wavelength; thus, one has

za(duration, y o + y l ) 2

with only collisions having E 2 AElo being averaged. Sometimes, it may be sufficient to use one of these bounds, but at other times an actual value may be needed. In such instances, we used the Weisskopf radius pa(yo,yl) to measure the interaction length for the transition yo + yl. This radius is simply the largest impact parameter for which a collision with a charge Z,e is “strong” in the aforementioned sense. When the trajectory is linear, we find

where angular integral factors of order unity have been ignored. The result for Coulomb trajectories (repulsive and attractive) is not much different, as long as the scattering angle is less than n/2 and AElo is small with respect to the collision energy. We have, then, with these limitations,

z,(duration, yo -+ 7 , ) =

which usually is intermediate to the two estimated bounds.

C. DYNAMIC PROPERTIES

Knowing how a plasma responds to an electromagnetic disturbance is basic to developing a model of atomic collisions in high-density plasma environments. This subsection summarizes results that are obtained when the


response is in direct proportion to the disturbance. When a homogeneous, equilibrium plasma experiences a disturbance at a spacetime point (r, t), its response at (r', t') is a function only of the separations (r' - r) and (t' - t ) . Moreover, in the linear response regime, the plasma's reaction to several (small) disturbances can be found by using the superposition principle.

Most of the time, it is convenient to work in Fourier, (k, a)-space instead of (r, t)-space. The transform convention adopted here is

+ m s - m B(k) = dr e-ik'rB(r), B ( o ) = 1 dt eiofB(t), (13)

where B is any function whose Fourier transforms exist, and where the r-integration extends over the volume V of the system. Transforms of real quantities B(r, t ) have the property B*(k, w) = B( -k, -w).

I . The Plasma Permittivity

Consider a one-component plasma (OCP), in which stability is maintained by a smooth, inert background of oppositely charged matter. In the linear response regime, when an external potential field (Dext(r, t)-which has Fourier components QeXt(k, 0)-is applied to this OCP, its density changes by a proportional amount,

(14)

Since (from Poisson's equation) the induced potential is (DFd(k, o) a &J,k, w ) cc Qex'(k, w), and the total potential is just 0: = @rd + Qext, it follows that (Dr is proportional to Q""'. The ratio

ga(k, w ) = [N,(k, w ) - N , ] a Qexl(k, 0).

1 -- Qy(k, o) Qext(k, o) - Ea(k, w )

serves to define the plasma permittivity, or dielectric response function, EAk, w).

For weakly coupled plasmas, E , can be determined from the Vlasov equation, and, when F,(p) is Maxwellian, it has the form


Here, "(5) is the plasma dispersion function, treated in detail by Fried and Conte (1961), and i2 = w/w, and K = kD, = k/k , are convenient dimensionless variables, with k, = 0,' being the Debye wavevector. Asymptotic expressions for <\Y(t) are

5 4 1 . 1 Thus, at low frequencies, w 4 k ( 2 0 / m a ) ' / 2 , the permittivity tends to the limit

&')(k,O) = 1 + K - ' ; (18)

at high frequencies w 4 k(20/ma)' / ' , it tends to the limit

Ef)(O, w ) = 1 - Q-'. (19)

At intermediate values of the frequency, Y must be obtained by numerical methods. We have made a small table of values of the RPA quantity IE(')(K, R)1'--which will appear in our scattering calculations-in Table I, for a considerable range of the variables R and K.

It is straightforward to generalize these OCP results to the case of two or more component species, when all are weakly coupled. One has

E(')(k, W ) - 1 = 1 [&f'(k, a) - 11. ( 2 0 ) a

When a plasma is strongly coupled, the E,'S must be approximated, and the additive relation, Eq. (20), no longer holds (Ichimaru, et a/., 1985).

TARLF 1

THE RPA QUANTITY Ido)(K, R)I2 FOR A ONE-COMPONENT PLASMA

log R

log K - 2 -1 0 1 2

- 2 6.54 + 7" 1.04 + 4 1.30 - 6 9.78 - 1 9.98 - 1 - 1 1.02 + 4 6.59 + 3 1.00 - 3 9.80 - 1 1.00 + 0

0 4.00 + 0 3.98 + 0 2.20 + 0 9.79 - 1 1.00 + 0 1 1.02 + 0 1.02 + 0 1.02 + 0 1.01 + 0 1.00 + 0 2 1.00+0 1.00+0 1.00+0 1.00+0 1.00+0

' 6.54 + 7 = 6.54 x lo7


2. Test Charge Screening

An elementary but important application of the foregoing ideas is the determination of the electric potential surrounding a “test charge” Z,e, introduced with constant velocity v* into a plasma. This test particle is the source of a charge-density disturbance Qext(r, t ) = Z,ed(r - v, t), whose Fourier components are Qex‘(k, o) = 2 n Z , e d ( o - k-v,) . The results given above allow one to calculate the potential which includes the contribution of the test charge itself plus that of the perturbed plasma,

Qto‘(r, t) = + dk[&(k, k - v*)] - exp[ik *(r - v,t)]. (21) (2n) ‘ S

For a weakly coupled, electron-ion plasma, three limiting, steady-state cases can be specified, each of which follows from simplifications in the expression for do)(k, w).

(i) High frequencies: o* = k - v , % o, + w,;

(ii) Intermediate frequencies: o, % o* + 0,;

r

(iii) Low frequencies: o, $=- o, % 0,;

r

In Eq. (22c), we have introduced the composite Debye length D, whose definition is

Thus, k i = D-’ = k,2 + k,2 in a two-component plasma. The r- and t- dependence of Qtot is quite complicated if the test charge’s motion does not allow one of these strings of strong inequalities to be satisfied. Some examples of this are described by Chen et al. (1973).

While our focus is on atomic physics in weakly coupled plasmas, it will be instructive to have a simple description of screening when r > 1, so that some insight concerning events in strongly coupled plasmas can be developed. Such a description is provided by the ion-sphere (or Wigner-Seitz) model (Ichimaru, 1982), in which the electron density everywhere in the


plasma is taken to be uniform. Then, within a distance R , of a given ion, that ion plus its Z charge-neutralizing electrons combine to give a potential

with @IS(r > R,) = 0. This cellular model has been greatly embellished, but a discussion of that work is outside the scope of this paper. The interested reader is referred to recent articles on Thomas-Fermi and density-functional theories (e.g., Dharma-Wardana and Taylor, 1981 ; Gupta and Rajagopal, 1982; Dharma-Wardana, 1982).

Three important “collisional” applications of these screening concepts bear mention, even though they do not involve atomic excitations. First, there is the influence of screening on plasma conductivity: in this process, elimination of the long-range Coulomb interaction between two charged particles reduces their momentum-transfer cross section to a finite value. Some recent work in this subject can be found in papers by Rozsnyai (1979), Boercker et al. (1982), and Ichimaru and Tanaka (1985). Second, there is the enhancement of thermonuclear reaction rates in the (strongly coupled) plasma of degenerate stellar cores. This topic was pioneered by Salpeter (1954), who first pointed out that the negative charge density of electrons in the vicinity of two colliding ions tends to enhance their cross section for fusion; up-to-date values for these enhancement factors can be found in Schatzman’s (1987) review. Third, there is the slowing of a test charge by the electric field gind = -VQind it induces in the plasma. The resulting loss of kinetic energy per unit path length involves the plasma permittivity E(k, kav,), and in the combined limits of high velocity and weak coupling one obtains the familiar Bethe-Bloch result. Some discussion of changes in the rate of energy loss due to quantum mechanical or strong coupling effects can be found in publications by de Ferrariis and Arista (1984), Yan et al. (1985), and Deutsch (1987).

3. Density Fluctuations and Structure Factors

Classical quantities in a macroscopic system fluctuate about their mean values whenever the temperature 0 > 0. Of particular importance for plasma studies are the fluctuations Na in each component’s density, whose relevant properties are described by the density autocorrelation function,

<N,(r,, t l W a ( r 2 , Q> = (N,(r = rl - rt, t = 1, - t2)R,(o, o>>, that measures the extent to which, on average, fluctuations at one spacetime point are related to those at another.

I I


The power spectrum in (k, w)-space of this autocorrelation function defines, to within a constant factor, the so-called dynamic structure factor,

1 - - ~ S i r n d t eiot [dr e-ik'r(na(r, t)R,(O, 0)). (25) 2nN, - m

Structure factors are central to the statistical analysis of many-body phenomena. They are real, and are positive definite for all finite values of k and w. When a system is invariant under space and time translations, two different Fourier components of fluctuations are, on average, uncorrelated. The general statement of this result is

(Na(k, w)IQ(k', 0')) = (27~)~S(k - kf)6(w - wf)NaSaa(k, W ) (26)

(Landau and Lifshitz, 1981). In describing fluctuation phenomena in a plasma, the corresponding

results for an ideal gas provide a useful basis of comparison. By starting with the linearized Boltzmann transport equation, one obtains the formula

which applies to any equilibrium distribution Fa of noninteracting particles. When the Maxwellian distribution Ff) is used, Eq. (27) yields

What is S,,(k,w) for an OCP? Although the derivation of fluctuation spectra in plasmas is lengthy (see, for example, Lifshitz and Pitaevskii (1981), Sec. 51, for a careful treatment), the answer is simply stated,

In fact, this result could be expected from an argument involving the test particle concept we introduced earlier: One at a time, every mobile charge in the plasma is treated as a test particle, say, NY'(r, t ) = 6(r - vl t ) for the charge with velocity vl, This perturbation induces changes in the plasma density, and the situation can be described in terms of the plasma permittivity by Ng"'(k, o) = NP'(k, u)/e,(k, 0). Viewed together with its response, the bare test particle is said to become "dressed." According to Rostoker and Rosenbluth (1960), every charge is at once a test particle and part of the dressing of all other test particles. The net effect is that a weakly coupled


plasma behaves as a collection of noninteracting, “dressed” charges. Thus, the description of fluctuations in an ideal gas applies, after the replacement m,(k, w ) -+ R,(k, w)/Ea(k, w ) is made. Eq. (29) is a direct consequence of this.

The situation is more complicated when the plasma has two or more components, because the response frequencies w, can differ greatly. However, when the coupling is weak, the structure factors of individual species can still be expressed in terms of the ideal gas functions SE and the RPA permittivities E:) (Ichimaru et al., 1985). For the (weakly coupled) two-component plasma one has

and an analogous equation for &. (For brevity, the k- and o-dependence of all quantities is suppressed in Eqs. (30) and (32).)

Structure factors describing the correlations between different interacting species can be defined by an obvious generalization of Eq. (25). Because of the equality (N,(r, t)mb(o, 0)) = (ma@, o)m,( -r, - t ) ) and the fact that these density-density correlation functions are real, we can write

sab(k, = sba(k, 0,

- - j“dte’”‘ jdre-i“’r(#a(r, t)m,(O, 0)). (31) 2 . J K - m

If these species are weakly coupled, Tab < 1, then the preceding equation reduces to (Ichimaru et al., 1985)

Now we can determine, for example, the autocorrelation function for fluctuations &r, t) of a plasma’s internal electric potential. Poisson’s equation leads to the expression

(ib(k, w)G*(k’, 0’)) = - (&(k, w)&*(k, o’)), (“k:>‘ (33)

with

&(k, w ) = 1 Zaema(k, w ) (34) a

being a Fourier component of the total charge-density fluctuation. It follows directly that

($(k, w)di*(k‘, w’)) = r$Y(27t)’d(k - k)6(w - w‘)N,,,S,,(k, w), (35)


where N,,, = 1, N , is the total mean density of plasma particles, and where

defines the charge-charge structure factor. For the two-component, electron- ion plasma one has

(Again, all k- and o-dependence has been suppressed.) Readers interested in structure factors for strongly coupled one- and two-

component plasmas are referred to the articles by Cauble and Duderstadt (1981), and Cauble and Boercker (1983), respectively, and works cited therein. For our purposes, the weak-coupling results should suffice as long as the plasma T‘s do not exceed unity, because it turns out that very large k- values ( K % 1) are most important in our applications, and structure factors are less sensitive to in that regime.

D. STATIC PROPERTIES

I . Radial Distribution Functions

As we have seen, the permittivity E(k, 0) specifies the space- and time- dependent response of a plasma to electromagnetic disturbances. Often, though, one is interested only in the equilibrium properties of the plasma. In such cases, the static structure factors,

suffice. For instance, the radial distribution function gab(r) = Nb(r)/Nbr which measures the actual density of “b” particles at a distance r from an “a” particle, relative to their mean density, is given by the Fourier transform of Sab(k) :

In a two-component, electron-ion plasma, both g,(r) and gee(r) exhibit behavior similar to that of the OCP radial distribution functions (see, for example, Rogers et a/., 1983), and tend to zero as r + 0; however, gez(r) diverges exponentially as r -+ 0. These statements, as well as the OCP trends at small separations, are direct consequences of the Coulomb interaction: one

114 Jon C. Weisheit

can consider the radial distribution functions to be Boltzmann factors, gab(r) = exp[- Kb(r)/@], where V,, is an interaction energy related to the mean force between the pair (a, b). Additional information on radial distribution functions in two-component plasmas can be found in the work of Hansen and McDonald (1978), Weisheit and Pollock (1981), and Dharma- Wardana, Perrot, and Aers (1983), but the emphasis in these papers is on plasmas with strong coupling.

2. Microfields

Even when a plasma is charge-neutral on a macroscopic scale, on the microscopic level, local charge imbalances and, hence, electric fields occur. The expected strength of this field 8 is another of the plasma’s important static properties. In particular, the distribution P ( 8 ) of field strengths controls the quasi-static Stark broadening of spectral lines.

The problem of determining a field strength distribution was first investigated by Holtsmark in 1919, for the case of noninteracting particles. (Chandrasekhar’s 1943 article is a more contemporary reference.) The Holtsmark (r = 0) expression for an OCP is

PH(f l ) = ($) Jomx(sin x)exp[ - dx,

where fl = R/b, is the strength of the field relative to the “normal” field 8, = 1Z,el/Ri. For plasmas with r > 0 there is no simple answer, and the calculation of P(8) by at least partly analytical techniques is still a subject of considerable interest. Major theoretical developments can be found in the works of Baranger and Mozer (1959) and Mozer and Baranger (1960); Hooper (1966 and 1968), and Iglesias and collaborators (1983 and 1985). Monte Carlo methods also are well-suited to this kind of problem, but computing machine time increases markedly as I’ decreases, and becomes excessive for r -= 1/2.

In Fig. 2 we plot OCP microfield distributions for r = 0 (Holtsmark), r = 0.12 (Hooper, 1966), and = 1 (Iglesias et al., 1983; DeWitt, 1987). Note that the most probable field strength (in units of 8,) gets smaller as r increases, because Coulomb repulsion inhibits, to a greater degree, inter- particle distances much smaller than average ( N Ra).

The calculation of microfields in electron-ion plasmas is complicated by the disparity of the response frequencies, we and 0,. The usual argument made is that the electrons behave as an OCP, moving rapidly with respect to the background of positive ions. Thus, the OCP results are often called “high- frequency” microfields. The “low-frequency” microfields arising from the ions have distributions with weaker average fields, because the more mobile


FIG. 2. Plasma microfield distributions, in terms of the normalizing field strength 8, =

IZ,le/R:. The solid curves are OCP data labelled by their r-value, or by H (Holtsmark). For r = 1, the triangular points represent Monte Carlo results (DeWitt, 1987) and the solid curve represents “APEX” results (Iglesias et al., 1983). For r = 0.12, the solid curve represents Hooper’s (1966) hydrogen plasma data for the high-frequency microfield, while the dashed curve represents his (1968) data for the low-frequency microfield

electrons tend to localize about, and shield, individual ions (see Eq. 22b). (The first (second) paper in each of the three pairs of microfield references cited above pertains to high- (low-) frequency field distributions.) To illustrate the differences that occur, we also show a low-frequency microfield distribution in Fig. 2: the dashed curve represents Hooper’s (1968) results for the protons in a hydrogen plasma at = 0.12.

111. Excitation Models for Dense Plasmas

We are now equipped to investigate atomic collisions in a dense plasma environment. This situation is radically different from that of the “textbook” scattering event where, in isolation from all else, two initially separate systems interact and then become separate again as an infinite amount of time transpires. We must expect any computable model of the influence of a dense plasma on atomic excitations to represent a substantial simplification of this many-body problem.

116 Jon C. Weisheit

A. QUANTUM STATES OF THE TARGET

One gross effect of the plasma environment on ionic states-continuum lowering-has already been mentioned: for every bound state of an isolated ion there is a density beyond which that state becomes part of the continuum when the ion is in a plasma. Conversely, at any density there is an uppermost bound state with principal quantum number nmax. We used the ion-sphere potential, Eq. (24), to obtain the limit referred to in Section II.A, nmax =

( Z R , / ~ U , ) ’ / ~ . A much more elaborate treatment of excited states in a dense plasma has been published by Rogers (1986). He finds that only low-lying states, with n << nmax, are relatively unaffected by the plasma. All our subsequent discussions, therefore, are limited to densities such that the target states involved in a transition are well below the effective, lowered continuum.

Even small perturbations can be important, however. In particular, the microfields just described give rise to familiar Stark effect phenomena. Because the correlation time among ions greatly exceeds that among electrons, the quasi-static ionic microfield causes the basis states of an ion in a plasma to be admixtures of (shifted) states of the isolated ion. In the special case of a central potential, the mixing is described exactly by coefficients of the transformation from spherical to parabolic coordinate eigenstates (Hughes, 1967); otherwise, perturbation theory must be used to obtain approximate mixing coefficients.

Here, we ignore issues connected with the small level shifts, although these pose interesting and controversial questions of their own. (See, for instance, papers by Dharma-Wardana et al., 1980; and Cooper et al., 1986.) Also, we temporarily put aside complications due to the mixing of levels by a quasistatic microfield in order to isolate the problems pertaining to electron-ion excitation dynamics in a dense plasma. After that discussion, the microfield will be resurrected and its effects briefly considered.

B. WEAK ELECTRON-ION INTERACTIONS

In this section we deal with interactions that typically are weak. Consider a particular ionic transition, y o -, yl, that is induced by inter-

action with a charge Z,e. If, for a given “target” ion, the interactions with different particles of the species “a” were well separated in time, then the notion of distinct collisions would be valid and the cross section oa(yo, yl) could be obtained from the first-order Born formula. Further, and what is more important here, there would be no complicating effects due to the


plasma environment. Our earlier discussion of test charge screening showed that Debye lengths effectively measure the interaction range among plasma particles. This unfortunately means that, at any instant, the number of charges Zae interacting with the target is of order NaD;. In weakly coupled plasmas this number is huge, and so the target experiences myriad simultaneous interactions. We have no a priori reason to exclude most (or any) of them from consideration, even though some of the interactions are surely much weaker than others. Therefore, we abandon the goal of determining cross sections and, instead, ask what is the rate of transitions in a target ion subjected to the stochastic potential field of the surrounding plasma.

It will be possible to utilize our knowledge of plasma fluctuations to answer this question, but only if the net interaction with all N a = N , Y of the plasma’s particles of that kind,

raB(Yot ~ 1 2 t ) = - e ( ~ , I S a ( r 9 ~)IYO), (41)

characteristically is weak, so that the transition rate still can be calculated with first-order perturbation theory. In this case, though, the strength of the fluctuating interaction pa, which does not vanish in the limits t -+ f co, must be measured with respect to the finite time interval At N z,(correlation), which is the average time it takes a charge Z,e to pass through the interaction region. (This same issue arises in the theory of spectral line broadening; see, for example, Lewis (1961), and Smith and Hooper (1967).) We used the normal field 8, to make an order-of-magnitude estimate of the interaction strength for a dipole transition, and find that the net interaction with the electrons usually is weak, while that with the ions is not:

I (3[(332 Therefore, only interactions with electrons are considered in the remainder of this section.

Vinogradov and Shevelko (1976) were the first to derive the rate of atomic excitation in the random field Se of the plasma electrons. Briefly, their derivation begins with the first-order expression,

for the excitation amplitude, where hw,, = AElo is the transition energy. Then, the potential field is expanded in Fourier components, Se(k, w ) =

(4ne/kz)me(k, w). Finally, the transition probability per unit time We is

118 Jon C. Weisheit

computed from the averaged value of 1 ~ ~ 1 ~ . By using Eq. (26) one eventually obtains the integral,

d ~ ( ~ 0 3 ~ 1 ) = (Iue(yO3 ~ 1 ; t)12>

which involves the dynamic structure factor of the electrons, evaluated at the transition frequency wI0, and the atomic form factor (yl leik“lyo).

Now, let Jh be the target’s total angular momentum, M h be its projection onto a specified quantization axis, and q be all the other quantum numbers needed to identify the atomic state, viz. y = qJM. The degenerate target substates can be accommodated, and the equation above cast into a convenient form, by the introduction of Bethe’s generalized oscillator strength,

in the limit k --t 0, the generalized oscillator strength tends to the dipole (absorption) oscillator strength f ( q O J o , qlJl). We have, then,

K ( q o J o , ~ 1 J 1 )

with tl and c being the fine-structure constant and the speed of light. At very low densities, the excitation probability We must reduce to N , times

an excitation rate coefficient (ud). To identify the effective cross section d in this limit, it is helpful first to define the differential (plane-wave) Born cross section for momentum transfer hk in an electron-ion excitation collision. From Inokuti (1971) we obtain

where hko = mevO is the incident electron momentum. Substitution of SS) into Eq. (46) and use of Eq. (27) leads to

wf’(qOJO 9 qlJ1) = dv0 uOFf’(vO)d(qOJO 4 lJl), (48) s with


Thus, the effective differential cross section due to fluctuations in a low density plasma and the differential plane-wave Born cross section are the same; the total cross sections are unequal only because of different integration limits on k. (Recall that the range for crpwB is from k, - k , to k , + k , . ) But, as Vinogradov and Shevelko (1976) noted, when ko is very large one has (k, - k , ) N (ki - k:)/2k0 = wlo/u0, and aPWB tends to 5.

We will not dwell on the numerical examples given by Vinogradov and Shevelko, as they involve only a hypothetical form factor and a one- component (electron) plasma. Moreover, their work considers situations in which [w,,z,(duration)] > 1, and these we contend are unphysical. Instead, we illustrate important aspects of the process of excitation via plasma fluctuations by examples involving hydrogenic states. For these, there exist analytic expressions for f ( y o , y l ; k). Pollock and Weisheit (1985) reported some preliminary calculations of this type.

fine-structure transition in Ne+9, all normalized to the low-density value [Wp)/N,] = C1.31 x cm3/sec], determined from Eq. (48). The solid curve represents the full result for (a few) neon ions in a two-component hydrogen plasma at 0 = (&Z2Z, = 340 eV, and the dotted curves represent contributions from the separate terms in S, , involving StJ and Sg (see Eq. (30)). Note that the term involving S c ) results in a transition rate almost identical to that occurring in an electron OCP (dashed curve). The arrow at the bottom of the figure marks the electron density at which a,, =

(wlo/w,) = 1. Unless N , is almost this high, the plasma environment cannot modify the excitation rate coefficient. The onset of the large, proton-related contribution at much higher densities reflects the increased coupling between electrons and protons, which enhances electron correlations at frequencies near w,. This coupling of electrons and ions prevents the asymptotic, a:, scaling of the (OCP) excitation probability, mentioned by Vinogradov and Shevelko, from being realized.

We performed analogous calculations, also at 0 = ($)Z2Z,, for two other isoelectronic ions, Ar+17(AE = 4.75 eV) and Kr+35(AE = 76.0 eV), and in both cases found the results to be very similar to those for Ne+9. Indeed, by plotting the normalized excitation probabilities as functions of Qlo instead of N,, the curves lie almost on top of one another. The generalized oscillator strength is the same in all three of these instances,

In Fig. 3 we plot several excitation probability curves for the 2s,/? +

but the abrupt diminution at ka, 2 2 here is characteristic of the k- dependence of generalized oscillator strengths for all allowed transitions (see Inokuti, 197 1, for several hydrogenic illustrations). Generalized oscillator

120

0.005 1 .o

Jon C . Weisheit

Tee

0.01 0.05

- TCP S,,

1 I

20 21 22 23 24 log (NJ1 cm3)

FIG. 3. Density dependence of the probability of excitation by plasma electron fluctuations, normalized to the low-density value, for the 2s,/, -. 2 p , / , transition in Ne+9. The dotted curves indicate separate contributions to this probability, for a two-component plasma (TCP) of electrons and protons at 0 = 1100 eV. The solid curve is their sum, and the dashed curve shows the normalized probability for a one component plasma (OCP) of electrons, also at 1100 eV. The electron-electron coupling constant re. is indicated at the top of the figure, and at the bottom of the figure the arrow marks the electron density at which hw, = A&, = 0.45 eV.

strengths for multi-electron atoms can exhibit local minima at small values of ka, (because the atomic form factor changes sign). Even then, we expect the qualitative features shown in Fig. 3 to be representative of plasma effects on electron-induced dipole excitations.

Generalized oscillator strengths for higher multiple transitions (El, 1 > 1) also decrease rapidly when ka, > Z, but in addition tend to zero as k2”’ when k + 0. Again, several hydrogenic examples are given in Inokuti’s (1971) paper. We chose the quadrupole excitation 2 3 P , + 23P2 in He-like Ar+I6


( A E = 2.41 eV) as our example of a forbidden transition, and used hydrogenic radial functions to obtain

Calculations were made for (a few) argon ions in a hydrogen plasma at 1000 eV, and in an electron OCP at the same temperature.

The excitation probability for this E2 transition decreased a negligible amount from its limiting value, Wf) = (3.00 x lo-” cm3/sec) N , , even at the highest density considered, N , = ~ m - ~ . The Sfi-related contribution to the two-component plasma’s value of We was negligible at all densities, too. The qualitative differences between these results and those we found for allowed transitions are due solely to the different small-k behavior of the generalized oscillator strengths in the two cases: The Sz-related term is important only when ka, is small, but when the transition is dipole-forbidden the noted k-dependence of f ( y o , yl; k) serves to make the integrand in Eq. (46) relatively small in this region. And, when ka, ‘u Z (the region in which the dipole-forbidden generalized oscillator strength is relatively large), the electron permittivity $‘)(k, w,,) is very nearly unity. These statements should apply generally to forbidden transitions with small energy defects, because their generalized oscillator strengths all exhibit a qualitatively similar k- dependence.

In all of the computations described to this point, dynamic plasma response was incorporated through the RPA permittivities $)(k, w). We decided to perform some calculations in which static screening was imposed through the use of @(k, 0) and @(k, 0), because only in this limit do there exist simple potentials for the effective interaction between charged particles in a plasma (Equations (22a,b,c)). We found that electron OCP values of We for both allowed and forbidden transitions change very little; this is also true for the SfJ-related contribution to See in the excitation rate calculations for two-component plasmas. The reason for this agreement is that K , = kD, S O , , for almost the whole range of k-values important to each transition rate integral, and therefore &f)(K,, Qlo) N &f)(K,, 0). On the other hand, this static screening approximation is poor for the Sg-related contribution: in this term, K , is not much larger than (w,,/w,) for the most important part of the integrand in Eq. (46). This makes a large difference in We-values computed for allowed transitions, but a negligible one in We-values computed for forbidden transitions, because in the later instance the Sf;-related term is unimportant.

122 Jon C. Weisheit

Thus, we come to the surprising conclusion that a steady-state response approximation is satisfactory for treating excitation by electron fluctuations, as long as the plasma electrons and ions are only weakly coupled, and Q,, is small. In fact, when the electron-ion coupling is strong enough that the Sti-related term is important, calculations based on Eq. (46) probably are unreliable anyway, because we initially assumed that the interaction strength, which is proportional to z(correlation), is weak [cf. Eq. (42)]. This assumption is fundamental to the whole scheme but, for example, in Fig. 3 it clearly has broken down when N e is as large as ~ m - ~ . Therefore, we recommend that the Sti-related contribution be dropped, and transition rates due to plasma electron fluctuations be determined from just the electron-OCP structure factor. It accurately reproduces the part of the excitation probability that does not violate the weak-interaction constraint. This neglected contribution somehow needs to be taken into account when the excitation rate due to strong interactions between ions is determined.

Altogether, then, when the typical electron-ion interaction is weak and R,, = ( o l O / w e ) I 1, we have the modified excitation rate formula,

(52)

Rates of excitation by plasma electron fluctuations are essentially equal to their low-density values, Ne(uoe(yo , y,)), when Q,, > 1 .

C. STRONG ELECTRON-ION INTERACTIONS

We now consider interactions that are strong and cannot be treated by first-order methods.

The number of simultaneous weak interactions with particles of species “a” that a given “target” ion experiences has been estimated as N,D: . Similarly, the number of simultaneous strong interactions is approximately Nap: , where p a is the Weisskopf radius defined in Eq. (1 1). When this latter number is small, Nap: < 1 , each strong interaction is isolated in time and takes place against a backdrop of the ongoing weak interactions. Thus arises the notion of a collision perturbed by the plasma environment. In contrast, when N a p : > 1 , there are overlapping strong interactions and the quantum- mechanical time evolution of the target is not directly related to a binary collision matrix. This difficult physics problem is unsolved-a fact that forces us to limit our analysis to plasma conditions in which strong interactions with electrons or ions occur only sequentially. The remainder of this section


deals only with electron-ion interactions; ion-ion interactions will be considered in Section 1II.D.

The relative magnitudes of certain timescales and lengths again suggest the plasma’s expected influence on a strong collision. Because pe Q D, here, the average time it takes an electron to transverse the strong interaction region is much less than the time s,(correlation). One is interested, therefore, in the plasma’s response when 51 1: (DJp,) is large. But, even then, the ratio (R/kD,) N (llkp,) is of order unity for the most important wavenumbers, k 5 Z/ao. From Table I we see that Is(K, n)I2 N 1 when both K and R are large, which leads us to conclude that strong electron-ion interactions are little affected by the background plasma.

However, we again adopt the static-screening model for situations in which R,, I 1. The important reason for doing this is that it permits us to use effective two-body potentials within the framework of collision theory to treat consistently the weak as well as the strong electron-ion interactions. To establish this point, let the Debye expression

e2 exp( - y) V(r, r’) = ~

Ir - r’I (53)

be the energy of interaction between a target electron at r and an incident plasma electron at r’. When the interaction is strong, the screening factor, which will be no smaller than exp( - p e / D e ) , will have no effect. But, when the interaction is weak, the Born approximation will be valid. By performing integrations first with respect to r - r’ and second with respect to r, it is found that the differential Debye-Born (DB) cross section is just that given in Eq. (47), divided by the factor [l + k D , ) - 2 ] 2 = ( s@)(k , 0)l2:

Then, by changing the order of integration in the transition rate expression, we obtain (after considerable manipulation)

where ( i )meufo = h2(k i - kf) /2m, = AElo. Except for the second factor in the exponential, this is the same as Eq. (52) . This difference can be traced to

124 Jon C. Weisheit

the fact that, in the derivation of Eq. (52), the conditions for energy and momentum conservation are related only through the Dirac delta function 6(w,, - ksv). Therefore, at a given wavenumber k, almost any velocity can contribute to the process of excitation through fluctuations.

We compared several electron excitation rates calculated according to Eq. (55) to ones calculated according to Eq. (52), for situations in which R,, I 1. Differences between the results calculated from plane-wave Born cross sections and those calculated from the plasma fluctuation formula were no more than a few percent. Thus, whether the interactions leading to a particular excitation are weak or strong, Born cross sections obtained from static-screened potentials will yield the appropriate excitation rates. More sophisticated collision models should be employed when a transition is dominated by strong interactions, but it is reasonable to expect that any such model will also provide a consistent treatment of the weak electron-ion interactions. We believe this is sufficient justification for the use of static- screened potentials to study electron-ion collisions in weakly coupled plasmas-when R,, 5 1.

The first calculation of plasma effects on electron-ion collisions was reported by Hatton et al. (1981), who used a Debye potential and the Born approximation to study 1 s-2s, ls-2p, and 2s-2p electron-impact excitations in hydrogenic ions. Their investigation revealed that screening effects could be substantial, especially near threshold, and it prompted further work : Deb and Sil(l984) made analogous Born calculations for other resonance transitions in He+. Davis and Blaha (1982a) made distorted-wave calculations for 1s-2s and 1s-2p transitions in Ne+9, in which many-body exchange and correlation effects were included in the effective potential for the bound and the incident electron; however, there was no screening of the interaction between these electrons. In a companion paper (Davis and Blaha, 1982b), the interaction between these two electrons was taken to be the Debye-screened Coulomb expression, Eq. (53). Pundir and Mathur (1984) also made distorted-wave calculations, and adopted a Debye-screened potential to study excitations in He+. Whitten et al. (1984) made distorted-wave and close-coupling calculations for transitions among the Is, 2s, and 2p states in He+, Ne+9, and Ar+I7; both Debye and ion-sphere screening models were employed.

Unfortunately, the plasma's dynamic behavior was not fully considered in these papers, and we now believe that such static-screening calculations are incorrect for transitions with Rlo > 1 (such as resonance excitations in hydrogenic ions). Further, in some of these papers, the Debye screening factor was given the form exp( - r'/De), instead of exp( - I r - r'I/D,). Because the resulting excitation rate does not reduce to the fluctuation formula, Eq. (52), when the interaction is weak, this too is incorrect. As Hatton et al. (1981) noted, the effct of screening can be considerably overestimated when


this wrong form is employed. On the positive side, however, the more elaborate treatments of the collision dynamics have provided useful information (cf. Whitten et al., 1984). For example, trajectory effects are relatively unimportant-it is the form assigned to the plasma screening that matters most. Also, as with unscreened electron-ion collision events, distorted-wave techniques often suffice, but plane-wave Born formulae can give poor results, especially near threshold.

D. ION-ION INTERACTIONS

It is now time to recognize the existence of an ion microfield that is quasistatic during electron-ion interactions. To gauge the extent to which this field modifies our discussion in the previous sections, we compare the relative magnitudes of three energies: h o e , AE,,, and the linear Stark term Vij(Stark) = e(d.(y,Irly,)J. Here, y, and y j represent the closest pair of adjacent states, either or both of which belong to the transition in question. It follows from Eqs. (5 ) , (8) and (1 1) that

For the plasma conditions of interest to us, this ratio is much less than unity. But AEij does not exceed AE,,, which itself must be less than h o e for significant screening effects to occur. Therefore, we need only consider the ratio ViJStark)/AEij. When it is very small, we can safely ignore the microfield's influence; when it is of order unity, transition states are strongly Stark mixed; when it is large, even the identification of a transition in terms of unperturbed basis states is unrealistic. In this last regime: all adjacent states will be statistically populated by rapid collisional transitions; and all spectral lines arising from these Strark-mixed states will be blended. Consequently, we can concentrate on situations where I/,,(Stark) - AEij Q h o e .

It is reasonable in these situations to calculate matrix elements in the regular atomic representation {Iy)} and then to make the transformation to the Stark representation (parabolic coordinates) { I y [ S ] ) } , as proposed by Whitten et al. (1984); much of the Racah algebra involved, including the averaging with respect to the angle between d and the collision axis, may be found in the paper by Greene et al. (1975). This procedure yields a cross section a(yOIS], yl[S]; 8) for a specified field strength, 8. Then, a final averaging with respect to field strengths gives an effective cross section

(57)

126 Jon C. Weisheit

It is clear that this Stark mixing will be most important when it gives some part of an allowed transition’s strength to a transition that is (dipole) forbidden in the field-free limit. In such a case, the forbidden transition will become not only stronger, but also more sensitive to plasma screening.

One study, involving Ne+9 transitions, of these various ion-microfield effects has been made very recently (Perrot, 1988). These calculations support the general statements presented here, and also reveal that the effective cross section, Eq. (57), is very nearly equal to the cross section a(y,[S], yl[S]; 8) computed for one-half of the normal field 8,. It will be interesting to see if calculations for other transitions (and close-coupling calculations for more complex targets) concur with this finding.

Besides providing a quasi-static microfield, 8, in which electron-ion interactions take place, ions can also cause transitions. In particular, when AE,,/@ is small, ion-ion and electron-ion collisions tend to be competitive. Therefore, the transitions for which plasma screening may influence electron- ion excitation rates are also the ones for which ion-ion excitations may be important. It has already been pointed out (Eq. 42) that the dynamic interaction of a “target” ion with all other ions in the plasma is usually too strong to be treated by first-order perturbation theory; even individual ion- ion interactions (unscreened) are too strong to be treated this way (Seaton, 1964; Walling and Weisheit, 1988).

We can anticipate using the methods of collision theory if strong ion-ion interactions do not occur simultaneously, that is, if p z N , -= 1. For unscreened ions, this constraint is much more restrictive than it is for electrons, because p , N Z(rn,/m,)1/2pe. Typical values yield Z 3 N , < 1019 ~ m - ~ , a limit so low as to be uninteresting here. Fortunately, on account of the difference in relaxation times, electrons do efficiently screen ions and thereby decrease pz to some extent. We already pointed out that ion microfields in low-T plasmas can be computed accurately from a model in which each ion has an (electron) Debye-screened potential, as in Eq. (22b). Moreover, in the strong-screening limit (r B l), a stationary ion’s potential approaches that of the ion-sphere model, Eq. (24). In this limit, the effective p , must reduce to R,, and strong ion-ion interactions again become sequential. In fact, Scheibner, et al. (1987) showed that the ion-sphere model provides a good approximation to Monte Carlo determinations of the screened ion-ion interaction, V,,(r) =

-O[ln gzzt(r)], even in plasmas with Tzz. - 1. All these comments support the use of static-screened interaction potentials in ion-ion collision studies as well.

In the paper just mentioned, Scheibner et al. published the first study of dense plasma effects on inelastic ion-ion collisions. Their calculations were for transitions among the n = 2 fine-structure levels of hydrogenic ions He+, Nef9, Ar + 17, and Fe+25. Cross sections were determined from semiclassical,


close-coupling equations. These authors derived and used two screened ion- ion interactions Vzz,(r), one based on the ion-sphere model and the other, on the Debye model:

In the equations above, R, = [3(Z + 2))/47~N,]~'~ is the ion-sphere radius in the united-ion limit (r -+ 0), and Ze and Z'e are the net ionic charges.

Scheibner, et al. (1987) found that Debye screening actually enhances cross sections, with respect to their unscreened values, at very low collision energies because closer encounters are possible. (Perhaps this near-threshold enhancement, which was not observed in their ion-sphere calculations, is a manifesta- tion of the $';-related contribution that was discarded in Section 1II.B.) For both models, however, a substantial reduction occurs at most energies because of the diminished strength of the interaction. Then, the ion-sphere expression, Eq. (58), yields cross sections that typically are 10 to 30% smaller than those computed from the Debye expression, Eq. (59), with D, =

RJtarget). Also, as in the case of electron-ion interactions, the quadrupole 2p,,2-2p3,2 transition was less affected by screening than were the dipole ones.

Although Scheibner et al. (1987) made the effort to extend their results by considering Z-scaling trends, other investigations need to be made before much can be said generally about the influence of a dense plasma on ion-ion collisions. In this regard, it is encouraging that a fully quantum mechanical calculation of excitation in screened proton-Ar + l 7 collisions has just been completed (Zygelman and Dalgarno, 1988). This work confirms the applicability of semiclassical methods in such scattering calculations.

IV. Conclusion

We do not have the space here to apply the present ideas to other collisional phenomena in dense plasmas; particularly interesting among these are bremsstrahlung and dielectric recombination. We suggest that readers wishing to pursue either topic first consult the following references.

Bremsstrahlung in dense plasmas: DeWitt (1958); Grant (1958); Zhdanov (1977); Rozsnyai (1979); Kim et a!. (1985); Totsuji (1985).

128 Jon C. Weisheit

Dielectronic recombination in dense plasmas: Weisheit (1975); Jacobs et al. (1976); Zhdanov (1979); Grigoriadi and Fisun (1982); Jacobs et al. (1985); Rzazewski and Cooper (1986).

It must be obvious that the lack of specific data in our subject is a serious handicap. It is unfortunate that it is not yet (and may never be) possible to sort out details of complex ICF or X-ray laser experiments and from them to infer the influence of a dense plasma on individual atomic processes. In consequence, this paper may be considered more a prospectus than a review, for it may have generated more questions than answers. Still on .the author’s list of important unanswered questions are:

( I ) What are the largest r-values for which weak-coupling (RPA) quantities can be used in atomic excitation problems?

(2) In weak electron-ion interactions, do calculations with actual form factors for complex target ions exhibit the expected excitation rate characteristics?

(3) How can the neglected electron fluctuation term (related to Sg) in two-component plasmas be incorporated into calculations of excitation caused by strong interactions?

(4) To what extent can data on spectral line shapes, which are formally related to the charge-charge structure factor S,(k, o) (e.g., Dufty, 1969), be helpful in resolving issues and ambiguities in the existing theory?

( 5 ) In strong interactions (electron-ion and ion-ion), is there a better, but still workable, approximation than static screening by the background plasma?

(6) Can Seaton’s (1964) procedure for bounding first-order excitation probabilities in (unscreened) ion-ion collisions be extended, so that the ions’ structure factor S d k , w) can be used to treat screened ion-ion interactions in the context of plasma fluctuations?

(7) What is a dense plasma’s influence on “collisional” ionization (say, from Rydberg states)? And, is a modified density of continuum states (cf. More, 1985) of practical importance here?

(8) What is the importance of microfield gradients in perturbing target fine-structure states that are not dipole-coupled? And, how can these be treated, if need be?

(9) Are the strong (> megagauss) magnetic fields sometimes generated in dense laboratory plasmas of consequence to inelastic atomic processes? (LO) What surprises await in the strong coupling plasma regime?

We hope that there will be progress to report on many of these issues by the time someone next ventures to survey the subject of atomic excitation in dense plasmas.


ACKNOWLEDGMENTS

I t is a pleasure to be part of a volume commemorating Alex Dalgarno’s contributions to physics. He has been a colleague and friend during my entire career.

1 have had several collaborators in my published work that is discussed in this article: N. Lane, E. L. Pollock, B. Whitten, K. Scheibner, and G. Hatton. I wish to thank all of them for their separate roles in the development of various topics. Also, J. Cooper has enthusiastically listened to and criticized my own ideas on this subject through their long gestation. In addition, Livermore colleagues D. Boercker, R. Cauble, H. DeWitt, C. Iglesias, and F. Rogers shared comments and unpublished results; their help is gratefully acknowledged, too. Finally, I wish to thank Mrs. R. Jensen for her skill and patience in the production of this manuscript.

My research has been performed under the auspices of the U.S. Department of Energy, and supported by its contract # W-7405-Eng-48 to the Lawrence Livermore National Laboratory.

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Stewart, 3. C. and Pyatt, K. D. (1966). Astrophys. J . 144, 1203. Tawara, H. and Kato, T. (1987). At. Data & Nuc. Data Tables. 36, 167. Totsuji, H. (1985). Phys. Rev. A. 32, 3005. Vinogradov, A. V. and Shevelko, V. P. (1976). Sou. Phys. JETP. 4, 542. Walling, R. S. and Weisheit, J. C. (1988). Physics Repts. 162, 1. Weisheit, J. C. (1975). J. Phys. B ; Atom. Molec. Phys. 8, 2556. Weisheit, J. C . (1984). In Applied Atomic Collision Physics Vol. 2 (C. F. Barnett and M. F. A.

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Vol. 20 (D. Bates and B. Bederson, eds.). Academic Press, New York, New York.

Harrison, eds.). Academic Press, New York, New York.

Gruyter, Berlin, West Germany.


ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL. 25

PRESSURE BROADENING AND

LINE SHAPES LASER-IND UCED SPECTRAL

KENNETH M. S A N D 0 Department of Chemistry Uniiwsity of Iowa Iowa Ciry, Iowa

SHIH-I CHU Department of Chemistry Uniaersrty o# Kan.ra7 Lawrence, Kansas

I. Atomic Line Shape Theory in the Weak Field Limit . . . . . . A. One-Perturber Line Shapes . . . . . . . . . . . . . . B. Statistical Mechanics of the Density-Dependent Line Shape. . .

11. Spectral Line Shapes in Strong Fields. . . . . . . . . . . . . A. Multiphoton Absorption Spectra and Quasi-Energy Diagram .

B. Multiphoton Dissociation of Small Molecules: The Inhomogeneous

C. Multiphoton and Above Threshold Ionization in Intense Fields .

Acknowledgments . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

ential Equation Approach of Dalgarno and Lewis . . . . . .

D. Intensity-Dependent Resonance Light Scattering . . . . . .

. . . 133

. . . 135

. . . 142

. . . 146

. . . 141 Differ- . . . 152 . . . 154 . . . 156 . . . 160 . . . 160

I. Atomic Line Shape Theory in the Weak Field Limit

Atomic spectral lines, due to transitions between electronic states of atoms, are not infinitely sharp, but are broadened by various mechanisms. Natural, Doppler, pressure, and Stark broadening all contribute to the shape of an atomic spectral line (Margenau and Watson, 1936). Natural broadening is due to the finite lifetime of at least one of the states involved in an electronic transition. It is important only at low temperatures and pressures, but it i s the only mechanism that persists even for an isolated, stationary atom. Doppler broadening is due to the higher frequencies of radiation absorbed or emitted by an atom moving toward an observer than by one moving away. The random component of velocity in the direction of the observer found in an equilibrium gas thus leads to broadening and a Gaussian line shape. Pressure broadening is due to collisions of the active atom with perturbing atoms,

133

Copynght Q 1988 by Academic Press, Inc All rights of reproduction in any lorm reserved

r n n r r ..........

134 Kenneth M . Sando and Shih-I Chu

molecules, or heavy ions. Stark broadening is due to fluctuating electric fields due mainly to electrons in a plasma.

In this section, we will emphasize pressure broadening because of its importance in understanding atomic interactions. Pressure-broadening is the dominant broadening mechanism in high pressure neutral gases. We will not cover the special problems that arise in self-broadening (broadening of an atomic spectral line by another atom of the same kind), therefore, some of what we say will be limited to foreign gas broadening.

It is useful to think of each of the broadening mechanisms as producing a “line shape function.” The line shape function is normalized to unity and can be regarded as a probability distribution. An important point is that if two broadening mechanisms are independent, the line shape resulting from the operation of both mechanisms is simply the convolution of the two line shapes resulting from each mechanism separately (Jablonski, 1945). Thus, the simple Lorentzian line shape of natural broadening can be convoluted with line shapes resulting from all other mechanisms as long as the radiative lifetimes of the atomic states are not affected by the other mechanisms. The simple Gaussian line shape of Doppler broadening can be convoluted with the pressure-broadening line shape as long as velocity-changing collisions are infrequent or of small effect (Ward et a/., 1974; Berman and Lamb, 1970).

The theory of pressure-broadening has had a long history which has been well-covered in previous reviews (Ch’en and Takeo, 1957) and monographs (Breene, 1961,1981), including a recent, comprehensive review by Allard and Kielkopf, 1982. In this review we will not attempt to be comprehensive, but we will try to give a summary of current theory.

The theory of pressure-broadening is a mixture of dynamics and statistical mechanics. A clean separation of the dynamics of individual collisions from the statistical mechanics has been made by Royer (1974, 1978, 1980) based upon the concepts of Jablonski (1949, Anderson and Talman (1956), Baranger (1958a, 1958b, 1958c), and Fano (1963), among others. In foreign gas broadening, we think of an active atom surrounded by perturbing atoms (perturbers). Therefore, a two-body collision becomes a one-perturber collision. Two-body collisions can also occur between perturbers, but these are generally neglected, along with all three-body (and higher) collisions. There- fore, the role of statislid mechanics in the theory of pressure-broadening is to express the N-perturber density-dependent line shape in terms of one- perturber line shapes. These one-perturber spectra are important in their own right. They find application in diatomic spectroscopy, where they are referred to as “continuous spectra” (Sando and Dalgarno, 1970), in the theory of excimer transitions (Sando, 197 l), in the determination of interatomic potentials (Hedges et al., 1972), and in atomic and molecular processes in planetary and stellar atmospheres (Dalgarno, 1973; Dalgarno et al., 1970;

LASER-INDUCED SPECTRAL LINE SHAPES 135

Allison et al., 1971; Stephens and Dalgarno, 1972; Dalgarno and Sando, 1973; Butler et al., 1977; Uzer and Dalgarno, 1979a, 1979b, 1980; van Dishoeck and Dalgarno, 1983; Cooper et al., 1984; van Dishoeck et al., 1984). In Section LA, we discuss one-perturber line shapes and diatomic continuous spectra. Then, in Section I.B, we briefly describe the statistical mechanics of putting these one-perturber spectra together to form the density-dependent line shape.

A. ONE-PERTURBER LINE SHAPES

The theory of the one-perturber spectrum (Sando and Dalgarno, 1970; Herman and Sando, 1978) is that of two-body dynamics in the presence of a weak electromagnetic field with frequency in the vicinity of an atomic transition. The difference between the frequency of the field and that of the atomic transition is often referred to as the “detuning” and given the symbol A. The units of A in this section and in the statistical mechanics section are always cycles/second. The dynamics may be treated at various levels of approximation. Quantum mechanics is capable of exact results (given accurate interatomic potentials), but classical dynamics may give useful results despite inherent approximations (Sando and Herman, 1983). We will first present theories of the one-perturber spectrum in the two-state approximation, i.e., two diatomic molecular electronic states correlating with two atomic states. This approximation is strictly valid only in the rare case of a collision-induced transition between two atomic ‘S states, but it is often accurate in the wings (large detunings) of allowed atomic lines as long as the molecular states are well separated. After presenting some examples of applications of the two-state approximation, we will briefly discuss some approaches to the coupled-state problem.

I . Quantum Mechanical Theory in Two-State Approximation

Line shape functions are probability distributions of unit area. For notational convenience, however, we will use unnormalized one-perturber line shapes, which we denote E(”(A) for emission and A(’)(A) for absorption. Here A = v - v o is the detuning and the superscript “(1)” indicates one perturber. Normalized line shapes are simply given by E“’(A)/E, and A(”(A) /Ao, where E , and A , are the line strength factors for emission and absorption equal to the average value of the square of the transition moment function (11’). In the case of a constant transition moment (independent of internuclear distance), often assumed for the pressure broadening of allowed


atomic lines, E , = A, = p z , Observable spectra are obtained from unnormalized line shape functions by multiplying by the appropriate Einstein coefficient. Thus, the diatomic continuous spectrum for stimulated emission (in photons per unit energy interval per unit time) is

and the absorption coefficient is

where n, is the number density of active atoms in state a, nb is the number density for state b, and np is the number density of perturbers.

The principle of detailed balance is important in atomic line shape theory (Nienhuis, 1973), both as a criterion that a valid theory must satisfy, and as a tool to simplify calculations. If the atoms are in thermal equilibrium with respect to translational motion, then

A(A) = exp( g ) E ( A ) , (3)

This relationship between the absorption and emission line shapes is a direct consequence of detailed balance and is valid as long as electronic to translational energy transfer is sufficiently slow that translational equilibrium is maintained. Eq. (3) applies to the N-perturber, density-dependent line shape, as well as the one-perturber line shape.

The diatomic continuous spectrum is a sum of contributions from three types of transitions: bound-free, free-bound, and free-free. Bound-bound transitions lead to discrete molecular line spectra. These molecular lines will appear superimposed upon the continuous spectra and will not be discussed here.

The expression for the bound-free line shape function, written for emission, in the two-state case is (Sando, 1971; Herman and Sando, 1978)

'%i = 11 P u b J S J J ' I < E a , J'Ipl'b, J > 1 2 . (4) Ub J J'

Here, h is Planck's constant, PVbJ is the probability of finding a molecule in electronic state b in vibrational-rotational level ob, J , and S j j , is the Honl-London factor. The free I&,, J') and bound lob, J ) state wave functions both satisfy the radial Schrodinger equation


Here, ,u is the reduced mass of the diatomic, V ( R ) is the interatomic potential for the diatomic electronic state in question, J is the quantum number for rotation of the diatomic, E is the energy, and I E , J ) represents either the free or bound state wave function. The bound state wave function is normalized to unity, as usual, and the free state wave function is assumed to be energy normalized (Child, 1974) so that it has the asymptotic form (for large R )

where K = ( 2 , u ~ / h ~ ) ' / ~ and 6, is the phase shift for the collision. It is relatively easy to prepare bound states in a non-equilibrium distribu-

tion, for example, in a resonance fluorescence experiment. Therefore, the probability P,, , need not represent an equilibrium population. In the case of translational equilibrium

where the translational partition function

2npkT 312

QT = (7) .

Then the number of mdecules in a particular vibrational-rotational state

= nbnPPVbJ' (9)

The corresponding free-bound line shape can be calculated simply by first calculating the bound-free line shape with equilibrium bound state populations and then calculating the free-bound line shape from detailed balance

The P , Q, and R branches, important in bound-bound spectroscopy, lead only to a slight broadening in continuum spectroscopy. Therefore, for computational convenience, the Honl-London factor is often replaced by S,,, = 6,,.(2J + l), so that Q-branches only are used.

The expression for the free-free line shape function, written for emission, in the two-state case is (Herman and Sando, 1978; Julienne et al., 1976)

0%. 3) .

Here, translational equilibrium is assumed and only Q-branch transitions are considered. The free-state wave functions are energy normalized, as before. Rigorous treatment of rotational angular momentum requires coupling with electronic angular momentum (to be discussed later). In the two-state


approximation, we require AJ = 0 (only Q-branch) to prevent spurious contributions at small detunings that arise from the improper asymptotic behavior of the wave functions. Conservation of total energy, of course, requires that hA = E~ - E,. The absorption line shape can be determined from the emission line shape by detailed balance.

Eq. (10) is satisfactory for calculating the intensities of collision-induced transitions (Sando and Dalgarno, 1970; Julienne et al., 1976), for which the transition moment p(R) goes to zero at large internuclear distances, R. For broadening of allowed atomic lines, however, p ( R ) approaches a constant equal to the atomic transition moment at large R , and the free-free matrix elements diverge. It can be shown (Royer, 1974) that EFJ(A) diverges as A-' as A + 0. This mathematical divergence has physical and computational consequences. Physically, it arises because the one-perturber spectrum represents the spectrum that would be observed for one perturber and one active atom in a container of macroscopic size. Of course, the perturber and active atom are nearly always distant from each other, so the intensity at line center (unperturbed line) is infinitely greater (actually greater by a factor of V, where V is the volume of the container) than the intensity in the line wings. This results in a finite width of the Lorentzian core of the pressure-broadened line, and will be discussed at greater length when we cover the density-dependent line shape. The computational consequence of the divergence is that we calculate not simply .@&A), but AzEyJ(A), which is finite at A = 0. Here, we present a simple expression that retains the internuclear distance dependence of the transition moment function p(R). We write p(R) = p + jl(R), where ji(R) goes to zero at large R and ii is a constant equal to the atomic transition moment. Then (Sando and Herman, 1983)

where 6V(R) = G(R) - t ( R ) is the difference potential between the interatomic potentials involved in the transition.

2. Classical Path Theory in Two-State Approximation

Because there is no classical correspondence to transitions between quantum states with different potential energy surfaces (Nienhuis, 1973), any classical path model is, to some extent, ad hoc. The central question (which has no rigorous answer) is: What potential energy surface controls the classical trajectory? Four choices have been proposed: straight-line trajector-


ies (potential energy is chosen to be zero) (Takeo, 1970; Kielkopf, 1976), trajectories on the initial state potential (Atakan and Jacobsen, 1973; Erickson and Sando, 1980), trajectories on the average potential (Sando and Herman, 1983; Riley, 1973), and surface-hopping trajectories (Lam and George, 1982). We will not discuss surface-hopping models here, but will present a model that can be used with any choice of a single potential energy surface to control the classical path. The model is consistent with the principle of detailed balance if either straight-line trajectories or trajectories on the average potential are chosen.

Derivations of classical path methods (Anderson and Talman, 1956; Tsao and Curnutte, 1962) usually intertwine the two-body dynamics with the statistical mechanics of line shape theory. Here (following Sando and Herman, 1983), we choose to keep these aspects of the line shape problem separate to simplify and to clarify the connections between classical path methods of line shape theory, quantum mechanical methods, and experimental diatomic continuous spectroscopy. Though not obvious, the end results presented here are equivalent to those from the earlier literature (Allard and Kielkopf, 1982).

Whereas quantum mechanical line shapes are most easily expressed and calculated in the frequency domain, classical path formulas are more readily obtained in the time domain. They result from solutions of the time- dependent Schrodinger equation using a trajectory which is a classical path on a potential energy surface. Thus, the transitions are treated quantum mechanically, but the paths are strictly classical.

In order to satisfy detailed balance, we define a new unnormalized line shape function f(')(A), intermediate between that for emission and that for absorption, such that E(')(A) = exp( - hA/2kT)Z(')(A) and A(')(A) =

exp( + hA/2kT)f(')(A). We do not distinguish between free-free and free- bound processes. Energy conservation determines whether the final state is free or bound. In parallel with the quantum mechanical formulae and for the same reasons, we need an expression for A2Z(')(A). In frequency space,

h2A2f(')(A) = h2(hQT)-' dc exp - (25 + 1)12ziAF(A)12. (12) j O m (;;) J

Here, F(A) is the Fourier transform of the time derivative of a time-dependent signal

m

271iAF(A) = exp( - (27ciAt)j(t) dt , - 0 3

where

140 Kenneth M . Sando and Shih-1 Chu

For each collision energy E and each angular momentum J (or equivalently, impact parameter), a trajectory R(t) is determined. The above expressions are then readily evaluated. Straight line trajectories have been used most often (Allard and Kielkopf, 1982), but trajectories on the average potential probably give better results. Comparisons of results with those of quantum mechanical calculations show good agreement (Sando and Herman, 1983).

Classical path methods require less computation time (by a factor of 5 to 10) than quantum mechanical methods, but with the speed of modern computers, this is unimportant. For the broadening of an atomic line by atomic perturbers, quantum mechanical methods may as well be used. Where classical path methods are still valuable, and are likely to remain so, is in more complicated systems, such as the broadening of an atomic line by molecular perturbers.

3. Quasistatic Theory in Two-State Approximation

The quasistatic formula (Kuhn, 1934) of pressure-broadening can be derived by stationary phase integration either from the quantum mechanical or classical path formulae. The formula is exeedingly simple. For emission,

Here, R , is a point of stationary phase, a point at which the classical Franck-Condon principle hA = 6 V(Rs) is satisfied. The quasistatic formula is extremely easy to evaluate and gives reasonable agreement with results of more sophisticated methods (Herman and Sando, 1978). The quasistatic formula does not have the proper A W 2 divergence at line center. It is, therefore, unsuitable for determining the width and shift of the density- dependent line shape.

4. Applications of One-Perturber Theory in Two-State Approximation

One-perturber theory in the two-state approximation is valid for intensities in the wings of atomic lines whenever the detuning is large compared to the inverse of the collision duration and the diatomic potential curves are well separated. The same theory is valid for diatomic bound-free continua and for collision-induced transitions with well-separated states. There are many applications that fall into these categories, and consequently many relevant publications. Rather than attempt to give an inclusive list, we will give a sampling to show the variety of types of applications that exist.


In quantum mechanical theory, there have been calculations of self- broadening of hydrogen in absorption (Dalgarno and Sando, 1973), of the hydrogen continuous molecular emission spectrum (Stephens and Dalgarno, 1972), of “excimer” emission spectra (Sando, 1971), collision induced spectra (Julienne et al., 1976), and bound-bound-free and free-bound-free resonance- fluorescence spectra (W. T. Luh et al., 1986). Diatomic continuous spectra show characteristic intensity oscillations (Tellinghuisen, 1984) that can be accurately reproduced only in a quantum mechanical calculation. These oscillations may arise from extrema in the difference potential (satellite bands) (Sando and Wormhoudt, 1973), from reflection structure (Condon, 1928) seen in emission from a bound state to a repulsive state, or from other sources.

Classical path theory has rarely been used to calculate one-perturber spectra, but quasistatic theory is widely used. Quasistatic theory is valuable for rapid, semiquantitative line shape determinations, but is most useful for inverting observed line wing spectra to obtain interaction potentials (Hedges et a/., 1972). It is one of few methods available for obtaining interaction potentials between excited atoms. Quasistatic spectra show no quantum oscillations and diverge at satellite bands, but comparison with quantum mechanical calculations have made these deviations reasonably predictable (Herman and Sando, 1978; Pontius and Sando, 1983).

5. Close-coupled Theory of the One-Perturber Line Shape

Why is it necessary to consider a close-coupled theory of the one-perturber line shape? In atomic scattering theory, we seek a representation in a basis that is discrete in all degrees of freedom, except the internuclear distance, R (Reid and Dalgarno, 1969, 1970; Mies, 1973). The basis, therefore, represents the electronic and rotational motion. The problem is that there exists no single R-independent basis for which the interaction potential is diagonal at all R. The collision dynamics cannot be treated as a series of separate collisions in each molecular state, each governed by a single potential energy curve, but rather a potential energy matrix must be used and the dynamics for nearby potential energy curves coupled.

There exists a representation that is strictly diagonal at infinite internuclear distance, a direct product representation involving atomic electronic eigenfunctions and eigenfunctions for the rotation of one atom about the other. This might be called the atomic channel state representation. This representation may be satisfactory for weak collisions, but is unsuitable for close collisions because the dimensionality of the potential energy matrix must be very large to accurately represent strong interactions. There exists a representation for which the potential energy matrix is diagonal at each internuclear


separation, the adiabatic representation (Royer, 1974). The problem here is that in order to achieve the diagonal potential, the basis itself must be R- dependent. The coupling between states merely moves from the potential energy to the kinetic energy. The adiabatic representation is useful over limited ranges of R, for example in bound-free transitions and, perhaps, in classical path methods where kinetic energy couplings are handled more easily. Considerable recent success has been achieved with a representation intermediate between the extremes of the atomic channel state and adiabatic representations. This representation might be called the molecular channel state representation (Mies, 1973). The basis is weakly R-dependent, so kinetic energy couplings may often be ignored, and the dimensionality of the potential energy matrix can be taken to be small with good accuracy, equal to the number of close-lying molecular electronic states.

To convert the equations given earlier for the quantum mechanical and classical path two-state approximations to coupled-state equations, we must convert all quantities into matrices. In the classical path method, the exponential in Eq. (14) must be time-ordered. The determination of the elements of the interaction potential matrix requires a knowledge of potential energy curves either from theoretical Born-Oppenheimer calculations or from analysis of experimental data and a careful analysis of the angular momentum coupling that occurs as the atoms separate. Asymptotic boundary conditions for the wave-vectors also require careful consideration. We refer the reader to papers in which these problems are clearly discussed (Mies, 1981; Julienne and Mies, 1984, 1986; Julienne, 1982; Mies and Julienne, 1986).

In applications, the one-perturber absorption line shape for the Sr resonance line (Julienne and Mies, 1986) and for the Na resonance line (Vahala et al., 1986), both perturbed by rare gases, have been calculated with quantum mechanical close-coupled theory, and the broadening of the Cs resonance line by Xe has been calculated in a classical path formalism that incorporates concepts of close-coupling (Allard e t al., 1974).

B. STATISTICAL MECHANICS OF THE DENSITY-DEPENDENT LINE SHAPE

The theory of the density-dependent (or N-perturber, where N is large) line shape consists of combining one-perturber line shapes in such a way as to properly reproduce the effects of multiple collisions that occur in a gas. The pressure regime we are considering ranges from very low pressures (convolution with the Doppler line shape may be necessary) up to pressures of about 20 atmospheres in the perturbing gas. At higher pressures, perturber-


perturber interactions and three-body collisions begin to play a significant role (Erickson and Sando, 1980; Evensky and Sando, 1985).

The statistical mechanics of atomic line shapes (Royer, 1974, 1978, 1980) is somewhat unique in that different regions of the line profile (different detunings) arise from rather different physical processes and show different density dependence. Intensities in the line wings result from strong collisions in which the distance between the perturber and the active atom becomes small. Because the probability of two simultaneous strong collisions is low, intensities in the fine wings are well described by one-perturber theory. The observed intensity is directly proportional to the perturber density. Intensi- ties in the line core result from weak collisions. Because the probablility of two weak collisions occurring within the radiative lifetime of the active atom is large, a multiple-collision theory must be used. The observed intensity at line center is inversely proportional to the perturber density. Historically, line wings were described by a one-perturber theory (usually quasistatic theory) and line cores were described by a separate theory, the “impact theory” (Lindholm, 1945; Foley, 1946). “Unified theories” that describe the entire line profile with a single equation developed gradually with important contributions from Jablonski (1949, Anderson and Talman (1956), Baranger (1958a, 1958b, 1958c), Fano (1963), and Szudy and Baylis (1975). There are at least three “unified theories.” Royer (1974) has shown the relationships between them and has shown that the statistical mechanics of the density- dependent line shape and the two-body dynamics of the one-perturber line shapes can be treated as distinct problems.

The general theory includes the possibility that distinct atomic lines may overlap at sufficiently high perturber density. A subset of that theory is the theory of isolated lines. The distinction is similar, but not identical to the distinction between coupled-state theory and two-state theory for the one- perturber line shape. A transition between ‘ S and ‘ P atomic states will involve coupled-state two-body dynamics, but not overlapping lines. Because it is simpler and well-understood, we will first describe the theory of isolated lines. Then we will briefly describe some of the still controversial problems connected with overlapping lines.

1. Statistical Mechanical Theories for Isolated Lines

The discussion in this section will be given for emission. Absorption line shape formulas are analogous. The reader is reminded that the units of A, as well as the shift and width, are cycles/second. The line-shape formulas given here are not necessarily normalized.

The density-dependent line shape depends upon A2EC1)(A) and upon the line shift parameter. Causality requires the line shape functions to be


complex. The function A2E(')(A) as given in Section 1.A is real. Kramers- Kronig relations or alternatively, Fourier transforms, relate the imaginary part to the real part. Information about the singularity in the imaginary part of E")(A) is, however, not contained in A2E(l)(A). The singularity is directly related to the line shift, and we must perform a separate calculation of the line shift parameter. The value of the shift parameter is the same as the impact theory shift given by Baranger (1958a),

d = -(2nQ,h)-l dE exp ~ (25 + l)sin[2(6! - &)I, (16) j O m (;:) J

where 6; and 8j are the phase shifts in the upper and lower states, both for a collision energy equal to E . Alternatively, in the classical path approximation (Anderson, 1952),

d = (2nQ,h)-' jomdE en($) J (25 + 1)sinPh-' /omAVIR(t)]dt]. (17)

The one-perturber dipole autocorrelation function C,(t), or rather its second derivative C,( t ) , is required in some of the line shape formulas (Royer, 1974, 1978, 1980; Erickson and Sando, 1980). It is related to A2E(')(A) by a Fourier transform

m

C , ( t ) = -4n2 f exp(2niAt)A2E")(A)dA.

IC(0)l = 2nE,d - 9 m jomCl(r)dt.

(18)

It is also convenient to have the value of the first derivative at t = 0, a purely imaginary number of magnitude

- m

(19)

The assumptions of completely uncorrelated perturber motion and pair- wise additivity of perturber-active atom interactions lead to the independent perturber approximation of Jablonski (1943, Anderson and Talman (1956), and Baranger (1958a, 1958b, 1958~). Because the broadening by each perturber is independent of that due to the others, the normalized one- perturber line shapes can be convoluted. A density expansion is performed and the equations are converted to the time domain because convolutions in the frequency domain become simple products in the time domain. The density-dependent line shape is then the Fourier transform of a correlation function

m

exp( -2niA~)C(z)dz, (20)


where

and n is the number density of perturbers. The assumption that strong collisions of the radiator with different

perturbers are disjoint in time leads to the binary collision approximation derived by Fano (1963) from a memory-function formalism. In one sense, the binary collision approximation can be regarded to be an approximation to the independent perturber approximation because it incorporates the non- overlapping (in time) collisions approximation, in addition to the approximations already inherent in the independent perturber approximation. O n the other hand, it can be argued that the binary collision formalism is more consistent in that it includes all interactions to first order while the independent perturber formalism allows multiple perturbers to interact with the active atom while ignoring all perturber-perturber interactions. The formula is that of a Lorentzian with a frequency-dependent shift and width. The frequency-dependent width parameter is

7 1 ~ 2 ~ ( 1 ) ( ~ )

EO B l ( 4 =

while the frequency-dependent shift parameter is given by

The density-dependent line-shape is

The physical assumptions in the Szudy-Baylis (1975) theory are not as clear as in the independent perturber and binary collision theories, but the formula is very similar in form to the binary collision formula. Frequency- independent shifts and widths replace the frequency-dependent values in the denominator to give

n(A - nd)2E(1’(A - nd) (A - + (nw)’

E(A) =

Three theories of the “unified line shape” have been presented. There is no generally accepted reasoning for preferring one over the other, however, there


is a region of densities over which they should all be valid. This region is defined by the Weisskopf (1933) radius, R , = (w/nij)'/*, where ij is the average velocity. If the density is sufficiently low that, on average, there is less than one perturber within a volume about the active atom defined by the Weisskopf radius, then all three theories should be valid for isolated lines and should agree with each other. Computational evidence confirms this prediction (Erikson and Sando, 1980; Evensky and Sando, 1985). At higher densities, for isolated lines, comparisons with simulated spectra show that the Jablonski-Anderson-Baranger theory is remarkably accurate, whereas the Fano and Szudy-Baylis theories begin to break down, as expected.

2. Statistical Mechanical Theories for Overlapping Lines

When a photon is emitted at a frequency in the region of overlapping lines, it is not possible to determine by measurement the line with which the photon is associated. Therefore, interference between photons of the same frequency, but associated with different atomic transitions, is possible. The one-perturber line shape function then becomes a matrix of dimension equal to the number of overlapping lines. The matrix replaces the line shape function in the density-dependent line shape formulas described above. The exponential in the independent perturber approximation (Eq. (20)) must be time-ordered. The Szudy-Baylis theory includes no interference effects. The Anderson- Jablonski-Baranger theory and the Fano theory incorporate interference effects in different ways. To our knowledge, no computational test has been done to determine whether interference effects are important, and if they are, which theory best incorporates them. The question of how to treat overlapping lines remains one of the most interesting incompletely resolved problems in atomic line shape theory.

11. Spectral Line Shapes in Strong Fields

The study of spectral line shapes in the presence of intense laser fields covers a wide range of subjects. Prominent examples are collisionless multiphoton excitation (MPE), ionization (MPI) and dissociation (MPD) of atoms and molecules, collisional redistribution of radiation, optical collisions, velocity-changing collisions and collision kernel, resonance fluorescence, multiple-wave mixings, and many others. As it is impractical to provide any general review in a short article, we shall confine ourselves in the following to the discussion of only a small subset of intense field phenomena, particularly those involving multiphoton transitions.


A. MULTIPHOTON ABSORPTION SPECTRA AND QUASI-ENERGY DIAGRAM

In the presence of strong fields, the spectral lines can undergo power- broadening, ac-Stark shifting, and Autler-Townes splitting. Fig. 1 shows an example of the effect of high intensity IR radiation on the position and line shape of the S ( 3 ) rotational transition of molecular hydrogen (Rahn et al., 1980). In a well-known experiment on the OCS molecule, Autler and Townes (1955) found that a microwave transition line could be split into two components when one of the two-levels involved in the transition was coupled to a third by a strong radio-frequency field. The optical analog of the Autler-Townes effect was confirmed by Gray and Stroud (1978) and Ezekiel and Wu (1978), using an atomic beam of sodium and a CW laser. An example of the experimental demonstration is shown in Fig. 2 (Gray and Stroud 1978). The first laser ( A ) resonantly pumps the 3'S,,,(F = 2, mF = 2) -+ 32P,,2(F' = 3, mF, = 3) transition of sodium atom, and the second laser ( B ) probes the absorption from 32P,,,(F' = 3, mF, = 3) to 4'D,,,(F" = 4, mFer = 4). At weaker pump fields ( I A ) , a single peak was observed, whereas for stronger fields a doubly peaked structure was obtained. The splitting increases as the intensity of laser A increases. For multi-level systems, multiphoton absorption (MPA) spectral patterns are more complicated. In the following we discuss some theoretical techniques for MPA study.

0 '1 x laser on

1034.0 1034.2 1034.4 1034.6 1034.8

RAMAN SHIFT (cm-') FIG. 1. The effect of high-intensity infrared radiation on the position and line shape of the

S(3) rotational transition of molecular hydrogen. Crosses represents CARS data obtained in the presence of 1.06-pm radiation from a Q-switched Nd:YAG laser; circles show data in the absence of this field. The solid and dashed lines are smooth curves drawn through the data points. (From Rahn et al., 1980.)

Kenneth M . Sando and Shih-I Chu

I, = 5.3 mW/cm2

1 I

I 1

-60 - 4 0 -20 0 20 40 60

FIG. 2. Autler-Townes absorption doublet observed in sodium. Laser A (pump field) is exactly on resonance with the 32S, ,2(F = 2, mF = 2) + 32P,i2(F' = 3, my = 3) transition. Laser B (probe field) scans over the 32P,i2(F' = 3, mp, = 3) + 42D,,,(F" = 4, mF" = 4) transition. The splitting increases as the intensity of Laser A increases. 6, is the detuning of Laser B from resonance. (From Gray and Stroud, 1978.)

Various perturbative and nonperturbative methods have been developed for the numerical computation of multiphoton excitation of finite-level quantum systems (for recent reviews, see Dion and Hirschfelder, 1976; Feld and Letokhov, 1980; Delone and Krainov, 1984; Chu, 1985, 1986, 1988). Most of the theoretical works were performed within the semiclassical framework. Namely, the systems are treated quantum mechanically while the fields are treated classically. It has been shown for example that the semiclassical Floquet theory (Shirley, 1965) leads to results equivalent to the fully quantized theory in strong fields.


The analysis of strong-field MPA spectral line shapes can be greatly facilitated by the introduction of the so called quasi-energy (Shirley, 1965) or dressed-atom energy (Cohen-Tannoudji and Haroche, 1969) diagram. The dressed-states are the eigensolutions of the combined system of atom and pump fields. The result yields a picture of the energy level structure of the “dressed” atom and hence a physical understanding of the absorption and emission spectrum. As first pointed out by Shirley (1965) in his semiclassical two-level study, any time-dependent periodic Hamiltonian can be transformed into an equivalent infinite-dimensional time-independent Floquet Harniltonian i?,. The quasi-energies are merely the eigenvalues of the Floquet Hamiltonian. As an example, the structure of the Floquet Hamilto- nian I?, for MPE of multi-level diatomic molecules in a monochromatic field (with frequency o) is shown in Fig. 3. The Hamiltonian A, is composed of the diagonal blocks, of type A, and off-diagonal blocks of type B. EL:’ are the unperturbed vibration-rotational energies and b , v, j . are the electric dipole couplings. Fig. 3 shows that A, possesses a block tridiagonal structure with only the number of w’s in the diagonal elements varying from block to block. This structure endows the quasi-energy eigenvalues and eigenvectors of fi, with periodic properties.

The MPA spectral line shapes are frequency, intensity, and level structure dependent. Fig. 4 shows a line shape analysis of a three-level system ( E , < El < E,) characteristic of the lowest three vibrational levels of H F molecule driven by a monochromatic field (Chu et al., 1982). The quasi- energies are shown in the upper portion, and the one-photon (solid lines) and two-photon (dotted lines) absorption curves are shown in the lower portion. The unperturbed state most closely associated with a particular quasi-energy state (QES) is used as a label for that QES. Each section of Fig. 4 corresponds to a different combination of physical parameters.

Column A: IE, - El 1 = IE, - E , ) Row a : Vol > V1,-“Normal Autler-Townes Splitting” Row h: V,, = Vl,-“Symmetric S-hump” Row c: V,, < V,,-“Inverted Autler-Townes Splitting”

Row a : V,, > I/,,-“Asymmetric Autler-Townes Splitting” Row h: V,, = Vl,-“S-hump” Row c: V,, < V,,-“Induced Transitions.”

Column B: JE, - Ell < ]El - E , ]

By resorting to these six quasi-energy diagrams, one can explain most of the nonlinear spectral features such as power broadening, dynamical Stark shift, Autler-Townes splitting, hole burning, S-hump behaviors, and so forth, found in intense field multiphoton spectra. Likewise, by examining the features found experimentally, one can obtain qualitative information about the relative dipole coupling strengths vj and detunings of the levels involved.


... n = 2 n = l n = O n = - 1 n = - 2

Where

A =

and

B =

v = o v = l

n’ = 2

n ’ = 1

n’ = 0

n’ = -1

n ‘ = - 2

v’ = 0

v’ = 1

v’ = 0

v ’ = 1

FIG. 3. Structure of the time-independent Floquet Hamiltonian for the nonperturbative treatment of multiphoton excitation of diatomic molecules.


0.5-

. ... ,

3920 3980 4040 3920 3980 4040

3920 3980 4040

m 0

t-

0. 3920 3980 4040

0.5-

3920 3980 4040

0.5-

0.5 . . . . 3920 3980 4040

FREQUENCY (CM- 1 ) FIG. 4. Spectral line shape analysis of a three-level system (E, i El < E2) undergoing

one-photon (solid curves) and two-photon (dotted curves) transitions driven by a monochromatic field. The absorption line shapes depend upon the frequency, relative level spacings, and electric dipole coupling strengths, and are closely related to the quasi-energy avoided crossing patterns. See text for details. (Reprinted with permission from Chu et al. 1982.)


The study of MPA spectra is an essential step towards quantitative understanding of MPE/MPD of small and large polyatomic molecules. Discussion of the multi-level quasi-energy diagram and MPA spectra can be found in recent reviews (Feld and Letokhov, 1980; Chu, 1985).

B. MULTIPHOTON DISSOCIATION OF SMALL MOLECULES: THE INHOMOGENOUS DIFFERENTIAL EQUATION APPROACH OF

DALGARNO AND LEWIS

While more than 100 molecules-from three to 62 atoms in size-have been observed undergoing collisionless multiphoton dissociation (MPD) (Bloembergen and Zewail, 1984), MPD from a ground vibrational level of a diatomic molecule has never been observed due to the low density of states and the vibrational anharmonicity. On the other hand, MPD from high vibrational levels of diatomic molecules can be achieved rather efficiently. Using CO and CO, lasers, Carrington et al. (1983), for example, have observed two-photon dissociation of HD' from ui = 14 and 16 respectively. Such high-resolution spectroscopic studies can provide accurate structure information near the dissociation limit. In contrast, spectroscopic information about vibrational excited states of polyatomic molecules is largely unavailable.

For weaker field MPD processes, perturbative techniques have been developed such as the Green function method of Bunkin and Tugov (1973). However, there remains the difficulty of carrying out the explicit summation over the complete vibrational intermediate states in a converged fashion even for the simplest molecules like H z . To circumvent this difficulty, the inhomogeneous differential equation (IDE) method .of Dalgarno and Lewis (1955) has recently been extended (Chu et al., 1983) for implicit numerical evaluation of the infinite sum over vibrational intermediate states. The method was found to be powerful for both non-resonant and near-resonant MPD calculations. It has been applied to two-photon dissociation (TPD) of vibrationally excited H l (Chu et al., 1983) and HD' (Laughlin et al.;1986) molecules.

The H: TPD cross sections (H:[lsa,(uiji)]*H + H) were found to be very small, as expected, for low-lying vibrational levels, but increase rapidly with increasing vibrational quantum number, v i . The cross sections are largest at the two-photon dissociation thresholds and exhibit monotonically decreasing profiles with increasing photon frequency. In contrast, the spectral pattern for the heteronuclear HD' is rather different as it possesses a


substantial dipole moment and allows resonant photoabsorption to intermediate vibration-rotational levels of the lsn, electronic state. Fig. 5 shows the first four allowable resonances and their interference structures corresponding to the TPD processes of HDf:

w w lsng(ui = 14, ji = 0) -, lsag(u, j = 1) + 2pa,(k, j , = 0, 2),

where v( = 17, 18, 19, 20) are the vibrational quantum numbers of intermediate levels. As IDE is a perturbative approach, the TPD cross sections correspond to the weak-field results and are independent of the intensity of the laser fields. Further, the cross section becomes infinite at each exact resonance position due to the vanishing of the energy denominator.

In the presence of strong laser fields, MPD cross sections and resonant profiles become intensity dependent and nonperturbative methods are required for proper treatment of such processes. Descriptions of various strong field approaches can be found in the review by Chu (1985).

Y z

u w rn

E rn rn 0

WAVELENGTH( MICRON) FIG. 5. Total two-photon dissociation cross section a?’ from the (ui = 14, j i = 0) level of the

Isa, electronic state of HD+. The vibrational quantum numbers ( u ) of intermediate resonance states are also indicated. (Reprinted with permission from Laughlin et al. 1986.)


c. MULTIPHOTON AND ABOVE THRESHOLD IONIZATION IN INTENSE FIELDS

Recently, high-power laser techniques have added a new dimension to atomic spectroscopy, termed “above threshold ionization” (ATI) (Agonstini et al., 1979; Kruit et al., 1983; LomprC et al., 1985; Cooke and McIlrath, 1987), which has been the subject of intensive studies, both experimental and theoretical. When atoms are irradiated by lasers with powers as high as 10’3-1015 W/cm2, the emitted electron can absorb (N, + S) photons, where N, is the minimum number of photons required to ionize the atom and S = 0,1,2,. . . . Thus the electron energy spectrum consists of a series of peaks evenly spaced by an amount equal to the photon energy. For high intensities, these spectra have displayed a number of unexpected features (see Fig. 6 for an example):

(1) Peak Switching. For lower intensities, the height of the peaks rapidly decreases with increasing S. However, for higher intensities the most pronounced peak corresponds to some value of S different from zero. Simultaneously, the spectrum extends to higher values of S.

(2) Peak Suppression. With increasing intensity, first the lowest energy peak (S = 0), and then the next-to-lowest peaks (S = 1, 2,. . .), one after the other can become completely suppressed.

Several theoretical models have been proposed to account for these observations (Muller et al., 1983); Mittleman, 1984; Bialynicka-Birula, 1984; Edwards et al., 1984; Deng and Eberly, 1985; Crance and Aymar, 1980; Szoke, 1985; Reiss, 1987). An ab initio nonperturbative study of MPI/ATI of atomic hydrogen has been performed (Chu and Cooper, 1985), using an extended version of the L2 non-Hermitian Floquet Hamiltonian method (Chu and Reinhardt, 1977). It was found that the ionization potential (for w < 0.5 a.u.) is intensity dependent and increases with increasing intensity. The ionization potential can be defined as

(27)

where E,(F) ( < 0) is the field-dependent perturbed ground state energy obtainable from the real part of the complex quasienergy, and

Eth(F> = Eos, + l E R ( O I 3

is the average quiver kinetic energy (also known as the pondermotive potential) picked up by an electron of mass m and charge e driven sinusoi- dally by the field. Since, in the limit of high quantum numbers, a Rydberg


4

3

2

1

0

3 n

fn c 2 c.

z 3 . 1 m a w o <

J < 33 c.

UJ 2

1

0 A

12nJ 4 0 Pa

2 - . &

T * %

I

0.0 2.0 4 . 0 6 . 0 8.0 10.0

ELECTRON ENERGY ( a V FIG. 6. Intensity-dependent AT1 electron energy spectrum from MPI of xenon gas by the

Nd-YAG laser (1.064 pm). The first peak corresponds to eleven-photon ionization to the P,,,, and twleve-photon ionization to the P , , , continuum. The laser intensity is raised from the bottom to the top in the figure. (Reprinted with permission from Kruit et al. 1983.)


electron becomes a free electron, the continuum threshold is shifted up by the amount equal to I,,,, Electrons traversing a laser beam scatter elastically from regions of high light intensity by the pondermotive potential. Thus, an electron with an energy less than I,,, cannot escape from the Coulomb potential and is trapped. This appears to be the basic mechanism responsible for the disappearance of the lowest photoelectron peaks at higher fields observed in the experiments (Fig. 6). Peak suppression has been observed to be significantly more effective for circular polarization. More recently, Monte Carlo classical trajectory calculations have also been performed (Chu and Yin, 1987; Kyrala, 1987), providing additional insights regarding the dynamical evolution of the electron movement in the MPI/ATI processes.

D. INTENSITY-DEPENDENT RESONANCE LIGHT SCATTERING

Multiphoton processes can be divided roughly into two types. The first type are those involving net absorption of two or more photons by the atoms or molecules, such as the MPE/MPD/MPI processes discussed in Sections 1I.A-C. In these processes, spontaneous emission and collisional damping do not play any significant role and can often be ignored. The second type are those which involve repeated absorptions and emissions of photons by the atoms or molecules, such as the resonance fluorescence processes. Radiative and collisional dampings play an essential role in the second type of processes.

The first complete theoretical treatment of the resonance fluorescence spectrum from a two-level atom irradiated by a strong monochromatic light was performed by Mollow (1969). Following this work, numerous theoretical treatments of the resonant light scattering processes have appeared (for reviews, see Swain, 1980; Cresser et al., 1982; Loudon, 1983). The spectrum of the scattered light is related to the Fourier transform of the first-order correlation function of the atomic operators. For low laser intensities, the atom remains very close to its ground state and behaves like a classical oscillator. The light is therefore scattered elastically, and for a monochromatic driving field, one observes a sharp spectrum at the same frequency as the driving field. As the intensity of the exciting light increases, the atom spends more time in the upper state and the effect of the vacuum fluctuations due to spontaneous emission comes into play. An inelastic component enters the spectrum, and the magnitude of the elastic scattering component is correspondingly reduced. The spectrum gradually broadens as the Rabi- frequency, Q, increases until R exceeds r/4(1/r = Einstein A coefficient); then sidebands begin to appear. The spectrum of the scattered light now splits into three well-separated Lorentzian peaks consisting of a central peak,


centered at the driving field frequency with a width r/2 and having a height three times that of two symmetrically placed sidebands, each of width 3r/4 and displaced from the central peak by the Rabi frequency. An experimental demonstration of this Mollow symmetric triplet spectrum is shown in Fig. 7 (Grove et al., 1977). Here, a circularly polarized CW dye laser with a linewidth less than 250 kHz was used to excite the 32S, ,z(F = 2, mF =

2) + 32P,,z(F' = 3, mF, = 3) transition of sodium in an orthogonally propagating atomic beam. With sufficiently high laser intensity, three peaks in the fluorescence spectrum could be readily observed, in agreement with the theoretical prediction. The light scattering spectral pattern shows strong dependence on the intensity and detuning of the incident light beam.

- 100 - 50 - 0 5 0 100

Frequency [MHzl FIG. 7. Resonance fluorescence spectrum of Na for the transition between the hyperfine

levels 32S, i2(F = 2, mp = 2) and 32P,i2(F = 3, mp = 3). The theoretical lineshape (smooth curve) is also shown here for comparison. (Reprinted with permission from Grove et al. 1977.)

158 Kenneth M. Sando and Shih-I Chu

Carlsten et al. (1977) performed a series of experiments in which the collisional relaxation rates exceed the radiative decay rates. It was found that the strength of the central peak is independent of the relaxation mechanism if the spectral widths are ignored. On the other hand, the weights of the sidebands do depend on the particular type of relaxation mechanism. The theory of the spectrum of the quantized light field has recently been reviewed by Cresser (1983).

The discussion so far has centered on one-photon induced resonant light scattering. Ho et al. (1986) developed a Floquet-Liouville super-matrix (FSLM) approach for nonperturbative treatment of multiphoton-induced resonance fluorescence spectra in very intense polychromatic fields. By extending the many-mode Floquet theory (Ho et al., 1983; Chu, 1985), the time-dependent Liouville equation for the density matrix of quantum systems undergoing (radiative and collisional) relaxations can be transformed into an equivalent time-independent non-Hermitian FLSM eigenvalue problem. This allows a unified treatment of nonresonant and resonant, one- and multiple- photon, steady-state and transient phenomena in nonlinear optical processes, beyond the conventional rotating wave approximation. Fig. 8 shows the prediction of the FLSM study of the fluorescence power spectrum of a two- level ( E , c Eb) system driven by an intense monochromatic field. The incident field frequency, wL, is tuned at the shifted three-photon resonance (i.e. E , - E , E 3 4 . Strong triplet fluorescence spectra appear at two locations o E wL and 3wL (Figs. 8(a) and 8(b)) and a much weaker triplet appears at o z 5wL (Fig. 8(c)). Particularly interesting is the strongly asymmetric three-peak structure near w z 0,. These intensity-dependent fluorescence power spectral patterns are really strong field effects and can be determined by a few dominant super-eigenvalues of the FLSM. At a much lower field, only those nearly resonant super-eigenstates are mixed. Intense fluorescence light can appear only at w s 301, and possesses a Mollow-type symmetric three-peak appearance. The FLSM approach also has been extended to the study of intensity-dependent nonlinear optical susceptibilities and multiple-wave mixings (Wang and Chu, 1987).

Further information on the nature of the fluorescent light may be obtained via the second-order correlation function of the light, defined as (Glauber, 1963)

where E ( + ) ( t ) and E'-)(t) are the positive and negative frequency components of the electromagnetic field, respectively. g(')(z) can be considered to be a measure of the probability that a second photon will be measured at time t + z in a light beam, after the detection of the first photon at time, t. The


I

3 c-

owl-

D.wO 1WO 1050 2 w o 2050 2100 2150 2

FREOUENCY,w 1.0

FIG. 8. Predicted fluorescence power spectrum f(o), near (a) w E w,,, (b) w 3w,, and (c) w 5 0 , , for a system of two-level atoms (E , < Eb) driven by an intense monochromatic field with frequency wL tuned at the three-photon resonance ( E b - E, GZ 303. The inset in each figure shows the schematic fluorescence cascade diagram. (Reprinted with permission from Ho et al., 1986.)


second-order correlation function of the light in resonance fluorescence has been studied by Carmichael and Walls (1976) and others. In the steady state, the result for the saturated atom (a + r) is

g(’)(z) = (1 - e(-3rT)/4 cos nz), (30)

exhibiting damped oscillations at the Rabi frequency, a. The unusual features of this correlation function are that it begins at zero and increases. This phenomenon, called “photon antibunching,” is due to the quantum nature of the light and has no classical analog. It has been confirmed by experimental observations (Dagenais and Mandel, 1978).

ACKNOWLEDGMENTS

The work of S.I.C. was supported in part by the Department of Energy (Division of Chemical Sciences), by the American Chemical Society-Petroleum Research Fund, and by the John Simon Guggenheim Fellowship.

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Reid, R. H. G. and Dalgarno, A. (1969). Phys. Rev. Lett. 22, 1209. Reid, R. H. G. and Dalgarno, A. (1970). Chem. Phys. Letters 6, 85. Reiss, H. R. (1987). J. Opt. SOC. Am. B 4, 726. Riley, M. E. (1973). Phys. Rev. A 8, 742. Royer, A. (1974). Can. J . Phys. 52, 1816. Royer, A. (1978). Acta Phys. Pol. A 54, 805. Royer, A. (1980). Phys. Rev. A 22, 1625. Sando, K. M. (1971). Mol. Phys. 21,439. Sando, K . M. and Dalgarno, A. (1970). Mol. Phys. 20, 103. Sando, K. M. and Herman, P. S. (1983). Spectral Line Shapes., Vol. 2, p. 497, Walter de Gruyter,

Sando, K. M. and Wormhoudt, J. C. (1973). Phys. Rev. A 7, 1889. Shirley, J. H. (1965). Phys. Rev. 138, B979. Stephens, T. L. and Dalgarno, A. (1972). J. Quant. Spectrosc. and Radiat. Transfer 12, 569. Swain, S. (1980). Adv. At. Mol. Phys. 16, 159. Szoke, A. (1985). J . Phys. B 18, L427. Szudy, J. and Baylis, W. E. (1975). J. Quant. Spectrosc. Radiat. Transfer 15,641. Takeo, M. (1970). Phys. Rev. A 1, 1143. Tellinghuisen, J. (1984). J. Mol. Spectrosc. 103,455. Tsao, C. J. and Curnutte, B. (1962). J. Quant. Spectrosc. Radiat. Transfer 2,41. Uzer, T. and Dalgarno, A. (1979a). Chem. Phys. Letters 63, 22. Uzer, T. and Dalgarno, A. (1979b). Chem. Phys. Letters 65, 1. Uzer, T. and Dalgarno, A. (1980). Chem. Phys. 51,271. Vahala, L. L., Julienne, P. S., and Havey, M. D. (1986). Phys. Reo. A 34, 1856. van Dishoek, E. F. and Dalgarno, A. (1983). J. Chem. Phys. 79, 873. van Dishoek, E. F., van Hemert, M. C., Allison, A. C., and Dalgarno, A. (1984). J. Chem. Phys. 81,

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Vol. 63, p. 127. Academic Press, New York, New York.

620.

New York, New York.

5709.

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 25 1 1 MODEL- POTENTIAL METHODS C. LAUGHLIN Mathematics Department University of Nottingham Nottingham, United Kingdom

G . A . VICTOR Center .for Astrophysics 60 Garden Street Cambridge. Massachusetts

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . 163 11. Development of Model Potentials . . . . . . . . . . . . . . . 164

A. Long-Range Asymptotic Form ofthe Model Potential . . . . . . . 165 B. Corrected Forms of the Model Potential . . . . . . . . . . . . 168

173 A. Atomic Energy Levels . . . . . . . . . . . . . . . . . . 173 B. Oscillator Strengths and Lifetimes . . . . . . . . . . . . . . 177 C. Photoionization. . . . . . . . . . . . . . . . . . . . . 180 D. Relativistic Effects . . . . . . . . . . . . . . . . . . . 183

IV. Molecular Model Potentials . . . . . . . . . . . . . . . . . 186 A. Form of the Model Potential for Two-Core Systems . . . . . . . . 187 B. Applications of Molecular Model Potentials . . . . . . . . . . . 188 References . . . . . . . . . . . . . . . . . . . . . . . . 190

C. Effective Operators for Model-Potential Wave Functions . . . . . . 171 111. Applications of Model Potentials . . . . . . . . . . . . . . . .

I. Introduction

Many investigators have developed quantitative theoretical methods based on the idea of “core” electrons and “valence” electrons to calculate quantum mechanical properties of many-electron atomic and molecular systems. The terms pseudopotential, model potential and optical potential are used to describe these methods. Although they do have different meanings, authors sometimes ignore the distinctions between the names. The distinction between “core” and “valence” electrons may be a matter of choice; it depends on the desired accuracy of the model. What the methods have in common is that they attempt to accurately describe the correlated motions of the valence electrons and reduce the dimensionality of the original many-electron problem by simplifying the treatment of the valence electron- core electron interactions.

163

Copyright 0 1988 by Academic Press, Inc. All ngbts of reproduction in any form reserved.

ISBN n-i2-oow25-n

164 C . Laughlin and G. A . Victor

Asymptotically, the leading terms of the valence-core interaction have a simple analytical form and usually can be modelled reliably. At short range, on the other hand, the valence-core interaction is extremely complicated and its nonlocal and energy-dependent nature makes it difficult to approximate in terms of local operators. Nevertheless, it appears that many properties of the system do not depend sensitively on the detailed form of the short-range part of the valence-core interaction, which can be replaced effectively by a local potential. Semi-local (angular-momentum dependent) and nonlocal potentials may be employed for improved accuracy, though the use of nonlocal terms will increase the complexity of the calculation.

Some treatments are ab initio in that no experimental data are employed in constructing an effective Hamiltonian for the valence electrons, while others are semi-empirical in that they use experimental data (usually energy levels) to determine parameters that occur in the effective Hamiltonian.

In this chapter, we are concerned with model potentials, though results of pseudopotential approaches will sometimes be mentioned. For our purposes, it is sufficient to classify a pseudopotential as a potential containing a short- range repulsive potential whose purpose is to circumvent core-valence orthogonality requirements, while a model potential, which ignores these requirements, will have valence-electron eigenfunctions corresponding to occupied core states.

Pseudopotential and model-potential methods have been reviewed by Weeks et al. (1969), Bardsley (1974), Dalgarno (1975) and Hibbert (1982). A monograph by Szasz (1 985) gives an extensive review of pseudopotential theory.

Much of the work to date has been concerned with one- or two-valence electron systems having a spherically symmetric core or cores. The molecular work has, in the main, been limited to diatomics. Because of the success of the methods in producing energy levels, potential energy curves, transition probabilities and scattering cross sections of useful accuracy, the literature in the field is very extensive. We have consequently had to be selective in our choice of material for this review; our bias is towards those aspects of model- potential theory and its applications that have been developed by Alex Dalgarno and his co-workers. We have also attempted to avoid overlap with the previous reviews mentioned above. Unless indicated otherwise, atomic units will be used.

11. Development of Model Potentials

Consider an atomic system consisting of a distinct spherically symmetric core plus a number of outer (valence) electrons. The distinction between core electrons and valence electrons is, of course, somewhat arbitrary; in practice,

MODEL-POTENTIAL METHODS 165

we shall limit our considerations to systems in which a very few (usually one or two) electrons are designated as valence. The central idea is to construct an approximate Schrodinger equation for the valence electrons alone and so reduce the computational complexity of the problems of making quantitative predictions of properties and interactions of the original many-electron system. This valence-electron Schrodinger equation will depend on the core through an effective potential V , the model potential for the system, whose asymptotic form we now seek to derive. Our analysis follows closely the method introduced initially in a significant paper by Bottcher and Dalgarno (1974) and later comprehensively extended by Peach (1983).

A. LONG-RANGE ASYMPTOTIC FORM OF THE MODEL POTENTIAL

If the Hamiltonian for the unperturbed core electrons is denoted by H , then the total (exact) non-relativistic Hamiltonian H for the system may be expressed as

H = H , + T + V , , (1)

where T is the kinetic energy operator for the valence electrons and V , includes their interaction with the core. Bottcher and Dalgarno (1974) assume the existence of a model potential V whose eigenvalues match the energies E , of the Rydberg levels,

(2)

and construct V by treating A V = V , - V as a perturbation. Let aC and 6, be, respectively, the eigenfunctions and eigenvalues of the

core Hamiltonian H , . We now solve the Schrodinger equation For the system,

(3)

( T + V ) Y , = ErYr,

H x = ( H , + T + V + AV)X = E,x,

by treating H , = H , + T + V as the zero-order operator whose eigenfunctions a,'€", satisfy

f f o ( @ , ~ , ) = (8, + E J P C ~ J (4)

We restrict our attention to those solutions of Eq. (3) that correspond in zero-order to states in which the core is in its (spherically symmetric) ground state, denoted by the subscript c = 0. Thus, IErI will be an ionization potential for the valence electrons.

Introducing the core projection operator, P = ~ @ , o ) ( ~ o ~ , and its orthogonal complement, Q = 1 - P, allows the Schrodinger Eq. (3) to be written as

166 C. Laughlin and G. A. Victor

Setting }Y) = (@oJx) in this equation reduces it to

( T + V + U)yI = EY, where E = EA - 8, and

Eq. (6) will be the same as the model Schrodinger Eq. (2) for the valence electrons, provided U = 0, a condition that allows an expression for the model potential V to be developed. This may be achieved by making the formal expansion

1 1 1 QAVQ, + (8) - - _ _ 1 - 1

Q(H - EA)Q - D + QAVQ - D D

where D = Q(H, + T + V - EA)Q. If we now make the fundamental assumption that the core excitation energies, 18, - djc1, are large compared to valence transition energies, JE, - E,I, we can further expand 1/D as

Then, retaining only the first two terms on the right-hand side of Eq. (8), the condition U = 0 results in

where

and

J‘B = (@,-,I KG,[(E - T - V)GClkAVGc[(E - T - V)G,]k“v,~Q~). (13)

In Eq. (12) and Eq. (13), G, is the Green’s function for the unperturbed core defined by

For an atomic system consisting of N , core electrons and N , valence electrons, the potential V , is given by


where ri j = Jri - rjJ and ri and rj denote, respectively, the position vectors of core and valence electrons relative to the nucleus of charge 2. The long-range forms of the terms in the potential V, Eqs. (10)-(13), may be obtained by employing the conventional multipole expansion of V, in terms of spherical harmonics and making the assumption that ri < r j . It is easy then to show that

where a(') is the static 2"pole polarizability of the core (Dalgarno, 1962).

written as The term V\2) in Eq. (12) is the first non-adiabatic correction to V . It can be

(18)

where P ( I ) is a dynamic 2I-pole polarizability of the core. The derivation of Eq. (18) has been described in detail by Peach (1983) and subsequently discussed by Norcross (1983).

A second non-adiabatic correction to V is given by the term Vi2) in Eq. (12), and it may be approximately evaluated to give (Peach, 1983)

1 " N u 1 Vl" = j c (A + 1)(21 + wo) 1 y21+4'

I = 1 j = 1 J

1 z . ' j

(A + 2)Ej + (A + 3) + O ( r j 2 ) , (19)

where y(* ) is a dynamical correction to a(')). Vi2) is energy dependent; for the case N u = 1, E j = E and Z j = Z - N , , but for N u =- 1, Ej and Z j cannot be given rigorous definitions, though they appear to be related to detachment energies and an effective core charge for the valence electrons.

The third-order static contribution Vb3d to V has been evaluated approximately by Peach (1983) to yield

where 42, 1, 1) and ~(1, 2, 1) are hyperpolarizabilities for the atomic core.

168 C . Laughlin and G. A . Victor

For a one-electron system we have, with an obvious change of notation,

+ O(r - *), (21) 28(Z - N,)yi + 242, 1, 1) + ~ ( 1 , 2, 1)

r7 +

while for a two-electron system, neglecting energy-dependent terms and terms that decay more rapidly than r-6,

The last term on the right-hand side of Eq. (22) is the “dielectric” term (Chisholm and Opik, 1964).

B. CORRECTED FORMS OF THE MODEL POTENTIAL

In the previous section, we presented the long-range form of the interaction of the outer valence electrons with the tightly bound core electrons. In practice, it is necessary to modify the potential I/ to attempt to give a more realistic description of the short-range interactions, which include exchange forces. Several modifications may be introduced. First, because all the terms in the multipole expansion for the long-range potential V are strongly singular at the origin, it is necessary to introduce cut-off functions Wn(r/rc), where rc is a cut-off parameter usually chosen to be of the order of the core radius. Many different forms of W,(x) have been employed in applications, e.g., 1 - exp( -x”), xZn/(x2 + I>”, 1 - (1 + x + x2/2! + ... + x”/n!)exp( -x> and (1 - exp( -x))”. Some authors treat r, as a variable parameter to be chosen to reproduce accurate experimental data (see, for example, Weisheit and Dalgarno, 1971a; Norcross, 1973; and Norcross and Seaton, 1976).

A second modification to V is to replace the monopole term in Vg), Eq. (16), by a more realistic core potential, possibly including nonlocal exchange terms. One possibility is to evaluate Eq. ( 1 I) for VL”, with a,, the Hartree- Fock isolated core wave function (Bottcher and Dalgarno, 1974). Other forms have been used (see, for example, Bardsley, 1974; Norcross and Seaton, 1976; Valiron et al., 1979; and Szasz, 1985).

Another modification is to introduce into V additional short-range correction terms that are chosen empirically to accurately reproduce reliable


experimental or theoretical data for the system or its ions. Typical examples are the use of observed term values for a one-electron system to construct a model potential for the corresponding two-electron system, and the use of atomic term values in the construction of molecular model potentials.

It may be pointed out that, because we have not orthogonalised the solutions of Eq. (2) to occupied core states, the lowest solutions of this equation will correspond to occupied core states and therefore will violate the Pauli exclusion principle. In practice, solutions of Eq. (2) are constructed that are orthogonal to core orbitals (Valiron et al., 1979) or to approximate core orbitals (Victor and Laughlin, 1972) and so have an appropriate nodal structure. In a pseudopotential approach (Szasz, 1985), the orthogonality requirement is replaced by a short-range repulsive term in the potential V so that Eq. (2) does not have solutions corresponding to occupied core states, even though the lowest solution is then nodeless.

A question remains concerning the number of terms to retain in the expansion of V in Eq. (lo), suitably modified by cut-off functions as described above. As Drachman (1979) has pointed out, the expression on the right- hand side of Eq. (10) is an asymptotic series, not a convergent series, and caution in its use must therefore be exercised. The number of terms that should be retained in principle will often differ from the number which it is possible to include in practice. For example, the values of the polarizabilities a(’), p‘”’, and so forth are not usually well known. In fact, some of these quantities have been treated as parameters and chosen in an empirical manner (Weisheit and Dalgarno, 1971a; Victor and Laughlin, 1972; Nor- cross, 1973). As a test, consider the simple systems H- and He, treating them as effective one-electron systems with a spherically symmetric core consisting of a single K-shell electron. For such systems, all terms up to order r - 7 have been evaluated (Dalgarno, 1962; Seaton and Steenman-Clark, 1977; Drach- man, 1979) with the result that

where

with Z being the nuclear charge and E the Rydberg electron energy. The cut-off functions W,(x) in Eq. (23) have the form

W,(x) = 1 - exp( - x”). (25)


The explicit forms of the model potentials uM are

where V,, and X,, represent, respectively, the static and exchange interactions of the outer electron with a fixed hydrogen-like 1s core orbital. The subscript k on V, indicates that the first k terms (k = 1, 2, 3,4 and 5 ) on the right-hand side of Eq. (23) were retained. The short-range potentials Uis'(r) (S = 0 for singlet series, S = 1 for triplet series), whose analytical forms are (ao + a,r + u2r2)Cbr , were chosen to reproduce observed He term values as accurately as possible.

Table I gives results for helium lskp3Po scattering phase shifts in various approximations and compares them with accurate variation-perturbation results of Stewart (1979). It can be observed that, for this simple system, the results progressively deteriorate as terms beyond the first three in the expansion of V, Eq. (23), are included and that little improvement is obtained by using V3 rather than V,. All the model potentials, uM, gave accurate representations of the bound-state spectra (though it is again preferable to omit the energy-dependent and r - 7 terms) and only information on bound- state spectra was used to construct the various uM. The model-potential phase shifts become less accurate as the energy increases because the 2s2p3Po Feshbach resonance, which lies at kZ = 2.48 Ryd. (Stewart, 1979), cannot be included in the current model-potential formalism.

Calculations were performed also for Iskp'PO scattering phase shifts in H - and He and a similar pattern of results obtained. Analogous conclusions have

TABLE I

HELIUM ATOM lskp nPo SCATTERING PHASE SHIFTS

to

k 2

0.0 0.2 0.4 0.6 0.8 1 .o 1.2 1.4 1.6 1.8 2.0

- 1

0.1235 0.2222 0.2283 0.2324 0.2350 0.2366 0.2374 0.2375 0.2372 0.2365 0.2347

2

0.2148 0.2251 0.2321 0.2368 0.2398 0.2415 0.2428 0.2424 0.2420 0.2413 0.2403

3

0.2146 0.2246 0.23 15 0.2362 0.2392 0.2410 0.2419 0.2421 0.24 I 8 0.24 12 0.2403

4

0.2135 0.2214 0.2259 0.228 1 0.2287 0.2283 0.2272 0.2257 0.2240 0.2222 0.2208

5 Stewart (1979)

0.2135 0.2279 0.2367 0.2434 0.2480 0.2508 0.2526 0.2534 0.2536 0.2533 0.2527

0.2243 0.2314 0.2364 0.2397 0.2417 0.2429 0.2434 0.2434 0.243 1 0.2427

a t is the number of terms retained in the potential V of Eq. (23) (see text).


been reached by Drachman (1982) in his study of high lsnl Rydberg levels of helium with 1 = 3 and by McEachran and Stauffer (1983a,b) who concluded that only the dipole part of the polarization potential should be used in their calculations on low-energy elastic scattering of electrons from helium, neon and argon. Drachman (1982) also found that short-range effects could be more important than the r - 7 and r-' terms in the potential.

The same conclusions will not necessarily hold for other (larger) systems. Analogous tests for larger systems, where the model-potential method has much greater utility, would be more difficult to perform due to the lack of relevant accurate data. Eissa and Opik (1967) and Opik (1967a) find that dynamical corrections (the P I term in Eq. (23)) for alkali and alkali-like systems are generally less than a few percent of the total polarization energy. On the other hand, Vaidyanathan and Shorer (1982) find that dynamical corrections to be adiabatic core-polarization potential change the quantum defects of some highly excited singlet F and G states of calcium by as much as 50%; Vaidyanathan et al. (1982) measured 4snf1F, - 4sng1G, (n =

23 - 25) microwave transition frequencies in calcium, which agree with the dynamical model of Vaidyanathan and Shorer (1982) but which differ by a factor of approximately two from their adiabatic core-polarization model. It should be mentioned that an open-shell Ca+ core is used by Vaidyanathan and Shorer (1982), resulting in relatively large dynamical effects (due to correlations between the Rydberg and 4s valence electrons), much larger than would be anticipated for closed-shell-core systems.

The coefficient p, in Eqs. (21) and (22) may be expressed as (Opik, 1967b; Kleinman et al., 1968)

where fb, is the oscillator strength from the ground state to excited state c of the core. Equation (27) may be used to estimate the value of pl. Kleinman et al. (1968) give upper and lower bounds for pl, e.g., fil I ia , /AEc, where AEc is the excitation energy of the lowest state of the core to which dipole transitions are allowed.

The factor aq - 64, is negative for all systems for which accurate data on a4 and Dl exist. Gallagher et al. (1982) deduced a negative value of up - 68, for Ba' from their observed ratio frequency resonance transition wavelengths for highly excited (n - 20) G, H, I and K states of Ba.

c. EFFECTIVE OPERATORS FOR MODEL-POTENTIAL W A V E FUNCTIONS

It first was shown by Bersuker (1957) that the long-range polarization terms in the potential effect a modification to the electric dipole operator in the calculation of transition matrix elements with valence-electron wave


functions. Consider, more generally, a one-electron operator, D, which can be expressed as

D = C d(i) (28) i

and define NC N , ,

D, = C d(i), D, = 1 d ( j ) . i = 1 j = 1

Through first order, solutions of the Schrodinger Eq. (3) for the system may be written as

x1 = (1 + GCVC~@O~lJ1~ x 2 = (1 + GcK)@0Yuz.

(XIIDIX2) = ( @ o ~ u l l D + KGCD + D G c K l @ o ~ u * )

(30)

(3 1)

The matrix element of D, also through first order, is

which may be expressed as

(XlIDIX2) = ( ~ , , I ~ u I ~ u z ) (32)

if 8, is defined as (Bottcher and Dalgarno, 1974)

8, = Du + (@olDcl@o) + (@olKGcDc + DcGcKl@o)* (33)

This operator b,, correct through first order in AV, should be used in model-potential calculations of matrix elements of the operator D. An interesting and relevant case is when D is the multipole operator, so that d(i) = r?P,(Pi). Eq. (33) then provides

where a(') is again the 2'-pole static polarizability of the core. For the particular case of the dipole operator (A = l), the derivations by

Bersuker (1957) and Hameed et al. (1968) give LY( ' ) (~ ) , rather than a('), in Eq. (34), where a( ' ) (o) is the polarizability of the core at the transition frequency, w = IE,, - Euzl . The value of a(')(o) usually is not available; therefore, in practice, a(1) is used. The two values will not differ appreciably for systems in which the core excitation energies are large compared to valence transition energies (Hameed et al., 1968), which is the basis of the model-potential development. Mohan and Hibbert (1987) point out that this assumption is not valid for mercury if it is treated as a two-electron system.

In actual calculations, the polarization term in Eq. (34) needs to be cut off for small values of Ti. For the dipole operator, Weisheit and Dalgarno (1971b) found that amongst the class of cut-off functions W,(x) = 1 - exp( -x"), W,(x) was the most satisfactory.


111. Applications of Model Potentials

There have been many applications of model-potential methods in atomic and molecular processes. Some of the first applications were concerned mainly with the prediction of energy-level spectra and molecular potential energy curves, but other properties were soon investigated. Currently, the method is used widely in calculations of both bound-state and continuum- state properties of both atomic and molecular systems. We now review a limited selection of some of the applications.

For one-electron atomic systems, we denote the model potential by uM, which will contain some of the long-range polarization interactions of Eq. (23) as well as short-range local (and in some cases nonlocal) terms (cf. Eq. (26)). The two-electron equation can then be written in the general form (cf.

[ -1VZ - “2 (35) Eq. (22))

2 1 2 2 + uM(1) + uM(2) + V(12)Iy = Ey+

where V ( 12) should contain the dielectric term in addition to the Coulombic repulsion l / r , 2 .

A. ATOMIC ENERGY LEVELS

A useful application of the method is provided by two-valence-electron systems, to which we restrict the discussion in this section. Chisholm and Opik (1964) performed calculations on the 4s’ ‘S, 4p2 ‘S and 3d2 ‘S states of atomic calcium using a model potential in the form of Eq. (39, including the dielectric term. Similar procedures, but excluding the dielectric term, are adopted by Friedrich and Trefftz (1969). Later, Victor et al. (1976a) also used an approach based on Eq. (35) in their studies of calcium. Several authors (Laughlin and Victor, 1973; Norcross and Seaton, 1976; Laughlin et al., 1978; Muller et al., 1984) have reported model-potential calculations on beryllium, and many other two-electron atomic and ionic systems have been investigated also.

The main differences in the various approaches are in the one-electron model-potential terms uM(l) and in Eq. (35) and in the numerical techniques used to solve the two-electron Schrodinger equation. Thus, Norcross (1973, 1974) uses an 1-dependent potential uM(r) of the form

u d r ) = 41, r> + up(rc, r), (36)

where u(A, r ) is a scaled Thomas-Fermi statistical potential (Eissner and Nussbaumer, 1969) and up is a polarization potential. The scaling parameter, 1, and the cut-off radius rc in u p ( r c r r ) are determined empirically to give


agreement between calculated and experimental energies. In practice, different values of I and I , are required for each angular momentum I of the valence electron. Norcross and Seaton (1976) use an I-dependent potential given by

where uHF is a Hartree-Fock potential for the core, including the exchange interaction, derived from scaled Thomas-Fermi statistical potential orbitals. Victor and Laughlin (1972) and Laughlin and Victor (1973), on the other hand, employ a local I-independent potential u&) based on a Hartree-Fock core potential and including, as well as long-range polarization terms, an additional short-range correction term #(I ) chosen by a least-squares procedure to fit observed one-electron spectra. Laughlin (1983) has modified this approach by using a nonlocal potential, uM, to simulate the exchange interaction with the core. In Table 11, we demonstrate the differences between the energy levels for lithium obtained from a local (V,,,) and a nonlocal ( KX) potential for the Li' core. The Hartree-Fock 1s orbital for Li' (Roothaan et al., 1960) is used to construct vOc and Kx. The additional short-range correction terms referred to above in vOc and V,, are, respectively,

udr) = UHF + up(rcl I ) , (37)

ulOc = (- 3.247 - 0.1 1 lr)exp( - 3 .58~) (38)

u,, = (0.0147 + 0.0005r)exp( - 1.97r). (39)

Note that the errors in the eigenvalues of V,,, are substantially larger than the errors arising from V,,, particularly for the lower s and p states, and that u,, is a much smaller correction than u,,,. From a computational point of view, it is desirable to use local potentials but, due to the difficulty of modelling a nonlocal operator by a local one, higher accuracy is achieved with nonlocal potentials.

To solve the model Schrodinger Eq. ( 3 9 , Norcross (1974) and Norcross and Seaton (1976) expand Y in terms of products of one-electron eigenfunctions of uM and channel functions, and solve the coupled differential equations for the channel functions numerically. An alternative procedure (Victor and Laughlin, 1972) is a configuration-interaction expansion for Y in a basis of eigenfunctions of OM. Fairley and Laughlin (1984) have shown that to achieve proper convergence such expansions should include the positive-energy eigenvectors that result when uM is diagonalized using a discrete basis. These positive-energy solutions adequately simulate the continuum eigenfunctions of uMr without which the set would not be complete.

Model-potential energies of high accuracy for two-electron systems can be obtained, though not of as high accuracy as very refined ab initio calculations for small systems. Model-potential methods have the advantage that the computations are relatively straight-forward and inexpensive, even for large systems, and they allow accurate predictions for Rydberg levels.


TABLE I1

LITHIUM ATOM ENERGY LEVELS (am)

State K O / K X b Experimen tC

2s 3s 4s 5s

2P 3P 4P 5P 3d 4d 5d

-0.19819 -0.07428 -0.03864 -0.02366

-0.13009 - 0.057 19 -0.03 195 -0.02036

-0.05562 - 0.03 128 - 0.02002

-0.19818 - 0.074 18 -0.03862 - 0.02363

- 0.13024 - 0.05723 -0.03197 -0.02037

-0.05561 - 0.03 128 - 0.02001

-0.198 16 - 0.074 18 -0.03862 - 0.02364

- 0.13025 -0.05724 -0.03 198 -0.02037

- 0.05561 - 0.03 128 - 0.02001

Calculated with a local model potential (see

Calculated with a nonlocal model potential text).

(see text). 'Johansson (1959).

The model-potential approach can provide estimates of the positions and widths of doubly-excited autoionizing resonances. As an example, consider the 3pns 'Po ( n 2 4) and 3pnd ' Po (n 2 3) resonances in magnesium, which lie above the 3s2S threshold of Mg' and can autoionise to 3skp'PO continua. wave functions and energy levels for these resonance states may be calculated, in first order, by omitting the 3s orbital from the basis set (in a configuration-interaction expansion) or by orthogonalising the ns channel function (in the coupled differential equations) to the 3s orbital. In either case, the result is that the calculated wave functions will be orthogonal to 3skp 'Po continuum functions. A selection of experimental and theoretical results for the positions of these autoionising levels is presented in Table 111. There is considerable discrepancy between the experimental energies for the lower members of the 3pns 'Po series, which autoionise rapidly and, consequently, give rise to broad peaks in the observed spectra. The model-potential results of Laughlin and Victor (1973) agree well with those of Mendoza (1981) who followed the method of Norcross and Seaton (1976). The difference between these two sets of results is a measure of the shift caused by interaction of the discrete 'Po levels with the 3skp'PO continuum, which was not included by the former authors.

Finally, in this section we demonstrate the high accuracy that may be achieved in model-potential energy-level predictions. Laughlin (1983) and

TABLE 111

ENERGIES (ev) OF 'Po AUTOIONIZING STATES OF MAGNESIUM RELATIVE TO GROUND STATE

Experiment

Mehlman- Balloffett Martin Laughlin Bates

and Esteva Rassi Baig and and and and Esteva et al. et al. Connerade Zalubas Victor Altick Mendoza Chang

c 4 Level (1969) (1972) (1977) (1978) (1980) (1973) (1973) (1981) (1986)

3P4s 9.86 9.52 9.81 9.75 9.752 9.62 10.0 9.706 9.655 3p5s 10.93 10.86 10.97 10.92 10.917 10.90 11.1 10.91 10.898 3p6s 11.39 11.35 11.41 11.35 11.385 11.38 11.5 11.38 11.376 3p7s 11.62 11.60 11.64 11.61 11.614 I 1.60 11.62 11.611 3p3d 10.65 10.65 10.64 10.65 10.653 10.61 10.8 10.66 10.686 3P4d 11.26 11.26 11.26 11.26 11.254 11.25 11.4 11.25 11.276 3p5d 11.55 11.55 11.55 11.549 11.55 11.556 3p6d 11.71 11.70 11.71 11.706 11.71 11.712

a\


TABLE IV

TRANSITION WAVELENGTHS (nm) I N THE QUARTET SPECTRUM OF BE 11

Theory Experiment

Galan Froese- Mannervik Bentzen and Bunge Fischer Laughlin et al. et al.

Transition (1981) (1982) (1983) (1981) (1981)

2p3d 4Do-2p4f 4F 2s3d 4D -2s4f 4F0 2s3d 4D -2p3d 4F0 2s4f 4F0 -2p4f *F 2s3d 4D -2p3d “Do 2s2p 4Pa -2p2 4P

-2s3d 4D - 2 ~ 4 s 4S -2p3p 4P

437.18 437.8 433.01 433.2 351.08 349.9 340.60 341.4 338.06 337.94

233.1 86.79 75.52 71.38

442.5 434.9 35 1.4 341.3 337.94 229.54

86.82 75.64 71.44

437.1 1 432.96 35 I .05 340.54 337.99 232.46

86.7 1 75.44 7 1.42

Fairley and Laughlin (1984, 1985, 1987) have applied a model-potential method to the ls2snl and ls2pnl quartet levels of the lithium sequence, systems that provide the simplest possible applications for a two-electron model potential. The one-electron potentials used not only reproduce observed lsn13L term values to within experimental uncertainty, but also satisfy the more stringent test of reproducing the highly accurate triplet oscillator strengths of Schiff et al. (1971).

Transition wavelengths in the quartet spectrum of Be TI are presented in Table IV where they are compared with extensive ab initio configuration- interaction calculations (Galan and Bunge, 198 l), multi configuration Har- tree-Fock calculations (Froese Fischer, 1982) and available beam-foil spectroscopy results. The model-potential wavelengths, though not as accurate as those of Galan and Bunge (1981), compare very favourably with the multiconfiguration Hartree-Fock values. With the help of calculated transition probabilities, several problems in experimental assignments were resolved and some unidentified spectral lines were assigned (Laughlin 1982a,b, 1983; Fairley and Laughlin 1985).

B. OSCILLATOR STRENGTHS AND LIFETIMES

Despite their success in predicting the quartet spectra of lithium-like ions, model-potential wavelengths usually do not approach spectroscopic accuracy. Oscillator strengths are, in practice, much more difficult to calculate than

178 C. Laughlin and G. A . Victor

TABLE V

COMPARISON OF MODEL-POTENTIAL OSCILLATOR STRENGTHS WITH OTHER VALUES

Oscillator strength

System Transition Model potential Other

Li II

Li 11

Be 111

Be I l l

Be I 1

Be I c 111 o v Mg 1

Mg 1 Al I1

Si 111

Ca I

cu I

Ag I

Cd I

c s I

A u I

Hg I

ls2s 3s-ls2p 3PO

ls2s 3S-ls2p 3PO

ls3s 'S-ls3p 'PO

2s' lS-2s2p 'PO

2s' 'S-2szp 'PO

2s' 'S-2S2p 'PO

3s2 'S-3s3p 'PO

3s2 'S0-3s3p 3P7

3s' lS0-3s3p 'Pp

3s' 'S,-3s3p 'P:

4s' 'S-4s4p 'PO

4s zs,/,-4p zpp/,

-4P 'P4,' 5s 2s,/'-5p =Ppl,

-5P IP:/' 5s' 'S,-5sSp 'PT

ls2p 'Po-ls3d jD

ls2p' 'P-lsZp3d +Do

0.3083"

06241'

0.2137d

0.3563'

0.6191

1.372O

0.764O

0.513'

1.72"

2.11.10-6'

1.10.10-5'

2.85.10-"

1.822v, 1.76', 1.63'

0.214*

0432"

0 l98*

0.413"

1.319'

0.001w

0.340N, 0.354'

0.707N, 0.7@

0.148*

0.339*

1.174', 1.195'

0.0254s, 0.0234'

0.3079'

0.6243'

0.21 31'

0.3557b

0.611'. 0.626'

1.344', 1.38 f 0.12'. 1.341 f 0.047'

O.76Sh, 0.753 f 0.026L, 0.7% f 0.014'

0.515h, 0.47', 0.527 f 0.014'

1.75", 1.83 f O.M', 1.83 It 0.09' (2.1 f0.2).10-", (2.06 f 0.29).10-'

(1.04 f 0.05).10~'""

(2.67 f 0.16).10-5'

1.75 f O.OW, 1.79 f 0.03'

0.22B, 0.215 rt O.OIOc

0.43q 0.4is, 0.431'

O.24lF, 0.215", 0.196#

0.5Mp, 0.45', 0.459"

1.42 + OM', 1.12 f 0.OXK, 1.30 f 0.1L

0.00200 f 0.M)03M

O.35lQ

0.714Q

0.19', 0.18"

0.4lP, 0.39'

1.21L, 1.15u, 1.18"

0.0237 f O.MX)4'", 0.0249 f 0.0004x

Fiurley and Laughlin (1984)

bSchifTefal.(1971)

' Weiss (1967)

Laughlin (1983)

'Galan and Bunge (1981)

' Froese Fischer (1982)

Laughlin et al. (1978)

Sims and Whitten (1973)

'Hontzeasetd.(1972)

' Reistad and Martinson (I 986)

' Reistad et al. (1986)

' Pinnington el al. (1974)

Victor et al. (1976b)

" Froese Fischer (1975)

I) Kelly and Mathur (1978)

Liljeby et al. (1980)

' Laughlin and Victor (1979)

' Furcinitti et ul. (1975)

' Kwong et al. (1982)

"Johnson (1985)

"Kwongetal.(1983)

*Victor et al. (1976a)

Hafner and Schwarz(l978)

Hansen (1983)

* Kelly and Mathur (1980)

" Migdalek and Baylis (1978)

* Lvov (1970)

Hannalord and McDonald (1978)

' Bell and Tuhbs (1970)

'Curtis et al. (1976)

Penkin and Slavenas (1963)

Lawrence et al. (1965)

" Moine (1966)

' Migdalek and Baylis (1986)

' Lurio and Novick (1964)

' Baumann and Smith (1970)

Andersen and Sorensen (1972)

Byron et al. (1964)

' Norcross (1973)

Weisheit (1972)

Fabry and Cussenot (1976)

" Einfeld et al. (1971)

Migdalek and Baylii (1985)

' Mohan and Hihhert (1987)

" Lurio (1965)

Ahjean and Johannin-Gilles (1976)

Halstead and Reeves (1982)

Mohamed (1983)


energy levels, because they depend on off-diagonal matrix elements of the dipole operator. Experience suggests that refined model-potential calculations produce very reliable oscillator strengths. Apart from a few extensive calculations on small systems, rather few oscillator strengths are known with high precision, nor are there many precise measurements. We assemble in Table V some model-potential predictions and compare them with other accurate data to give substance to our claim that model-potential oscillator strengths are reliable.

The relative size of the correction arising from the dipole term in the modified dipole operator (cf. Eq. (34)) is of some interest. A trivial observation is that the magnitude of the correction is proportional to the dipole polarizability of the core, and so it will play an increasingly important role as the number of core electrons increases. A correction of approximately 1 % is obtained by Caves and Dalgarno (1972) for the oscillator strength of the 2s-2p resonance transition in lithium (cld = 0.1923), whereas a correction of approximately I5 % is calculated by Norcross ( I 973) for 6s-6p transitions in cesium (ad = 19.06).

It also is clear that the relative size of the correction will depend on the magnitude of the unmodified value, so that for weak transitions that, due to cancellation, have small dipole matrix elements, the modification to the dipole operator may become very important. In fact, factors of two or more have been found for some weak transitions (Weisheit and Dalgarno 1971b; Butler et al. 1984).

A somewhat disturbing feature is that the matrix element of the core- polarization correction to the dipole operator may be sensitive to the cut-off radius rc (Weisheit, 1972; Laplanche et al., 1983). Again, this is likely to be important only for weak transitions, but it has also been found to occur in bound-free transitions (photoionization).

Oscillator strengths or, equivalently, transition probabilities, have many important applications, for example, in deducing relative abundances in plasmas from the observed emission. Transition probabilities may be combined to give a radiative lifetime for an excited state. Radiative lifetimes can be measured in the laboratory (e.g., by beam foil or beam gas spectroscopy, laser excitation, Hanle effect, or ion traps), allowing a comparison between theoretical predictions and experimental measurements. Table VI shows such a comparison for core-excited quartet levels of lithium. The experimental measurements here were all carried out using the beam foil technique that has been found to efficiently populate quartet levels of three-electron ions (for a review see, for example, Pinnington, 1985). Some of the earlier measurements are not reliable but there is pleasing agreement between the latest experimental values (Mannervik 198 1 ; Mannervik and Cederquist 1983) and theoretical values. The spectral resolution of the beam foil technique is not high and

180 C . Laughlin and G . A . Victor

TABLE VI

RADIATIVE LIFETIMES FOR Li I QUARTET LEVELS (ns)

Model Other Level potential" theory Experiment

2s3s 4S 2s4s 2s5s 2s3p 4P0 2s4p 2s5p 2p3s

2s3d 4D 2s4d 2s5d 2s6d 2s7d 2s4f 4F0 2p4f 4F

2pz 4P

6.72 16.5 33.2

150 276 399

9.25 5.45 4.06 9.59

18.5 31.9 49.3 59.3 23.9

6.72b, 6.9'

270h, 140' 300h, 3 W

> SOOh

5.76b, 5.78' 4.1 Sb. 4.22'

60.6* 23.0h

7.7 k l.@, 9.7 k 0.7' 15.4 f OS', 10.4 f 2.0'

34 f 29

10.6 f O.Sd, 11.0 f 2.0', 11.8 f 0.2', 12.4 f 0.3' 5.86 f O.lSd, 6.4 k 0.3", 5.8 f 0.7', 6.5 k 0.3k, 7.0 2 2.01

9.6 f 0.6d, 5.9 f 1.0' 17.8 0.6# 26.8 f 1.6# 44.0 & 6.08

4.3 k O.ld, 4.5 f 0.4', 5.3 1.2'

a Fairley and Laughlin (1984); * Bunge and Bunge (1978);' Weiss (1967); Mannervik (1981); Bickel et al. (1969); ' Berry et al. (1972); Mannervik and Cederquist (1983); Bunge (1981); Bukow (1981); 'Gaillard et al. (1969); Ir Berry et al. (1971); ' Buchet et al. (1969).

the availability of reliable transition probabilities and excited-state lifetimes is a valuable aid in the verification of spectral assignments.

C . PHOTOIONIZAT~ON

Model-potential methods have been widely used in photoionization calculations. Bates (1947) estimated the photoionization cross section of atomic potassium using a one-electron equation for the continuum wave function that included the polarization potential -)ad/(r2 + r,2)2, and adjusted the polarizability ad of the K + core to try to bring calculated and measured cross sections into agreement. Subsequently, potassium (Weisheit and Dalgarno, 1971a; Weisheit 1972) and other alkali metals (Caves and Dalgarno, 1972; Norcross, 1973; Laughlin, 1978; Butler and Mendoza, 1983) have been studied within the model-potential framework. The nonzero minima in the sodium, potassium, rubidium and cesium cross sections result from the spin- orbit interaction (Seaton, 1951), and ionization of unpolarized alkali atoms by circularly polarized light at wavelengths near the minimum yields highly spin-polarized photoelectrons (Fano, 1969). Weisheit and Dalgarno (1971a) and Weisheit (1972) use precise experimental photoionization data on potassium, rubidium and cesium (Heinzmann et al., 1970; Baum et al., 1972)


to determine values for the effective core radii, rc, to be used in the dipole operator correction. Norcross (1973), in an extensive investigation of cesium oscillator strengths and ground-state photoionization cross section, used a two-parameter model potential, v M , of the form given by Eq. (36), to which a spin-orbit potential was added. In this case, the parameters A and rc were determined by fitting &he 6s, 6p,,, and 6p,,, eigenvalues of vM(r) to observed term values; different values of r , are required for the s ( r , = 3.333) and p ( r , = 4.132) states.

Despite their apparent simplicity and amenability to a model-potential treatment, there has been a perplexing failure in attempts to calculate alkali- metal atom cross sections that agree with experimental measurements. Consider sodium as an example. Results are presented in Fig. 1. All

1

I I I I I I 2 4 -

0 0 0

0

0

2 0 -

0

0

0 a * a 16- -

0 a

! 0 a

PHOTOELECTRON ENERGY ( Ryd)

FIG. 1. Photoionization cross section of the ground state of sodium. Open circles: experiment (Hudson and Carter, 1967); full circles: experimental results scaled by a factor of 0.7t; full curve: core polarization effects included in both the wave functions and the dipole operator; dotted curve: core polarization effects included in wave functions only; broken curve: core polarization effects omitted. The curves are the theoretical results of Butler and Mendoza (1983), from whom the figure has been adapted. (Reprinted with permission from IOP Publishing Ltd. 1983.)


calculations reproduce the correct qualitative shape of the experimental cross section for a range of energies above threshold, but no calculation provides any evidence for the 'hump' in the experimental value at higher energies. Use of the unmodified dipole operator gives good agreement with experiment near threshold but seriously underestimates the cross section above the minimum. Introducing the core-polarization correction to the dipole matrix element lowers the cross section at threshold but leads to a substantial increase above the minimum. Butler and Mendoza (1983) find that their results agree extremely well with the experimental results of Hudson and Carter (1967) when the latter are scaled by a factor of 0.71. It is of interest to note that the presence of Naz dimers, whose cross section is much larger than that of atomic sodium, could increase the cross section significantly (Chang, 1974). In view of the current situation, it is not surprising that there have been repeated suggestions that re-measurement of alkali-atom photoionization cross sections would be extremely valuable.

Ground-state photoionization cross sections for alkali and alkali-like systems are relatively small, rendering them sensitive to minor changes in the wave functions or to perturbations of the dipole operator. It is thus not surprising to find significant differences between the cross sections computed with and without the core-polarization correction to the dipole operator. One would expect some sensitivity to the choice of cut-off radius, rc , and such behaviour has been noted (Butler et al., 1984).

Photoionization of Rydberg states of alkali-atoms has been considered by Aymar et al. (1976, 1984), who used the parametric potential method introduced by Klapisch (1971). This is a one-electron model in which a central potential depending on a number of parameters is determined by minimising the root-mean-square deviation between observed and calculated energies of a selected set of levels. The np and nd cross sections are close to the corresponding cross sections for atomic hydrogen and they can be estimated with reasonable accuracy by use of quantum-defect theory (Burgess and Seaton, 1960).

Finally, in this section we mention very briefly the work of Saraph (1980) who has computed photoionization cross sections (and bound-state oscillator strengths) for several states of 0 IV. This is an example of a system with a non-closed-shell core and, though it may be viewed as a one-electron model- potential calculation, it is considerably more complex than the applications discussed so far. The six lowest terms (2s' 'S, 2 ~ 2 p ' * ~ P ~ , 2p2 'S, 3P, 'D) of 0 V are used as target functions in a close-coupling calculation of 0 IV wave functions. The 0 V wave functions are obtained from the atomic structure code SUPERSTRUCTURE (Eissner et al., 1974) in which the parameters of a statistical Thomas-Fermi potential are varied so as to minimise the energy. Core-polarization terms are not included explicitly in the potential.


D. RELATIVISTIC EFFECTS

It is clear from Sections III.A,B and C that model-potential methods can be applied very successfully to determine the valence structures and properties of light atomic systems. Many workers have studied methods to include relativistic effects in model potentials so that heavier systems could be investigated. Some of these methods include Breit-Pauli operators in a nonrelativistic calculation, while others are based on relativistic Dirac equations.

Semi-empirical model potentials, determined so that they reproduce observed valence energy levels, as discussed in Section ILB, will already include effects of relativity on the core electrons (e.g., relativistic core contraction). Alternatively, the effect of the relativistic contraction of the core electrons may be described by using Dirac-Fock orbitals to generate a core potential. This latter approach was used by Victor and Taylor (1983) in their model-potential calculations on the copper and zinc isoelectronic sequences.

Weisheit and Dalgarno (1971a), in a study of the Cooper minimum in the photoionization cross section of potassium, investigated several forms for a spin-orbit term and adopted

where

01 is the fine-structure constant and a is a screening constant. They find that an average value of Z - a = 17.33 reproduces the n = 4 - 8 fine-structure splittings to within 3 % of the observed values. Norcross (1973) and Theodo- siou (1 984) employ the parameter-free form

where E , ~ ~ is a valence-orbital energy. This form ensures that bo(r) has the correct behavior near the origin.

Laughlin and Victor (1974) and Laughlin et al. (1978) use perturbation theory to include the spin-orbit interaction bO( l ) + K0(2), where Go is given by Eq. (40), and spin-spin and spin-other-orbit interactions, in calculations on two-valence-electron systems. Fine-structure splittings in harmony with observed values are obtained for beryllium and magnesium and calculated intercombination-line oscillator strengths agree well with sophisticated ab initio values and with experiment (see Table V).


Ivanov et al. (1986) have used a model potential of the form

Z - 2[1 - e-2r(1 + r)] Zr

(43) 3br b2rZ b3r3 ) ] ]

- ( N , - 2 ) 1 - e - b r 1 +-+-+- [ ( 4 4 1 6

in a one-electron Dirac equation, where N , is the number of core electrons. The parameter b, which is a function of the one-electron quantum numbers n, 1, j and the nuclear charge Z , is obtained by making the calculated and measured energies coincide. The values of b thus obtained for several values of Z are fitted to a formula of the type

b, b2 b3

z zz z3 b = b, + - + - + -, (44)

which allows interpolation or extrapolation to other nuclear charges. Ivanov et al. (1986) present energy level results for 4Ej and 5Ej states of the copper isoelectronic sequence for 2 = 36 - 80. Wavelengths determined using interpolated values of the parameter b agree well with experimental results.

In a recent series of papers, Migdalek and co-workers (Migdalek, 1984; Migdalek and Baylis, 1985, 1986; Migdalek and Bojara, 1987) have carried out calculations of energy levels and oscillator strengths for the mercury and cadmium isoelectronic sequences, treating them as two-electron systems. They use a modified form of the Desclaux (1975) multiconfiguration Dirac Hartree-Fock code, in which a core polarization term

where

Fj = rj(rj + r,2)-3/2, (46)

is added to the Hamiltonian. Note that for N u = 2, V,, includes the dielectric term of Eq. (22). Two choices of the cut-off radius r, are investigated: firstly, it is taken equal to the mean radius of the outermost core orbital and, secondly, it is adjusted so that the experimental energy of the two-electron system (relative to the core ground state) is recovered. Better agreement with experimental oscillator strengths is achieved with the latter choice.

The transition matrix elements are corrected for core polarization by replacing the dipole operator D = 1;: rj by D + D,, where


Since different values of re are adopted for the initial and final states, an average value D, = f(DL + DE) is used to calculate the dipole transition matrix elements. I t is found that excitation energies and oscillator strengths agree much better with experimental data when core-polarization effects are included. For the higher members of the mercury isoelectronic sequence, the resonance 6lSo - 6lP: oscillator strength results of Migdalek and Baylis (1985) are still about 40 % higher than the experimental values. The disagreement is somewhat smaller for the 5lS, - 5lP: transitions in the more highly ionized members of the cadmium sequence (Migdalek and Baylis, 1986). It is possible that these discrepancies may result from difficulties in analysis of the experimental observations.

In earlier calculations on the one-electron copper, silver and gold sequences (Migdalek and Baylis, 1978), the cut-off radius r , and the dipole polarizability ctd were adjusted to bring the calculated and observed ionization energies into agreement. Again, the oscillator strength results obtained from this model are in much better agreement with the experimental values for the lowest 2S1,2-2P7,2, 3 / 2 and 2P:,z, 3,2-ZD3i2, 5,z transitions than are the results of calculations that ignore core polarization.

It may be mentioned here that care needs to be taken in the determination of the core cut-off radius, r, . If, as is often the case, it is chosen empirically to reproduce observed one-electron ionization energies, then the resulting value may be too small. A value of Y, that is too small will clearly overemphasise the role of core polarization in transition matrix elements and the contribution of the dielectric term to the energies of two-valence-electron systems.

For heavy atoms, relativistic effects cause contractions or expansions of the valence orbitals (Desclaux and Kim, 1975). In a calculation on mercury, Mohan and Hibbert (1987) simulate relativistic contraction by adding mass- correction (VMc) and Darwin (V,,,) terms to the model potential uM for the Hgc+ core. They use

and VM,(r) = --)@.'[&,lj - uM(r)]* (48)

where cnl j is an Hg' orbital energy. The core-polarization terms of Eqs. (45) and (47) used by Migdalek and Baylis (1984) and a spin-orbit operator of the form given by Eq. (40), with Z - a chosen to give the correct J = 0 to J = 2 fine-structure splittings for 6s6p3Po, are used by Mohan and Hibbert (1987). The calculated oscillator strengths from the ground 6s' 'So state to the 6s6p Py and 6s6p 'P: excited states are in reasonably good agreement with those calculated by Migdalek and Baylis (1984) and with the experimental values (see Table V).

186 C. Laughlin and G. A . Victor

A relativistic version of the parametric-potential method of Klapisch (1967) has been developed by Koenig (1972) and Klapisch et al. (1977). In this method, the electrons are assumed to move in a central potential U(r , a) depending on a set of parameters, a, and a zero-order Hamiltonian, which is a sum over one-electron Dirac Hamiltonians, is used. Each parameter describes the radial charge density in a complete shell of the ionic core; the optimal values of the parameters are obtained by minimising the total first- order energies of either the ground level or the ground complex of the spectrum. The method has been employed by Aymar and Luc-Koenig (1977) to study relativistic effects in transition probabilities in the magnesium isoelectronic sequence, and it has been used extensively by Klapisch and coworkers to study the spectra of highly ionized heavy atoms in laser-produced and tokamak plasmas (see, for example, Bauche-Arnoult et al. (1985) and Audebert et al. (1985)).

A relativistic R-matrix approach based on the Breit-Pauli Hamiltonian (Scott and Burke, 1980) was used by Bartschat and Scott (1985a,b) and Bartschat et al. (1986) in photoionization studies of mercury and barium. In the barium calculation (Bartschat et aZ., 1986), a two-electron model, including spin-orbit terms and core-polarization terms of the form -iq, W6(rr rc)/r4, where

is used to generate the initial (ground) and final (continuum) states. The dipole polarizability ad and l-dependent parameter r,(l) are adjusted to obtain good agreement between calculated and spectroscopic ionization potentials for LS states of Ba'. The resulting r,(Z) are in the region of 0.6, probably rather small considering that the mean radius of the outermost orbital of the Ba2+ ion is 1.927 (Migdalek and Baylis, 1987). The dielectric term is not included in the model and the effects of core polarization on photoionization cross sections are not discussed. The total ground-state cross section is found to be up to an order of magnitude larger than that measured by Hudson et al. (1970), though there would appear to be serious doubts about the calibration of the experimental measurements.

IV. Molecular Model Potentials

Ab initio quantum chemistry calculations represent one of the most computationally intensive areas of scientific research. At a given level of approximation, molecular calculations require substantially more computer


resources than atomic calculations because multicenter integrals are significantly more difficult to evaluate than single-center integrals and many nuclear geometries need to be considered, especially for polyatomic molecules. Consequently, many research groups have explored molecular pseudopotential and model-potential methods. The literature is too extensive to allow a comprehensive review, so we shall be selective and concentrate on small diatomic systems with one or two valence electrons. Thus, we shall omit discussion of some important papers, such as those by Stoll et al. (1984) and Hay and Martin (1985), where heavy molecular systems are treated using relativistic pseudopotential theory.

A. FORM OF THE MODEL POTENTIAL FOR TWO-CORE SYSTEMS

Consider a diatomic molecule with two spherically symmetric cores, A and B. Let R be the internuclear separation, and let Z , and z b be, respectively, the excess charges on cores A and B. The generalisation of the theory given in Section I1.A has been presented by Bottcher and Dalgarno (1974) and Peach (1983). The general case in which an arbitrary number of valence electrons interact with the cores A and B is too complex to present here; for a detailed account we refer the reader to Peach (1983). Instead, we write down the long- range forms of the electron-core and core-core interactions correct to the inverse sixth powers of the various distances for the important one-valence- electron system, viz.

v = v, + vb + vab + v n t . (51)

V , and Vb are the obvious generalisations of Eq. (21). The core-core interaction V,, is

where ad and a,, denote static dipole and quadrupole polarizabilities and c 2 ( 1 , l ) is the exact Van der Waals coefficient for the interaction of cores A and B (Dalgarno and Davison, 1966). The three-body term Knt is given by

where ra and r, are the positions of the electron relative to the nuclei of the cores A and B, respectively, and R is the position of nucleus B relative to nucleus A.

188 C . Lnughlin and G. A . Victor

Many applications have been concerned with a neutral one-valence- electron atom interacting with a neutral core B, in which case Z , = 1 and Z , = 0. The extension to a two-valence-electron molecule is straightforward. For such a system, dielectric terms of the form given in Eq. (22) for both of the cores are present in the interaction.

B. APPLICATIONS OF MOLECULAR MODEL POTENTIALS

Early applications were to one-valence-electron molecules with few-electron spherically symmetric cores. Dalgarno et al. (1970) carried out calculations on the 'X: and 'ZL states of Li;. Terms up to the inverse fourth power were retained in Eqs. (52) and (53), and the electron-Li'-core interactions V , and V, were modelled by the potential vM, where

with VHF(r) the static Hartree-Fock potential for Li'. The results were in good agreement with pseudopotential calculations by Bardsley (1970) and with ab initio calculations by Bardsley (1971). Later, Bottcher and Dalgarno (1975) studied more states of the same system employing a model potential with additional exponential terms u(r), where

u(r) = (-2.6664 + 1.2323r)exp ~ . (i;) Cut-off functions of the form

n ..-1 A

WXx) = 1 - exp( -x) C ___, t = 1 ( t - l ) !

( 5 5 )

were introduced, chosen to simplify the calculation of molecular integrals, the dipole and quadrupole cut-off parameters now having the common value 0.4.

Bottcher et al. (1971) carried out similar calculations for Na: and obtained the cross section for symmetric resonance. charge transfer, which can be expressed as an integral over the difference of the lowest two 'ZJ and 'Z; potential energy curves. Using similar methods, Bottcher and Oppenheimer (1972) obtained potential energy curves for the six lowest levels of NaLi' and calculated the nonresonant charge transfer cross section.

In a series of short papers, Cerjan (1975), Cerjan et al. (1976) and Kirby-


Docken et uf. (1 976) studied the results of model-potential calculations for molecular properties other than the potential energy curves. Core- and valence-electron contributions were compared for various one-electron operators and band oscillator strengths and photodissociation cross sections were obtained for Li l and Na:. The existence of the correct nodal structure in the model-potential valence-electron wave function is important for obtaining accurate values of many of the one-electron operators.

Model-potential calculations for X, II and A excited states of Na: and K,f have been presented by Henriet (1985). The dipole terms in Ynt, Eq. (53), are multiplied by cut-off functions, and the cut-off radii are determined to give good agreement with experimental molecular constants for the ground states. Dipole matrix elements for Z-E, Z-II and II-n transitions between excited states are evaluated using an effective dipole operator corrected for core- polarization effects (Bottcher and Dalgarno, 1975).

The generalisation of the method to the alkali-metal dimer, Liz, was carried out by Watson et a/. (1977). Care had to be taken to avoid collapse to the unphysical states that dissociate to 1s and n/. This was achieved by constructing anti-symmetrized trial functions from products of Li: eigenfunctions, excluding the iowest two Iso, and 1 so, eigenfunctions. Radiative lifetimes of the vibrational levels of the A'E; and the B'II, states of Li, have been reported (Watson, 1977; Uzer et al., 1978).

Additional one- and two-valence-electron molecular model-potential calculations were carried out for MgHe' and MgHe by Bottcher et al. (1975). The potential curves were used to calculate line broadening for Mg'('P) and Mg(' P and 3P) collisions with helium. Collision-induced transitions between fine-structure levels were calculated in the elastic approximation. Malvern (1978) extended the calculations to include Ca as the alkali earth and neon as the rare gas. Orlikowski and Alexander (1984) used these model potentials to calculate fine-structure transitions using close-coupling methods.

For a correct description of atom-rare-gas interactions, such as H-He and Na-Ne, Valiron et al. (1979) demonstrated the need to introduce nonlocal terms in the rare-gas model potential, V, (Eq. 51), in order to obtain wave functions that are consistent with the Pauli exclusion principle. Their procedure is equivalent to constraining the model-potential wave functions to be orthogonal to the rare-gas orbitals. The orthogonality condition involving the outer rare-gas orbitals is particularly important because virtual bound states associated with the rare-gas potential can lead to spurious molecular curve crossings. Pseudopotential approaches suffer from difficulties originating from the short-range repulsive potential introduced to simulate the orthogonality conditions. It appears to be necessary to use /-dependent pseudopotentials to obtain accurate phase shift results for electron-rare-gas scattering (Valiron et al. (1979)).


Bottcher (1973) developed a molecular model-potential method for open- shell cores and applied the method to calculate potential energy curves for HeH' and He:. In this treatment, the open shell has zero orbital angular momentum. Peach (1978) studied the potential curves of excited states of helium interacting with the ground state of helium or neon using similar methods. Some of the potential curves are not in good agreement with the calculations of Guberman and Goddard (1975), even at large internuclear separations. A study of an open-shell-core molecular model-potential method for cores with non-zero orbital angular momentum has been presented by Hennecart and Masnou-Seeuws (1985). The electron-open-shell-core interaction is treated using a method due to Feneuille et al. (1970) and the molecular model potential is based on the approach of Valiron et al. (1979). Potential curves and scattering cross sections are presented for Ne(2ps3s or 2ps3p) interacting with ground state helium or neon.

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2-EXPANSION METHODS M . COHEN Deparrmenr of Physical Chemistry The Hebrew University Jerusalem. Israel

1. Introduction . . . . . . . . . . . . . . . . . . 11. Z - ’ Expansion of Schrodinger’s Equation . . . . . . .

A. Zero-Order Solutions . . . . . . . . . . . . . B. First-Order Solutions . . . . . . . . . . . . . C. Approximate Solutions . . . . . . . . . . . . D. Expectation Values . . . . . . . . . . . . . . E. Off-Diagonal Matrix Elements . . . . . . . . . . F. External Fields: Double RSPT . . . . , . . . . .

111. The Screening Approximation . . . . . . . . . . . . IV. The Hartree-Fock Approximation . . . . . . . . . .

A. Nonequivalent Electrons . . . . . . , . . . . . B. Expectation Values of Single Electron Operators . . . . C. Atomic Oscillator Strengths . . . . . . . . . . .

V. Some Representative Results . . . . . . . . . . . . A. Energies . . . . . . . . . . . . . . . . . . B. Expectation Values . . . . . . . . . . . . . . C. Transition Elements . . . . . . . . . . . . . .

es, H yperpolarizabilities and Shielding Factors VI. Summary and Conclusions . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . .

. . . . . 195

. . . . . 197

. . . . . 198

. . . . . 199

. . . . . 200

. . . . . 201

. . . . . 202

. . . . . 203

. . . . . 204

. . . . . 206

. . . . . 201

. . . . . 208

. . . . . 209

. . . . . 210

. . . . . 210

. . . . . 212

. . . . . 214

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. . . . . 219

. . . . . 219

I. Introduction

The variational calculation of the ground-state energy of neutral helium (Hylleraas, 1929) was properly regarded as quantitative verification of the validity of Schrodinger’s wave mechanics for a many electron atom. Similar separate treatments of the isoelectronic ions LiII, BeIII, and so forth were equally successful, but Hylleraas (1930) then developed a particularly effective variational Rayleigh-Schrodinger (RS) perturbation theory (PT) in order to treat the entire isoelectronic sequence in a single calculation. For a two-electron atom or ion of nuclear charge Z, the procedure led to the energy formula, in Rydberg units,

E ( 2 ) = -22’ + 1.252 - 0.31488 + 0.017522-1 . . . (1)

and provided a firm theoretical foundation for nonrelativistic 2-expansions (more precisely, Z - ‘-expansions) of atomic energies.

195


196 M . Cohen

Equation ( 1 ) is a special case of a more general energy series

that displays the first few leading terms of an infinite series representation of E(Z) . Only the first two coefficients ( E , and El) can be computed exactly, although many of the higher order coefficients have been determined to very high numerical accuracy in some cases. Moreover, the radius of convergence of a series such as Eq. (2) is generally difficult to determine, while even within the circle of convergence one might expect to obtain quantitatively useful results only on the basis of very many coefficients, En. Remarkably, low order truncations of Eq. (2) and of analogous power series expansions of some other atomic properties (or simple functional forms derived from a few leading terms) frequently yield reliable results far away from the asymptotic limit (2 --f 00, corresponding to A 4 0).

Since Z-expansions are based on RSPT solutions of Schrodinger’s equation, a formal objection to their use is that relativistic and radiative effects must be taken into account for heavy systems (corresponding to large Z ) to achieve full agreement with experiment. Some extensions of the nonrelativistic theory have been undertaken (see for example, Layzer and Bahcall, 1962; Doyle, 1969; Mohr, 1985) and it is often sufficient to estimate relativistic and radiative effects by using low order PT (see, for example, Dalgarno and Stewart, 1960a; Aashamar, 1970) or, semi-empirically, by invoking the spirit of the nonrelativistic theory (for example, EdlCn, 1983). In the present work, we confine our attention to the nonrelativistic theory.

One spectacularly successful application of Z-expansions has been the systematic study of atomic oscillator strengths (see, for example, Wiese et al. 1966, 1969) where even departures from the predictions of the Z-expansion theory are well understood for the most part.

In the following sections, we begin with a brief review of the relevant RSPT and introduce some basic interpretative models. Applications to energies, diagonal, and off-diagonal matrix elements and external (field) perturbations are presented in turn. Because of its central importance in atomic theory, Z - expansions of the conventional Hartree-Fock (HF) approximation will be treated in some detail. Applications to Thomas-Fermi-Dirac theory (see, for example, Chen and Spruch, 1987, which contains many useful references) and to a large-dimensional analysis of electron correlation (see, for example, Herrick and Stillinger, 1975; Goodson and Herschbach, 1987) will not be considered here.

Throughout, we stress those features that make low order Z-expansions particularly useful. A few results of higher order calculations are summarized in tables.

2-EXPANSION METHODS 197

11. 2- ‘-Expansion of Schrodinger’s Equation

We employ Hartree’s (1957) atomic units (a.u.) throughout, and write the field-free nonrelativistic Hamiltonian of an N-electron atomic system of nuclear charge, Z ,

N

H ( N ; Z ) = - E i = 1

(3)

Here, ri denote the position vector of the ith electron relative to the atomic nucleus (assumed infinitely massive and fixed) while r i j = ri - rj. The uniform change of scale,

“i ri +. - Z’ (4)

yields directly

where

Thus, with energy in units of Z z a.u., the transformed Hamiltonian may be decomposed naturally into a form appropriate to RSPT treatment in which the expansion parameter A has a clear physical interpretation (the inverse nuclear charge). It is usual to expand the eigenfunctions and eigenvalues of H ( N ; A) in power series in A,

I ) (N; A) = I)o + A I ) l + E ( N ; A) = Eo + A E , + A2E2 + ...

+ . * .

where the successive pairs of corrections (I),,, E,,) are solutions of the standard RSPT equations:

(9)

(10)

( ~ 0 - + ( H I - ~ 1 ~ n - 1 = E m + n - m (n 2 2). (1 1)

(Ho - E0)I)O = 0

(Ho - E O M I + (Hl - ElMO = 0

m = 2

Although Eq. (8) embodies the traditional energy expansion, there has been a great deal of effort recently to rewrite the energy in alternative

198 M . Cohen

functional forms (see, for example, Stillinger, 1966; Brandas and Goscinski, 1970; Massaro, 1977; Dmitrieva and Plindov, 1981; Cohen and McEachran, 1981). These forms have usually been obtained ex post facto by fitting the leading coefficients, En, but Cohen and Feldmann (198 1) have shown that it is also possible to obtain a rational fraction representation of the eigenvalue E(N; 2) of H ( N ; Z ) ,

E , + AN, + A2N2 + ... 1 + AD, + A2D2 + . . *

E ( N ; 2) = Z 2

directly from Schrodinger's equation. Since we are concerned here mainly with low order truncations of Eqs. (7),

(8) and (12), we do not consider questions of convergence, ultimately appealing to comparison with experiment or highly accurate (usually variational) calculations to justify our procedures.

A. ZERO-ORDER SOLUTIONS

Because H, consists of a sum of N hydrogenic Hamiltonians, the zero order solutions t,ho and E , are known for any atomic system. t,ho is given universally by suitable linear combinations of antisymmetric products of one- electron hydrogenic functions (conventionally called orbitals, see below) and E , by a suitable sum of hydrogenic energies. The degeneracies of the hydrogenic spectrum, however, have the consequence that for very many states of most atoms, it is necessary to use degenerate perturbation theory in order to select an appropriate t,h, (Layzer, 1959). For example, the ground state of the beryllium isoelectronic sequence is described traditionally by the single configuration (ls22s2; ' S ) , but actually requires (in zero order) the linear combination

(13)

Here, the coefficients a, b are obtained by diagonalizing the first order perturbation operator

t,hO('S) = at,h0(ls22s2;'S) + bt,ho(ls22p2;'S).

Hl =cc;i7 i + j I J

in the two-dimensional basis consisting of the degenerate (hydrogenic) configurations ls22s2 'S and ls22p2 'S; Layzer (1959) has termed such a restricted finite basis a complex. The calculation of the coefficients a, b requires nothing more than the hydrogenic orbitals that enter into the

Z-EXPANSION METHODS 199

appropriate configurations, so that in all cases, the zero- and first-order energies are given exactly by the usual formulae

Eo = ($olHol$o>, El = ( $ O l H l l $ O > , (15)

provided that we now employ the symbol Jl0 to denote a normalised single configuration or a multiconfiguration (complex) wavefunction, as appropriate.

B. FIRST-ORDER SOLUTIONS

To proceed, it is necessary to solve the higher order RSPT equations, beginning with the first order problem of Eq. (10). This inhomogeneous equation is satisfied by any linear combination of the form

(16) $1 = $1 + W O

and i t is convenient to choose as “standard” solutions those combinations that satisfy the requirements

($o lJ lo> = 1, ( $ l l $ O > = 0. (17)

Once Jll is known, the second- and third-order energies are given formally by several equivalent expressions, including

E2 = ($lIHll$O> = - Q $ 1 l H o - Eel$,> (18)

Equation (1 9) emphasizes a general result demonstrated explicitly by Dal- garno and Stewart (1956): knowledge of the functions $o, $l,. . . , $n is sufficient to calculate all the energy coefficients up to E2n+l. However, if higher order corrections are also obtained, some additional formal identities may be derived from Eqs. (9) to (1 1). If we assume that the operators H o and HI are hermitian, then we find, for example,

E3 = <$~IHI - Eil$o> = -Q$zIHo - EoI$I); (20)

so, whenever approximations to $1, J 1 2 , . . . , are used in place of the exact solutions, the relative agreement between alternative expressions for the energies may serve as some indication of the probable accuracy of the calculated $1, tj2,. . . . It should be stressed, however, that agreement between the various expressions in Eqs. (18)-(20) provides only a set of necessary conditions for reliable $1, 4b2,. . . .

200 M . Cohen

C . APPROXIMATE SOLUTIONS

The two-electron nature of the perturbation H , makes it impossible to obtain an exact solution to any of the higher order RSPT Eqs. (10) or (11). Formal solutions, in terms of the complete set of hydrogenic eigenfunctions (which must include both discrete states and appropriate continua) are both slowly converging and inconvenient in general. But fortunately, there exist quite accurate approximate solutions.

Hylleraas (1 930) introduced the variational functional

whose Euler equation is seen to be Eq. (lo), and that satisfies

JZ(X1) = E2 + ( X l - $lIHO - EolX1 - $1). (22)

Equation (22) implies that, provided Eo is the ground state energy of H o , minimising J2(x1) with respect to an abritrary trial function x1 yields an upper bound to E , ,

J2(X1) 2 E2. (23)

With x1 calculated in this way, Hylleraas (1930) used Eq. (19) to estimate E, without claiming that this constitutes a bound. The Hylleraas functional Eq. (21), together with its higher order generalisations, have been employed routinely up to much higher order (see, for example, Midtdal et al., 1969; Aashamar, et al. 1970) on both ground- and excited-states of two electron systems. The choice of basis functions included in and its higher order analogues remains problematical, and even when the energy coefficients appear to converge smoothly, this may not be the case for other atomic properties calculated with the same approximate functions.

Although more complicated functionals than JZ(x1) are available that would also provide a lower bound to complement the upper bound of Eq. (22) (for example, Prager and Hirschfelder, 1963), they are more complicated to apply and do not seem to have been used widely in the context of Z - expansion theory.

A practical alternative to the Hylleraas functional method, introduced by Dalgarno and Drake (1969), employs the formal apparatus of eigenfunction expansions but within a finite dimensional basis of variationally determined “pseudostates” of H o . In a series of applications (including a number to doubly excited states), it has proved convenient to solve the successive RSPT equations exactly within the finite subspace spanned by the chosen basis, in principle to all orders. In practice, there will be some inevitable loss of accuracy with increasing order of PT, but this can be monitored effectively by

Z-EXPANSION METHODS 20 1

comparing alternative formal expressions (cf. Eqs. (1 8)-(20) above) for the successive energy coefficients.

D. EXPECTATION VALUES

A power series expansion of I ) (N;A) implies similar expansions of all atomic properties and is not restricted to the energy, E ( N ; A). For example, any homogeneous function of overall degree s of the coordinates ri and momenta Vi,

N N N

(L(” may include many body operators, so long as it remains homogeneous under the scale transformation of Eq. (4)), has expectation value

where, quite generally,

m = O m = O

In this form, the nth order corrections L,, S , require knowledge of I),,, but a particular choice of normalization of the higher order $, is always possible in order to set S , = 0 for all n 2 1. This choice is neither unique nor necessary, however, and Eq. (25) naturally has the functional form of a rational fraction, analogous to the energy, Eq. (12) above.

Now, as first shown by Dalgarno and Stewart (1956), the first order matrix element, L,, may be written equivalently in either of two forms (we assume S , = 0 for convenience):

(27)

(28)

The significance of this “interchange theorem” arises from the possibility of obtaining an exact solution, 41 of Eq. (28), when only an approximate solution I)l of Eq. (10) is available. This occurs whenever contains only one-electron operators provided that E , is nondegenerate (Hirschfelder et al., 1964). Even if 41 is also an approximation, numerical comparison of the differing forms of L may give a valuable indication of the probable accuracy of the calculated

Ll = 2(+1 IL l I )o> = x41 IH, I$o>

where 4t satisfies

w, - EOMl + ( L - L 0 ) I ) O = 0 3 (41 140) = 0.

202 M. Cohen

The second order matrix element may also be expressed analogously in either of the forms

L, = <$llLl$l) + 2<$,lLl$*) (29)

(30) = <$1IL- LOI$l) + 2(41lH, - E,l$l) + LOS,

so that, provided we choose S , = 0, L, may be calculated using two first order functions t,h1 and 41 without knowledge of the second order correction, $,. Although this process may evidently be generalized to higher order, it is clearly a much less desirable result than that obtained for the energy expansion. Not only are more solutions required through each order, but different equations must be solved for each operator L(’) of interest. Neverthe- less, exact values of Lo and L , sometimes provide suprisingly accurate estimates of (L). We return to this point below.

E. OFF-DIAGONAL MATRIX ELEMENTS

Matrix elements of an operator such as L(’) between two distinct states of H ( N ; Z ) with normalized eigenfunctions $ p and $q for example, may be written down analogously:

Here, we write (cf. Eq. (26) above) n n

Tn = C <$;IL(’)I$:-m), Sf: = C <$LI$:-m) (32) m = O m = O

with S: defined analogously; we note that Eq. (32) is not immediately in the form of a rational fraction. Interchange theorems, however, still allow us to rewrite the first order matrix element

with an analogous equation for 4:. It should be noted that Eq. (36) is different for each pair of states for any given L. As before, complications may arise if either EP, or E8 denotes a degenerate eigenvalue of H o .


F. EXTERNAL FIELDS: DOUBLE RSPT

Thus far, we have considered only the field-free Hamiltonian operator and exploited the consequences of applying a simple scaling transformation to H ( N ; Z ) . Both the “unperturbed” and “perturbation” operators of Eq. (6) are in reality components of a physical Hamiltonian that has been decomposed in this way only to simplify the treatment of Schrodinger’s equation. Now we extend the treatment to include additional interactions (for example weak external electric or magnetic fields). It is then necessary to treat a more general Hamiltonian

H”’(Z, u) = H ( N ; Z ) + UV, (36)

where V represents the operator of the “external” perturbing field and v is a parameter that measures its strength. If u is not too large, it is customary to employ RSPT to solve the eigenvalue problem in powers of the parameter v :

Y(Z, u ) = Y,(Z) + Uul,(Z) + ... &(Z, u ) = c , (Z) + U E 1 ( Z ) + ....

(37)

(38)

Provided that the limit u + 0 actually exists, it is clear that Y,(Z) + $ ( N ; A) and &,(Z) + Z 2 E ( N ; A), and we may reasonably expect to obtain analogous series expansions for ul,(Z), E,(Z) , and so forth.

Equations (37), (38) are completely analogous to Eqs. (7), (8), and differ from them only in that the expansion coefficients are now functions of an additional parameter (Z) rather than constants; when they are expanded in Taylor series, we obtain power series in two variables. In the important special case of a uniform electric multipole field, for which effectively

N

V = 1 rfPk(cos Oi) i = 1

(39)

and P,(cos Oi) denotes the usual Legendre polynomial of order k , application of the scaling transformation Eq. (4) to the complete H“’(Z, v ) yields

(40) H“’(Z, v ) + Z q H , + AH, + p F )

where now the field perturbation parameter, p, is given by

H , and AHl remain components of H ( N ; A) as in Eq. (6). The eigeniunctions and eigenvalues of H(” may now be expanded as double series

204 M. Cohen

or, by suitable regrouping of terms, they may be expressed as power series in A only (the coefficients now being functions of p).

The formal theory of two (or more) perturbations is greatly complicated by degeneracy effects and (simply as a practical consideration) the covergence rates of the expansions in Eq. (42) may be very different when they are viewed as functions of either A or p separately. Some of these problems have been reviewed carefully by Killingbeck (1 977).

Fortunately, measurable properties are closely related to the more “physical” expansions of Eqs. (38). For example (Drake and Cohen, 1968), the multipole polarizability, a, and hyperpolarizability, y, of atoms and ions in S states are directly related to the second- and fourth-order energy coefficients in the expansion of Eq. (38) (all coefficients of odd order vanish through parity considerations)

c i = -2E2(Z), y = -2EJZ) (43)

where (cf. Eq. (18) above)

= (Yl(Z)

E 4 m = (Y2(Z)

and

v I Yl(Z)> - tEZ(Z)W 1 I Y 1 ( a > . (45)

A closely related quantity is the multipole shielding factor, b, defined by Dalgarno (1962)

where

Thus, we are finally concerned with the calculation of 2-expansions of quantities such as a, p and y which are clearly analogous to the transition elements discussed above. The fact that V and V‘ are sums of one-electron operators usually allows efficient use of interchange theorems in these calculations, as before.

111. The Screening Approximation

Provided that the various 2-expansions actually converge, it is now only a matter of computational effort to calculate coefficients and so obtain reliable atomic properties. For most matrix elements, however, only the leading


(zero-order, hydrogenic) term can be calculated exactly, although exact first- order corrections can usually be obtained for sums of one-electron operators. Higher order corrections must be calculated using approximate wavefunctions, with inevitable loss of accuracy. So, it is entirely possible that estimates derived from a Z-expansion truncated at a relatively high order are no more reliable than those obtained from a lower order truncation, in spite of apparent numerical convergence of the coefficients.

Now, assuming that only the first two leading coefficients are known exactly, the expectation value of an operator such as L(’’ (cf. Eq. (25) above) is

(L‘”) = z-s[L, + AL, + 0(12)] (48)

and one possible way of estimating the contributions of higher order terms is to write

(L‘”’) - L,(Z - D L ) - s

L, SLO

This is the screening approximation (Dalgarno and Stewart, 1960b), and is derived most easily by rewriting the Schrodinger Hamiltonian of Eq. (3):

(49)

where

(50) dL = -.

(51) 1

H ( N ; Z) = - ? (i V: + - i = 1

The screening constant oL of Eq. (50) has the effect of making the expectation value (L(’)) stationary throughjrst order, and it is different for each state and each operator, L(‘).

It should be emphasized that H ( N ; Z ) remains independent ofa, and that it is perfectly possible to choose D in other ways, for example by requiring that some particular L(‘) be stationary through some higher order. This approach has been used recently in some energy calculations, but not in the context of Z-expansion theory (Arteca et al., 1984).

The first-order screening approximation result, Eq. (49), suggests that a very simple description of many atomic properties results from assuming a hydrogenic model of the atom but with an appropriately screened nucleus. Since the value of g L is calculated theoretically, and occasionally turns out to be negative in sign (“anti-screening”), it may be better not to ascribe physical significance to oL, but to recognize Eq. (49) as just one possible finite approximant representation of Eq. (48). A useful review, in the context of Pade approximants, was given by Brandas and Goscinski (1970). These considerations apply equally to off-diagonal matrix elements.

206 M. Cohen

For operators that are not sums of one-electron operators, L , and oL cannot be calculated exactly. It is always possible, however, to obtain an approximate L , by adopting the Hartree-Fock value, LyF. In principle, LyF (and indeed all higher order coefficients LFF, LyF, etc.) can be calculated exactly. For operators that are sums of one-electron operators, LyF and L, coincide (Cohen and Dalgarno, 196 1 a); in other cases, replacing L by LyF in order to estimate uL obviously cannot be justified rigorously, but it should yield reasonable estimates of (L(')) for sufficiently large Z .

IV. The Hartree-Pock Approximation

Formally, the field-free nonrelativistic unrestricted Hartree-Fock (UHF) approximation replaces the Schrodinger Hamiltonian H ( N ; 2) by a separable Fock operator

i = 1 i = 1 [: " I ri (52) N

F ( N ; Z ) = c F,(Z) = c - - vi" + - - vpff .

Here, the one-body effective potential Vpff is defined self-consistently in terms of the N orthonormal spin orbitals { u j ( r j ) ; j = 1,. . . , N } which are supposed to describe the appropriate electronic state of the system via a single Slater determinant (cf. Cohen and McEachran, 1980). The effects of various spin and equivalence restrictions lead to a variety of slight modifications of Vf", but do not alter the main result here. Essentially, the effect of VPff on an arbitrary spin orbital, u(ri) say, is given by

where the Coulomb and exchange potential operators are to be interpreted in the sense that

Thus, applying the usual change of scale to Fi(Z) yields at once

provided that the spin orbitals that contribute to V;ff are expressed in scaled variables, rJ2, rather than in natural (unscaled) variables. It is now clear that


each spin-orbital ui(ri) and orbital energy ei (the one electron eigenvalues of Fi(Z)) may also be expanded in powers of A:

ui = uio + AUi1 + A Z U i , + . . .

E i = Z2[EiO + A&(, + A2Ei, + . . .] and

(56)

(574

and the leading terms (uio and E ~ ~ ) are simply hydrogenic orbitals and energies. Moreover, the one body nature of VP" implies that successive terms in the expansion of each ui may, in principle, be calculated exactly to all orders (cf. Young and March, 1958). In practice, even the first order solutions are quite complicated (cf. Cohen, 1963) and for most low order calculations of atomic properties other than the energy, use of interchange theorems is preferable whenever possible.

Because all HF approximations emphasize the separability of the Fock operator, they usually specify a single configuration (rather than a complex configuration) for each many-electron state. Consequently, the zero-order N - electron HF wave-function is correct only for certain special cases, such as completely filled shells of electrons (s2, p6, d", etc.). It should be stressed that it is quite feasible to derive a generalized Fock operator by assuming a multiconfiguration complex; this yields the extended Hartree-Fock (EHF) approximation (Cohen and Parkinson, 1966).

Alternatively, it might be possible to expand the conventional Fock operator in a manner analogous to our earlier treatment of the Schrodinger Hamiltonian by writing

N

F ( N ; 2) = Z y F , + / IFl ) , F , = c v:". ( 5 8 ) i = 1

Whereas Eq. (55) implies that expansion in powers of A follows separation of variables, Eq. (58) implies that the order of these operations is to be reversed. A similar change of order in polarizability calculations is well-known to lead to significantly different results (for a review of uncoupled and coupled procedures, see Dalgarno, 1962). At all events, most HF 2-expansion calculations that have been reported are based on Eq. (55).

A. NON-EQUIVALENT ELECTRONS

A conceptually simple type of improvement of the standard (single configuration) H F approximation treats shells of "equivalent" electrons as if

208 M. Cohen

they were inequivalent. For example, if the helium ground-state is approximated by an open shell orbital function (designated lsls’ ‘ S ) rather than by the conventional closed shell HF function (designated Is2 IS), the two distinct (but nonorthogonal) orbitals denoted by u and v, have been shown to have expansions (Stewart, 1964; Coulson and Hibbert, 1967):

U, U = UO A 1 / 2 U 1 j 2 + I U , 13/ ’U3/2 + * * . . (59)

The need for half-integral powers of I in these expansions stems from the presence of overlap integrals such as (u I u ) in the appropriate HF equations. It seems likely than an analogous treatment of other shells of “equivalent” electrons, perhaps by the methods of Jucys (1967), would lead to orbital expansions involving other fractional powers. As in the helium case, however, the total wave-function contains only integral powers of I , so that all observables (energies, expectation values, etc.) have expansions in integral powers of A only.

B. EXPECTATION VALUES OF SINGLE ELECTRON OPERATORS

For any system described by a single Slater determinant, as well as for some others, the expectation value of an operator of the form

N

L = C Ii(ri) i = 1

is given by

This result is a straightforward consequence of the orthonormality of the HF orbitals ui(ri) and implies that for sums of single electron operators, the total expectation value is simply a sum of the separate orbital contributions. The interchange theorem of Dalgarno and Stewart (1956) may be applied to calculate exactly the two leading terms in the expansion of each separate term of the sum ( L ) (Eqs. (27) and (28) above are now single particle equations) and yields for a homogeneous operator of degree --n (cf. Eq. (24) above):

( 1 ) = Z”C1, + I l l + O ( I ’ ) ] . (62)

Although higher order terms can actually be calculated for this case, the two leading terms may be written


where now the orbital screening constant, IT,, is different for each occupied orbital, as well as for each operator, 1. Equation (63) often yields reliable estimates of individual orbital contributions to atomic expectation values in the HF aproximation scheme (Cohen and Dalgarno, 1961b).

c. ATOMIC OSCILLATOR STRENGTHS

For electric dipole transitions between two atomic states, the simplest form of transition matrix element involves the homogeneous operator

N

T = C ri cos ei, i = l

once again, a sum of one-electron operators. With a suitable choice of normalisation, the transition matrix element is simply

(65)

(66)

T,, = [ ($p [T l$q) l = Z- ' [To +ATl + A2T2 + ...I

= ZZIAEo + AAEl + A'AE, + ...I. and the excitation energy has the expansion, based on Eq. (2)

A& = E~ -

Thus, the oscillator strength can be expanded analogously (see, for example, Crossley, 1969):

(67)

and the leading term f o of Eq. (67) is, as usual, given by its hydrogenic value. Since the leading term of A& vanishes for transitions that do not involve a change of principal quantum number, (n, I ) + (n, 1 ? l), so thatf,, = 0 in such cases, it is often more useful to estimate f by calculating T,, from Eq. (65) (or some appropriate approximant derived from it), but to use the observed excitation energy, A&. These considerations apply particularly when HF orbital approximations are employed to calculate Tpq, which reduces rigorously to a single-electron matrix element only in special cases (see, for example, Cohen and McEachran, 1980).

For transitions for which f o does not vanish, a good representation of Eq. (67) may be obtained in some cases from the geometric approximation (for many references, see the review of Killingbeck, 1977)

2 3

f = - A & T ; , = f o + f i n + f i A z + ...

(clearly the [O/l] Pade approximant tof); in other cases, a similar representation based on ,f, and f 2 derived from rather precise variational wavefunctions is sometimes adequate. But, for the most part, rather simple

210 M . Cohen

approximate wave functions have been used to estimate Tpq, with inevitable loss of accuracy.

V. Some Representative Results

Because of the long history of the 2-expansion procedures described here, no attempt will be made to present a complete survey of results of all the relevant calculations. Instead, we give only a small sample of several different computations, in the hope that they are sufficiently representative of the much larger body of work done over the past sixty years.

A. ENERGIES

Historically the first, and still the most extensively treated, energy expansion is that for the ground state of the helium sequence. In Table I, we present the simplest screening approximation (all results are in Hartree a.u.), based on the two leading terms of Eq. (1) (which are exact). For comparison, we also list results based on the [0/5] Pad6 approximant to a reduced energy function (Cohen and Feldmann, 198 1)

TABLE I

HELIUM SEQUENCE (in atomic units)

GROUND-STATE ENERGIES (- E ) OF THE

2 2.84766 2.90369 2.90372 4 13.59766 13.65557 13.65557 6 32.34766 32.40625 32.40625 8 59.09766 59.1 5659 59.15660

10 93.84766 93.90681 93.90681 ~~

(1) Screening approximation,

- E = (Z - 0.3125)’

(2) [0/5] Pad6 approximant (Cohen and

(3) Aashamar (1970), essentially exact Feldman, 1981)

values

Z-EXPANSION METHODS 21 1

TABLE I1 ENERGIES ( - E ) OF THE ls22s2p1P STATES OF THE

BERYLLIUM SEQUENCE (in atomic units)

~

4 14.4226 14.4237 14.4269 - 14.4229 6 36.0252 36.0256 36.0269 35.9998 36.01 13 8 67.6444 67.6446 67.6453 67.6295 67.6307

10 109.2699 109.2701 109.2705 109.2599 109.2572 ~ ~~

(1) Eq. (704 (2) Eq. (70b) (3) Eq. (704 (4) 10th order RSPT sums (Watson and O'Neil, 1975) (5) Variational values (Hibbert, 1974)

as well as more precise values (Aashamar, 1970). These results are quite typical for small atoms; the primitive screening approximation is qualitatively highly successful and quantitatively useful with increasing Z , while a representation based on only a few additional computed RSPT coefficients is capable of very high accuracy.

The functional form of the representation chosen would appear to be much less important than the number of coefficients employed in deriving it. Thus, in Table I1 we give results for an excited state ( l s 2 2 s 2 p ' P ) of the beryllium sequence based on four computed coefficients only and derived from the following respresentations:

- E l Z 2 (1 - an)' 2Eo

a=- E E3A3 - = E o + E l l + E2L2 + -.

(Dalgarno and Stewart, 1960b);

(70b) 3E

- = Eo + E1A - E

Z 2 [bn + ln(1 - b l ) ] ; b = 3

2E2 (Cohen and McEachran, 1981); a%d

(Cohen, 1984). All these functions reproduce the RSPT expansion up to the term E 3 1 3 , but no further. The computed E , and E , were taken from the work of Watson and O'Neil (1975) and we give for comparison their 10th order RSPT sums as well as some fairly refined variational energies (Hibbert, 1974).

212 M . Cohen

B. EXPECTATION VALUES

The use of the screening approximation (or some other summation scheme) is not restricted to expectation values of single electron operators, even if the first order expansion coefficient is known only approximately. In the important special case of the interelectronic repulsion energy, Lowdin (1959) has shown that the combined use of the quantum mechanical virial theorem together with the Hellmann-Feynman theorem yields

aE (c ') = 2E - Z - i # j r i j az

(assuming that the energy is written in the form of a Taylor series); thus, only the energy expansion coefficients are required in this case. Moreover, the formal result of Eqs. (71) remains valid in the HF approximation scheme.

In Table 111, we present results for the helium sequence ground state. The first order screening approximation based both on the HF approximation and on accurate (variational) RSPT wavefunctions are compared with direct computations. The improved accuracy of the screening approximation with increasing Z is as expected; the reliability for low 2 may be fortuitous in this case.

TABLE 111 EXPECTATION VALUES ( 1 / r , 2 ) FOR THE

GROUND STATE OF THE HELIUM SEQUENCE

(in atomic units)

~ ~~

2 1.0280 1.0258 0.9347 0.9458 4 2.2780 2.2771 2.1847 2.1909 6 3.5280 3.5274 3.4347 3.4389 8 4.7780 4.7776 4,6847 4.6879

10 6.0280 6.0275 5.9347 5.9372

( 1 ) Screening approximation, HF approxi-

(2) Numerical HF values (Boyd, 1978) (3) Screening approximation using varia-

(4) Accurate values (Pekeris, 1959; Scherr

mation

tional E ,

and Knight, 1964)


TABLE IV EXPECTATION VALUES FOR NEUTRAL

HELIUM (in atomic units)

Operator (1) (2) (3)

r ;; 1.4153 1.4455 1.4648 r I-; 0.9347 0.9472 0.9458 r12 1.4304 1.4180 1.4221 4 2 2.5350 2.5203 2.5164

( 1 ) Screening approximation (2) [1/1] Pade approximant, Eq. (72) (3) Accurate values (Pekeris, 1959)

We emphasize that it may well be possible to obtain estimates more reliable than those of the first-order screening approximation by using other functional forms, particularly if a few additional expansion coefficients are available. One particularly appealing alternative for the expectation value (L‘”) is to adopt the [1/1] Pade approximant form

which has the appearance of the exact expansion, Eq. (25), truncated at terms of order A; this form requires knowledge of the leading three expansion coefficients only. In Table IV we illustrate the use of Eq. (72) by comparing results for a series of two-electron operators in neutral helium. As Z increases, the small differences between results based on two coefficients (the first-order screening approximation) and on Eq. (72) diminish rapidly.

For heavier systems, higher order coefficients are not generally available, but the first order screening approximation can be calculated both for separate orbitals and for complete atoms (within the HF approximation scheme). Table V, reproduced from the work of Cohen and Parkinson (1966), compares calculated orbital values of ( r ’ ) for four-, five- and six-electron atoms and ions with the results of direct numerical H F calculations. It is clear that the trends are as before, but the procedure is somewhat less reliable for the outer orbitals of neutral systems; clearly, the Z-expansions converge more slowly for 2s- and 2p-orbitals than for ls-orbitals. In this case, estimates of (xi :) derived from the screening approximation applied to the entire system and to separate orbitals are almost indistinguishable. Furthermore, use of a zero order complex (rather than a single configuration) also has little effect on the results for several single-electron operators. Unfortunately, there can be no guarantee that this will remain true in other cases.

214 M . Cohen

TABLE V

(in atomic units) CALCULATED VALUES OF ( r2> FOR 4, 5- AND 6-ELECTRON ATOMS AND IONS

ls2 2s2 'S

ls2 2s2 2p 2P0

Is2 2s2 2p2 3P

4 5 6 7

5 6 7 8

6 7 8 9

0.235 0.144 0.0967 0.0695

0.147 0.0987 0.0707 0.053 1

0.101 0.07 19 0.0539 0.04 19

0.233 0.143 0.0962 0.0693

0.143 0.0969 0.0698 0.0526

0.0972 0.0701 0.0529 0.04 13

6.95 8.43 - -

3.51 3.83 - -

2.11 2.21 - ~

1.41 1.45 - -

4.22 4.71 4.28 6.15 2.43 2.59 2.26 2.69 1.58 1.64 1.39 1.54 1.11 1.14 0.941 1.01

2.83 3.05 2.83 3.75 1.78 1.87 1.65 1.93 1.23 1.27 1.08 1.20 0.894 0.916 0.766 0.821

(1) Screening approximation (Cohen and Parkinson, 1966) (2) Numerical (calculated by P. S. Kelly)

C. TRANSITION ELEMENTS

Electric dipole transition matrix elements (which involve one-electron operators) are not given correct to first order by the H F approximation, but it is sometimes possible to calculate the first order Z-expansion coefficient exactly (as well as in the HF scheme) by use of interchange theorems (Cohen and Dalgarno, 1966).

For transitions in L- and M-shell atoms and ions, the effects of zero-order mixing have been treated in detail and shown to be much more significant than for expectation values (Cohen and Dalgarno, 1964; Crossley and Dalgarno, 1965). The dipole length transition element is given quite generally by Eq. (65), or in the screening approximation by

the screening constant opq now referring to two specified states; To is given as usual by the appropriate hydrogenic value for the one-electron transition (n, 0 --f (n', 1 * 1).

Equation (73) yields absolute multiplet strengths (related to Tiq) which are generally in good agreement with the corresponding numerical HF values,


whether or not zero-order mixing is considered. As Z varies, T,,, is much less liable to wild variations than the oscillator strength, f (the excitation energy AE is usually a more sensitive quantity), and this may explain the widespread success of Eq. (73).

In Tables VI and VII we present a small sample of L- and M-shell multiplet strengths derived from Eq. (73), compared with numerical HF values with and without limited configuration mixing (use of a complex for one of the states). These results should be quite typical, and in fact, estimates based on Eq. (73) are usually adopted for all ionized systems for which more direct computations are not available (Wiese et ul., 1966, 1969).

For a few transitions for which refined variational calculations have been performed, an alternative to Eq. (73) is again provided by low order Padt-like approximants. For the oscillator strength j , a representation of the form

TABLE VI VALUES OF u2 FOR L-SHELL 2s -2~

TRANSITIONS IN NEUTRAL AND IONIZED NITROGEN AND OXYGEN

Ion Transition ( 1 ) (2)

NI

NIII

NV 01

0111

ov

2s22p3-2s2p4 4s-4P

ZP-*S 2s2p4-2p5 2P-2P 2s22p-2s2p2 2P-2P 2s2p2-2p3 4P-4s

2P-2D 2S-2P

2s-2p 2S-2P 2s22p4-2s2pS 3P-3P

2s22p2-2s2p3 3P-3P

2s2p3-2p4 3P-3P

2s2-2s2p 'S-'P 2s2p-2p2 'P-'S

'D-'P

' D-'P

'D-'P

'P-'D

0.641 0.635 0.660 0.682 0.424 0.435 0.454 0.454 0.320 0.480 0.478 0.344 0.342 0.346 0.355 0.250 0.270

0.610 0.614 0.614 0.626 0.436 0.425 0.441 0.440 0.323 0.449 0.450 0.339 0.340 0.334 0.347 0.256 0.271

(1) Screening approximation (Cohen and Dal-

(2) Numerical single configuration HF values garno, 1964)

216 M. Cohen

TABLE VII ABSOLUTE MULTIPLET STRENGTHS FOR FeXV

Single configuration Complex

Transition (1) (2) (1) (2)

3~3p-3~3d 3~3d-3p3d 3~3p-3~3d 3~3d-3p3d 3s3p-32

-3pz 3pz-3s3p

-3p3d

3P-3D 2.251 2.302 3D-3P 1.385 1.399

'D-'P 0.468 0.470 'P-'S 0.915 0.937 'P-'S 0.307 0.314

'P-lD 0.737 0.768

'D-'P 1.532 1.564 ID-'P 0.025 0.026

2.077 1.560 1.449 0.628 0.769 0.348 0.599 0.006

2.068 1.633 1.438 0.689 0.757 0.341 0.575 0.003

(1) Screening approximation (Crossley and Dalgarno, 1965) (2) Numerical HF values (Froese, 1964)

or

yields excellent results for several transitions in the helium and lithium isoelectronic sequences (Cohen and McEachran, 1978).

D. POLARIZABILITIES, HYPERPOLARIZABILITIES AND SHIELDING FACTORS

Degeneracy effects are crucial in calculations of the effects of external fields. Thus, for the helium ground state (closed shell, ls2 IS) the leading term of the dipole polarizability is of order Z-4, whereas for metastable helium (open shell 1 ~ 2 s ' , ~ S ) the leading term is of order Z - 3 . This is a direct consequence of the degeneracy of the hydrogenic 2s- and 2p-levels, which are linked by the external (dipole) field. For ground state beryllium (whether described by the single configuration ls22s2 'S or by the complex including ls22p2 'S) the degenerate zero-order configuration ls22s2p 'Po is linked. By contrast, there is no degeneracy effect in the case of the quadrupole polarizability for Lshell atoms, but such effects will arise for M-shell systems.

Once these degeneracies have been taken into account properly, however, the leading terms can be calculated as before, although the computational


TABLE VIII DIPOLE POLARIZABILITIES OF THE

LITHIUM ISOELECTRONIC SEQUENCE

(in units of cm3)

3 27.3 21.9 25.1 4 4.00 3.55 3.65 5 1.25 1.15 1.16 6 0.542 0.510 0.509 7 0.282 0.268 0.267

(1 ) Screening approximation (HF values)

(2) Screening approximation (“exact” first order)

(3) Variational open-shell calculations (Flannery and Stewart, 1963)

effort grows rapidly with the size of the system. In Table VIII we reproduce some dipole polarizabilities for the lithium sequence (Cohen, 1966), both in the HF approximation and using a more exact first-order coefficient. It should perhaps be mentioned that the dipole shielding factor, which is given theoretically for any N-electron ion of nuclear charge Z by the formula (Sternheimer, 1954)

N Z

p = - (75)

yields 3.140/Z + O(Z-2) in the H F approximation, and 3/Z + O ( Z - 2 ) in the more precise treatment.

In Table IX, we reproduce some dipole polarizabilities for the sodium sequence (Cohen and Drake, 1967). In addition to the usual screening approximation, which appears to be inadequate for the neutral atom, we also give results based on a (semi-empirical) improvement, namely

c1 00 = -

Z - a0 a* =

( Z - a)3’

Equation (76) may be viewed as an extension of the Z-dependent screening parameter theory of Froese (1965) introduced for dipole transition elements. Here, we do not give H F results, since the leading coefficient in the expansion of p is 19.6 (in place of 11) indicating a serious loss of accuracy in this case.

218 M . Cohen

TABLE 1X DIPOLE POLARIZABILITIES OF THE

SODIUM ISOELECTRONIC SEQUENCE (in units of i O P 4 cm3)

11 6.39 24.12 20”, 21Sb 12 2.84 6.75 -

13 1.50 2.80 -

14 0.890 1.43 ~

17 0.273 0.352 -

26 0.035 0.038 -

(1) Elementary screening approximation

(2) Improved screening approximation, Eq. (76)

(3) Experimental values: “Salop et a!. (1961) bChamberlain and Zorn (1963).

Finally, in Table X, we reproduce the dipole hyperpolarizabilities (y) of the helium sequence ground states, which may be expected to be reliable. As will be seen, departures from the HF results are quite minimal in this case, whereas even the sign of y is sensitive to the accuracy of the approximation in other cases (Drake and Cohen, 1968).

TABLE X

HELIUM ISOELECTRONIC SEQUENCE (in atomic units)

DIPOLE HYPERPOLARIZABILITIES OF THE

2 44.0 28.5 24 3 0.271 0.209 4 0.946 (-2)” 0.783 (-2) 5 0.770 (- 3) 0.665 (- 3) 6 1.038 (-4) 0.920 (-4)

(1) Screening approximation, HF values (2) Screening approximation, exact values (3) Finite field PT (Cohen 1965) “0.946 x lo-*


VI. Summary and Conclusions

It is clear that the use of the Z-expansion methods described here is by no means exhausted. Although the gratifying success of the first-order screening approximation of the helium sequence (for all properties, not only the total energy) does not appear to extend to much larger atoms and ions, it seems very likely that only slightly more elaborate representations can achieve similar accuracy for other systems, provided that one or two more expansion coefficients can be calculated accurately for each property of interest. This goal should not be beyond the scope of current (and future) computational effort. It should be emphasized, in conclusion, that the results of the first- order screening approximation must ultimately become reliable for sufficiently large 2; nevertheless, we still require an improved theory for intermediate and low values of the nuclear charge.

REFERENCES

Aashamar. K. 1970. Nucl. Instrurn. Methods. 90, 236-8. Aashamar, K., Lyslo, G., and Midtdal, J. 1970. J . Chern. Phys. 52, 3324-36. Arteca, G. A,, Fernandez, F. M., and Castro, E. A. 1984. J . Math. Phys. 25,3492-6.

Brandas, E., and Goscinsky, 0. 1970. Phys. Rev. A 1, 552-60. Chamberlain, G. E. and Zorn, J. C. 1963. Phys. Rev. 129,677-80. Chen, Z. and Spruch, L. 1987. Phys. Rev. A 35,4035-43. Cohen, H. D., 1965, J . Chern. Phys. 43, 3558-62. Cohen, M. 1963. Proc. Phys. SOC. 82, 778-84. Cohen, M. 1966. Proc. Roy. SOC. A 293,365-77. Cohen, M. 1984. J . Phys. A 17, 1639-48. Cohen, M. and Dalgarno, A. 1961a. Proc. Phys. SOC. 77, 748-50. Cohen, M. and Dalgdrno, A. 1961b. Proc. Roy. SOC. A 261, 565-76. Cohen, M. and Dalgarno, A. 1963. Proc. Roy. Soc. A 275,492-503. Cohen, M. and Dalgarno, A. 1964. Proc. Roy. SOC. A 280, 258-70. Cohen, M. and Dalgarno, A. 1966. Proc. Roy. SOC. A 293, 359-64. Cohen, M. and Drake, G. W. F. 1967. Proc. Phys. SOC. 92.23-36. Cohen, M. and Feldmann, T. 1981. J . Phys. B 14, 2535-43. Cohen. M. and McEachran, R. P. 1978. Int. J . Quant. Chern. Symp. 12, 59-66. Cohen, M. and McEachran, R. P. 1980. Adu. A t . Mol. Phys. 16. (David Bates and Benjamin

Bederson, eds.). Academic Press, New York, New York, 1-54. Cohen, M. and McEachran, R. P. 1981. Chern. Phys. Lett . 84,622-6. Cohen, M. and Parkinson, E. M. 1966. Proc. Roy . SOC. A 292, 193-202. Coulson, C. A. and Hibbert, A. 1967. Proc. Phys. SOC. 91, 33-43. Crossley. R. J. S. 1969. Adv. At . Mol. Phys. 5. (David Bates and Benjamin Bederson, eds.).

Crossley, R. J. S. and Dalgarno, A. 1965. Proc. Roy . SOC. A 286, 510-8. Dalgarno, A. 1959. Proc. Roy . SOC. A 251, 282-90.

Boyd, R. J. 1978. J. Phys. E 11, L655-8.

Academic Press, New York, New York, 237-96.

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Dalgarno, A. 1962. Adv. Phys. 11, 281-315. Dalgarno, A. and Drake, G. W. F. 1969. Chem. Phys. Lett . 3, 349-50. Dalgarno, A. and Stewart, A. L. 1956. Proc. Roy. SOC. A 238, 269-75. Dalgarno, A. and Stewart, A. L. 1960a. Proc. Phys. SOC. 75,441-4. Dalgarno, A. and Stewart, A. L. 1960b. Proc. Roy. SOC. A 257, 534-40. Dmitirieva, I. K. and Plindov, G. I. 1981. J . Phys. B 14, 1095-102. Doyle, H. T. 1969. Adu. A t . Mol. Phys. 5, (David Bates and Benjamin Bederson, eds.). Academic

Drake. G. W. F. and Cohen, M. 1968. J . Chem. Phys. 48, 1168-77. EdlBn, B. 1983. Phys. Scr. 28, 483-95. Flannery, M. R. and Stewart, A. L. 1963. Proc. Phys. Soc. 82, 188-91. Froese, C. 1964. Astrophys. J. 140, 361-5. Froese, C. 1965. Astrophys. J. 141, 1206-21. Goodson, D. Z. and Herschbach, D. R. 1987. J . Chem. Phys. 86,4997-5008. Hartree, D. R. 1957. The Calculation of Atomic Structures. Wiley, New York, New York. Herrick, D. R. and Stillinger, F. H. 1975. Phys. Rev. A 11, 42-53. Hibbert, A. 1974. J . Phys. B 7, 1417-34. Hirschfelder, J. 0. Byers-Brown, W. B., and Epstein, S. 1964. Adu. Quantum Chem. 1, 255-374. Hylleraas, E. A. 1929. Z. Physik. 54, 347-66. Hylleraas, E. A. 1930. 2. Physik. 65, 209-25. Jucys, A. P. 1967. ln t . J . Quantum Chem. 1, 311-9. Killingbeck, J. 1977. Repts. Prog. Phys. 40, 963-1031. Layzer, D. 1959. Ann. Phys. 8,271-96. Layzer, D. and Bachcall, J. 1962. Ann. Phys. 17. 177-204. Lowdin, P.-0. 1959. J . Mol. Spectrosc. 3, 46-66. Massaro, P. A. 1977. J . Phys. B 10, 391-8. Midtdal, J., Aashamar, K., and Lyslo, G. 1969. Physica Noruegica 3, 163-78. Mohr, P. J. 1985. Phys. Rev. A 32, 1949-57. Pekeris, C. L. 1959. Phys. Rev. 115, 1216-21. Prager, S. and Hirschfelder, J. 0. 1963. J . Chem. Phys. 39, 3289-94. Salop, A., Pollack, E., and Bederson, B. 1961. Phys. Rev. 124, 1431-8. Scherr, C. W. and Knight, R. E. 1964. J . Chem. Phys. 40, 3034-9. Sternheimer. R. M. 1954. Phys. Rev. 96, 951-68. Stewart, A. L. 1964. Proc. Phys. Soc. 83, 1033-7. Stillinger, F. H. 1966. J . Chern. Phys. 45, 3623-31. Watson, D. K. and ONeiI, S. V. 1975. Phys. Rev. A 12,729-35. Wiese, W. L., Smith, M. W., and Glennon, B. M. 1966. Atomic Transition Probabikties Vol. I (US

Wiese, W. L., Smith, M. W., and Miles, B. M. 1969. Atomic Transition Probabilities, Vol. I1 (US

Young, W. H. and March, N. H. 1958. Phys. Reu. 109, 1854-5.

Press, New York, New York, 337-413.

Govt Printing Office, Washington, DC).

Govt Printing Office, Washington, DC).


1. 11.

111. IV.

V.

SCH WINGER VA RIA TI0 NA L METHODS DEBORAH k3 Y WATSON Deportment of Physics and Astronomy University of Oklahoma Norman, Oklohomo

Introduction . . . . . . . . . . . . . . . . . . . . . . . 221 Early Development. . . . . . . . . . . . . . . . . . . . . 222 Studies by Nuclear Physicists . . . . . . . . . . . . . . . . . 228

230 A. Application to Scattering and Photoionization . . . . . . . . . . 230

1. Electron-Molecule Scattering Studies . . . . . . . . . . . . 230 2. Photoionization Studies . . . . . . . . . . . . . . . . . 233

B. Applications to Bound and Resonance States . . . . . . . . . . 239 1. Bound State Studies . . . . . . . . . . . . . . . . . . 239 2. Resonance Studies. . . . . . . . . . . . . . . . . . . 242

Summary . . . . . . . . . . . . . . . . . . . . . . . . 247 References . . . . . . . . . . . . . . . . . . . . . . . . 247

The Schwinger Variational Method in Atomic and Molecular Physics . . .

3. Higher-order Expressions and Multichannel Extensions. . . . . . 236

I. Introduction

The importance of variational theory to the development of both theory and practical applications in physics is difficult to overestimate. Originally introduced by Fermat to formulate his principle of least action in the field of optics, variational theory was instrumental in laying the foundations of classical mechanics, resulting in the elegant Hamiltonian-Jacobi theory of mechanics. The Rayleigh-Ritz variational principle, which was originally developed to evaluate discrete eigenvalues in classical dynamics, has provided quantum theorists since the 1920s with a very powerful technique to study bound states of quantum systems. In contrast to the unqualified success of the Rayleigh-Ritz principle, variational principles in scattering theory have been plagued by problems. Introduced in the 194Os, they can be divided into two groups based on the equation of motion, or Euler equation: the Hamiltonian- based variational methods of Hulth2n (1944), Kohn (1948), and Rubinow ( 1 954) (H-KV methods) and the integral-equation-based Schwinger variational methods (SV methods) introduced by Schwinger in his lectures at Harvard University and subsequently published in 1947. These variational methods do not immediately provide bounds for scattering parameters in

22 1


ISBN 0-12-003825-0

222 Deborah Kay Watson

contrast to the energy bounds provided by the Rayleigh-Ritz method. The choice of appropriate trial functions and the systematic improvement of the results remain ambiguous in many practical applications. In addition, many of the methods are associated with the appearance of spurious singularities which can seriously hinder the accuracy of practical calculations. As a result, many authors have proposed new variations of scattering variational methods, investing considerable effort in the search for variational bounds, for systematic techniques of improving results, and for variational formulations that are absolutely free of spurious singularities. At the present time, it is still not clear that any single variational method is consistently the best choice.

The purpose of this article is to review the development of the Schwinger variational method from its introduction in 1947. Specifically, we will review the early development of the method with an attempt to understand its slow progress relative to the H-KV methods and the reasons for its re-emergence in the 1970s in both atomic and nuclear physics as a powerful and practical variational formalism. The accomplishments of the last decade in electron- atom/molecule scattering, photoionization, bound state and resonance state studies will also be reviewed. Many areas of interest will not be covered in this article, such as the relationships of the SV method to the method of Pad6 approximants, applications of the SV method to such fields as heavy-particle dynamics, multiphoton ionization and surface studies, the formal relationship between the Schwinger method and Hulthh-Kohn methods, variational bounds using the SV principle, and specific details of the numerical procedures developed for the applications. Several good review articles discuss some of these areas in more detail (Abdel-Raouf, 1984; McKoy et al., 1984; Lucchese et al., 1986).

11. Early Development

Despite their similar introduction dates, the development of the SV method has lagged considerably behind that of the H-KV methods. As demonstrated below, this discrepancy was not due to discouraging results from the early applications of the SV method, but was due primarily to the “difficult” integrals required by the SV method compared to the H-KV methods. Specifically, the H-KV methods require only Hamiltonian matrix elements which were being independently and rapidly developed for bound- state problems during the 1950s and 1960s. The SV expression for the T matrix,

SCHWINGER VARIATIONAL METHODS 223

requires the evaluation of matrix elements of the operators V G , where I/ is the potential of interaction, Go is the Green’s function operator, 1 cp,(k)) is a free wave, and Iq,) is the trial function. The determination of these matrix elements requires a double integration and cannot be done analytically for realistic atomic and molecular potentials. This fact significantly slowed the application of the SV principle, particularly in atomic physics, despite the early promising results of Blatt and Jackson (1949), Kato (1950, 1951), Altschuler (1953), Gerjuoy and Saxon (1954), and Bransden and McKee (1957).

The first application of the SV principle was published in 1949 by Blatt and Jackson. They successfully studied neutron-proton scattering below 10 MeV by using the SV principle to obtain an expansion for the phase shift in powers of the energy. They applied an iteration procedure suggested by Schwinger to improve the wavefunction and extrapolated their phase shift results to negative energy to obtain an estimate for the bound-state energy of the deutron. In 1950 and 1951, Kato proved that the SV principle gives an upper (lower) bound for the phase shift if the potential is purely positive (negative) and determined estimates of the error. He also obtained both upper and lower bounds for elastic electron-hydrogen scattering using a static potential (see Table I). The advantages of the SV principle were summarized by these authors (Blatt and Jackson, 1949; Kato, 1950; 1951) as follows :

(1) The Schwinger expression is homogeneous, i.e., multiplication of the trial function does not change the variational expression. Thus, the trial functions can have any normalization.

TABLE I UPPER AND LOWER BOUNDS FOR THE

PHASE SHIM FOR e-H SCATTERING USING A

STATIC POTENTIAL’

Lower Upper k Bound Bound Numerical

0 0 0 0.068 0.55565 0.55706 0.557 0.136 0.84749 0.84859 0.847 0.272 1.03383 1.03400 1.034 0.384 1.05776 1.05781 1.060 0.608 1.01851 1.01916 1.007 1.OOO 0.90438 0.90665 0.905

a Kato, 1951.


(2) The wavefunction appears only in conjunction with the potential, i.e. V ( r ) l q r ) . Thus, an error in the trial function is important only if it occurs within the range of the potential. This means that trial functions with the incorrect asymptotic form may be used, in contrast to the H-KV methods that are constrained to use functions with the correct asymptotic form.

(3) One can systematically improve the trial wavefunction by iterating with the integral equation to generate an improved wavefunction.

Additional advantages were to emerge as the method developed. In 1953 Altschuler reformulated the SV method to incorporate the use of

trial functions composed of linear combinations of basis functions. He used polynomial basis functions to study four model problems using the following potentials: square well, Yukawa, and the static potential for both one- and three-dimensional e-H scattering. He concluded from these studies that the Schwinger formalism furnishes phase shifts superior to the H-KV method for a given trial function. He also noted that the three-dimensional e-H results were better than the second-Born results which require the same amount of work.

A paper by Gerjuoy and Saxon in 1954 studied neutron-proton scattering at intermediate energies using a Yukawa potential both with and without exchange. They concluded that the Schwinger results were better than the Born approximation, particularly when exchange effects were included.

Multichannel Schwinger calculations were published in 1957 in a study of 1s-2s e-H excitation using a static potential. These results by Bransden and McKee showed smooth convergence toward the exact results as the number of polynomials included in the trial function was increased. They are compared in Table I1 to the available HulthCn results (Massey and Moiseiw- itsch, 1953), the distorted wave results (Erskine and Massey, 1952), and the exact numerical results (Bransden and McKee, 1956). Although certainly not a definitive test of the two methods, since both methods used inadequate basis sets, it is interesting that these early Schwinger results were so promising.

Other studies in the 1950s included the calculation of phase shifts for e+-H elastic scattering (Moiseiwitsch, 1958), 1s-2p e-H excitation (Kingston et al., 1959), and two studies that derived higher order SV principles including “hybrid” variational expressions which combined features of both the H-KV and SV methods. In 1958 Kolsrud demonstrated the derivation of an infinite number of higher Schwinger-like variational principles through successive iterations of the Lippmann-Schwinger equations, including a hybrid expression of the form:


TABLE I1 COMPARISON OF CROSS SECTIONS FOR THE

1s-2s EXCITATION OF HYDROGEN WITHOUT EXCHANGE

E(eV) DW” H-KVb SV‘ Exactd

10.2 0 0 0 0 13.5 0.239 0.076 0.189 0.204 19.4 0.118 0.041 0.094 0.102 30.4 0.045 0.020 0.040 0.0450 54.0 0.014 0.008 0.014 0.0155

‘ Distorted wave results (Erskine and Massey,

Hulthen-Kohn method (Massey and Moisei-

Schwinger variational method (Bransden and

Exact numerical solution (Bransden and

1952).

witsch, 1953).

McKee, 1957).

McKee, 1956).

where K, , is the Born term and the conventional denominator containing the Green’s function has been replaced by a denominator based on the Hamil- tonian. This form, which retains the ratio form of the Schwinger expressions while requiring only Hamiltonian matrix elements like the H-KV expressions, was later investigated more thoroughly by Takatsuka and McKoy (1981a). Kolsrud (1958) compared this expression and several of the higher- order Schwinger expressisons to the conventional Schwinger form and the Born approximation using an exponential potenial. His results demonstrated a general improvement in the results for higher-order expressions.

Moe and Saxon (1958) also derived the above hybrid expression which they called an “amplitude-independent form of Kohn’s principle.” Their work was motivated by the desire to avoid the Green’s function integral in the Schwinger formalism while retaining a ratio or amplitude-independent form that allows more flexibility in the trial functions. They concluded that such hybrid forms lead in general to complex phase shifts, i.e., that probability is not conserved. They also concluded that there was no criteria on which to choose the “best” variational expression based on decreasing higher variations, but suggested choosing forms such as the conventional SV expression which preserve probability, unlike some of the higher-order SV expressions and the hybrid expressions.

Given the success of the above studies, the scarcity of SV applications during the 1960s is surprising. This is particularly true since the year 1961 was a turning point in the development of scattering variational methods with the


discovery by Schwartz of anomalies or spurious singularities in the H-KV methods. The fact that the Schwinger variational expression was anomaly- free was not recognized at this time. In fact, Schwartz stated in his study of H- KV anomalies that the SV method would also be subject to spurious singularities (Schwartz, 1961a). Thus, the attempts to find anomaly-free methods were based on Hamiltonian formulations such as the least-squares method and the quadratic variational methods, which are anomaly-free, and other attempts including the Malik-Rudge variational method, the Harris-Michel variational method, the minimum-norm method, Nesbet’s anomaly-free method, an optimized minimum-norm method, the optimized anomaly-free method, and the optimized Kohn method, none of which are completely free of anomalies. (For a detailed analysis and comparison of these methods see Abdel-Raouf, 1984.)

During the next decade, the SV principle continued to be the subject of several very interesting studies. In 1961 Saraph applied the SV principle to zero-energy e-H scattering formulating the problem in terms of the radial- exchange equations. Her results using three-term polynomials compared very well to exact numerical integrations. A follow-up study by Saraph and Seaton (1962) used unconverged numerical solutions of the integro-differential equations as trial functions in both Kohn and Schwinger variational expressions. Their goal was to speed up the convergence of straightforward numerical iteration of the integro-differential equations. They applied the method to e-H scattering in the static-exchange approximation and concluded that the Kohn and Schwinger methods were comparable in accuracy; if wavefunctions were required, however, they preferred the Kohn method, since they did not obtain wavefunctions using their Schwinger approach. They also noted that the numerical work to calculate the Schwinger phase shift using n trial functions was less than that required for the Kohn phase shift with n trial functions.

In 1964 Sugar and Blankenbecler published an important paper that studied variational upper and lower bounds for multichannel scattering using the SV principle. They discussed two-particle scattering, two-particle coupled channels, and three-particle scattering, giving a numerical example of phase shifts for s-wave e-H scattering, neglecting exchange. They demonstrated for the first time the equivalence of the SV principle with the use of a separable expansion for the potential in the Lippmann-Schwinger equation. This relationship, which will be discussed below, was to be of central importance to the re-introduction of the SV method into both atomic and nuclear physics.

In 1966, Schwartz published a comparison study of the SV principle and the H-KV method for s-wave scattering from a Yukawa potential. In this study he retracted his earlier assertion that the Schwinger variational


TABLE 111

CONVERGENCE FOR SCHWINGER COMPARISON OF THE BASIS SET

AND KOHN SCATTERING LENGTHS USING A YUKAWA POTENTIALa

N b Kohn Schwinger

1 2 3 4 5 6 7 8 9

10

7.92 155 8.09387 7.91 142 8.00863 7.91139 7.91228 7.91138 7.91 178 7.91 138 7.91178

7.91 147 7.91 142 7.91139 7.91 138 7.91 138

Schwartz, 1966. * N is the number of basis functions.

Slater functions were used in the Schw- inger calculation and Slater functions plus long-range functions of the correct asymptotic form were used in the Kohn calculations.

principle suffered from spurious singularities. He also concluded, however, that the Kohn variational method converged faster than Schwinger (see Table 111). This conclusion was later criticized as misleading since different basis sets were used for the two methods (Lucchese et al., 1986).

Although the Schwinger variational method was largely ignored in atomic physics except for these isolated studies, developments in nuclear physics were laying a foundation for the re-emergence of the Schwinger method in conjunction with finite-rank approximations to the Lippmann-Schwinger equation. During the 1960s nuclear physicists were proposing theoretical formulations for nonrelativistic three-particle systems. A common component of many of these theories was the use of separable approximations for the two-body interactions. This reduces the three-body equations, which involve a set of coupled two-dimensional integral equations, to a coupled set of one-dimensional integral equations. The justification for the separable approximation is the knowledge that the T matrix in the neighborhood of a pole is known to be separable and that the T matrix is often dominated in the energy regions of interest by a finite number of bound and resonance states. Various separable expansions were proposed for the interaction potential as well as for the T matrix itself, the Green’s function, and the kernel, Go K of the


Lippmann-Schwinger equation. They included the Weinberg series (Wein- berg, 1963), which expands in terms of energy-dependent solutions, called Gamow states, of the homogeneous Lippmann-Schwinger equation (see also Fuller, 1969); the unitary pole approximation (Lovelace, 1964; Fuda, 1968); and the unitary pole expansion (Harms, 1970), which expand in terms of eigenfunctions of the homogeneous Lippmann-Schwinger equation. They were applied to nuclear reaction systems using simple potentials. It was eventually recognized that some of these separable expansions were equivalent to using a variational principle of the Schwinger type.

A study by Sloan and Brady (1972) was particularly instrumental in providing an increased understanding of the SV principle, especially its connection to the use of separable potentials in the Lippmann-Schwinger equations, and its freedom from spurious singularities. They calculated a variationally stable T matrix by employing a finite-rank approximation to the full Green's function,

T = V + VG'V N N

G'= 1 1 Goln)(nlGo - G o ~ G o I ~ ) - ' ( ~ l G o , n = l m = l

and proved that this T matrix is exact for potentials of finite rank. They also discussed the connection between this variational expression and the conventional SV expression and proved that the SV expressions are not plagued by the spurious singularities of the H-KV methods. Their argument can be stated briefly: since the Schwinger denominators have zeros only at the roots of the equation, 1 - VGo = 0, which is the condition for a true bound state or resonance pole of the T matrix, there is no reason to expect spurious singularities. Actually, the fact that the Schwinger T matrix is the exact T matrix for a separable potential is itself proof that no spurious singularities will appear.

The recognition of the anomaly-free nature of the Schwinger method and its equivalence to the use of a separable approximation in the Lipp- mann-Schwinger equation played a central role in the increased use of this method. In the next two sections we review the applications of the SV principle in nuclear physics and in atomic and molecular physics beginning in the early 1970s.

111. Studies by Nuclear Physicists

In this section, we have chosen to review the work of three research groups in nuclear physics who began using the SV principle in the 1970s. We will concentrate on their early work, which focused on separable potentials, and


their relationships to the SV principle; we will not attempt to cover their work through the present.

Ernst, Shakin, and Thaler, in a series of articles beginning in 1973, investigated the utility of different separable expansions in the Lipp- mann-Schwinger equation. They pointed out that it was possible to construct a rank-N separable potential that yields an exact T matrix on the energy shell and half-off the energy shell at N selected bound or continuum energies by using eigenfunctions of the potential (Ernst et a!., 1973a). They also demonstrated that the separable potential that leads to the SV principle,

V s e p = 1 Vln)(nl ~ l m ) - ’ ( r n l K

gives matrix elements identical to those of the true potential Vin p space, the subspace spanned by the basis set In):

n, m

P = c InXnl n

p v s e p p = p vp.

This same separable potential gives matrix elements identical to those of the true potential between p-space and q-space:

q = l - p

p V S e P q = p vq.

Ernst and his collaborators studied a separable approximation to a square- well potential (Ernst et d., 1973b) and later extended this work to off-shell T matrices (Ernst et aE., 1974). McLeod and Ernst (1978) tested the effectiveness of energy-dependent basis functions in the separable expansion.

Adhikari and Sloan also studied different separable expansions in the Lippmann-Schwinger equation using simple polynomials rather than energy- dependent functions or solutions of the kernel of the Lippmann-Schwinger equation (Adhikari, 1974). Their work re-emphasized the flexibility of the basis functions that can be used by Schwinger expressions. They also derived higher-order separable approximations of the potential, including the form used by Sloan and Brady and showed the connection of their separable expansions to the Schwinger variational principle (Sloan and Adhikari, 1974; Adhikari and Sloan, 1975a; 1975b). They performed calculations for the Yukawa potential, the Reid soft core ‘ S o potential, and the Malfliet-Tjon potential for s-wave nucleon-nucleon scattering. Their work was later generalized to fully off-shell T matrix elements (Adhikari, 1979; Tomio and Adhikari, 1980a) and to iterative solutions of multichannel three-body equations (Tomio and Adhikari, 1980b).

In the mid 1970s, Zubarev began to use the SV principle to study a number of problems using simple potentials, including low-energy nucleon-nucleon


scattering using a square-well potential (Zubarev and Podkopaev, 1976), charged-particle scattering on atomic systems such as e+-H using a static potential (Belyaev et al., 1979), high-energy scattering using an exponential potential (Zubarev, 1976a), nonstationary problems such as resonance scattering of neutrons (Zubarev, 1977), and binding energies of nuclei using a parameterized Gaussian potential (Zubarev, 1976b, 1978). He investigated the Schwinger formalism thoroughly, studying its equivalence to the separable expansion of the potential in the Lippmann-Schwinger equation, investigating the choice of optimal trial functions, and studying the convergence with respect to the number of trial functions by characterizing the error with the expression, I V - V ( N ) ( , where V(N) is the rank-N theory approximation to the potential (Zubarev, 1976~). Zubarev reviewed and expanded on the theory for successive iterations (Zubarev, 1976a, 1978), the theory for bound states and multichannel theory, the theory for the nuclear three-body problem and the nonstationary theory (Zubarev, 1978). He also investigated the problem of obtaining bounds on scattering parameters using the SV principle (Zubarev, 1976a).

Zubarev concluded from his work that, despite the need to calculate the VG, Vterm, the SV principle is a useful theory due to the flexibility allowed in the basis set compared to the H-KV methods. He states: “of all variational principles the best is the one that depends least on the choice of the trial function. Compared with the variational principles of Kohn, Hulthbn and Ritz, the Schwinger variational principle has this property.” He also noted that Schwinger is better for bound states than Ritz because it doesn’t approximate the kinetic energy, i.e., the asymptotic behavior is correct for all trial functions. This is certainly true for the determination of highly excited bound states. (See Section IV.B.l.)

IV. The Schwinger Variational Method in Atomic and Molecular Physics

A. APPLICATION TO SCATTERING AND PHOTOIONIZATION

1. Electron-Molecule Scattering Studies

The re-introduction of the SV method into atomic physics occurred slightly later and for different reasons than the nuclear physics studies. The major impetus did not come from nuclear physics or from efforts to find an anomaly-free variational method, but rather from attempts to use discrete- basis-set methods to describe scattering phenomena. The sophistication

SCHWINGER VARIATIONAL METHODS 23 1

obtained by this time for computing bound-state integrals, which had earlier contributed to the faster development of the H-KV methods, had also prompted the development of scattering methods that took advantage of this sophistication by using discrete basis functions to describe the continuum. This was a particularly attractive approach for the study of electron-molecule scattering since the noncentral potential of a molecule can be well described near the origin by discrete basis sets. The stabilization technique (Hazi and Taylor, 1970; Fels and Hazi, 1972), the Fredholm method (Reinhardt, 1970; Heller et al., 1973), the Stieltjes imaging method (Langhoff, 1973; Langhoff et al., 1976), the complex-coordinate rotation methods (Nuttall and Cohen, 1969; Bain et al., 1974; Moiseyev et al., 1978; McCurdy et al., 1980), and T matrix method (Rescigno et al., 1974) all used discrete-basis-function or L2 techniques to study continuum phenomena. In particular, the T matrix method of McKoy and coworkers used a separable approximation for the potential in order to solve the Lippmann-Schwinger equation for the T matrix in the Hilbert space defined by the basis. This T matrix was not variationally stable, but the results could be corrected for first-order errors by calculating a Kohn correction. In 1979 Watson and McKoy tested two discrete-basis-function techniques that were free of first-order errors for e-He scattering in the static-exchange approximation. The first method applied the Schwinger variational principle to the T matrix, evaluating the VGoV integrals by inserting a set of Gaussian functions around Go and performing the integral analytically. The second method made a finite-rank approximation to the full Green's function

G' = Go + G,VG'

or

G' = (1 - GoV)-'Go

and calculated

T = V + VG'V

similarly to Sloan and Brady's calculation. These two methods were compared to the original T matrix method with and without the Kohn correction. The Schwinger results and the results using the full Green's function were comparable in accuracy to each other and clearly superior to the uncorrected T matrix results. Furthermore, the results were more accurate at very low energy than the corrected T matrix results. This was attributed to the analytical evaluation of the bound-free integrals in the numerator (see Table IV).

This comparative study was followed by numerous applications of the SV principle by McKoy and coworkers to electron-molecule/ion scattering.


TABLE IV MATRIX ELEMENTS FOR S-WAVE SCATTERING OF ELECTRONS BY

HELIUM IN THE STATIC-EXCHANGE APPROXIMATION'

Uncorrected Corrected k(a.u.) T matrix T matrix Schwinger V + VG'V

0.0 1 f0.0428 -0.0353 -0.0146 -0.0131 0. I -0.00732 -0.144 -0.148 -0.134 0.5 - 0.743 -0.805 -0.820 - 0.804

a Watson and McKoy, 1979.

Numerical techniques to evaluate the VG, V integrals and the bound-free integrals in the numerator were developed and applications of electron- molecule scattering were made to e-H, (Watson et al., 1980; Lucchese et al., 1980), e-LiH (Watson et al., 1981) and e-CO, (Lucchese and McKoy, 1982a). These calculations used single-center expansion techniques and numerical integrations to evaluate the Schwinger expression in the static-exchange approximation. An iteration procedure was implemented which improves the initial basis set by using the Lippmann-Schwinger equation to generate a wavefunction at each stage. This wavefunction, which has long-range character, is used as an additional basis function in the subsequent run. If convergence is obtained, the solution is the exact solution to the Lipp- mann-Schwinger equation for the given potential. This iteration procedure worked very well on these single-channel applications, completely removing the ambiguity associated with the basis set choice that plagues most discrete- basis-set methods. Table V shows the convergence of e--H, K matrix elements starting from a single basis function (Lucchese et al., 1980).

TABLE V CONVERGENCE OF THE K-MATRIX USING THE

ONE DISCRETE BASIS FUNCTION FOR e--H, "g SCATTERING AT k = 0.5 a.u."

ITERATIVE SCHWINCER METHOD STARTING FROM

1 - 2.045 -0.0276 - 0.000372 2 - 1.552 0.0133 0.0163 3 - 1.548 0.0134 0.0163

a Lucchese et al., 1980. N is the iteration number.


2. Photoionization Studies

A very successful series of photoionizationJe--ion scattering studies were also carried out using the iterative Schwinger method for He (Lucchese and McKoy, 1979; Lucchese and McKoy, 1980), H, (Lucchese and McKoy, 1981a), N, (Lucchese and McKoy, 1981b; Lucchese et al., 1982), CO, (Lucchese and McKoy, 1982b, 1982c), and C,H, (Lynch et al., 1984). In these studies a two-potential formalism was employed to factor out the long-range Coulomb potential. The corresponding Lippmann-Schwinger equation is given by

*k = 4; + G C v , * k

where 4: is a Coulomb wave, G, is the Coulomb Green’s function, Vis the full potential, V , = - ZJr, Z, is the net charge and V , is the short-range potential:

V = v , + v , .

These calculations used the frozen-core approximation which assumes that the core orbitals do not relax after the ionized electron leaves. This approximation neglects electron correlation effects, but significantly simplifies the orthogonality considerations between the open-shell Hartree-Fock core and the continuum electron. The appropriate orthogonality constraints, which can become very complicated, can be incorporated using a generalized Phillips-Kleinman pseudo-potential or a Lagrange multiplier formalism. A detailed description is given by Lucchese et al. (1986).

Studies were made of the effects of the initial-state correlation by using both static-exchange and configuration interaction wavefunctions for the initial state and then comparing the length and velocity forms of the photoionization cross sections. Fig. 1 shows that in the region of a shape resonance for N , , the inclusion of initial-state correlation does bring the length and velocity forms of the frozen-core photoionization cross sections into closer agreement with each other and into good agreement with experiment. Careful studies were also made of the convergence of the single- center expansions since the resonant features of the cross section were found to be sensitive to the parameters of these expansions. For N , , good agreement with experiment was obtained for the photoionization cross sections in the channels leading to the X’C,‘, A 2 n U , and B2C: states of N:. Poor agreement with experiment was obtained, however, for the asymmetry


2-

0

121 I 1 I I I I

-

1 I I I I

Photon Energy ( e V ) FIG. 1. Comparison of dipole length and velocity photoionization cross sections of N ,

leading to the X'z: state of N i using different initial states: HFL, dipole length form using H F (Hartree-Fock) initial state and FCHF (frozen-core Hartree-Fock) final state; HFV, dipole velocity form using HF initial state and FCHF final state; CIL, dipole length form using CI initial state and FCHF final state; CIV, dipole velocity form using CI initial state and FCHF - final state; 0, experimental data (Plummer et a/., 1977); H, experimental data (Hamnett et a/., 1976).

parameter in the channel leading to the B2C: state of N:. Basden and Lucchese (1986) demonstrated that this disagreement was due to the neglect of coupling to other channels, particularly the (3aJ-l X 2 x l state of N:, which has a resonance. They performed a two-state coupled-channel calculation for the photoionization of N , leading to the (2a,)-' BzCJ and (30,)~' X2C8+ states of N : using the Pade approximant C-functional formalism (see below). The channel coupling strongly affects the asymmetry parameter in the (2aJ' channel, particularly in the energy region of the resonance in the (30,)- channel. Fig. 2 shows the quantitative agreement with experiment


I I I I 1

O-O' 20 2 i 30 3i 4il I

Photon Energy (eV) FIG. 2. Comparison of the exact Hartree-Fock results with the coupled-channel results for

the photoelectron asymmetry parameter for photoionization leading to the (2a,)-'B2C,,+ state of N i : ....., exact Hartree-Fock results of Lucchese et al., 1980; - , coupled-channel results using the dipole-length approximation (Basden and Lucchese, 1986); 0, experimental results of Southworth et a/., 1986; A, experimental results of Adam et al., 1983; 0, experimental results of Marr et al., 1979.

obtained with the coupled calculation compared to the single channel calculation.

An interesting feature in the CO, photoionization study was the existence of a narrow ko, shape resonance found by the iterative Schwinger method that is not seen experimentally. This feature is found by other theoretical methods including the continuum multiple scattering method (Swanson et al., 1980) and the linear algebraic method (Collins and Schneider, 1984). The Stieltjes-Tchebycheff moment theory approach (Padial et at., 198 1) also shows a resonance in the raw moment spectrum, although this structure is smoothed away by the moment inversion. This discrepancy between theory and experiment is currently not understood, although there is some experimental evidence for this resonance in the photoelectron angular distributions. At the time of this writing, Lucchese and coworkers are performing a coupled-state calculation for CO, photoionization to investigate whether the disagreement is due to coupling effects from other channels.

In addition to the iterative Schwinger method used for the above studies, two modified approaches were developed and applied to particular photoionization cases. A Pade approximant correction method was developed

236 Deborah K a y Watson

that uses a sequence of variationally stable Pad6 approximants to correct an original variational estimate. It was applied to the photoionization of CO (Lucchese and McKoy, 1983) and was found to converge more rapidly at certain energies than the iterative Schwinger method. A two-potential technique was applied to the photoionization of NO whose long-range dipole-moment potential is difficult to describe using discrete basis functions. This technique separates the potential into a longer-range local static term of the Hartree-Fock potential, V,, and the residual short-range exchange potential, V,

V = v , + v , .

The solution to the direct or static potential is then solved for numerically while the short-range exchange potential is described using a discrete basis set. This requires a “direct” Green’s function, G,, which is obtained using the integral equations approach of Sams and Kouri (1969a; 1969b). The use of this two-potential formalism was found to converge much faster than the regular iterative Schwinger method (Smith et al., 1983; 1984; 1985).

3. Higher-order Expressions and Multichannel Extensions

In addition to the above single-channel static-exchange studies, McKoy and coworkers have investigated higher-order SV expressions and extended their calculations beyond the static-exchange approximation. Takatsuka and McKoy (198 la; 1981b) investigated several higher-order SV expressions including the hybrid form originally introduced by Kolsrud and Moe and Saxon and a form which they called the functional also introduced by Kolsrud. Calculations using the C functional,

- fK$ = (Sil V ( S j ) + (SiI VG, VlS , ) + X i j

e = l+hi - si, where S i is the regular solution to the unperturbed Hamiltonian and l+hi is the full scattering wavefunction, demonstrated that this expression yields results comparable in accuracy to those obtained after one iteration of the Schwinger iterative method while requiring minimal additional work (see Table VI). It was also shown to be particularly effective for problems with long-range potentials such as the dipolar potentials of heterogeneous molecules, since the first and second Born terms are included explicitly (Lee et al., 198 1). This functional has also been used by Lucchese and coworkers in conjunction with the Pad6 approximant correction method (see below).


TABLE VI

ITERATIVE SCHWINGER METHOD: K-MATRIX ELEMENTS FOR e-LiH SCATTERING at k = 0.1"

CONVERGENCE OF THE €-FUNCTIONAL APPROACH COMPARED TO THE

Iterative Schwinger C-functional

Iteration 0 1 2 0 1 K l l 0.009 0.419 0.750 0.493 0.788

K,, -0.2 x -0.050 -0.053 -0.060 -0.051 K , , -0.1 x 10-4 0.012 0.071 0.017 0.090 K,, -0.2 x -0.361 -0.417 -0.364 -0.402

K , , - 0.00 1 -0.449 -0.544 -0.504 -0.500

K33 - 0.0 -0.023 -0.020 -0.022 -0.011

Lee rt al., 1981.

In 1981 Takatsuka and McKoy proposed a new variational principle for scattering theory that extends the SV principle beyond the static-exchange approximation to inelastic scattering. Their approach which is based on the Lippmann-Schwinger equation for the total wavefunction rather than on coupled equations allows the configurations for the closed-channel components to be selected easily without the use of an optical potential. Configura- tions for both open and closed channels are treated equivalently and the method is free from the divergences that arise in optical-potential methods. Specifically, they use an operator P to project the Lippmann-Schwinger equation onto the open-channel space. This removes the closed channel and continuum components from the Green's function which are then recovered using a projected Schrodinger equation. The resulting equation contains information about the closed channels without explicitly using the closed- channel Green's function. The corresponding variational functional,

where Sn is the solution of the unperturbed Hamiltonian, GP, is the projected outgoing-wave Green's function, and A = E - HNfl, can be evaluated completely analytically if the trial function is expanded in a Gaussian or plane wave basis set and a set of Gaussians is inserted around the Green's function. Thus, the method can be extended easily to target molecules of arbitrary geometry.


The first application studied elastic e-H scattering (Takatsuka and McKoy, 1981~). The results demonstrated the rapid convergence of the method with respect to the basis set in the closed channel. It yielded phase shifts that could be more accurate than those of the close-coupling approximation even with the same state expansion, especially if the basis set was roughly optimized. (This method does not use an iteration procedure to improve the initial basis (see Table VII).) Takatsuka and McKoy also argued that their variational principle would give accurate phase shifts with fewer basis functions than the Kohn principle and would show faster convergence to the exact close-coupling values with respect to the number of states than Kohn-type principles when both are applied directly to the close-coupling equations.

This method has now been applied to elastic scattering for a number of diatomic and polyatomic molecules including H, (Gibson et al., 1984), N, (Huo et al., 1987), CH, (Lima et al., 1985b), CO (Lima et al., 1985c), and H,O (Brescansin et al., 1986). An interesting result of the e-CH, calculation was its failure to produce a Ramsauer minimum in the differential cross section at the static-exchange level. This feature has been found experimentally and by static-model-exchange-polarization studies (Abusalbi et al., 1983). A later study, which extended this calculation beyond the static-exchange, verified that this minimum is a polarization effect and obtained the first ab initio determination of the Ramsauer minimum (Lima et al., 1987). For N,, careful studies of the effects of radial and angular correlation have been made as well as the effects of basis set, orbital representation, and the closed channel configurations. A number of inelastic calculations have also been completed for transitions in H, (Lima et at., 1984; Lima et al., 1985a) and N, (Huo et al., 1987a; Huo et al., 1987b). Two open channels were found to give good agreement with experiment for differential cross sections for H,, while for N,

TABLE VII SINGLET PHASE SHIFTS FOR ELASTIC e-H COLLISIONS

Methods k 2 = 0.01 0.09 0.25 0.49 0.64

Static-exchange (CCa*b) 2.396 1.508 1.031 0.744 0.651 1s-2s-5 (CC) 2.529 1.657 1.155 0.875 0.823 1s-2s-2p-5 2.532 1.663 1.162 0.881 0.832 1s-2s-q (Schwingerr 2.550 1.684 1.179 0.895 0.818d “exact”e 2.556 1.696 1.201 0.930 0.887

CC denotes the close-coupling calculation.

Takatsuka and McKoy, 1981c. 6, = 0.821 if another p-type basis function is added. Shimamura, 1971; Schwartz, 1961b; Abdel-Raouf and Belschner, 1978.

’ Burke et nl., 1969.


as many as five open channels have been required. The important channels are dependent on both the incident energy and partial wave, suggesting that successful inelastic calculations for molecules other than H, will be very challenging. Studies in progress at the time of this writing include two-state calculations for excitations in both CO and H,O (Huo et al., 1987~).

B. APPLICATIONS TO BOUND AND RESONANCE STATES

As a final application of the SV principle, let us discuss the determination of bound and resonance states from the poles of the Schwinger T matrix. This subject has its roots in the elegant S matrix theory of Newton (1961) and Humblet and Rosenfeld (1961) which provides a formal definition of bound and resonance states as poles of the S matrix. This formal definition has served as the basis for practical applications in conjunction with the SV expression whose ratio form allows one to determine bound-state poles along the negative real-energy axis or resonance poles in the fourth quadrant of the complex momentum plane by searching for the zeros of the Schwinger denominator. A number of studies that have been completed recently are described in the following two sections.

1. Bound State Studies

The study of bound states using the SV principle, which is normally considered a scattering variational principle, dates back to the first application of the SV principle by Blatt and Jackson (1949) in which they determined a bound state of the deuteron by extrapolating their phase shift results to negative energy. The more general variational treatment of eigenvalue problems by converting them into integral equations has been known in the mathematical literature since the works of Bubnov and Galerkin (see Michlin, 1962). The procedure is as follows:

The Schrodinger equation,

W O + Ul+) = El$),

can be rearranged to yield

( E - Ho) I+) = Vl+>

or

I + ) = ( E - H o ) - ' W f r >

I$> = -GoVI+)

which can be written


where Go is the Green’s function operator ( H , - E ) - ’ . This equation is the homogeneous Lippmann-Schwinger equation. Multiplying by V gives the Bubnov-Galerkin eigenvalue equation:

Vl$> = - VG,VI$).

( $ l V + VG,VI$)=O

Projecting with ( $ 1 , one obtains

which is the condition for a bound or resonance state and is also the condition for the poles of the T matrix.

Sugar and Blankenbecler (1964) explored this aspect of the SV method and Singh and Stauffer (1975) and Zubarev (1978) presented explicit formulations of the SV principle for bound-state problems. Zubarev applied his formalism to the binding energies of light nuclei using simple Gaussian potentials (Zubarev, 1976b, 1978). Because of the difficulties associated with the Green’s function and the unqualified success of the Rayleigh-Ritz method, however, the SV method was seldom applied to nuclear or atomic and molecular bound-state problems. The numerous resonance methods that have been proposed have also been based almost exlusively on Hamiltonian methods.

In 1980 Maleki and Macek demonstrated that the SV method was preferable to the Rayleigh-Ritz method in the determination of quantum defects for Rydberg states. These highly excited, diffuse states are not described well by the Rayleigh-Ritz variational method, which uses discrete basis functions to describe their highly oscillatory character. As pointed out by Zubarev, the SV method treats the kinetic energy numerically through the use of a Green’s function rather than putting it on a basis. While this is not so important for the lowest bound states, it becomes critical for an accurate description of the long-range character of highly excited states. Maleki and Macek (1979) applied the SV principle to a Yukawa potential and to the Rydberg states of helium using completely analytical techniques (Maleki, 1981).

In 1983 Watson extended their work into a general numerical method applicable to any atom or molecule. Applications were made to the Rydberg states of lithium in the static-exchange appoximation (Watson, 1983); to the s, p , and d Rydberg states of lithium, sodium, and potassium using a model potential to include polarization (Snitchler and Watson, 1986); and to the Rydberg states of beryllium using a multichannel formalism (Snitchler and Watson, 1987). These studies demonstrated the numerical stability of the method up to extremely high principal quantum number (la - 80) compared to alternative methods. Very accurate quantum defects were obtained for an entire Rydberg series with a single small basis set and grid. Table VIII

SCHWlNGER VARIATIONAL METHODS 24 1

TABLE VIII QUANTUM DEFECTS FOR POTASSIUM

/ = 0 I = 1 1 = 2 n Calc Expt" db Calc Expt' A Calc Exptd A

4 2.212 2.230 0.018 1.753 1.766 0.013 0.203 0.203 0.000 5 2.180 2.199 0.019 1.730 1.735 0.005 0.230 0.231 0.001 6 2.179 2.190 0.011 1.722 1.725 0.003 0.247 0.245 0.002 7 2.179 2.186 0.007 1.718 1.720 0.002 0.252 0.254 0.002 8 2.179 2.184 0.005 1.716 1.718 0.002 0.261 0.260 0.001 9 2.179 2.183 0.004 1.714 1.716 0.002 0.262 0.263 0.001

10 2.179 2.182 0.003 1.714 1.715 0.001 0.266 0.266 0.000 15 2.179 2.181 0.002 1.712 1.713 0.001 0.270 0.272 0.002 20 2.179 2.181 0.002 1.711 1.712 0.001 0.274 0.274 0.000 25 2.179 2.180 0,001 0.274 0.275 0.001 30 2.179 2.181 0.002 0.276 0.276 0.000 35 2.178 2.180 0.002 0.276 0.276 0.000 40 2.180 2.180 0.000 0.277 0.277 O.OO0 46 2.179 2.181 0.002 0.276 0.276 0.000 50 2.179 2.180 0.001 0.276 0.276 0.000 55 2.173 2.178 0.005

"Thompson et a/., 1983 for n # 5; Risberg, 1959 for n = 5. Absolute difference between Schwinger and experiment. ' Risberg, 1959 for n = 4 - 8; Lorenzen and Niemax, 1983 for n > 9.

Thompson et al., 1983.

compares quantum defects for potassium with experiment. Two different methods were developed to search for the poles (Goforth et al., 1987). The initial search procedure used a Green's function that is constructed from the regular, f k , and irregular, f:, solutions to the Coulomb equation at negative energy :

ut(r) = 0 r

L J

where W, is the Wronskian of fd and fe+ and G , is bounded asymptotically since

lim f t ( k , r ) = 0.

This results in a bounded wavefunction when the bound-state energy is reached,

r - t m

I$> = - G , V $ h


which causes the denominator (I) I V + VG, V1$) to vanish. This procedure requires an individual search for each pole. An alternative procedure suggested by Greene et al. (1979) defines a negative-energy reaction matrix, K , using an unbounded smooth Green’s function constructed by continuing to negative energy the Coulomb functions that define the positive-energy K matrix. These Coulomb functions remain linearly independent even at hydrogenic energies and are unbounded asymptotically. Quantum defects, pLd, can be obtained from this reaction matrix, K, at any energy using

= tan-’(K,) = npLd

and fit to a quantum-defect curve. Bound states occur at the intersection of this curve with the equation:

n = n*(E) + p(E)

where n is the principal quantum number and n* = Z , / O is the effective quantum number. This method permits the determination of very accurate quantum defects for an entire Rydberg series from a small number of energy points by taking advantage of the continuity of properties at both positive and negative energies. The results can also be systematically improved by iteration using

I$> = 14c) + G?KII)> = 14,) + G?Kl4,)

where is a regular Coulomb wave. This unusual bound state equation has an inhomogeneous term I bC) and an unbounded Green’s function and is identical to the Lippmann-Schwinger scattering equation used above threshold.

2. Resonance Studies

The determination of resonance energies and widths using the SV principle is a fairly recent application of this variational principle and offers an intriguing alternative to the many Hamiltonian-based resonance methods. The diversity of resonance methods reflects the different manifestations of a resonance including the rapid variation of the phase shift used by Breit-Wigner methods, the purely outgoing asymptotic behavior of the resonance wavefunction employed by the complex rotation (Nuttall and Cohen, 1969; Bain et al., 1974; Moiseyev et al., 1978; McCurdy et al., 1980), Siegert (Siegert, 1939; Bardsley and Junker, 1972; Isaacson et al., 1978; McCurdy et al., 1981), and complex R matrix method (Schneider, 1981), as well as the vanishing of the T matrix denominator used by the Schwinger method. The first use of the SV principle to study resonance states was by


Domcke (1982). He investigated the analytic properties of the SV expression for the T matrix and proved that the Schwinger principle provides variational bound, virtual, and resonance energies. He applied the method to an attractive square-well problem obtaining accurate energies with a small basis set.

This work was extended by Watson (1984; 1986a; 1986b) to multichannel resonance problems. The multichannel extension of the SV expression used by Watson is straight-forward. The Schwinger denominator, D,,, becomes a matrix over channel indices LX, /?.

D a , = va/as + (vG,+ V a p

where I/., is the interaction potential between channels LX and p and the Green’s function matrix, Go, is diagonal in the channel indices. The diagonal elements are constructed from the regular and irregular bessel functions, F,(k,, r ) and H t ( k , , r ) which, for open-channel s-wave scattering, have the form :

Fo(k,, I ) = sin k,r

H,+(k,, r ) = eikar

where the open channel momentum is k, = k: + ik6 = ( E - E,)’” and E , is the threshold energy for channel LX. For closed channels, F&,,r) and H,+(k,, r ) have the form:

Fo(k,, r ) = sinh(lc,r)

H,+(k, , I ) = e-Kur

where K , = ( E , - E)’”.

each channel, I u , , ~ ) , LX = 1, N , ; i = 1, N , , The trial function, 14t), is expanded in terms of basis functions from

Nc Ne

I$,> = C C ca, i Iua, i ) , a = l i = l

resulting in the following elements of the multichannel denominator: r 1

The condition for a resonance state is obtained by setting up the eigenvalue equation,

~ ( u a , i l [ ~ ~ +F ~ y ~ : y y l / y p Iup, j )cg, j=o, 8. j 1


which has a nontrivial solution if and only if the determinant of coefficients is zero :

r 1

The complex energy, E, which causes the determinant to vanish, yields both the real energy, E, , and the width, r,

iT r 2

E = E --.

Watson applied this formalism to a multichannel problem consisting of three coupled square wells. The results, shown in Table IX, demonstrated impressive convergence compared to the Siegert results (McCurdy and Rescigno, 1979) and the complex R matrix results (Schneider, 1981). This study also pointed out the difference between Breit-Wigner widths, which assume a unitary S matrix, and the width obtained from a pole of the S matrix which is, in fact, not unitary at the complex resonance energy.

In addition to the total width, Watson obtained partial widths and properly normalized resonance channel wavefunctions (1986a; 1986b). The complete characterization of a resonance requires not only the resonance energy and total width, but also the complete set of partial widths that are associated with the decay probabilities into different open channels. Despite their importance and the determination of very accurate total widths by various resonance methods, very few partial widths have been calculated.

TABLE IX CONVERGENCE STUDIES FOR THREE RESONANCE

METHODS

~

Schwinger" N = 4 N = 5 N = 6 R matrixb

Siegert' N = 10

N = l O N = 20 N = 25 Exact

Case B

E , r 91.2705 0.1530 91.2694 0.1458 91.2694 0.1457

91.2694 0.1458

91.3312 0.1436 91.2716 0.1458 91.2702 0.1458 91.2694 0.1457

~~

Case D

E, r 93.7054 1.1196 93.7026 1.1112 93.7026 1.1111

93.7022 1.1154

93.8010 1.1346 93.7068 1.1134 93.7042 1.1122 93.7026 1 . 1 1 1 1

"Watson, 1984. ' Schneider, 1981.

McCurdy and Rescigno, 1979.


This has been due primarily to the difficulty in obtaining the necessary information about the channels themselves, such as the T matrix channel residues, the channel eigenphases, or properly normalized resonance channel wavefunctions. Although several approaches for normalizing resonance wavefunctions had been introduced, these methods had not been widely applied or understood. Watson investigated five different approaches for normalizing resonance wavefunctions and obtaining the corresponding partial widths. These methods included those based on the definition of a resonance state in terms of the pole of the total Green’s function (More and Gerjuoy, 1973; More, 1971a; 1971b) or T matrix (Watson, 1986a); a method based on a study of adjoint resonance states in an adjoint space (Hokkyo, I965), which was shown to yield partial widths equivalent to the T matrix partial widths; a normalization that defines formal partial widths which sum precisely to the total width (Humblet and Rosenfeld, 1961); the simple branching ratio method, which also assumes that the partial widths sum to the total width (McCurdy and Rescigno, 1979); and Breit-Wigner methods (Macek, 1970). Very interesting relationships between the methods were demonstrated and a new expression for the normalization constant was derived:

i = 1 i+ j

where w i are the nonzero eigenvalues of the denominator of the T matrix (there is one zero eigenvalue, wj = 0) and D is the finite-basis representation of the denominator. This complex, energy-dependent expression shows clearly the relationship of the normalization to the rate at which the denominator goes to zero as the pole is approached. While the Schwinger method is obviously ideally suited to the normalization of the resonance wavefunction as defined above, this normalization expression was shown to be equivalent to a wavefunction expression that should be simpler to apply for any resonance method that can obtain accurate resonance wavefunctions.

where ro is the range of the potential. Partial widths obtained using the Schwinger T matrix residues, ya,


TABLE X COMPARISON OF PARTIAL WIDTHS

Partial Case" widths

A l-1

l-2

C r, rz

E r1

r 2

Schwinger T matrix residues

0.001053 0.001 5 19 0.1461 0.1064 0.0674 4.9068

Formal (Humblet and

Rosenfeld) Branching

ratio ~

0.001053 0.001 519 0.1458 0.1062 0.0754 5.4613

0.001053 0.001 519 0.1458 0.1062 0.0751 5.4613

The potential parameters for each case are given in Watson, 1984. Each case, A-E, supports a resonance state of increasing width below the third threshold.

or equivalently using the golden-rule formula

r u = 21($uIvIqr>12

where I), is the total scattering wavefunction and qr is the properly normalized resonance wavefunction, are compared in Table X to the formal partial widths defined by Humblet and Rosenfeld and the branching ratio partial widths, which both assume that the partial widths sum to the total width. Since the partial widths sum to the total width only in the limit r -, 0, i.e., for narrow widths, this results in errors that increase as r increases.

Although the results for resonances using the SV principle are presently limited to model problems, the Schwinger method appears to have a number of interesting advantages compared to previous methods. Like the complex rotation, Siegert, and complex R matrix methods, the Schwinger method is based on the formal definition of a resonance as a pole of the S matrix, however, it searches for this pole explicitly and thus avoids the diagonalization of complex Hamiltonians and the ambiguity surrounding the study of eigenvalue behavior needed by these complex methods. At each energy guess, it requires the evaluation of the determinant of the T matrix denominator, which is much faster than matrix diagonalization. This approach provides a systematic way to search for resonances and requires no corrections to the resonance parameters such as those needed by the Feshbach and Breit- Wigner methods. The energies are variational and a Green's function is used to enforce the correct asymptotic behavior resulting in very minimal requirements for the basis functions. No scattering solutions or resonance wavefunctions must be calculated to obtain resonance parameters, although these are easy to obtain if desired. This approach is also particularly well suited to


determining partial widths and properly normalized resonance wavefunctions, a difficult goal for many resonance methods.

V. Summary

The development of the SV method as a viable, practical method of studying scattering processes and bound and resonance states has been extremely rapid in the last decade, taking advantage of the sophistication of available computing techniques and the tremendous body of existing work based on the H- K V method. The results have led to interesting comparisons with previous theories, new applications to diatomic and polyatomic systems, and contributions to the general development of variational theory. The questions that remain unanswered, however, are the same questions that have plagued scattering variational methods since their introduction: namely, the search for useful bounds, the choice of appropriate trial functions, and the systematic improvement of results. The Schwinger method has been demonstrated to be an amazingly versatile theory, giving rise to many modifications such as the iterative Schwinger method, the use of Pad6 approximant corrections, the development of two-potential formalisms to handle long-range potentials numerically and short-range potentials with discrete basis functions, the C-functional formalism, and the multichannel variational principle of Takatsuka and McKoy. Although the goal of a scattering variational method that guarantees convergence has not been achieved, the development of these hybrid theories, which employ both numerical and basis function techniques and which are free from spurious singularities, continues to offer quantum theorists the hope that such a goal is attainable.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 25 l l FINE-STR UCTURE TRANSITIONS IN PROTON-ION COLLISIONS R. H . G. REID Departmen1 of Applied Mathematics and Theoretical Physics The Queen’s University of Belfast Belfast, Northern Ireland

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . 251 11. Semiclassical Calculations . . . . . . . . . . . . . . . . . . 255

A. Introduction. . . . . . . . . . . . . . . . . . . . . . 255 B. First-Order Approximations . . . . . . . . . . . . . . . . 255 C. Modified First-Order Approximations . . . . . . . . . . . . . 257 D. Close-Coupled Calculations . . . . . . . . . . . . . . . . 258

111. Close-Coupled Quanta1 Calculations . . . . . . . . . . . . . . 261 A. Introduction . . . . . . . . . . . . . . . . . . . . . . 261 B. Work of Faucher . . . . . . . . . . . . . . . . . . . . 262 C. Work of Dalgarno and Co-Workers . . . . . . . . . . . . . 263

IV. Summary . . . . . . . . . . . . . . . . . . . . . . . . 265 References . . . . . . . . . . . . . . . . . . . . . . . . 265

I. Introduction

We consider the proton-induced fine-structure transition

H + + xq+ - b H + + x$+ (1)

where X z + denotes an ion with net charge, 2. Data on many such processes are needed in the study of astrophysical and

laboratory plasmas. Early interest focused on ions whose ground state belongs to a muitiplet, such as Fe XIV, where Transition (1) is a means of exciting the forbidden lines seen in the solar corona (Seaton, 1964). Interest has extended to intramultiplet transitions in excited terms, however. Thus, in the analysis of intensity-ratios of emission-lines to estimate densities and temperatures in plasmas, it is necessary to consider the proton-mixing of all the multiplets included in the calculation of the state-populations. Dupree (1978) has reviewed the multiplets in the Be-sequence (mainly C I11 and 0 V) and the B-sequence (mainly 0 IV, Si X, Mg VIII and S XII) that are involved in the UV and x-ray emissions from the sun. Dufton and Kingston (1981)

25 1

Copyright Q 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

ISBN 0-12-003825-0

252 R. H. G . Reid

have reviewed the atomic processes, including Transition (l) , necessary to analyse solar emission lines. The same considerations apply to laboratory plasmas. The atomic processes leading to the forbidden-line emission of highly ionized species in Tokamak plasmas has been discussed by Feldman et al. (1980) and Bhatia et al. (1980). The diagnostic role of Fe XVIII has been considered further by Keenan et al. (1987) and Keenan and Reid (1987).

In astrophysics, added impetus has been given to the study of fine-structure excitation of not-so-highly charged ions by the IRAS observations. (For a review, see Beichman (1987).) For example, Pottasch et al. (1986) have used the observed infrared lines of Ne 11, Ne 111, Ne V, Ne VI, S 111, S IV and Ar 111 to derive abundances, and electron densities and temperatures in planetary nebulae (although only electron-impact excitation is used in their analysis).

We shall consider only calculations where the effects of proton collisions are confined to the mixing of states within a multiplet (or, at most, within a configuration). Thus, even though the study of a line-ratio involves an allowed transition with a multiplet upper (or lower) term, the proton excitation of the allowed transition is discounted. This may be a reasonable approximation if the excitation energy of the allowed transition is much larger than the fine-structure splitting. There are cases, however, where it is inappropriate to regard the proton-mixing as isolated in this way. For example, in hydrogen-like ions, a calculation of 2p,,, to 2p,,, excitation must include the 2s,,, state also, because of the small 2p,,,-2sI,, splitting. Accurate calculations for hydrogen-like ions have been made by Zygelman and Dalgarno (1987).

In plasmas, excitation can be caused by electrons and protons. Seaton (1955) noted that the proton cross sections will exceed the electron cross sections when the impact energy is well in excess of the excitation energy. It is for this reason that, for fine-structure excitation, protons play an important role. For example, Chevalier and Lambert (1970) found that observed line- ratios in coronal Ca XV could not be explained if the proton fine-structure mixing was omitted. Although direct excitation between states within a multiplet may be more efficient when caused by protons than by electrons, however, indirect excitation by a two-stage process involving a state outside the multiplet may be the dominant process. For example, in their study of Fe XI11 in the solar corona, Flower and Pineau des For&ts (1973) showed that, while protons dominate the direct excitation of the 'PI and 3P, levels above a certain temperature, the indirect processes of collisional excitation to' the 3s3p3 and 3s23p3d configurations, followed by radiative decay, are so important that the statistical equilibrium is not sensitive to the cross sections for the direct process.

The impact energies for which Transition (1) is important are determined by the ionization equilibrium in the plasma (cf. Allen and Dupree, 1969). For

FINE-STRUCTURE TRANSITIONS IN PROTON-ION COLLISIONS 253

a particular plasma temperature, each element is found predominantly in one particular degree of ionization, this degree increasing with temperature. Thus, there is only a certain temperature range for which a particular ion occurs. Usually, this corresponds to energies up to, and slightly beyond, the energy at which the cross section for Transition (1) is maximum. For example, the temperature range for Fe XIV is about 1 x lo6 K to 7 x lo6 K, corresponding roughly to proton impact-energies from 100 eV to 1000 eV.

Theoretically, Transition (1) has been viewed as rather distinct from other types of fine-structure changing collisions because of several of its features: the Coulomb repulsion in the relative motion; the simplicity of the proton as a perturber; and the high impact energies for which the cross sections are needed. Extensive use has been made of semiclassical models. Accurate quanta1 calculations have been made also. These are of interest not only in their own right but also because they give insight into the accuracy of the semiclassical results.

The objectives of this review are to indicate the scope of the available results and to place the various theories in perspective, hopefully supplement- ing the account given by Dalgarno (1983). Table I lists the calculations in chronological order, indicating the method used and the ions considered. We

TABLE I

BY PROTON IMPACT CALCULATIONS OF CROSS SECTIONS OR RATES FOR FINE-STRUCTURE EXCITATION OF IONS

Reference Method" Termb Ion'

Seaton (1964) Bahcall & Wolf (1968)

Reid & Schwarz ( I 969) Bely & Faucher (1 970)

Masnou-Seeuws & McCarroll (1 972)

Landman (1973, 1975)

Sahal-Brechot (1974)

Faucher (1975)

sc- 1 sc-cc

sc-cc sc- 1

sc-cc

sc-cc

sc- 1

0

3p zP 2p zP 2pz 3P 3p zP 3pz 3P 3p zP 2p zP 2p5 ZP 3p ZP 3p5 zP 2pz 3P 3p zP 3pz 3P 3p zP 3pz 3P 3p 2P 3pz 3P 3p2 3P

Fe13+ C + Nz+ N + Si+ S 3 + SZ +

s3 +

C + (iso. seq.) Cal +

Ne' (iso. seq.) Cr"+ Si+ (iso. seq.) Nil5+ Ar' (iso. seq.) Gels+

Fe13+ FeIZ+ FeI3+ FeI2+ FeI3+ Fel2+ Fel2+

Nf O Z f Si8+

Table (Continued)

254 R. H . G . Reid

TABLE I (Continued)

Reference Method" Term' Ion'

Malinovsky (1975)

Mason (1975) Faucher (1 977)

Kastner (1977)

Landman (1978) Landman & Brown (1979)

Kastner & Bhatia (1979)

Faucher et al(1980) Landman (1980)

Doyle et a1 (1 980)

Feldman et al (1980) (for Cr, Fe, Ni) and

Bhatia et al(l980) (for Ti)

Heil et a l (1982) Heil et al(1983) Landman (1985) Keenan & Reid (1987)

sc-1

sc-1 Q

sc-1

sc-cc sc-cc

sc-1

SCIQ sc-cc

sc-cc

sc-1

Q Q sc-cc sc-cc

2s2p 3P 2p2 3P 2p2 3P 2p2 3P 3p2 3P 2p 2P 2p2 3P 2p4 3~

2p5 2P

3s3p 3P 3p 2P 3p2 3P 3P3

2s2p 2p2 3P

3s3p 3P 2p2 3P 2p4 3~ 2p5 2P 3p 2P 3p2 3P 3p5 2P 2p2 3P 2p4 3~

3p4 3P 2s2p 3P

2p2 3P 2s2p 3P

2P2 2p 2P

2s2p2 2P3 2P2

2~2p3 2P3 2P4 2p5 ZP 2p 2P 3p 2P

2p5 2P 2 ~ ~ 3 s 3P

SC-I, first-order (or modified first-order) semiclassical; SC-CC, close-coupled semiclassical;

If the configuration alone is shown, more than one term was considered. Q, close-coupled quantal.

' "(iso. seq.)" indicates a run of the iso-electronic sequence.


shall discuss the theories in the order in which they developed historically. Thus, we first consider semiclassical (i.e., classical-path) theories, starting with the first-order theory and approximations based on the first-order theory, before going on to consider close-coupled theories. We shall not discuss the general aspects of the collision-calculations, however, as these are well described in standard texts.

II. Semiclassical Calculations

A. INTRODUCTION

In semiclassical calculations, the position of the proton relative to the ion is taken to be a classical, time-dependent trajectory R(t). Thus, the electronic state of the ion, which is treated quantally, is subjected to a time-dependent perturbation during the collision.

When a multipole-expansion is made of the electrostatic interaction between the proton and the ion, the monopole term is just Ze2/R. Hence, in a semiclassical theory, the classical path followed by the proton is a hyperbolic, Coulomb trajectory. The leading term of the expansion, which can couple states in the same multiplet, is the quadrupole term

(if R > ri)

where ri is the position of the ith electron from the nucleus and the sum is over all the ion’s electrons. Indeed, within a configuration, states can be coupled only by the even terms in the multipole expansion. Also, the states within a P- multiplet can be coupled only by the quadrupole term. This quadrupole term is taken as the sole interaction in semiclassical theories. Interaction (2), however, is valid only for R > ri and so will not apply when the proton penetrates the ion’s electron cloud. Semiclassical calculations have used ad hoc procedures to deal with this penetration in close collisions.

B. FIRST-ORDER APPROXIMATIONS

At low energies, the Coulomb repulsion keeps the colliding partners sufficiently far apart that collisions at all impact parameters are weak. Thus, the interaction need be treated in first-order only.

256 R. H. G. Reid

A very detailed account of first-order semiclassical theory as it applies to Coulomb excitation of nuclei has been given by Alder et al. (1956), and their results for electric quadrupole excitation apply directly to proton-induced fine-structure excitation. Their Table 11.3 gives the cross section as a function of impact energy, and from their Table 11.8 the transition probability as a function of impact parameter can be deduced at several energies. They also discuss the first-order quantal treatment (the Coulomb-Born approximation). Their Table 11.5 gives the cross section in this approximation in a way that manifests the disparity between the semiclassical and quantal results.

Alder et al. (1956) also give a symmetrized version of the semiclassical theory that takes account of the different relative velocities in the initial and final channels. They noted that the agreement between the semiclassical and quantal cross sections is greatly improved when the symmetrized form is used. Adapted to Transition (l), the symmetrized expression for the cross section Q in atomic units (a;) is

Q j + j , ( E ) = 134 Z - 2 M B ( S L ; J + J’)Ej(<)q2 ( 3 4

with

and

where M is the reduced mass in AMU (thus, usually close to unity), E is the barycentric energy in the initial channel in eV, and ~ j , j is the excitation energy in eV. Some values of the factor B(SL; J + J’ ) arising for ions with open p-shells are:

where ( r ’ ) , is the expectation value of rz for the p-orbital. The function f(t) is given in Table 11.3 and in Fig. 11.5 of Alder et al. (1956), from which it can be seen that f ( Q decreases roughly exponentially with increasing t for 5 2 0.8. Thus, Q ( E ) decreases exponentially with decreasing E below a certain value, given from Eq. (3b) with t = 0.8. At the other extreme, f(0) = 0.895, and so for high energies Q ( E ) cc E . Clearly, this behaviour with increasing E is unrealistic, and it unfortunately turns out that the energies for which the cross sections are needed extend well above the range where the first-order results are reliable.


C. MODIFIED FIRST-ORDER APPROXIMATIONS

In his calculation on Fe XIV, Seaton (1964) used an approximation based on the first-order theory with a simple adaptation to allow for strong collisions. The reason that first-order theory can still play a role is that, for any energy, the first-order transition probability is reliable for sufficiently large values of the impact parameter, p. Seaton considered three energy regions and used a different prescription for the transition probability 9 ( p ) in each region. In the low energy region, the first-order transition probability 9 ’ (p ) remains less than one half for all p. Seaton thus took 9 ( p ) = @(p) for all p so that the cross section is given by Eq. (3).

For intermediate energies, there is an impact parameter p1 such that 9 ’ (p ) > 1/2 if p < p l , indicating that the first-order theory is not valid for p < pl. Seaton took 9 ( p ) = P’(p) for p > pl, and 9 ( p ) = 1/2 for p < p l .

The high energy region is the region where there is an impact parameter po (> p l ) below which the proton penetrates the ion’s electron cloud, so that the long-range form of Interaction (2) is no longer valid. Seaton took the criterion for po to be that the distance of closest approach equals the mean radius of the 3p-orbital. He took 9 ( p ) = 9’ (p) for p > po and Y ( p ) = Y’(p,) for p < po.

Seaton’s results are shown in Fig. 1 . They compare not unfavourably with later calculations. Certainly, Seaton’s analysis gave the correct insight into the collision process, and identified where improvements were needed, namely, close-coupling for energies above the low-energy region, and proper treatment of close collisions in the high energy region. In Seaton’s calculation of Fe XIV, the boundary between the intermediate energy region and the high energy region occurs at about 6500 eV, which is well in excess of the energy at which the cross section is maximum (see Fig. 1).

In a modification of Seaton’s approximation, Sahal-Brkchot (1974) took p1 to be such that P‘(p l ) = (25’ + l)/w where w is the total number of states in the multiplet. He then took 9 ( p ) = (25’ + l ) / w for p < p l , the assumption being that the states of the multiplet are completely mixed by a collision with p < p l . When applied to 3P multiplets, this assumption leads to an estimate for the cross section for 3P0 -+ ’P1 transitions, which is zero in first order.

Bely and Faucher (1970) have made extensive calculations for ’P multiplets using an approximation based on the first-order transition probability, 9’ (p) . To ensure that the probabilities remain physically reasonable, they use a “unitarizing” procedure, taking 9 ( p ) = @[l + 8’/4]-2. Their calculated rates for Fe XIV agree with Seaton’s to within a few percent.

Kastner (1977) and Kastner and Bhatia (1979) used the symmetrized first- order cross section for low energies. For high energies, their cross section is based on an expression given by Bahcall and Wolf (1968) which erroneously

258 R. H . G. Reid

has an E-”’ behaviour. Since they do not consider any approximation for intermediate energies, their cross sections have an unrealistic cusp where their high and low energy approximations meet. The method of Kastner and Bhatia has also been used by Feldman et al. (1980) and by Bhatia et al. (1980). Keenan and Reid (1987) have noted that, for Fe XVIII, the rate given by Kastner and Bhatia differs significantly from the close-coupled result.

D. CLOSE-COUPLED CALCULATIONS

In close-coupled semiclassical formulations, the transition probabilities are determined from the numerical solution of coupled differential equations ( c j Dalgarno, 1983), and so, no first-order approximations are made. Thus, the close-coupled method removes the uncertainty in the intermediate energy region.

The matrix elements of the quadrupole Interaction (2), needed for close- coupled calculations in p ’P and p2 3P multiplets (in which all other electrons are in closed-shells), are shown in Table 11. We note that the matrix elements are partitioned into two sets. Also, the R-dependence of all the elements is given by a common factor u2(R). When only the long-range Interaction (2) is used,

u2(R) = - e 2 ( r 2 ) , R - 3 .

The matrix elements for a p5 ’P multiplet differ from the p 2P case only in an overall sign change. Similarly, the elements for a sp 3P or a p4 ’P multiplet differ from the p2 3P case only in an overall sign change, whereas a sp5 3P multiplet has the same elements as p2 3P.

The quantization axis of the states used in deriving Table I1 is perpendicular to the collision plane. This is adequate for calculating J + J’ cross sections. When calculating cross sections for transitions between magnetic sub-levels, however, a space-fixed quantization axis is required. The space- fixed states are related to the states used in Table I1 by a rotation matrix. Hence, the calculation can still be performed using the basis of Table 11, and the transition amplitudes for the space-fixed basis can be derived by use of rotation matrices. (Gordeyev et al, 1969; Masnou-Seeuws and Roueff, 1972).

As Table I shows, there have been many close-coupled semiclassical calculations. The methods used by the various authors are essentially equivalent. We shall not attempt a complete resume of all the work, but rather we shall note some relevant points.

Table I shows that the first close-coupled calculations were made by Bahcall and Wolf (1968). They solved the coupled-equations only for a few energies for each ion, however, and then used a fitting formula that, whatever

TABLE I1 MATRIX ELEMENTS I N LS-COUPLING FOR THE QUADRUPOLE INTERACTION IN ATOMIC UNITS,

WHEN THE QUANTIZATION AXIS IS PERPENDICULAR TO THE COLLISION PLANE

* The ionic state IJM) is shown above each column. The same labelling applies to the rows. y = ezig, where 4 is the polar angle of R measured from the position of closest approach.

i 4 P m 1 P b

N VI W

260 R. H . G. Reid

its merits for intermediate energies, is incorrect for high energies. They also suggested a simple method for estimating excitation rates by interpolating between low-energy and high-energy formulae. Thus, they estimated rates for several ions of significance in the solar corona. This procedure, however, has been largely discredited (Faucher, 1977).

Reid and Schwarz (1969) noted that the energy at which the close-coupled results merge with the low-energy, first-order approximation corresponds to 5 21 0.7, where 5 is defined in Eq. (3b). This result has been confirmed by subsequent calculations.

The first close-coupled calculation for Fe XIV was made by Masnou- Seeuws and McCarroll (1972). To make the comparison with the quanta1 results of Heil et al. (1983) unambiguous, we have repeated the Masnou- Seeuws and McCarroll calculation using the value of ( r 2 ) , used by Heil et al. (1983), namely, 0.5510 a;. The results are shown in Fig. 1. Compared to the results of Seaton (1964), the cross section is reduced by 8 % at 500 eV, rising to 14% at 900eV.

Masnou-Seeuws and McCarroll(l972) emphasized that it is more correct to view the coupling as due to the difference between the adiabatic potentials W,(R) and W,(R). Accordingly, the common radial-factor u2(R) should be gCW, - Wn) for the p ’P case, and gCWn - WJ for the p2 3P case. In their actual calculations, they used the long-range Interaction (2) only, and unfortunately, there is a sign error in their calculations for the p2 3P case, which caused significant errors (cf: Landman 1975, 1977).

There have been differences in how the various authors have dealt with close collisions at high energies, when the proton can penetrate the ion’s electron cloud. Reid and Schwarz (1969) modified the R-dependent factor u2(R) so that it was roughly constant for R 5 ( r2 ) i ’2 . Landman (1973,1975, 1978, 1980, 1985) and Landman and Brown (1979) have followed Seaton’s high energy prescription of taking constant probabilities for p < p l . Doyle et al. (1980) modified u2(R) by replacing (rZ),R-3 by the expectation value of r: r F 3 for a suitably scaled hydrogenic 2p-orbital, where r < and r , are the lesser and greater of r and R, respectively. They noted that such modification of the short-range form of u2(R) has little effect on the cross section, except at high energies.

A noteworthy feature of the work of Landman (Landman, 1973, 1975, 1978, 1980, 1985, and Landman and Brown, 1979) is that he has considered intermediate coupling, rather than pure LS-coupling, in the ionic states. The various multiplets within a configuration are admixtures of different LS- multiplets. Hence, although the quadrupole Interaction (2) cannot couple multiplets in LS-coupling if AS # 0 or if AL is odd, it can couple such multiplets in the intermediate coupling scheme. Thus, by including all the multiplets arising from the configuration, Landman has obtained the cross


sections not only for transitions within multiplets but also for transitions from one multiplet to another. For example, in Fe XI11 (2p2), he has obtained the rates for 3P + 'D and 3P .+ 'S transitions. Regarding the effect of intermediate coupling on transitions within a multiplet, Landman (1975) made calculations using LS-coupling for the 3P multiplet of Fe XI11 and compared the results with his intermediate coupling results. Differences of over 20 were observed, which suggests that it may be important to consider intermediate coupling even for transitions within one multiplet. Also, in intermediate-coupling calculations on Ni XIII, Landman (1980) found that the restriction of the close-coupling to the 3P multiplet alone decreased the cross sections by up to 15 %.

Another feature of Landman's work is that he has used a symmetrized version of the semiclassical coupled-equations (cf. Alder and Winther, 1966). It is unlikely, however, that symmetrization has a significant effect in the intermediate and high energy regions.

Rates for proton-induced transitions between terms have also been given for S X by Kastner and Bhatia (1979), for highly ionized Cr, Fe and Ni by Feldman et a/. (1980), and for highly ionized Ti by Bhatia et al. (1980).

111. Close-Coupled Quanta1 Calculations

A. INTRODUCTION

In quantal calculations, the motion of the proton relative to the ion is treated quantum mechanically. The problem reduces to solving coupled second-order differential equations with R as the independent variable. These radial equations are obtained when, in the time-independent Schrodinger equation, the wave function for the complete system is expanded in a basis made by coupling the angular part of the relative motion to the electronic states of the ion. The basis may incorporate the Coulomb aspect of the radial motion. Also, distorted or quasi-molecular electronic states may be used.

In the quantal calculations, the interaction has been treated more accurately than in the semiclassical calculations. This is partly because the question of the short-range interaction comes to the fore in solving the radial equations. It may also reflect a greater computational commitment. The use of a more accurate interaction is not an intrinsic part of a quantal calculation, however, and quantal calculations also can be made with only the Coulomb repulsion and the long-range quadrupole interaction. Conversely, the semiclassical methods can be adapted to use more accurate interactions (Masnou-Seeuws and McCarroll, 1972).

R. H. G. Reid

B. WORK OF FAUCHER

The first close-coupled quantal calculations for Transition (1) were made by Faucher (1975) for Fe XIII. He adapted the method used for electron-ion collisions by Bely et al. (1963). Thus, the interaction was not confined to the long-range quadrupole term, but rather used the expectation value of r’, r ; ’. Compared to electron collisions, however, the large mass of the proton completely changes the character of the calculation. First, a much smaller step-size is needed in solving the radial-equations, and second, many more partial waves are required. Faucher had difficulty achieving accuracy with the computational power at his disposal. Also, he solved the equations only for some values of 1 (the angular momentum of relative motion) and interpolated for other l-values. For large 1, he used the 1-4 asymptotic form of the partial collision strengths.

Faucher and Landman (1977) have made a detailed comparison between the quantal results of Faucher (1975) and the semiclassical results of Landman (1 975). Quanta1 transition probabilities (as functions of impact parameter) were derived from the partial collision strengths, and these were compared with the semiclassical transition probabilities. They found that, in the low energy region, where the proton does not come near the ion, the transition probabilities are identical. At higher energies there are significant differences between the quantal and semiclassical transition probabilities below a certain impact parameter. In the cross sections, these differences have a significant effect for energies in and above the region where the cross sections are maximum. Faucher and Landman showed that these differences are due to the difference in the interaction being used in the two calculations, however, and are not due to the difference between the quantal and semiclassical treatments of the collision. They showed this by performing quantal calculations with the long-range quadrupole interaction only. The results so obtained were identical to the semiclassical results for all impact parameters.

The conclusion that no significant error is caused by using a semiclassical treatment of the collision is very important, because semiclassical calculations are computationally much less demanding. But as Faucher and Landman (1977) point out, this conclusion may not apply to less highly ionized cases, where the impact energies of interest are significantly lower.

Faucher and Landman (1977) attributed the difference between the quantal and semiclassical cross sections to the short-range part of the interaction. Thus, significantly, their work implies that the use of the expectation value of r: r;’ in place of the pure long-range quadrupole form affects the cross sections at energies as low as the region where the cross sections are maximum. This energy region is well below that suggested by the


high-energy-region criterion of Seaton (1964). Indeed, Faucher and Landman noted that the differences were significant even when penetration of the 3p charge-density distribution was still slight.

C. WORK OF DALGARNO AND CO-WORKERS

Close-coupled quanta1 calculations have been made for 0 IV (2p ’P) by Heil et al. (1982), and the same method has been applied to Fe XIV (3p ’P) by Heil et al. ( 1 983). Their collision formulation followed the quasi-molecular approach of Mies (1973), adapted to allow for the Coulomb nature of Transition (1). As significant as this treatment of the collision, however, is that they calculated the interaction energy accurately. They evaluated the ’ll and ’C adiabatic potentials of the collision system in a way that not only gives the short-range interaction accurately, but also gives the polarization term in the long-range form, in addition to the quadrupole term. Thus, in their calculation of the cross sections

3 e2( r2> , a, - aZ WE- W n N ---++++.. 5 R3 2 ~ 4

where a, and clE are the perpendicular and parallel polarizabilities of the ion. The authors comment on the validity of various approximations. Heil et al.

( 1982) note that the Coulomb-Born approximation gives accurate partial cross sections above a certain value of 9 (the total angular momentum of the collision system); thus, the computational procedure that they recommend for its speed and reliability is to use close-coupling at low $, giving way to Coulomb-Born at higher $. The $-value above which the Coulomb-Born approximation is accurate depends on energy, of course, and it is reliable for all f at sufficiently low energies, corresponding to the low energy region of Seaton (1964). Heil et al. (1982) and Heil et al. (1983) both comment adversely on the possibility of saving time by using either a unitarized form of the Coulomb-Born approximation or the “elastic” approximation (cf. Dal- garno and Rudge (1964), Wofsy et al. (1971), Bottcher et al. (1975)).

The results of Heil et al. (1983) for FeXIV are shown in Fig. 1. These results, with the accurate interaction, should be the most accurate to date. Also shown in Fig. 1 are their results obtained using the long-range quadrupole interaction only. Compared to the latter, the results with the accurate interaction are reduced by 6 % in the region of the maximum.

Of interest is the comparison between the results of Heil et al. (1983) using the long-range quadrupole interaction alone and the semiclassical close- coupled results using exactly the same coupling interaction (also shown in Fig. 1). Contrary to what was found by Faucher and Landman (1977), the

R. H . G. Reid

I .2 I I I I I I

1.0

0.8

0.6

0.4 - -

0.2

-

-

-

- -

- 100 300 500 100 90 0

E ( e V ) FIG. 1. Total cross section for Fe XIV 3p ’P,,, + 3p 2P3,z, induced by proton impact: curve

H1, close-coupled quanta1 calculation with accurate interaction (Heil et a/, 1983); curve H2, same as H 1 except long-range quadrupole coupling only; curve X, close-coupled semiclassical calculation with long-range quadrupole coupling only; curve S, modified-first-order semiclassical calculation (Seaton, 1964).

results are not “identical”, but nevertheless, they are close, especially for energies between 600 eV and 800 eV. The divergence of the results above 800 eV is puzzling, however.

As far as understanding Transition (1) is concerned, perhaps the most important conclusion of Heil et al. (1983) concerns the relative importance of electron-cloud penetration at short-range and polarization at long-range. They found that, compared with calculations using the long-range quadrupole interaction only, the main correction in the intermediate energy range comes from the polarization and not from the penetration. Only at high energies does the short-range interaction have a significant effect, in keeping with the physical description of Seaton (1964).

This conclusion is at variance with that of Faucher and Landman (1977), who inferred that the cross sections are altered even at intermediate energies when the long-range quadrupole interaction is modified by use of the expectation value of r< r ; 3 . Indeed, the changes to the cross sections reported by Faucher and Landman (1977) are similar in magnitude and energy dependence to the changes reported by Heil et al. (1983) in going from the long-range quadrupole interaction to their accurate interaction.


IV. Summary

The description of the process given by Seaton (1964) in terms of three energy regions is confirmed, except for uncertainty about the significance of electron-cloud penetration.

In the low energy region, first-order results are sufficiently accurate for all impact parameters (partial waves). At very low energies, it may be necessary to use the Coulomb-Born approximation, although the symmetrized semiclassical formula may suffice because of its surprising accuracy, noted by Alder et al. ( 1 956).

The high energy region is where electron-cloud penetration becomes significant. According to Seaton’s criterion, with confirmation by the results of Heil et al. (1983), this region occurs at energies well in excess of where the cross sections are maximum. If this is so, then electron-cloud penetration has little influence on excitation rates at the temperatures relevant to physical environments. The results of Faucher and Landman (1977), however, suggest that electron-cloud penetration is significant at much lower energies.

In the intermediate energy region, first-order formulae cannot be used for all the impact parameters (partial waves), although for any energy there is an impact parameter above which they are reliable. Any patching of the first- order theory cannot be relied on (such as unitarizing or assuming a mean transition-probability). Thus, in this region, close-coupling must be used. For highly ionized species at least, little error is caused by using the semiclassical formulation rather than the quanta1 formulation. For the interaction, Heil et al. (1983) have emphasized the importance of including the polarization terms. Also, while there is some debate about its effect, the short-range interaction should take account of electron-cloud penetration. Finally, in the light of Landman’s work, it may be important to allow for departures from pure LS-coupling.

REFERENCES

Alder, K. and Winther, A. (1966). Coulomb Excitafion. Academic Press, New York, New York. Alder, K., Bohr, A,, Huus, T., Mottleson, B. and Winther, A. (1956). Rev. Mod. Phys. 28,432-542. Allen. J. W. and Dupree, A. K. (1969). Astrophys. J . 155, 27-36. Bahcall, J. N. and Wolf, R . A. (1968). Astrophys. J . 152, 701-29. Beichman, C. A. (1987). Ann. Rev. Astron. Astrophys. 25,521-63. Bely, 0. and Faucher, P. (1970). Astron. Astrophys. 6, 88-92. Bely, O., Tully, J. A,, and Van Regemorter, H. (1963). Ann. Phys. (Paris) 8, 303-21. Bhatia, A. K., Feldman, U. and Doschek, G. A. (1980). J. Appl. Phys. 51, 1464-80. Bottcher, C., Cravens, T. C., and Dalgarno, A. (1975). Proc. R. SOC. A346, 157-70. Chevalier, R . A. and Lambert, D. L. (1970). Solar Phys. 11,243-57.

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Dalgarno, A. (1983). In Atoms in Astrophysics (P. G. Burke, W. B. Eissner, D. G. Hummer and I.

Dalgarno, A. and Rudge, M. R. H. (1964). Astrophys. J. 140, 800-2. Doyle, J. G., Kingston, A. E., and Reid, R. H. G. (1980). Astron. Astrophys. 90, 97-101. Dufton, P. L. and Kingston, A. E. (1981). In Advan. Atom. Molec. Phys. 17, (D. R. Bates and B.

Bederson, eds.). Academic Press, Orlando, Florida, pp. 355-418. Dupree, A. K. (1978). In Advan. Atom. Molec. Phys. 14. (D. R. Bates and B. Bederson, eds.).

Academic Press, New York, New York, pp. 393-431. Faucher, P. (1975). J. Phys. E: Atom. Molec. Phys. 8, 1886-94. Faucher, P. (1977). Astron. Astrophys. 54, 589-92. Faucher, P. and Landman, D. A. (1977). Astron. Asfrophys. 54, 159-61. Faucher, P., Masnou-Seeuws, F., and Prudhomme, M. (1980). Astron. Astrophys. 81, 137-41. Feldman, U., Doschek, G. A,, Cheng, C. C., and Bhatia, A. K. (1980). J. Appl. Phys. 51, 190-201. Flower, D. R. and Pineau des For& G. (1973). Askon. Asfrophys. 24, 181-92. Gordeyev, E. P., Nikitin, E. E., and Ovchinnikova, M. Ya. (1969). Can J. Phys. 47, 1819-27. Heil, T. G., Green, S.. and Dalgarno, A. (1982). Phys. Rev. A. 26, 3293-8. Heil, T. G., Kirby, K., and Dalgarno, A. (1983). Phys. Rev. A. 27,2826-30. Kastner, S . 0. (1977). Astron. Astrophys. 54, 255-61. Kastner, S. 0. and Bhatia, A. K. (1979). Astron. Astrophys. 71, 211-3. Keenan, F. P. and Reid, R. H. G. (1987). J. Phys. E: Atom. Molec. Phys. 20, L753-7. Keenan, F. P., Mohan, M., Baluja, K. L., Berrington, K. A., and Hibbert, A. (1987). Phys. Lett. A.

Landman, D. A. (1973). Solar Phys. 31, 81-9. Landman, D. A. (1975). Astron. Astrophys. 43, 285-90. Landman, D. A. (1978). Astrophys. J. 220, 366-9. Landman, D. A. (1980). Asfrophys. J. 240,709-17. Landman, D. A. (1985). J. Quant. Spectrosc. Radial. Transfer. 34, 365-71. Landman, D. A. and Brown T. (1979). Astrophys. J. 232, 636-48. Malinovsky, M. (1975). Asfron. Astrophys. 43, 101-10. Masnou-Seeuws, F. and McCarroll, R. (1972). Asiron. Astrophys. 17, 4 4 - 4 . Masnou-Seeuws, F. and Roueff, E. (1972). Chem. Phys. Lett. 16, 593-7. Mason, H. E. (1975). Mon. Not. R. Astr. SOC. 170,651-89. Mies, F. H. (1973). Phys. Rev. A7. 942-57. Pottasch, S. R., Preite-Martinez, A., Olnon, F. M., Mo, J.-E., and Kingma, S. (1986). Astron.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 25 l l ELECTRON IMPACT EXCITA TION R. J. W. HENRY Department of Physics and Astronomy Louisianri State University Baton Rouge, Louisana

A. E. KINGSTON Departmeni (f Applied Mathematics and Theoretical Physics The Queen's University of Bevast Belfast, Northern Ireland

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 267 11. Close-Coupling Approximation . . . . . . . . . . . . . . . . 268

111. Convergence of the Close-Coupling Expansion . . . . . . . . . . . 272 A. Results for Atomic Hydrogen . . . . . . . . . . . . . . . . 273 B. Results for Atomic Helium . . . . . . . . . . . . . . . . . 276

IV. The ERect of Resonances on Electron Excitation Rates . . . . . . . . 282 A. Resonances in Electron-Hydrogen Scattering . . . . . . . . . . 283 B. Resonances in Electron-Helium Scattering . . . . . . . . . . . 285

V. Inner Shell Excitation Autoionization . . . . . . . . . . . . . . 289 VI. Resonances in Cu . . . . . . . . . . . . . . . . . . . . . 298

References . . . . . . . . . . . . . . . . . . . . . . . . 300

I. Introduction

As a consumer of atomic physics data in aeronomy and astrophysics, Professor Alexander Dalgarno has stimulated many colleagues to calculate or observe interactions of electrons with atoms and ions. In his early days, Alex m .tde significant contributions to electron collision theory in his study of the elastic and inelastic scattering of electrons by He+ (Bransden and Dalgarno, 1953; Bransden et al., 1953). He was also the first to study the simultaneous excitation and ionization of helium by electron impact (Dal- garno and McDowell, 1955). One of his earliest papers (Dalgarno, 1952) was on the photoionization of argon, followed by an extensive calculation on the photoionization of oxygen (Dalgarno and Parkinson, 1960; Dalgarno et a/., 1964). It is also interesting to note that he returned to this field and published a paper on the electron excitation of the fine structure levels of hydrogen-like ions (Zygelman and Dalgarno, 1987).

267

Copynght 0 1988 by Academic Press, Inc. All rigbls of reproduction in any form reserved.

I C R N n-17-MlQ1C n

268 R. J . W Henry and A. E . Kingston

Scientists in many areas of research are increasingly interested in accurate electron impact excitation rates for atoms and ions. At present, these rates are very difficult to measure and much of our information on them comes from theoretical calculations. Considerable advances have been made in the calculation of electron excitation rates, and in this chapter we review work in four areas: in Section I11 we study the convergence of the close-coupling expansion using recent work on H and He; in Section IV we consider the effect of resonances on electron excitation rates in H and He; in Section V we discuss inner shell excitation-autoionization contributions to total ionization, with examples for Li-isoelectronic and Na-isoelectronic sequences; finally, in Section VI we present some recent results on the electron scattering of a complex atom, copper.

11. The Close-Coupling Approximation

Much of the progress in calculations on electron impact excitation in the 1970s occurred because of improvements in computational techniques and in computing equipment. A number of methods have been developed in the efficient solution of the close-coupling approximation equations. A review of some of the computational methods was presented by Burke and Seaton (1971). We will describe briefly some general programs that have been developed to investigate collisions of electrons with atoms and ions of arbitrary complexity in LS-coupling. First, we describe the general close- coupling approximation, following the notation and description of Henry (198 1).

Let the nonrelativistic Hamiltonian for a positive ion with N electrons and nuclear charge 2 be H ( 2 , N ) .

The Schroedinger equation that describes the electron-positive ion system is

(1)

where E is the total energy of the system, and Yo is the wave function for the ( N + 1)-electron system. The Schroedinger equation is to be solved subject to the boundary conditions of an incoming wave in channel “a” and an outgoing wave in all other open channels. An open channel is one that is energetically accessible to the incident electron. Otherwise, the channel is called closed. Wave function Y, is expanded in a set of N-electron target wave functions, xi. The expansion coefficients are functions of the coordinates of the scattered electron:

H ( 2 , N + l)Y, = EY,

i

where the operator d antisymmetrizes the total function, and the symbol x,

ELECTRON IMPACT EXCITATION 269

denotes the space and spin coordinates of the nth electron. We take i to denote all the quantum numbers of the system.

In principle, the set of functions X i in Eq. (2) is complete. In practice, however, different expansion procedures result when specific choices of the functions X i are used. A more general expansion for the complete wave function is used extensively in practice. Let Qj be an ( N + 1)-electron bound state function. Then, in place of Eq. (2), we write

I

Ya(x1, . . ., xN+ 1) = 1 Xi(x1, ., XN)@ia(XN+ 1) i = 1

J

+ 1 Qj(xI,...,XN+1)ci (3) j = 1

where the functions Oi, and coefficients c j are to be determined. The functions Qtj have the same total orbital angular and spin moments and parity as Ya, and they decrease exponentially at large radial distances. They are referred to as correlation functions and they serve a dual purpose. One-electron functions +j(x) are used frequently to construct the functions Xi and mi. Some Qti are included to remove a constraint on the total wavefunction Yo which occurs when orthogonality of one-electron orbitals, (c#J~, Oi) = 0, is imposed. Other Qj permit an improved description of short-range electron correlation and should lead to more rapid convergence of the expansion in Eq. (3).

The close-coupling approximation results when trial wave functions given by Eq. (3) are truncated at Z target terms. It is a procedure that requires the solution of a set of coupled integrodifferential equations for the radial functions Fio(r) in 0, and coefficients cj. We will refer to it as ZCC, where I has the value of the number of target states used in the expansion. Exchange potential terms are assumed to be included.

The asymptotic form of the scattered radial function is normally taken to be given by

where F~,(r)r~mk!’2{sin(Ci + q ) 6 , + cos(li + zi)p:,} (4)

( 5 ) ii = kir - lin/2 + qi In(2kir) + arg T(Ii + 1 - iqi).

Application of the Kohn variational principle leads to the requirement that the following equations have to be solved for Fi, and cj, and the reactance matrices, K , extracted from the solutions of

I J

[(hi - k?)dir + Vii. + Wii.]Fi., + 1 U i j c j + 1 pLlaPii = 0 (6a) i’ = 1 j= 1 i

1 I


The quantities in Eq. (6) are as follows:

(i) h is a diagonal matrix element with elements

d2 l i ( l . + 1) 22 h . = --++'---* ' dr2 r2 r (7)

(ii) Vii, is a local direct potential, with asymptotic form

Vii,(r)rzm 1 aijAr-'- + exponentially decreasing terms. (8)

(iii) Wii, is a nonlocal exchange potential operator that has exponentially decreasing terms asymptotically.

(iv) The U j i , are known functions that involve matrix elements of the Hamiltonian operator with the bound target functions and the correlation terms.

(v) The H,. are known constants that involve matrix elements of the Hamiltonian operator with the correlation terms.

(vi) The pAa are undetermined . Lagrange multipliers, representing each orthogonality constraint imposed on the F,.

In the early 1970s, Seaton and colleagues developed a computer program that allows many configurations to be retained in the close-coupling expansion, including configuration-interaction terms in the description of the target wave functions. The integrodifferential equations are converted to a set of linear algebraic ones and the program is called IMPACT (Seaton, 1974a; Eissner and Seaton, 1972; Seaton, 1974b; Crees et al., 1978). Burke, Robb, and colleagues (Burke, 1973; Burke and Robb, 1975; Berrington et al., 1974; Berrington et al., 1978) have developed a program, RMATRX, which is based on the R-matrix method of nuclear physics (Wigner and Eisenbud, 1947). Henry and colleagues (Smith and Henry, 1973; Henry et al., 1981) have developed a noniterative integral equation method (NIEM) that is based on work of Sams and Kouri (1969). Collins and Schneider (1981, 1983) converted the integrodifferential equations first to a set of integral equations and then to a set of linear algebraic (LA) ones.

The above methods have replaced standard techniques such as Numerov (Fox and Goodwin, 1949) or DeVogelaere (DeVogelaere, 1955) which were used to solve coupled sets of integrodifferential equations (Conneely et al., 1970; Ormonde et al., 1970). These standard methods and others are still used for solution of a single (integro-) differential equation, however. For example, Numerov (Fox and Goodwin, 1949) is used to obtain the Green's functions required in NIEM; RMATRX used the DeVogelaere method (DeVogelaere, 1955) to obtain numerical orbital functions necessary to construct basis states; and IMPACT uses a Fox-Goodwin technique (Norcross and Seaton,

1 = 1

ELECTRON IMPACT EXCITATION 27 1

1976) or a fourth-order Runge-Kutta method (Ralston, 1965) in the asymptotic region.

In IMPACT, the coupled integrodifferential equations are solved using radial functions that are tabulated at a sparse set of grid points. Finite difference approximations for the differential and integral operators are used to ensure good stability for the solution of the resulting linear algebraic equations. Beyond some radius r = a, where all exchange and bound orbitals may be neglected, there remains only a set of coupled ordinary differential equations in which the direct potentials include combinations of inverse powers of r (Eq. (6)). Matching of the asymptotic solutions to the functions obtained in the inner region yields the radial wave functions and K-matrix by, effectively, a matrix inversion at each energy.

In RMATRX, the ( N + 1)-electron wave function is expanded in a finite set of energy-independent basis functions in the inner region r < a. These basis functions have a specified logarithmic derivative at r = a, and they are used to calculate the logarithmic derivative of the collision wave function on the boundary. A single matrix diagonalization yields information for a specified energy range. When this is combined with the asymptotic solutions, the K-matrix elements may be obtained.

In NIEM, coupled integrodifferential equations are transformed to coupled integral equations and solutions are written in terms of Green’s functions. For r < a, exchange contributions are included and each exchange term yields an additional inhomogeneous equation that must be solved. The method propagates solutions outward from the origin with only one matrix inversion required at r = a for each energy to obtain a set of coefficients. Beyond r = a, only a homogeneous solution of coupled integral equations is required.

The linear algebraic method (LAM) of Collins and Schneider (1981) involves the conversion of a set of coupled integral equations to a set of LA equations by imposing a discrete quadrature on the integrals. The resulting set of matrix equations then can be handled by standard linear systems methods. Beyond r = a, they use standard propagation schemes such as the R-matrix propagator of Light and Walker (1976).

The various methods for solving sets of coupled integrodifferential equations have the following advantages or disadvantages. Some of the limitations may be overcome as improved programs are developed. Standard numerical techniques were judged to be inadequate due to limitations of both speed and storage. This resulted in the development of other programs. Programs IMPACT, RMATRX, and LAM are particularly well suited for those electron-ion problems in which the internal region is fairly small. Otherwise, in IMPACT and LAM, too many grid points may have to be used and, in RMATRX, too many basic functions may have to be used to span the


space. Program NIEM does not suffer from this limitation; thus, it may be more useful for calculations involving diffuse excited state cases. In NIEM, however, each additional exchange term or correlation function generates another inhomogeneous equation that must be solved so that the computing time increases rapidly for NIEM. Thus, LAM, IMPACT, and RMATRX may more easily treat elaborate CI target wave functions, since these can produce both many exchange terms and correlation functions. Program RMATRX has an additional advantage in dealing with a case in which calculations are required at many energies, since in the internal region a matrix diagonalization of the ( N + 1)-electron Hamiltonian is performed only once owing to the energy-independent nature of the basis functions. This advantage may be reduced when IMPACT and LAM are run on a vector machine such as CRAY, since matrix inversion is inherently faster than diagonalization and is vectorized more easily.

All the programs can be considered as different numerical techniques to solve the set of close-coupling equations. A point that is sometimes over- looked is that none of the methods contain any more physics than the other methods. All programs can be considered as numerical methods to evaluate the scattering wave function at a boundary r = a, where the exchange and correlation functions can be neglected. Then, any numerical procedure can be used to obtain the reactance matrix K by solving the coupled set of differential equations that are valid for r > a.

In the past few years, much of the effort in speeding up the programs has concentrated on techniques to evaluate more rapidly the angular momentum algebra and radial integrals involved in constructing the various interaction potentials, and in the solution of the coupled differential equations in the asymptotic region.

111. Convergence of the Close-Coupling Expansion

In the close-coupling expansion, Eq. (3), the total wave function of the atom and free electron is written in terms of a sum of the wave function of the target system. This sum includes a sum over all bound state wave functions of the target and also an integral over the continuum states of the target. Since we can only include a finite number of terms in the expansion of the total wave function, it is important to study the convergence of cross sections as the number of terms in the expansion is increased.

Few systematic studies of the convergence of the close-coupling approximation have been carried out. With increased computing power, however, it is now becoming possible to study this convergence. Most of these studies


have been carried out on atomic hydrogen and helium, and we will concentrate on these atoms.

In this work, it is convenient to write the cross section Q(i +j) for a transition from a lower state i to an upper state j in terms of the dimensionless and symmetric effective collision strength

Q(i - j ) = wi k'Q(i + j ) (10)

where Q(i + j) is in mi, wi is the statistical weight of the state i, and k' is the wave number of the incident electron. For a Maxwellian distribution, the effective collision strength is given by

where E j is the energy of the scattered electron, k is Boltzmann's constant and T is the electron temperature in OK. If E , is in eV and T is in "K then Ilk = 11604.52. This then gives the electron excitation rate q(i + j ) as

(12) 8.629 x

exp( kl)y(i - Eij -+ j ) cm3 s- 1 q(i + j) = wi Tl i2

and the electron de-excitation rate q( j -+ i) as

8.629 x q ( j + i) = y(i - j ) cm3 s- '

w j T ' I 2 (13)

where E i j is the energy difference between state i and state j.

A. RESULTS FOR ATOMIC HYDROGEN

For the electron energies, 0.75 5 E I 0.888 ryd, the excitation cross sections for the 1s - 2s and 1s - 2p transitions in hydrogen have been calculated to a high accuracy by Taylor and Burke (1967) using a total wave function with twenty correlation terms. H and He' (Burke and Taylor, 1969) are the only systems for which we have such accurate excitation cross sections. By comparing their accurate results with cross sections obtained from a three-state ( 1 s, 2s, 2p) and a six-state (Is, 2s, 2p, 3s, 3p, 3d) close- coupling expansion (Taylor and Burke, 1967), they were able to study the convergence of the close-coupling results. Their results are given in Figs. 1 and 2 where we also include the results from a recent fifteen-state (n =

1,2,3,4, and 5) calculation (Pathak et al., 1988a). In the case of the 1s 4 2s cross section there is good agreement between the three- and six-state calculation over a limited energy range from 0.7 to 0.83 ryd; below and above this energy range the three-state results are higher than the six-state results.

274 R. J . W! Henry and A. E. Kingston

.- W 0.1 1 1

0.75 0.80 0435 Electron Energy ( ryd)

FIG. 1. Cross section for electron excitation of the 1s -+ 2s transitions in atomic hydrogen. Theoretical close coupling calculations: with fifteen states ---; with six states - -; with three states - -- -. Accurate calculations of Taylor and Burke (1967) -.

Electron Energy (ryd) FIG. 2. Cross section for electron excitation of the 1s -+ 2p transitions in atomic hydrogen

Theoretical close coupling calculations: with fifteen states ---; with six states - -; with three states -. Accurate calculation of Taylor and Burke (1967) -.


The inclusion of the n = 4 and 5 states in the fifteen-state calculation only reduces the cross section somewhat at both high and low energies. We would not expect that the inclusion of higher bound states in the close-coupling expansion would change the cross section significantly. Comparing the 1 s + 2s close coupling results with the very accurate correlation calculations of Taylor and Burke, we find that below 0.85 ryd the fifteen-state results are in error by approximately 8 %, and the six-state results are only slightly less accurate, but the three-state results can be in error by up to 25 % at high and low energies.

The convergence of the close-coupling expansion for the 1s + 2p transition is not as obvious as that for the l s + 2 s transition. Unlike the l s - + 2 s transition, the 1s -, 2p results for the three- and six-state calculations are not in good agreement even over a limited energy range; the cross sections differ by between 10 and 20%, with the largest differences a t low energies. The fifteen-state calculation reduces the cross section by a further 7 % at high and low energies, but by only 4 % at 0.8 ryd. From these calculations, it is difficult to judge if the 1s + 2p cross sections will decrease further if further bound states are considered. Preliminary calculations from a ten-state (n = 1,2,3, and 4) R-matrix calculation, however, (Pathak, Berrington, and Kingston, 1988a) lie only slightly above the fifteen-state calculation. We conclude that the inclusion of further higher states will not alter the 1s + 2p cross section significantly in this energy region. Comparing the 1s + 2p close-coupling results with the accurate correlation results, we find that the error in the fifteen-state results is small at low energies, but increases as the energy increases, the error being 3 %, 10 %, and 15 % at 0.76,0.8, and 0.85 rydbergs. At the same energies, the error in the six-state calculation is 12 %, 16 %, and 17 %, respectively, and the error in the three-state calculation is 45 %, 25 %, and 30%.

Since the results of Taylor and Burke (1967) for the excitation of the 2s and 2p states are very accurate, and the fifteen-state results represent most of the contribution to the cross sections from the bound state wave functions in the close-coupling expansion, the difference between the Taylor and Burke results and the fifteen-state results must represent the contribution from the integration over the continuum in the close coupling expansion. For the 1s -+ 2s transition the contribution from the continuum is fairly constant at about 0.07 mi, except at very low energies. The contribution from the contiuum to the Is -+ 2p cross sections is very small at low energies, but increases to a maximum of approximately 15% at the highest energy considered here.

Some allowance can be made for the contribution from the contiuum by including suitable pseudo-states in the close-coupling expansion. For example, Burke 3 al. (1969) have shown that the 1s-2s-2p results for low energy

276 R. J . W Henry and A . E. Kingston

elastic scattering can be improved greatly by the inclusion of a 5 state which is chosen so that the free electron moves in the correct long range polarization potential. Pseudo-state calculations were also carried out for the 1s + 2s and 1s + 2p excitation cross sections at low (Geltman and Burke, 1970) and intermediate energies (Burke and Webb, 1970), but it is difficult to judge the accuracy of these calculations. Some information on the accuracy of pseudo- state expansions has been obtained by Poet (1978) who has derived an accurate solution for a simplified calculation on 1s -+ 2s electron scattering by hydrogen. He compared his results with results obtained for the same problem by Burke and Mitchell (1973) using pseudo-state expansions. Poet concluded that with one pseudo-state the ls-2s cross section was accurate to about 20% but extra pseudo-states did not improve the results because of wide pseudo-resonances. Burke et al. (1981) have shown, however, how T- matrices can be smoothed over pseudo-resonances to give smoothed cross sections. It is clear that future calculations will use pseudo-states to account for continuum contributions to the close-coupling expansion.

B. RESULTS FOR ATOMIC HELIUM

The first R-matrix calculation on the electron excitation of helium considered the 1 IS, 23S, 2'S, 23P, and 2lP states (Berrington et al., 1975). This was followed by an eleven-(n = 1,2 and 3) state calculation (Berrington et al., 1985) and by a nineteen-(n = 1,2,3 and 4) state calculation (Berrington and Kingston, 1988a). Each of these calculations represented the biggest calculation of its type that could be carried out with the available programs and computers. A study of these calculations can give us a good indication of the convergence of the close-coupling expansion for the bound state targets, but it tells us nothing about the contribution to the close-coupling expansion from the continuum. Only in the case of hydrogen ls-2s and ls-2p can we estimate the importance of the continuum from the correlated wave function calculations and also from pseudo-state calculations, but such calculations are not yet available from helium.

I . Transitions in Which An 2 1

If the energy between a group of states is small compared with the energy to all other states, we would expect that the close-coupling expansion should converge rapidly for transitions between these close-lying states. For atomic hydrogen, the transition energy for the 1s and 2s or 2p states is much larger than the energy between the n = 2 and higher states. Hence, we ebtain rather


0.02 li:l 0.01

18 20 22 24 26 28 30 32

Electron Energy (eV1 FIG. 3. Cross sections for electron excitation of atomic helium (a) 1's + 2% and (b)

1 ' S + 2's. Theoretical close coupling calculations: with nineteen states -; with eleven states - -; with five states . . . .

slow convergence of the close-coupling expansion. Similarly, for An 2 1 transitions in atomic helium, we would also expect slow convergence of the close-coupling expansion. But as states with the same n lie close together, we would expect that for An = 0 transitions, the close-coupling expansion will converge rapidly.

Figures 3 and 4 compare the 1 'S + 23S, 2lS, 23P and 2lP cross sections obtained from the five-, eleven-, and nineteen-state close-coupling calculations. This suggests that the close-coupling expansion converges quite rapidly.

If, for example, we consider the 1's -+ 23S cross section, the five-state calculation agrees with the eleven- and nineteen-state calculation up to the emergy of the 2' P state, the highest state included in the five-state calculation. Above the 2l P threshold, the five-state results differ considerably from the other calculations. Similarly, the eleven-state calculation agrees with the nineteen-state calculation up to the energy of the 3lP state, the highest state included in the eleven-state calculation. Even above the 3 l P threshold, however, the eleven- and nineteen-state calculations give results that are quite close. A similar pattern occurs from excitation from the 1 'S to the 2'S, 23P, and 2l P states with the five-state calculation giving good agreement with the larger calculation up to the 2lP threshold. This region of agreement is gradually decreased as we go from the 2's to 23P excitation. For excitation to the 2'P, the five-state results do not even agree with the larger calculations at the 2l P threshold.

R. J . W Henry and

w 008 - -

(a) 1k-23~ 0

-

20 22 24 26 28 30 32

A. E. Kingston

20 22 24 26 28 30 32

Electron Energy (evl FIG. 4. Cross section for electron excitation of atomic helium (a) 1 ' S + Z3P and (b)

1 ' S + 2l P. Theoretical close coupling calculation: with nineteen states -; with eleven states -..-..- : with five states ... .

FIG. 5. Effective collision strengths for electron excitation of atomic helium, 1's + 2% and 1 'S + Z3P. Theoretical close coupling calculation: with nineteen states -; with eleven states - - _ : with five states . . . .


For many applications, it is more important to study the convergence of the rate coefficient, Eq. (12) or effective collision strength, Eq. (11). In Figs. 5 and 6, we plot the effective collision strengths for the 11S-23S, 2lS, 23P, 2 lP transitions. For both the 1'S-23S and 2's transitions there is good qualitative and quantitative agreement between the five- and the eleven-state results up to about 10,000K. For the 11S-23P and 2'P transitions up to about 10,000 K, however, the shape of the five-state calculation is different from that of the eleven-state results. The eleven- and nineteen-state results are in good agreement for all four transitions up to 10,000 K, and we expect the inclusion of further bound states in the close-coupling expansion would not make a significant change to the nineteen-state results up to 10,000 K. At the highest temperature considered here, 30,000 K, the inclusion of the nine n = 3 states reduces the 1'S-23S, 2'S, 23P and 2 lP effective collision strengths by 15 %, 25 %, 32 %, and 25 % respectively, while the inclusion of a further eight n = 4 states reduces the same effective collision strengths by a further OX, 6%, 10% and 22%, respectively. Clearly, the results for the l'S-2'P transition have not converged at 30,000 K, but the results for the 1 'S-23S and 2lS transitions have converged, while the 1 'S-Z3P results may only be a few percent from the converged results for bound states.

' . 0 8 ~ j 0.06

0 10000 20000 30000 Electron Temperature O K

FIG. 6 . Effective collision strengths for electron excitation of atomic helium 1 'S --* 2lS and I 'S -+ 2'P. Theoretical close coupling calculation; with nineteen states -; with eleven states - ~ -; with five states . . . ,


Berrington and Kingston (1 988a) have also considered transitions between the n = 1 and ~t = 3 states and also among the n = 2 and n = 3 states. These calculations show similar qualitative trends as those described above for n = 1 to n = 2 transitions. The eleven- and nineteen-state results agree only at very low temperatures; At higher temperatures, the two calculations diverge significantly. This emphasizes the fact that the inclusion of states with principle quantum number n in the close-coupling expansion will not give good results for transitions to these states, but will give good results only over a limited range for n - 1 states. Because of this, Berrington and Kingston

loo00 20000 3OOOO Electron Temperature OK

FIG. 7. Etrective collision strengths for electron excitation of atomic helium, 2's -+ 2'S, 2's -+ 23P and 2% -+ 2lP. Theoretical close coupling calculation: with nineteen states -; with eleven states - --; with five states . . . .


I I

(1988a) did not publish results for transitions to the n = 4 levels, although they may be obtained from their nineteen-state calculation.

/ '

2. Transitions in Which An = 0

For these transitions, we would expect that the close-coupling expansion would converge rapidly. Using the five-, eleven-, and nineteen-state R-matrix results, we can study this convergence. Figs. 7 and 8 display the effective collision strengths for transitions between the four n = 2 levels of helium,

50-

40 -

30 5 -

13) c a, L

Ln

t 0

c

._ 3" - 20- 0 0

.-*-.-.-.-.-.- I / + [ , [ , 0

0 10000 20000 Electron Temperature O K

000

FIG. 8. Effective collision strength for electron excitation of atomic helium, 2% + 23P, 2 ' s + 2 ' P and 2'P + 2'P. Theoretical close coupling calculation: with nineteen states -; with eleven states - - -: with five states . . . .

282 R . J . W Henry and A. E . Kingston

calculated using five-, eleven-, and nineteen-states in the close-coupling expansion. Comparing the five-state results with the eleven- and nineteen- state results, it is seen that the five-state results are reliable at low temperatures for all of these transitions except the 23S-21S.

At higher temperatures, the five-state calculations do not agree with the larger close-coupling calculations, but the convergence for the 23S-21S, 21S-23P and 23S-21P transitions in Fig. 7 differ significantly from those in Fig. 8. The eleven-state calculations for the three transitions in Fig. 7 at T = 30,000 K are considerably lower than the five-state calculations, but they are only slightly higher than the nineteen-state calculation. We can conclude that the addition of further bound states in the close-coupling expansion will not change the effective collision strength significantly. For the 23S-23P, 2'S-2'P, and 23P-21P transitions, however, (Fig. 8), the nineteen-state calculations are so much smaller than the eleven-state calculation at T =

30,000 K that we cannot claim that the calculations have converged.

IV. The Effect of Resonances on Electron Excitation Rates

The first identification of resonance states in helium was made more than sixty years ago (Compton and Boyce, 1928; Kruger, 1930) when broadened emission lines in the ultraviolet were attributed to these quasi-bound states. Shortly afterwards, in 1934 Whiddington and Priestly observed resonances in electron collision experiments. Rudd (1964) has observed the same resonances in heavy particle collision experiments.

Early electron excitation calculations using such simplified approximations as the Born approximation or distorted wave approximation were unable to take account of these resonances. The first realistic electron excitation calculations were carried out in the 1960s by Burke and his colleagues (Smith and Burke, 1961; Burke and Schey, 1962; Smith et al., 1962; Burke et a!., 1967; Ormonde et al., 1967) using the close-coupling expansion. These calculations on the electron excitation of H and He' were the first calculations to show the importance of resonances in electron excitation.

With increasing computing power and further developments in collision theory, it has been possible to extend these calculations to a large number of systems. In this section, we will review recent studies of resonances in the electron excitation of hydrogen and helium. These two atoms provide a useful basis to study other atoms and ions; we will show that for helium a finite number of resonances can make significant changes to electron excitation rates but for hydrogen an infinite number of resonances only changes electron excitation rates by a small amount.


A. Resonances in Electron-Hydrogen Scattering

There have been a very large number of calculations in the electron excitation of atomic hydrogen. This is partially due to the importance of hydrogen in astrophysics, and also partially due to the fact that for atomic hydrogen the wave functions of the atomic target states are known exactly.

One of the earliest achievements of close coupling calculations in electron hydrogen scattering was the identification of the ' S resonance which is about 0.7 eV below the n = 2 excitation threshold (Smith et al., 1962). Since that time there have been extensive studies on the positions and widths of resonances in hydrogen (Pathak et al., 1988b) and these have been confirmed in a small number of cases by measurements (Warner et al., 1986 and Williams, 1976). A paper by Pathak et al. (1988b) reports results from a fifteen-state close-coupling calculation in which all of the n = 1,2,3,4, and 5 states are included. This paper lists six resonances below the n = 2 threshold, thirty-one resonances between the n = 2 and n = 3 thresholds, and ninety- four resonances between the n = 3 and n = 4 thresholds. This list is not exhaustive, since it is assumed that all levels of hydrogen with the same principle number have the same energy. Hence, the collision calculations will give resonance series which have an infinite number of resonances. Gailitis and Damburg (1963a,b) have shown that for a resonance series that converges to a particular threshold,

where R is a constant for the series, r n and Tn+ are the widths of the nth and (n + 1)th resonance in the series, and en and en+ represent the energies of the corresponding resonances below the threshold. The ratio R for different resonance series may be obtained from the eigenvalues of a matrix that can be constructed from the coefficients of the centrifugal and dipole terms in the close-coupling equations between the channels opening at the threshold under consideration. For the n = 2, 3, and 4 thresholds, the values of R have been obtained by Gailitis and Damburg (1963a,b) and by Herrick (1975). There is very good agreement between these values of R given and the values of R obtained by Pathak et al. (1988b) from their fifteen-state close-coupling results.

Although atomic hydrogen has this rich array of resonances, it is not necessary that these resonances will contribute significantly to electron excitation rates. For example, in atomic hydrogen with widest resonance is the 'S' resonance at 9.56 eV which has a width of 0.47 eV. It only arises in elastic scattering and it is at such a high energy it would only have a very


small effect when the cross section is integrated with a Maxwellian distribution to give an elastic scattering rate.

At low electron energies, the cross sections for the electron excitation of 2s and 2p states of hydrogen from the ground state are dominated by two features (see Figs. 1 and 2). Unlike any other atom, these excitation cross sections for hydrogen are not zero at the excitation threshold and the cross sections have a ' P" resonance of width 0.0023 ryd at an energy of 0.001 34 ryd above the excitation threshold. It has been suggested by Callaway and McDowell (1983) that for low electron energies (0.75 I E I 0.85 ryd), the excitation cross sections (in nag) can be written in the form

Q(ls + 2s) = 1.3 x 6 ( E - E,) + 0.13 + 0.89(E - 0.75) (15) and

Q(ls + 2p) = 1.6 x 6 ( E - E,) + 0.16 + 2.O(E - 0.75) (16)

where E is the energy of the incident energy in rydbergs, E , is the position of the resonance and 6 is a delta function. In Eq. (15) and Eq. (16) the first term represents the 'Po resonance, the second term the finite threshold, and the third term the threshold increase that is proportional to the energy of the electron after the collision. The excitation cross sections proposed by Callaway and McDowell (1983) have been integrated over a Maxwellian distribution by Aggarwal (1983) to give the effective collision strengths (Eq. (11)) given in Fig. 9. In his calculation, Aggarwal did not include the

-

51.0 - Ol c 2! c v , -

c 0 in .- .-

QI - 0.0. I

1.0 2.0 3 0 4.0 5.0 Log (Temperature)

FIG. 9. Effective collision strengths for electron excitation of atomic hydrogen 1s + 2s and 1s -+ 2p. Theoretical results of Aggarwal(1983): without 'Po resonance -; with 'Po resonance

~~


contribution from the lP'' resonance. We have estimated the contribution from this resonance by integrating the tabulated results of Taylor and Burke (1967) over the resonance.

For both of the transitions, the effective collision strength y tends to a nonzero value at low temperatures; this behaviour is common for ions, but for all atoms except hydrogen the effective collision strengths tend to zero at T -+ 0. If we exclude the resonance contribution, the increase in y(ls + 2p) and y(ls -, 2s) with temperature is common for both atoms and ions with the y for the optically allowed l s + 2 p transition varying as log T at high temperatures, while the y for the optically forbidden transition Is -+ 2s tends to a constant value at high temperatures. The maximum contribution of the 'Po resonance to y(ls -+ 2p) occurs at 300 K; this temperature is approximately the energy above the excitation threshold where the resonance has its maximum value (0.001 7 ryd). The resonance increases the effective collision strength by about 60 % from 0.34 to 0.54. To transform this effective collision strength to an excitation rate coefficient, Eq. (12), we must multiply by exp( - E,/kT) at T = 300 K. This gives such a small rate coefficient that an underestimate of 60 % would have little practical consequences. In contrast, for the de-excitation rate, Eq. (13), we do not have the exponential factor and the effect of the resonance would be quite important.

The excitation cross sections to the 2s and the 2p states have an infinite number of resonances converging to the n = 3 and higher thresholds, but these resonances are so narrow and lie so far above the excitation threshold that they will not contribute significantly to excitation rates. Similarly, excitation rates to higher states of hydrogen and between excited states of hydrogen will not, in general, be greatly affected by these narrow resonances. They may be affected significantly, however, by resonances that lie just above an excitation threshold. To date, the 'Po resonance at 0.75134 is the only resonance that is known to lie close to a threshold. Further calculations may reveal other resonances that lie close to thresholds.

B. RESONANCES IN ELECTRON-HELIUM SCATTERING

The theoretical electron excitation cross sections of hydrogen are unique and they differ in many ways from the cross sections for other atoms. The theoretical calculations take the energy levels with the same principal quantum number to be equal. Thus, the excitation cross sections for atomic hydrogen are nonzero at the excitation threshold, and there can be an infinite number of resonances in a resonance series.

The excitation cross sections of atomic helium are much more like other atoms than atomic hydrogen. Excitation cross sections in helium tend to zero


at the excitation threshold, and there are only a finite number of resonances between the excitation thresholds.

In Table I, we list the position and width of some of the resonances of helium that were obtained in a nineteen-state close-coupling calculation (Berrington and Kingston, 1988b). The 'S' resonance at 19.375 eV is only 0.0117 eV wide: it is too small and at too high an energy to contribute significantly to the rate for elastic scattering. The 2P0 resonance at 20.14 eV, however, lies so close to the 2% threshold (19.82 eV) and is so wide, 0.497 eV, that it has a very large effect on the excitation cross section to the 2% state. This is illustrated in Fig. 3 where we plot the five-, eleven- and nineteen-state results of Berrington et al. (1975), Berrington et al. (1985), and Berrington and Kingston (1988a). The resonance is seen as a very large peak just above the excitation threshold; this peak is followed by a second smaller peak, the 'De resonance which lies at 20.89 eV and is 0.5 eV wide. This 'De lies just above the 2% threshold and, in Fig. 3, it is seen that this resonance has a very marked effect on the excitation cross section from the ground state of helium to the 2% state.

In Fig. 4 we also plot the five-, eleven- and nineteen-state results for the excitation cross sections from the ground state of helium to the 23P and 2lP states. It is seen that the cross section has a change of slope just above the threshold. This is most marked in the eleven- and nineteen-state calculations.

TABLE I RESONANCES IN ATOMIC HELIUM

Theoretical Experiment"

e--He Energy Width Position Width Position symmetry levels (meV) (eV) (mev) (ev)

2Se 11.67 19.375 2% 19.82

2's 20.62 P" 497 20.14 780 20.27

2Se - 20.62 2De 495 20.89

z3P 20.87 2'P 21.22

2s' 26 22.439 36 22.47 2PO 39 22.606 38 22.64 2De 45 22.645 20 22.70 2P0 15 22.715 22.79 2Se 1 22.716 22.72

2Fo 53 22.849 2De 37 22.865

3% 22.72


2Se P"

2D' 2F0 2Se

2D' 2PO

F" 2G' 2Ds 2 Po 2F0

2Sc 2Fo

Po 2DC 2Fo 2De 2Ge 2Sc Po

2De 2F" 2Gc

2Se 2Dc 2 p"

2Ho 2Se

2De 2F" 2De 2F"

2Ge 2 Po F"

2 S e 2Gc

2De 2PO

7 10

26 60 5 6 5

12 24 22 33 4

3's 22.92

33P 23.01

33D 23.074 3 lD 23.075 3'P 23.09

154 3

43 53 31 37 28 5 3

2 20 40 14 5

12 4

0.8

4 4 4 6 7 7 7 2 3 2

0.3

4% 23.59

4 ' s 23.674

43P 23.71

22.875 18 22.89 22.9 14

22.934 20 22.93 22.952 32 22.99 22.956 23.00 23.00

23.019 23.028 23.056 23.059 23.073

23.434 50 23.445 23.474 23.483 40 23.53 23.486 23.567 23.570 25 23.57 23.579 23.581 23.584

23.594 23.607 23.625 23.63 1 23.660 23.664 23.670 23.671 18 23.67

23.675 23.676 23.683 23.686 23.693 23.698 23.703 23.703 23.703 23.705 23.706

' Brunt et al. (1977).

288 R. J . W Henry and A . E. Kingston

The nearest resonance that lies above the 23P threshold could not have a significant effect on the threshold cross section, as it is only 0.026 eV wide and is 1.4 eV above the z3P threshold. The change in slope of the cross section appears to be associated with the 2'P threshold. The 2lP cross section has a linear threshold behaviour and does not appear to be altered by the 'De or the 'Se resonances, which are, respectively, 0.3 eV below and 1.22 eV above the 2lP threshold. For the five-state calculation, the linear dependence in the 1 'S-2'P cross section extends to 3 or 4 eV above the threshold.

The cross section in Figs. 3 and 4 have been integrated over a Maxwellian distribution to give effective collision strengths, Eq. (1 l), for excitation of the 23S, 2lS, 23P, and 2lP states of helium from the ground state. These are displayed in Figs. 5 and 6. At low temperatures, the collision strengths to the 23S and 2lS states are significantly larger than those to the 23P and 2'P states. The enhancement of these effective collision strengths for the 23S and 2% states is mainly due to the two large low-lying resonances. The 2P0 resonance lies 0.32 eV above the 23S threshold and produces a broad maximum in the effective collision strength at T = 5000 K. Similarly, the 'De resonance which lies 0.27 eV above the 2lS threshold adds a large contribution to the 2lS effective collision strength, centred at T = 5000 K. This resonance contribution added to a linear background gives the effective collision strength as plotted in Fig. 5. Without resonances, we would expect these effective collision strengths to be approximately the same size as those for the 23P and 2lP states obtained with the five-state calculation.

At low temperatures, the shape of the 23P effective collision strength is determined by the change in slope of the excitation cross section at the 2'P threshold. The almost linear behaviour of the 2l P effective collision strength suggests that resonances are not important for this transition.

Close-coupling results for transitions between the n = 2 levels of helium have been given by Berrington et al. (1985, 1988a). The only resonance that should be seen at low energies in these cross sections is the large 'De resonances at 20.89 eV. This is because 'Po resonance lies below the 2% threshold, and the higher resonances lie too far above the n = 2 levels to be important. The 23S-21S cross section is found to have a large peak at low energies; this is associated with the 2De resonance and greatly enhances the 23S-21S effective collision strength (Fig. 7). The effective collision strength for the two optically allowed transitions, 23S-23P and 2lS-2lP, are almost linear over a large temperature range (Fig. 8), and so they cannot be greatly affected by resonances. The other spin forbidden transitions, 23S-21P, 21S-23P, and 23P-21 P, have cross sections that rise very steeply from the threshold and this is probably associated with the 'DC resonances.

Berrington et al. (1985) and Berrington and Kingston (1988a) have also calculated electron excitation cross sections for transitions from the n = 1


and n = 2 states of helium to the n = 3 state. Table I shows that there are a large number of small resonances lying below the n = 3 and n = 4 levels. A study of the eleven-state results of Berrington et d. (1985) shows that the three cross sections from the l'S, 2%, and 2's states to the 33S state are dominated at low energies by the resonances lying between the 3jS and 3lS levels. Similarly, the transitions from the l'S, 23S, 2lS to the 3lS states are greatly affected by the resonances between the 3lS and 33P thresholds. A similar effect is seen for the 33P state, but as we go to the higher 33D, 3lD, and 3'P states, the resonance effects tend to become less pronounced.

Results for transitions to higher states of helium have not yet been studied in detail, but we would expect a similar pattern as for the n = 2 and n = 3 levels with the cross sections being altered by a large number of resonances lying between and below the levels. Since the energy gap between the states with the same principle quantum number will become smaller, however, we may find that the effect of resonances on the effective collision strengths will also become smaller.

Although at present we have only detailed results for atomic hydrogen and helium, we would expect that the large effects that resonance has on the excitation rates of helium are typical for atoms. Resonance that will contribute significantly to excitation rates will have to be large and lie close to the excitation threshold. There is as yet no simple method of determining the position and widths of these resonances for an arbitrary atom. It will be necessary to carry out calculations for each atom if we are to obtain low energy electron excitation rates accurately.

V. Inner Shell Excitation Autoionization

Ionization by electron impact is a complex process, since a large number of mechanisms can cause the ejection of electrons from bound states of atomic systems. The direct ionization cross section becomes progressively smaller as the charge state of a given ionized atom increases. Thus, indirect pathways to ionization begin to compete with direct ionization and even to dominate the ionization of many highly charged ions. The most important indirect ionization mechanism is the excitation-autoionization process. In it, the incoming electron excites an inner-shell electron, leaving the ion in a core- excited state that can subsequently lose its energy by ejection of a more loosely bound electron from an outer shell. An example of this process is

e + 2p63s -+ e + 2p53s3p

+ e + 2p6 + e.

290 R . J . W Henry and A. E. Kingston

Another significant contribution to electron impact ionization comes from the temporary capture of the incident electron with simultaneous excitation of an inner-shell electron. An example of this resonance-excitation-double autoionization (REDA) process is

e + 2p63s -+ 2p53s3p31

+ 2p53s2 + e

--f e + 2p6 + e.

The quasi-bound excited state of the complex may also decay by auto- double-ionization, in which two electrons are ejected simultaneously. An example of this resonance excitation auto-double-ionization (READI) process is

e + 2p63s + 2p53s3p31

+ e + 2p6 + e.

Mechanisms that involve resonances and that may appear to be exotic pathways to ionization have been predicted to make measurable contributions for some ions. A particular inner-shell excitation can produce an abrupt jump in the ionization cross section at the threshold for the excitation process, where the cross section is finite and often at a maximum. Thus, careful measurement of the energy dependence of an ionization process can provide quantitative information about excitation processes as well. The resonances that converge on the newly opened threshold, however, sometimes mask the step-like features. Further, the finite width of the energy resolution in the experiment also smooths out some of the predicted behaviour.

The lithium isoelectronic sequence has been a prime candidate for extensive study by experiment and theory. Experimentally, there are few metastable ions in the beams; emission lines are prominent also. Theoretically, the electronic structure is simple and target wave functions may be represented simply and accurately. Assuming that the ionization calculations of Younger (1980, 1981a, 1981b) (slightly renormalized in some cases) provide the best direct ionization values, excitation cross sections can be extracted by subtracting the direct ionization from the measured total ionization cross section. The resultant 1s’ 2s + ls2s21 excitation cross section can be compared with theory. The first measurements on ionization of Li-like ions of C IV, N V, and 0 VI were made by Crandall et al. (1978, 1979). Subsequently, measurements have been made on Be I1 by Falk and Dunn (1983) and on B I11 and 0 VI by Crandall et al. (1986). Excitation calculations in a six-state close-coupling approximation have been made by Henry (1979) for C, N, and 0, and reported in Crandall et al. (1986) for Be and B.


TABLE I1 TOTAL EXCITATION CROSS SECTIONS

AT THRESHOLD FOR

Is2 2s- 1 ~ 2 ~ 2 1 IN cmZ

Six-state close-coupling Experiment

u.b

Be I1 9.3 20. f 8. B I11 4.1 4.0 & 1.0 c IV 2.24 2.3 f 0.7 N V 1.27 1.6 & 0.4 0 VI 0.74 0.8 k 0.3

Crandall et nl. (1986). Henry (1979).

Table I1 gives the total excitation cross secton at threshold for 1s’ 2s -, ls2s21 in lo-’’ cmz obtained in a six-state close-coupling calculation and deduced from experiment. Excellent agreement is found between theory and experiment, except for Be 11. The shape of the cross section versus energy also is found to be in very good agreement, as can be seen from an example for B I11 given in Fig. 10. There, the points represent measurements of Crandall et al. (1986). A semiempirical formula by Lotz (1968, 1969) and the distorted wave prediction by Younger (l981a) are shown for comparison. The inset shows the energy region where excitation-autoionization should contribute. Within the inset, the Younger theory of direct ionization has been multiplied by 0.90 and the arrows indicate the energies for inner shell excitation of a 1s electron to the nl orbital indicated. The upper curve in the inset adds the six-state close-coupling calculations of excitation of 21 substates to the scaled direct ionization.

For the Be I1 case, the discrepancy between the deduced experimental cross section and the coupled-state calculations may be indicative of relatively stronger coupling between states in this lowest charge state. In this case, the off-diagonal terms in the reactance matrix are relatively large compared to those found in other Li-like systems. This may indicate a need to include more coupled states in order to converge the close-coupling expansion.

Cross sections for 0 VI are given in Fig. 11. This figure is similar to Fig. 10, with the addition that the open circles represent a less accurate experiment reported by Crandall et al. (1979). Also, Younger’s direct ionization calculation has been multiplied by 1.07 to obtain the best fit to data between 400 and 550 eV. A significantly better fit to all of the experimental data below 440 eV, however, is obtained without any renormalization of Younger’s results. This

292

8

7

6 - (u

5 5

$ 4

f 3

s -

2

I

0

R . J . W Henry and A. E . Kingston

ORNL-DWG 8340842R2

FIG. 10. Electron impact ionization of B 111. Points are measurements of Crandall et al. (1986). The Lotz (1969, dashed curve) prediction and the distorted wave prediction of Younger (1981a, solid curve) are shown for comparison. Within the inset, the Younger theory of direct ionization has been multiplied by 0.90 and the arrows indicate the energies for inner-shell excitation of a Is electron to the nl orbital indicated. The upper curve in the inset adds six-state close-coupling calculations of excitation of 21 substates (Crandall et at., 1986) to the direct ionization.

would then suggest that the feature in the data near 440 eV may be due to decay via auto-double-ionization (READI). Thus, there is some ambiguity in sequences in which inner-shell excitation-autoionization can occur.

Several assumptions underlie the comparison of experiment with theory. One is that there is no inteference between the direct ionization and indirect excitation-autoionization channels. Another is that all the intermediate complexes decay via autoionization. The former has been tested by Jakubow- icz and Moores (1981). They found no appreciable interference modification of the total cross sections on allowing for both channels in the target wave functions. In the latter case, radiative stabilization becomes important only for heavier ions. Branching ratios for autoionization are close to unity for the light ions considered by Crandall et al. (1986).

In 1965, Goldberg et al. suggested that the excitation-autoionization process might be of importance in highly ionized atoms. This is especially


ORNL-DWG 85-11648R

I , I I ' " 1 T 10

0

05+

100 200 500 1000 ELECTRON ENERGY (a)

FIG. 11. Electron impact ionization of 0 VI. Solid points are measurements of Crandall et a/. (1986). Open circles are experiment from Crandall ef a/ . (1979). The distorted wave prediction of Younger (1981, solid curve) is shown for comparison. Within the inset, the Younger theory of direct ionization has been multiplied by 1.07 and the arrows indicate the energies for inner-shell excitation of a 1s electron to the nl orbital indicated. The upper curve in the inset adds six-state close-coupling calculations of excitation of 21 substates (Henry, 1979) to the direct ionization.

probable for the Na-isoelectronic sequence, where there is one easily remov- able 3s electron to contribute to direct ionization, but there are eight L-shell electrons that can contribute readily to inner-shell excitation. Goldberg et al. considered contributions from only 2p63s + 2p53snd transitions in Fe XVI. On assuming that all of the excited ions would autoionize, they found that this indirect contribution was about twice as large as direct collisional ionization. On including other indirect ionizations, but still assuming unit branching ratios, Bely (1967) found that the inner shell excitation process was about five times as important as the direct one. Cowan and Mann (1979) showed that excitation-autoionization effects would still dominate total ionization for Fe XVI, even after allowing for radiative decay as a competitor to autoionization. La Gattuta and Hahn (1981) made an even more

294 R. J . K Henry and A. E. Kingston

0.5 I I I I

Electron Energy (eV) FIG. 12. Electron impact ionization of Fe XVI. The dashed curve is direct ionization (Lotz,

1969); the solid curve is excitation-autoionization added to the direct part: the cross-hatched area is REDA added to the two other contributions (La Gattuta and Hahn, 1981).

surprising prediction that resonance-excitation double ionization (REDA) would dominate the total ionization over a relatively small energy range as is given in Fig. 12. Direct ionization is given by the dashed curve; the solid curve is excitation-autoionization added to the direct part and the cross- hatched region is REDA, averaged over 20 eV energy bins, and added to the other two contributions.

Considerable progress has been made experimentally on ionization cross sections for the Na isoelectronic sequence. Martin et al. (1968) reported work on Mg I1 which showed no evidence of excitation-autoionization. Crandall et al. (1982), however, reported measured absolute cross sections for Mg 11, A1 111, and Si IV. Their ionization data showed some anticipated excitation- autoionization structure in addition to a smooth direct ionization contribution. For S VI, C1 VII, and Ar VIII, Howald et al. (1986) found that the background count rate, due to the presence of autoionizing metastable ions in the ion source, may be large enough to prevent the accumulation of meaningful statistics. This effect was expected to decrease with increasing charge, however, and does not appear to be a problem for Fe XVI. Gregory et al. (1987) reported absolute cross sections for Fe XVI for the energy range from 630 to 1000 eV with an energy spread in the beam of 2 eV.

Figure 13 compares measurements for Fe XVI of Gregory et al. (1987) with a distorted wave calculation of Griffin et al. (1987) and a twelve-state close-


0.4

0.3

0.2

0.1

(u - E Oa5* _c ' - ,-

-

- -

- - - -

V OD 0

Electron Energy (eV) FIG. 13. Electron impact ionization of Fe XVI. The dashed curve is direct ionization (Lotz,

1969); the dot-dashed curve is excitation-autoionization calculated in a distorted wave approximation (Griffin et al., 1987) added to the direct part; the solid curve is a twelve-state close- coupling calculation by Tayal and Henry (1987) which has been added to the direct part.

coupling calculation of Tayal and Henry (1987). The indirect contributions have been added to the direct ionization results of Lotz (1969). Calculations of Tayal and Henry (1 987) include excitation-autoionization and the effects of REDA averaged over 2 eV energy bins. In contrast to the predictions of La Gattuta and Hahn (1981), the large enhancement of the cross section due to the REDA process is neither observed nor present in the close-coupling calculation. The measurements do indicate, however, that excitation- autoionization contributes approximately four times the direct cross section. This is in good accord with predictions based on measurements through Si IV by Crandall et al. (1982). Their studies revealed that the relative importance of indirect processes increased with increasing charge.

For A1 111, distorted wave calculations of Griffin et al. (1982) indicated that the largest contributions to excitation-autoionization were due to the 2p63s + 2p53s3p excitations. This largest predicted step, however, effectively is absent in the measurements of Crandall et al. (1982). Henry and Msezane (1982) showed that this apparent discrepancy can be explained by the REDA mechanism. They performed a three-state close-coupling calculation to estimate the effects of resonances. They included the 2p63s, 2p53sz, and 2p53s3p states and examined the energy region below the 2p53s3p threshold. Fig. 14 illustrates the position and shape of some of the resonances that have an inner-shell vacancy and that can decay via double autoionization resulting

0.8

0.4

0.2

0.0

1 1

3s) 5.5 5.6 f

1 -I ,GAILITIS AV

/ E;,(3p)

70 80 90 I00 I10 E (eV 1

FIG. 14. Electron impact ionization of Al 111. Upper Jigure: Model calculations of resonances (Henry and Msezane, 1982). These resonances decay dominantly by double autoionization. The Gaussian average is with an electron energy spread of 2eV. Lowerfigure: Lower solid curve is distorted wave calculation of direct ionization (Younger, 1981), normalized to experiment at 70 eV by multiplying by 0.65. Dashed curve is distorted wave excitation of Griffin et a!. (1982) added to Younger's scaled results. Upper solid curve is close-coupling calculation of Henry and Msezane (1982) added to Younger's scaled results. Dash-dotted curve is estimated close-coupling result including both excitation-autoionization and resonances like those of the upper figure. Open circles represent measurements of Crandall et at. (1982).

296


in net ionization of A1 111. In this case, the effect of these resonances is to smooth out the expected, abrupt jumps at the inner-shell excitation thresholds. In the energy range 73-79 eV (5.4-5.8 ryd), the resonances provide a small enhancement of the ionization cross section which obscures the 2p-3p threshold. Fig. 14 gives also a comparison of calculated and measured cross sections for A1 111. The lower solid curve is a distorted wave calculation of direct ionization by Younger (1981a,b) normalized to experiment at 70 eV by multiplying by 0.65. The dashed curve is a distorted wave calculation of Griffin et al. (1982) added to Younger’s scaled direct ionization results. Upper solid curve represents the two-state close-coupling results of Henry and Msezane (1 982) added to Younger’s direct ionization results. The dash- dotted curve between 73 and 80 eV is an estimated close-coupling result on including both excitation-autoionization and REDA like those of the upper figure.

Studies such as the above for Na-like ions give credence to the concept that the indirect processes should and can be included in calculations of electron impact ionization of ions. It is still a formidable task, however, and the predicted cross sections are of unknown reliability for untested cases.

Although space limitations do not permit a full discussion, we also mention briefly work on the excitation-autoionization contributions to electron ionization of Ca 11, Ti IV, and Mg-like ions. For Ca 11, close-coupling calculations by Burke et al. (1983) and distorted wave ones by Griffin et al. (1984) show strong coupling effects for the excitation 3p64s + 3p53d4s. Agreement with the crossed-beam results of Peart and Dolder (1975) is quite promising, but, in the threshold region, theory predicts large resonance structures which are not yet observed. For Ti IV, close-coupling calculations by Burke et al. (1984) and Msezane and Henry (1985) and distorted wave results of Bottcher et al. (1983) show coupling effects at the 20 % level for the excitation 3p63d + 3p53d2, which agrees reasonably well with measurements of Falk et al. (1983). For the magnesium isoelectronic sequence, close- coupling and distorted. wave calculations have been made by Tayal and Henry (1986) and Pindzola et al. (1986), respectively. As has been found in all systems to date, close-coupling results are lower than those in a distorted wave calculation. In addition to calculating excitation-autoionization from the ground state configuration, however, Pindzola et al. (1986) made calculations from metastable states in the 2p63s3p configuration. It is believed that the experimental results of Howald et al. (1986) for S V and Cl VI are dominated by ionization from these metastables. Agreement between the metastable calculation and experiment is excellent. This possibility of metastables being present in the ion beam presents an additional complication to comparisons between theory and experiment for electron impact ionization of highly charged ions.

298 R . J . W Henry and A . E . Kingston

VI. Resonances in Cu

Scheibner et al. (1987) calculated electron impact excitation cross sections of atomic copper in the 0.1 to 8.0 eV range. They found that the cross sections exhibit a rich resonance structure and that some transitions are dominated in this energy range by resonances. They used target wave functions developed by Msezane and Henry (1986), who compared their four-state close-coupling calculations in the energy range 6 to 100 eV with measurements of Trajmar et al. (1977). Four states were used to describe the target; 3d1'4s, 3d1'4p, 3d94s2, and 3d1'4d.

Calculations of Scheibner et al. (1987) yield many results characteristic of inelastic scattering problems. Fig. 15 gives total and some partial cross sections for elastic scattering of electrons from the ground state of Cu. The 'S partial cross section remains finite at zero energy, where its magnitude, 49n a:, is determined to some extent by the existence of the bound state of Cu-. Fig. 15(b) clearly shows that the total elastic cross section is dominated by the 3P0 symmetry, which exhibits a large peak near 0.3 eV. This is identified as a 3d1'4s4p 3P0 shape resonance of Cu-. An analogous, but much weaker, resonance occurs in the 'Po channel around 0.5 eV (feature "a" of Fig. 15(a)). Two other broad resonances (3dg4s24p)'Po and 3P0 account for the shoulder at 2.3 eV (e.g., feature "b" of Fig. 15(a)). Last, a cusp (feature "c" of Fig. 15(a)) appears in the 'Po partial cross section at the calculated threshold, 3.55 eV, for the (3d''4p)'Po state. A cusp can appear at a threshold when the scattered electron has zero angular momentum.

The 'D partial elastic cross section of the ground state also exhibits a variety of features. At 3.5 eV, the peak labelled feature "e" is identified as being due to a (3d"4p2)'D Feshbach resonance. Just above this resonance, where the 2S-2Po channel becomes open, the 'D partial cross section shows a dip (feature "f") which is due to flux conservation. Last, Fig. 15(a) shows two other broad features in 'D symmetry. The first is a broad (3d1'4s4d) shape resonance, which appears as a shoulder near 2.0 eV (feature "d"). The second is a cusp (feature "g") at the (3d1'4d)'D threshold at 5.93 eV.

For applications such as a copper vapour laser, the electron temperature will usually be below 5 eV, and the relevant momentum-transfer cross section will be dominated by elastic collisions. Consequently, the low energy features that are prominent in the elastic cross section, in particular the (3d''4~4p)'*~PO shape resonance, will be important also for the correct determination of momentum transfer, and hence, for electron transport in copper vapor laser discharges. A more detailed description of the copper atom via use of improved target states will probably lead to more precise positions and widths of some of the resonances. This, in turn, will effect the


20.0

co 16.0 k - E .o 12.0

x g 8.0

*E

CI u

v) v)

0

m -

2 4.0

0.0 0.0 2.0 4.0 6.0 8 .O

Energy (eV)

400

- NO

m 300

I

C 0

u .- + 3 200

g u) v)

0

100

0 0.0 1 .o 2.0 3.0 4.0 5.0 6.0

Energy (eV) FIG. 15. Total and partial cross sections for elastic scattering of electrons from copper. (a)

Partial cross sections: 'S (solid curve); 'P" (short dashed curve); 'D (long dashed curve). Letters label the Cu-resonances and other structures discussed in the text. (b) Total (solid curve) and 3Pa partial (dashed curve) cross sections. Thl, Th2, and Th3 indicate the calculated thresholds for the ( 3 d ' 4 ~ ~ ) ~ D , (3d'04p)ZP9 and (3d'04d)2D states, respectively.


low energy behaviour of the electron scattering cross section. The above information, however, gives a good flavour of the effect of resonances on a scattering process in a heavy atomic system.

REFERENCES

Aggarwal, K. M. (1983). Mon. Not. R. Ast. Soc. 202, 15. Bely, 0. (1967). Ann. d’Astrophys. 30, 953. Berrington, K. A. and Kingston, A. E. (1988a). J. Phys. 8. (in press). Berrington, K. A. and Kingston, A. E. (1988b). J . Phys. B. (in preparation). Berrington, K. A,, Burke, P. G., Chang, J. J., Chivers, A. T., Robb, W. D., and Taylor, K. T.

Berrington, K. A., Burke, P. G., and Sinfailam, A. (1975). J. Phys. B. 8, 1459. Berrington, K. A,, Burke, P. G., LeDourneuf, M., Robb, W. D., Taylor, K. T., and Vo Ky Lan,

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C. McDowell, eds.) Plenum Press, New York, New York.

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 25 I I RECENT ADVANCES IN THE NUMERICAL CALCULATION OF IONIZA TION AMPLITUDES CHRISTOPHER BOTTCHER* Physics Division Argonne National Laboratory Argonne, Illinois

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . 303 11. Formal Solutions of the Stationary Schrodinger Equation . . . . . . . 305 111. The Boundary Function Method . . . . . . . . . . . . . . . . 308 IV. Path Integral and Semiclassical Methods . . . . . . . . . . . . . 31 1 V. Calculations on a Two-Dimensional Model . . . . . . . . . . . . 315

VI. Calculations in Three Dimensions . . . . . . . . . . . . . . . 320 Acknowledgment . . . . . . . . . . . . . . . . . . . . . 321 References . . . . . . . . . . . . . . . . . . . . . . . . 321

I. Introduction

The calculation of electron impact ionization remains an unsolved problem of the greatest theoretical and practical importance. Some years ago Bottcher (1 985) reviewed approaches using wavepackets and the time dependent Schrodinger equation. The present article deals with progress in going beyond the limitations of wavepackets by directly solving the stationary Schrodinger equation. In particular, it is possible to probe the threshold region in great detail without analytic approximations.

This paper will focus on the simplest case where the target is a hydrogen atom

(1)

At impact energies below about 50 eV one must describe the dynamics of two electrons moving freely in a correlated fashion. Direct scattering models such as the Born and distorted wave series are clearly inadequate. The essence of

e + H(ls)-+e + H+ + e.

* Permanent address: Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 3783 1-6373.

303

Copyright Q 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

ISBN 0-12-003825-0

304 Christopher Bottcher

the problem is contained in the Schrodinger equation for the ' S part of the wave function,

+ - -- -- - E Y(rlr2f3) = 0. r: r:) )

The nucleus is supposedly infinitely massive and placed at the origin. The electrons are at distances r1 and r 2 , and subtend an angle 8 at the origin. Theoretical methods may productively be tested using a two-dimensional reduction of Eq. (2),

Of course, Eqs. (2) and (3) must be supplemented by appropriate scattering boundary conditions. Atomic units will be used throughout.

Given a numerical technique for the solution of Eq. (2) or Eq. (3), the first problem to address is the validity of Wannier's predictions on the behavior of ionization cross sections near threshold (Wannier, 1953; Rau, 1971; Peterkop, 198 1). For completeness these predictions will be summarized here.

Let the energies of the electrons in Eq. (1) be written

e(E + I) + H + e(E) + H + + e(E - E ) (4)

where I is the binding energy of H . Then, near E = 0, the cross section can be writ ten

where Q is the total cross section, and 8 is the angle between the two outgoing electrons. Wannier (1953) predicted from classical considerations that

Q(E) 1: C1E1.1269... (6)

and that

wheref(x) N constant for all x(0 < x < 1) and g(8) peaks around 8 =

an opening width with

A8 N C2E'I4. (8)

NUMERICAL CALCULATION OF IONIZATION AMPLITUDES 305

In a series of papers, Temkin (1974, 1980, 1982) has offered a critique of the Wannier theory, which can be summarized as follows. He suggests that as E / E -, 0, P must have a different, rather complicated behavior, determined by quantum mechanics. Furthermore, as E/I becomes small enough, this behavior must dominate the cross section.

This paper will be confined to describing progress on one approach to this problem, making no attempt to review the experimental and theoretical literature. The reader may consult significant papers, e.g. Crothers (1986). The author thinks it is fair to state that no synthesis has yet emerged.

This article represents an attempt to isolate the mathematical essentials of a physically significant problem, and to pursue them wherever they lead. The author believes this philosophy or style characterizes much of the work of Alex Dalgarno, whom we honor as he approaches his 60th birthday.

11. Formal Solutions of the Stationary Schrodinger Equation

We will now establish the formal apparatus needed to calculate scattering amplitudes, which is no more than a generalization of the theory of Frauenhofer diffraction in elementary optics.

Ionization amplitudes are conveniently expressed in hyperspherical coordinates,

rl = R cos a, 1, = R sin a. (9)

For the purpose of exposition, I will concentrate on the model defined by Eq. (3), thus omitting the angle 8. Imagine a wavepacket moving asymptotically so that both electrons are unbound with momenta

The energy above threshold

E = ~ I C ’ = &(kf + k i ) .

It follows from Eq. (10) that the asymptotic hyperspherical angle satisfies

k2

k l tan a = -

whence

k , = IC cos a, k, = K sin a. (13)


The energies of the ejected and scattered electrons in Eq. (4) are then

E = ) k : = E sin2 a

and

E - E = ) k ; = E COS’ a.

In quantum mechanics, the wave function has the asymptotic form, when r , or r2 or both are large,

The incoming wave

4i = F ( T , ) ~ ~ z )

where u is the initial state of the target, satisfying

and P is, in general, a distorted wave, satisfying

1 d2

+ (1 + x l )e ikr l . (18) I F k - 1/2 , - ikrc i The initial wavenumber is given by

k 2 = 2(E + I )

(E - H)Y = 0,

( E - H i ) 4 i = 0,

(19)

and T,, is the elastic (distorted wave) T-matrix. Thus, if Eq. (3) is written as

(20)

(21)

H = H i + K (22)

and the incoming wave satisfies

we can write

where


Returning to Eq. (15), the ionization probability in a segment (a, a + Am) is given by

d P - A@ = I T(a)I2 Aa. da

From Eq. (14), the probability in an energy interval (E, E + AE) is given by

dP A& d P d E [&(E - &)]"2 Z' -A& =

Our object is now to find the asymptotic form of the scattered part of Y , defined as

A T = Y - q$

( E - H) AY = V , + i .

(26)

(27)

which satisfies

Formally, Eq. (27) is solved in terms of the outgoing Green function of the full Hamiltonian, which is the solution of

( E - H)G'+'(EIR, R,) = 6"'(R - R 0). (28)

A source at R, radiates Gf+), so that if one adds the contributions for each part of the source represented by the RHS of Eq. (27), one obtains

AY(R) = d2R,G'+'(EJR, R,)K$JR,,). (29) s This key equation is not a tautology. It reduces the problem to that of radiation from a finite source. While the full dynamics of the three-body problem have still to be incorporated in G(+) , a solution can be constructed guided by the concept of causality, viz. that outside the source, propagation must proceed in only one direction.

The sufficient condition that a scattering amplitude, as introduced in Eq. (15), can be extracted from Eq. (29) is that for large R( 9 R,), G ( + ) can be factored into an outgoing wave and part independent of R,

G'+'(EI R, R,) - W ) ( E , alR,)O'+'(E, a ] R). (30)

The form of the outgoing wave may be inferred from the form of Eq. (3) in hyperspherical coordinates,


where C is a dynamical charge

(32) I 1 1

C(a) = ~ + __ - sin a cos a (sin a + cos a)'

At large R, the solutions of Eq. (31) are (apart from a factor p1I2) Coulomb functions F , or G,(q, p), with r] = C/K and p = KR. Thus, we can take

O(+)(E, alR) = (KR)- ' /~ exp (33)

This result is accurate to order R-'12, and it is only useable at extremely large values of R, as we shall see in Section V.

Inserting Eq. (30) into Eq. (29) we find that

T(a) = <@(-)(a) I V, I + i > * (34)

The asymptotic form of Eq. (33) was first derived by Peterkop (1981) and by Rudge and Seaton (1965), who also derived integral expressions for the ionization amplitude. Earlier authors have usually employed exact initial states, and asymptotic final states, in contrast to Eq. (34). It is instructive to compare Eq. (30) with the analogous result in the absence of Coulomb interactions; in 3D space,

exp(ik1R - R,I exp(ikR) - exp( - ik . R,) IR - RoI R (35)

Thus, we can think of a(-) as an ingoing regular solution, that can be constructed numerically, if not analytically.

111. The Boundary Function Method

We must determine G(+), the full outgoing Green function defined by Eq. (28), in a domain of R extending from R, to extremely large distances at which the factorization of Eq. (30) becomes valid. The problem can be separated into three parts, corresponding to the following division of space:

(I) Atomic zone. R lies within a few atomic diameters of the origin. R, is always in this zone.

(11) Coulomb zone. R extends from a few atomic diameters to a distance between lo3 and lo6 a,.

(111) Far zone. R + co, beyond 11.


In Zone I, all interactions are strong; let us use integral equations or path integrals in this region. In Zone 11, the Coulomb interaction varies slowly but cannot be treated perturbatively; let us use semiclassical methods in this case. In Zone I11 all interactions are weak, and Eq. (30) is valid.

It is neither obvious nor trivial to prove that a solution exists joining Zones I and 111. Let us begin by establishing an integral equation valid within the boundary dividing Zones I and 11, denoted by Z. It will turn out that some knowledge is required of the solution on C. In the following section, it will be shown that this knowledge is supplied by semiclassical solutions valid in Zone 11.

In place of Eq. (28), let us consider the more general problem with an arbitrary source on the RHS,

( E - H)* = x. (36)

I require the Green function of the unperturbed kinetic energy

( E - H,)Gb+’(R, R,) = 6(R - Ro).

Gb+)(R, R,) = H ( ~ ) ( K J R - RoI).

(38)

(39)

The solution of Eq. (38) is a Hankel function

It is worth noting that for R N R,, G(+) itself must have the form of Eq. (32) with K replaced by

i? = ( 2 ( E - V(R,J)}”z. (40)

If we apply Green’s lemma to

J d z R ( $ ( E - H)Go(R, R’) - G d E - HI$),

and use Eq. (36) and Eq. (38), we find that

we use customary abbreviations, e.g.

Gox = d2R’Go(R, R’)x(R) s (42)

(43) ( F ) , = 1 ds F(R). z

3 10 Christopher Bot tcher

Distance along the arc, C, is denoted by s while n is normal to C. The structure of Eq. (41) may be concisely expressed as

where L is a linear operator in the space of + within X, and $,; p is a source depending on x and (Jt,b/an),. It can be shown that L has no zero eigenvalues, so that Eq. (44) has a unique solution. If (&,b/an), were moved to the LHS, the new L would have a zero eigenvalue. Thus (J$/dn), must be supplied independently by an equation of the form

In Section IV, it is shown that solutions can be constructed in Zone I1 which provide such a relation on C, and which satisfy Eq. (30) in Zone 111. Thus, the existence of solutions of Eq. (28) or Eq. (36) with outgoing boundary conditions is established.

The reasoning leading to Eq. (41) is called “the boundary function method”, by applied mathematicians, and it is extensively applied in, for example, fluid mechanics (Fletcher 1984). The significance of Eq. (41) is more than theoretical, for it may provide the only way of solving Eq. (36) in Zone I. Suppose we imagine the two operators

LH = ( E - H) and LG = 1 - GoV

represented on a discrete lattice {Rp}. The matrix [LtJ is diagonally dominant, and for increasingly fine meshes, its spectrum is bounded. The matrix [LfJ has neither of these properties.

In consequence, a numerical representation of LG, by discretization or expansion in basis sets, will lead to rapidly convergent solutions of Eq. (36). Numerical representations of LH, for example, as provided by the Kohn variational method, may not converge at all. This disconcerting statement is supported by observing that the convergence of

(LH)- ‘P,

if expanded in eigenfunctions of H o , is similar to that of Go, which we understand analytically.

The convergence of Go is problematical because it has a singularity as x + 0 ( x = R - Ro). As a function of the dimensionality, d, of R,

G o - ( ~ I d = 1

InX d = 2

NUMERICAL CALCULATION OF IONIZATION AMPLITUDES 3 1 1

The convergence of Go in terms of eigenstates of H , labelled by d integers n is like that of

i.e., divergent for d 2 2. The singularities in Eq. (46) do not disturb LG, since they are integrable:

jIXld-'Go(X)dR

always exists. The solution of Eq. (41) is complicated in practical terms, and the program

of calculating accurate solutions is not yet complete. Fortunately, the threshold regime can be discussed without such detailed knowledge.

IV. Path Integral and Semiclassical Methods

The difficulties of constructing solutions of Eq. (28) in Zone I1 is that most approaches, including that of Section 11, require a knowledge of the solution on an outer boundary-precisely what one is trying to establish! The apparent circularity of reasoning can be broken by invoking the concept of causality; intuitively, the solution of Eq. (30) can only propagate outwards beyond Zone I. Causality must always be imposed on a second order differential equation in the form of an additional postulate, as illustrated by the well-known retarded potentials of electromagnetism. Thus, let us factor the Schrodinger equation, Eq. (3), into two first-order equations.

One must identify one variable that determines the causal ordering, e.g., R. Then Eq. (31) is readily written

where Arad is an operator, not containing a/aR, and

Another equation of the form of Eq. (47) may be obtained in a coordinate system based on classical orbits, which is particularly suited to constructing the function in Eq. (28).

Consider a sheaf of orbits emanating from a point R, (denoted by A in Fig. 1). It is possible to map continuously the whole asymptotic plane R by a set of


-

I 10 lo2 lo3 lo4 lo5 lo6 FIG. 1 . Classical orbits of energy 1.22 eV above threshold emanating from the points: (a)

( R , G() = (1,45"); ( b ) ( R , a) = (2,20"). The scale preserves tl but is logarithmic in R.

orbits of the same energy, each labelled by a different starting condition q. Then {Ro, E , q} are the four required constants of motion. If 5 is the distance along one orbit from A, 5 and q provide a new set of coordinates. It seems natural to use 5 as a causal coordinate. To write H, in terms of 5 and q we need a metric, provided by inspecting two orbits differing slightly in initial conditions, q and q + Aq, At a distance 5, the arc normal to either orbit has length w(5, q)Aq. Thus, the metric is

(49) dXZ = d t 2 + W2dqZ.

NUMERICAL CALCULATION OF IONIZATION AMPLITUDES 3 13

If there is no potential, and A is the origin, 5 = R, q = a and w = R. The Laplacian in terms of c, q follows from Eq. (49). The Schrodinger equation becomes

($ + A$c) Y = 0

where

(“sc” stands for “semiclassical,” though we have as yet made no assumptions regarding the validity of classical mechanics).

An equation such as Eq. (50) may be factored into

(& + iB)( & - iB)Y = 0

without approximation, provided the operator B satisfies

aB B2 + A$c + i - - . a t (53)

Then, an exact outgoing solution of Eq. (52) is given by the ordered exponential (or path integral),

N

Wt) = n exp[iB(tm) A t I Y ( t * > (54) m = 1

where t, = to + mA5, tN = 5 and N + co. More concisely

W5) = 0 exp[ i s:, B(r)dz]Y(tJ (55)

The ingoing solution is obtained from Eq. (55) if iB is replaced by - iB*. The general solution of Eq. (52) is a combination of both ingoing and outgoing waves. It is an instructive exercise to show, using Eq. (53), that Eq. (55) exactly conserves flux.

To obtain a semiclassical approximation, let us suppose that laB/a<I < A:=, so that

Referring to Eq. (51), we write


replace a/ag by iK, and drop a/aq. Then Eq. (56) becomes

aK

whence, to the same order,

(59) i a

2 a t B N K + --ln ( o K ) .

Then Eq. (55) becomes

The subscript O0 refers to a quantity evaluated at to. We shall subsequently write

Z(q) = Y o ( ~ K ) A ’ 2 . (61)

Obviously, the phase is an action integral,

r: S(5 ,r t ) = jcoKd5. (62)

The amplitude (wK)- ( ’ /2) is precisely that required to conserve the flux across an element wAq,

oKI Y I2Aq = constant. (63)

Expressions similar to Eq. (60) have a long history (cf. Brink, 1985). The drawback of semiclassical expressions is that when o + O , they have a singularity, usually integrable. Such phenomena are often found at the boundary of a classically allowed region, and can be interpreted physically as caustics or rainbows. It would be desirable to have the ability to go beyond Eq. (60) by directly solving Eq. (53). The operator A, in either the form of Eq. (48) or Eq. (51), must be discretized in the space CI or q. Is is then necessary to calculate the square roots of operators in the complex domain. A program is underway and appears viable.

It is instructive to glance at the limit opposite to the semiclassical, in which diffraction dominates. Then we retain the “sideways” terms in a/aq, dropping the terms in aw/ag, so that


This is similar to the equation describing the propagation of a laser beam of finite cross section. In either case, the solution spreads out transversely by diffraction, in a manner govered by a diffusion equation.

V. Calculations on a Two-Dimensional Model

Let us now apply the semiclassical solution (Eq. (60)) to the two- dimensional Schrodinger equation (Eq. (3) or Eq. (30)). If one again imagines a set of trajectories emanating from the point Ro, as illustrated in Fig. 1, an outgoing wave solution is given by

where

Reference should be made to Eqs. (57) and (61). The condition that the semiclassical approximation be valid certainly holds in Zone 11, but one does not yet know whether Eq. (65) joins onto a solution of Eq. (28) in Zone I. Let us argue that it can be so joined with a proper choice of the function Z(q), as yet undetermined.

From Eq. (65), we have at once

a quantity independent of 2. This is a relation of the form of Eq. (45) required to close Eq. (41). Knowing $ inside X, Z(q) can be read off from IClz, the limit as R .+ X. Thus, we can join the solutions in Zones I and 11. The connection with Zone 111 is straightforward. At very large distances the classical trajectory becomes a ray with a+ constant, a,. The action can be written as a constant phase shift plus a Coulomb part,

(68) s + 44 + S,(R> a,)

where

S,(R, a) = K(R, a)dR IoR


is known analytically. It is also the case that K + IC and

0 + Y(r1)R

so that Eq. (65) has the factorization of Eq. (30) with

and

Z , Y and A all depend on R, as well as q. The variable q is mapped onto a, by following the classical orbit, and hence, is mapped onto the energy of the ejected electron using Eq. (14),

E = E sin' a,.

Without a subscript, a should usually be understood as a,. The program of constructing a Green function, satisfying Eq. (28) and the

asymptotic condition of Eq. (30) is now complete. The scattering amplitude (Eq. (34)) follows. As was noted earlier, a complete numerical solution can be circumvented to extract the threshold behavior.

Close to R,, G ( + ) is given by Eq. (39), which can be compared with Eq. (65) to yield,

Z ( g ) N einl4 (73)

This suggests that 2 is a slowly varying function of q. We choose q = a,, the direction of initial velocity,

(74)

in the notation of Eq. (40). Then Fig. 1 shows that near threshold (E =

1.22 eV), the ionizing trajectories 0 < a < 4 2 originate from a small range of a,; in Fig. l(a), 44.04" < a, < 44.08". As E -+ 0, this opening angle contacts at least as fast as El.'. Thus, if Z is continuous in a,, Z can be taken as a constant. It turns out that near threshold, Y has a dependence on E and a, (hence E ) independent of R,,

k, = E cos a,, k, = 2 sin a,

The phase is almost independent of E and a,,


Hence, the scattering amplitude

where

In short, the threshold behavior may be summed up in terms of the differential probability introduced in Eqs. (24) and (25),

- ICiI' - dP dE W(E, E / E ) [ E ( E - E ) ] 1'2*

(79)

The threshold behavior of ionization in the two-dimensional reduction (Eq. (3)) has been reduced to calculating a function W(E,y). Apart from normalizing factors, W is just o(c, q) , the separation of adjacent classical orbits radiating from a single point. For illustration, we choose orbits radiating from two points, one on the classical saddle (R,,, a) = 1, 45") and the other not, ( R , , a) = (2,20"), as in Figs. l(a) and l(b), respectively. The function W can be accurately extracted only if the orbits are integrated to an immense distance. In Fig. 2, dP/dE is calculated from Eq. (79) for E =

0.012 eV and in Fig. 3 for 0.54 eV. The results are clearly independent of R,, apart from normalization.

In accordance with Wannier's analytic predictions (1953), dP/de is constant to a few percent for 20" < c1 < 80°, corresponding to 0.1 < E/E < 0.9. Wannier's theory is based on a parabolic approximation to the potential which should not be valid outside this range.

Different behavior, in the form of an integrable caustic, does indeed appear at c1 < 20"and > 80". The singularity is fitted numerically by E - ' . ~ ' ; the exponent can be explained by considering orbits close to the r,-axis (e.g., A D in Fig. 1). As CI -+ 0 and R + 03, p : N 2/r2, so that o - p z - (ra)-'/' - &-'I4. Thus, from Eq. (79) we have

Of course, a fully quanta1 solution based on Eq. (55) would replace the cusp by a finite peak.


a D \ W -0

r I I I I I 1 I

0.5 - -

-

I

II

0. I 0" 20° 4 0" 60" OoO

c1

FIG. 2. Variation of dP/de with a for two energies: I, 0.54 eV; 11, 0.012 eV. Based on orbits emanating from (R , a) = (1,45").

I I I I I I -

I

II

0.2 I I I 6 I I I 0" 20" 40" 60" 80"

a FIG. 3. Same as Fig. 2, based on orbits emanating from (R, a) = (2,20").

It is instructive to plot the exponent defined by

a [ = E -In rz)

aE

where the derivative is taken with E/E constant. Fig. 4 shows [ as a function of u for the same cases chosen in Fig. 2. Again, Wannier's value 5, = 1.1269.. . is reproduced accurately for 20" < u < 80". As ct + 0, [ -+ a value N 0.75, consistent with Eq. (80).


I 1 I I I I 8

1.2- - W

The integrated probability

g( E ) = joE de

has an exponent, plotted in Fig. 5 as a function of E . Over a wide range of energies from 10 MeV to several eV, c I c,, at such a level that the difference would not be easy to detect experimentally. As E becomes very small, moves

0.91 I I I

lo3 102 16' E

FIG. 5. Power law exponent of the total Probability vs. energy (in Hartree units). Wannier's value is indicated by the horizontal line, W


closer to unity. What really happens at E + 0 may be difficult to predict, since the increasing prominence of the caustics suggest that the semiclassical treatment is breaking down. The scale in Fig. 5 is built in by choosing R , - a few a,. If the initial state 4i were excited, the scale energy would decrease.

We conclude that Wannier's theory is highly accurate for E / E > 0.1 and a wide range of E > 10 meV. If these conditions are not fulfilled, Temkin's critique is borne out.

VI. Calculations in Three Dimensions

Let us conclude by presenting some calculations based on the same principles as those of Section V, but using the full Hamiltonian of Eq. (2). In addition to rl and r z , we have 8, the angle between rl and r2.

For clarity, consider orbits originating from the configuration rl = r z , Bo = K. To obtain a final 8 different from K, we introduce initial velocities such that

= tan y o . r,B +I

At yo = 0, 8 = K always. If y o = K - 6, where 6 is a small number, however, a little thought shows that the electrons will collide almost head on, reflect, and

t 4

_I, 4 0"

FIG. 6. Final deviation from 180", 180" - 8 vs. initial parameter yo, defined in Eq. (83). The curves are for two energies: I, 0.14 eV; 11, 0.54 eV.

NUMERICAL CALCULATION O F IONIZATION AMPLITUDES 32 1

I .6-

0.4

7-

t

m

a N 0.8

7J

0.4- r

0 lhOo 148 150" 160" 170" 180"

I I 1 1 I I I

e FIG. 7. Variation of d2P/de0 vs. 0 for the two energies considered in Fig. 6.

again end up at 6 = 7c. Thus, at some yo in [0, n], n - 8(yo) must go through a maximum value, n = 6,. The function o in Eq. (65) is proportional to

so that the threshold cross section should exhibit a caustic at 6 = 6,. Accurate calculations of 6(yo) at two energies are shown in Fig. 6. For

small y o , 7c - 8 1: c E " ~ , in accordance with Wannier's theory, but the maximum opening angle satisfies n - 6, N c'E''' fairly closely. The caustics appear in plots of a2P/&88, shown in Fig. 7. They are integrable singularities - (8, - 6)-'14 and should be removed in the next approximation beyond the semiclassical.

ACKNOWLEDGMENT

This work was supported by the U.S. Department of Energy, Office of Basic Energy Science, under contract W-3 1-109-ENG-38.

REFERENCES

Bottcher, C. (1985). Advances in Atomic and Molecular Physics (D. R. Bates and B. Bederson,

Brink, D. M. (1985). Semiclassical Methods in Nuclear Scattering. Cambridge University Press, eds.) 20, Academic Press, Orlando, Florida, 241.

Cambridge. England.


Crothers, D. S. F. (1986). J . Phys. B. 19,463. Fletcher, C. A. J. (1984). Computational Galerkin Methods. Springer Verlag, New York, New

Peterkop, R. (1981). J . Phys. B. 14, 513. Rau, A. R. P. (1971). Phys. Rev. A . 4. Rudge, M. R. H. and Seaton, M. J. (1965). Proc. Roy . SOC. (London) A283,262. Temkin, A. (1974). J . Phys. B. 7 , L450. Temkin, A. (1980). Phys. Reo. A. 22, 324. Temkin, A.

(1982). Phys. Rev. Lett. 49, 365. Wannier, G. H. (1953). Phys. Reo. 90, 817.

York.


l l THE NUMERICAL SOLUTION OF THE EQUATIONS OF MOLECULAR SCA TTERING A . C. ALLISON Department of Computing Science University of Glasgow Glasgow, Scotland

I. Introduction . . . . . . . . . . . . . . . . . . . . . . 323 11. Numerical Methods . . . . . . . . . . . . . . . . . . . . 324

111. Close-Coupled Equations . . . . . . . . . . . . . . . . . . 327 A. Initial Conditions . . . . . . . . . . . . . . . . . . . 328 B. Boundary Conditions . . . . . . . . . . . . . . . . . . 329

IV. Solution Following Methods. . . . . . . . . . . . . . . . . 331 V. Potential Following Methods . . . . . . . . . . . . . . . . 335

VI. Adiabatic and Diabatic Representations . . . . . . . . . . . . . 336 VII. Propagators . . . . . . . . . . . . . . . . . . . . . . 337

VIII. Summary . . . . . . . . . . . . . . . . . . . . . . . 338 References . . . . . . . . . . . . . . . . . . . . . . . 339

I. Introduction

A full quanta1 formalism for the problem of rotational excitation of a diatomic molecule was presented by Arthurs and Dalgarno (1963). In this seminal paper, they treated the molecule as a rigid rotator impacted by a structureless particle and derived the relationship between the scattering amplitudes and the S matrix, defined by the asymptotic solution of the close coupled equations

Fi,(R) = c Vi,(R)Fi@). 1 i'

l i ( l i + 1) R2

[$ + kf - ~

Here, l i and ki represent the orbital angular momentum quantum number and wave vector in the ith channel and V is the potential matrix. The subscript I represents the initial state, while the possible final states are represented by i'.

This formalism has stood the test of time in that only minor modifications are necessary to the theory to include the cases of spherical top, symmetric

323 Copyright 0 1988 by Academic Press, Inc.

All rights of reproduction in any form reserved. ISBN 0-12-003825-0

324 A. C . Allison

and asymmetric top molecules and impact by another molecule. Scattering by a vibrating rotator is similarly handled and many other descriptions of physical processes lead to sets of equations of the same type. Numerical methods for the solution of Eq. (1) are the subject of this paper.

Earlier attempts to solve this equation numerically had centered on various approximations such as the Distorted Wave or Born Approxima- tions, (Mott and Massey, 1965) but in the mid 1960s the search was on for a fast, accurate and robust numerical method for the direct solution of Eq. (1). Many algorithms were developed and many claims of “best” were made in the ensuing ten years. This was a time of extremely rapid growth in the area of computational physics and computing generally, and it was quickly realised that a good method imbedded in a package that was easy to use, reliable and well-documented was much more attractive than a more efficient algorithm without these advantages. This argument does not invalidate the search for better algorithms, but it does mean that any new method will have to be proven significantly better before it would replace the algorithm at the heart of one of the major packages.

Two landmarks occurred in 1979. The first was the publication of a review by Secrest (1 979), an excellent and comprehensive classification of existing methods that remains relevant today. The second was a workshop organised by the now defunct National Resource for Computation in Chemistry, which brought together some of the people working actively in the field. The goals of the workshop were to identify which of the existing computer codes for solving the coupled equations of quantum molecular scattering perform most efficiently on a variety of test problems. The results (Thomas et al., 1981) concluded that no one method was best and that hybrid methods consisting of a method with special properties applicable in the region of small R, combined with a method that exploited the known asymptotic behaviour of the solution would give the best overall performance.

The aim of this article is to give some understanding of the underlying numerical concepts that have been used in the solution of the close-coupled equations and some of the details by which various algorithms have been tuned to high efficiency.

11. Numerical Methods

The radial form of the Schrodinger equation in its simplest form, appropriate for single channel scattering, is

SOLUTION OF THE EQUATIONS OF MOLECULAR SCATTERING 325

with boundary conditions

Y(0) = 0

y ( R ) - sin(kR + q ) ; ( 3 )

where y ~ , the phase shift, is to be determined. This equation and its variants have attracted the attention of the best

mathematicians and physicists for well over half a century, to the extent that a vast body of theory is available to aid in the understanding of the behaviour of Eq. (2). Preeminent among available texts is the volume by Mott and Massey ( 1 965). In Chapter IV of their book, Eq. (2) is rewritten as an integral equation in the following way:

Let y (R) = u(R) + g(R) (4)

where u ( R ) is the solution, regular at the origin, of the equation

Then R

g ( R ) = u(R) u(R’)V(R’)y(R’) dR‘

R

- v(R) j, u(R’)V(R‘)y(R’)dR’ (6)

where u(R) is another independent solution of Eq. (5 ) , chosen such that the Wronskian of u ( R ) and u(R) is unity.

One may approach the problem from a purely numerical point of view, however, and appeal to any of the well-known texts on numerical analysis (Fox, 1962; Lambert, 1973; Lapidus and Seinfeld, 1971; Shampine and Gordon, 1975; Press et al., 1986). In these texts one would find a number of methods for solving equations of the form

subject to the boundary conditions that the solution y has known values yo and y , at two values of the independent variable R , and R,.

One of these methods would be relaxation where, if a reasonable first guess to the solution is available, say u,, then successive corrections can be found by solving a linear differential equation. If the original differential equation is linear, then the solution is obtained in a single step and Eqs. (4) and (6) above are a good example of this numerical concept.

The linear problem is usually solved numerically by dividing the range [ R , , R,] into N equal intervals of size h, replacing the derivatives by a finite

326 A. C . Allison

difference representation and writing the resulting simultaneous linear equations in the form of a matrix of size close to N x N . To handle the semi-infinite boundary condition, however, a very large number of intervals is required. The conventional wisdom is that, for the multichannel case, the size of the matrices becomes prohibitively large. This understanding might profitably be questioned as computer power and ability to handle large data sets increases.

A second approach is to transform the boundary value problem into an initial value one by guessing an extra condition at R , and proceeding step by step “shooting” from R , to R,. The usual recommendation is to let z = y’, thus transforming the equations to first order as

y’ = z

z’ = f ( R , Y , z ) (8)

with initial conditions y = yo and z = s (a guess) at R = R, . The calculated solution that now depends on s will not, in general, satisfy

the boundary conditions, but an iterative process may be defined that hopefully will adjust the value of s until the solution matches the outer boundary condition. Thomas (1979, 1982) has explored this idea for the solution of the molecular scattering equations, but there are as yet unsolved difficulties.

For linear equations of the form

y’ = z

Z’ = B(R)y + A(R)z, (9)

the required solution is obtained by taking a linear combination of solutions with initial guesses s1 and s2, shooting into the asymptotic region and matching to the value at R,. This approach is used, in some measure, by the vast majority of practitioners in the field.

For Eq. (2), however, A ( R ) = 0 and B(R) has a functional form of k2 - W(R), where the potential W ( R ) varies rapidly at small values of R, has an extremely well-behaved behaviour for large values of R and eventually tends to zero. Eq. (9) now can be written

Y” = B(R)y, (10)

and it is the absence of the first derivative and remarkable properties of B(R) that set the Schrodinger equation apart from a more general equation such as Eq. (9) and has motivated the search for efficient algorithms specifically designed for this problem.

The matrix form of Eq. (9) is now

Y’ = D(R)Y (1 1)


and our standard texts would indicate that, if D(R) were independent of R, equal to Q say, then the solution is obtained easily by first finding the matrix T such that TQT-’ = L, where L is a diagonal matrix. The uncoupled equations Z’ = LZ, where Z = TY, are then solved in terms of the eigenvalues of Q and the required solutions found from Y = T-’Z. Of course, if Q were an approximation to D(R) valid over some interval, then the above process would give an analytical solution to Eq. (9) over that interval. The above concept has been extensively used in those methods which are classified under the “potential following” heading.

In most cases, a multichannel description of the scattering process is necessary, so we return to our original Eq. (1). Much ingenuity, skill and determination has been exhibited in the search for efficient, robust numerical algorithms for the solution of this equation. Practically all approaches, however, depend heavily on one or more of the basic numerical and mathematical concepts that have been presented, in their most basic form, in the above section.

111. Close-Coupled Equations

The close-coupled equations may be written

where Ii and ki represent the orbital angular momentum quantum number and wave vector in the ith channel.

In matrix form, Eq. (12) is

p$ + k2 - W(R) F(R) = 0 1 (13)

where W(R) = 12(R) + V(R) , k2 and 12(R) being diagonal matrices. The boundary conditions are

F i , ( R ) = 0 at R = 0 (14)

Fir(R) - k,;”’{exp[- i(k,R - +Z,Z)]~~, - exp[i(kiR - $li7~)]Sir}, (15)

and the final result is the scattering or S matrix which contains all the information necessary to describe the scattering process. 6 , is the Kronecker delta function.

328 A. C . Allison

Typically, one is interested in scattering from an initial state I to a set of final states specified by the summation over i’. Thus, the required information is contained in a single column of the S matrix. Channels for which k: > 0 are open have oscillatory asymptotic behaviour while those for which k: < 0 are closed behave asymptocially as decaying exponentials. It may be thought that, given a set of linear second order differential equations with specified boundary conditions, the numerical solution would be straightforward. Nothing could be further from the truth: first, the boundary conditions are complex and second, the solution is afflicted with instabilities at both boundaries.

The S matrix is complex, so, in order to avoid the use of complex arithmetic, the problem may be reformulated in terms of a standing wave solution involving the real symmetric reactance matrix, R.

The boundary conditions become

Fi,(R) - k,li2(sin(k,R - &,n)bi, + cos(kiR - &n)Ri,} (16)

for open channels, and

Fi,(R) - k ; li2(exp( I k , I R)6, + exp( - I ki I RIR,,} (17)

for closed channels. The relationship between S and R is

S = (I + iR)(I - iR)- (18)

To find a single column of S now requires all the elements of the square matrix R. The subscript I must now include all initial and final states.

A. INITIAL CONDITIONS

In problems of molecular scattering, the elements of the potential matrix are strongly repulsive in the neighbourhood of the origin, and it is very often impossible to start the integration at the origin. In this nonclassical region, the solutions of the equation will be very small, and one finds that there is a range of values of R such that integrations started at any point in this range lead to R matrix elements which are effectively unchanged. Thus, to avoid unnecessary integrations, one wants to choose the starting value R , as far from the origin as possible, commensurate with indistinguishable variation in the R matrix elements. The choice of the position of R , varies from problem to problem and more details may be found in Secrest (1979).


Thus, the initial conditions (Eq. (14)) are modified to be

Fi,(R) = 0 at R = R,. (19)

The most commonly used strategy now becomes clear. We start the integration of the N equations at R , and shoot into the asymptotic region using N different guesses for the first derivative or equivalent at R, giving rise to a set of calculated solutions represented by the matrix Y.

None of these equations will, in general, satisfy the boundary conditions, but some linear combination given by Y . C, where C is a matrix of coefficients, must represent the required solution. All that is required is that these solutions be linearly independent. A prerequisite for this is that the initial guesses are themselves linearly independent, guaranteed by the obvious choice of Y’ = I at R , . Unfortunately, in this region solutions are growing at different rates and some of the weakly growing ones may be many orders of magnitude smaller than the dominant solution. Since each y, contains some component of the most strongly growing solution, the tendency will be for each column to look like the dominant solution. The matrix of solutions then will be extremely ill-conditioned and the linear independence required for matching in the asymptotic region will be lost.

This problem may be solved by using a stabilisation technique in which the solutions are transformed into a different basis producing an orthogonal matrix via a Gram-Schmidt process or an upper triangular matrix as given by Gordon (1969). An elegant discussion of the stabilisation problem is contained in the latter reference.

B. BOUNDARY CONDITIONS

The boundary conditions (Eqs. (16) and (17)) are conveniently written

where K . . = d..k. LJ 13 1

M i j = SijkiRjli(kiR) k? > 0

= S i j exp(lk,IR) k? < O

k; > 0

= Sijexp(-IkilR) k? < 0.

N i j = SijkiRnli(kiR)

where jl(kiR) and n,(k,R) are the Bessel and Neuman functions respectively spherical.

330 A. C. Allison

The calculated solution may be written

Y = M . A + N . B .

So, equating Y . C with F, we obtain the relation

R = K'/ZB. A-lK-112

The matrices A and B are found by either matching to the solution and derivative at the point R , or by matching to the solution at two points R, and R,. In the latter case,

A = - NbM,)-'(N,Yb - NbY,) (23)

B = (N,Mb - NbM,)-'(MbYo - May,). (24)

Note that numerical difficulties will arise from the increasing exponential term appearing in the definition of the matrix M in Eq. (20). Again, different techniques have been developed, stabilisation as described above can be used or the exponentials scaled in a suitable way (Johnson, 1973).

If the integration is carried out far enough, we may deal only with the R submatrix appropriate to open channels and obtain an S matrix of similar dimension from Eq. (18).

Boundary conditions (Eq. (20)) are only valid for values of R, where elements of the potential matrix have become negligible. If the potential has a long range form of R - 3 or R - 4 say, one might have to integrate out a very long way. In this case, one should use the asymptotic solutions for long range potentials first presented by Burke and Schey (1962) and adapted to molecular collisions by Brandt and Truhlar (1973). McLenithan and Secrest (1984) have recently discussed the case of the centrifugal term not being of the form l(1 + 1).

To recap, from R , the solutions may be propagated through the numerically difficult nonclassical region to a mid-range with no particular problems, into the asymptotic region where the open channel solutions oscillate with a frequency depending on the energy and the closed channel solutions are exponentially decreasing-all controlled by the potential matrix whose elements have the behaviour described in Section 11.

It is these observations that led to the classification of the methods first defined by Secrest (1979) into two main groups.

(a) Solution following methods-where the matrix F(R) of solutions is approximated by a numerical technique similar to those described at the start of Section 11.

Potential following methods-where a functional approximation to the potential matrix W(R) leads to analytical solutions of Eq. (1) in a series of segments.

(b)


IV. Solution Following Methods

Most of the standard methods for the solution of ordinary differential equations will fall into this category, the best of which have been imbedded in high quality software packages such as NAG and IMSL available from the major libraries. Some of these, for example the SLEIGN code of Bailey, Shampine and Gordon (1978) have been specifically targeted at the Schrod- inger equation. All of these packages will work perfectly well, but it is the author’s experience that attempting to solve a new system frequently produces unusual behaviour which must be investigated and understood before deciding to use a package.

In this section, three methods that have a special place in the development of the field will be described. All lead to simple algorithms for the solution of the equation

y“ = -fy. (25)

For ease of presentation, the methods will be presented in terms of a single equation-the extension to a set of equations is straightforward and details may be found in the various references.

Foremost among the special methods is that of Numerov (1924), used in this context by Allison (1970). The formula is

with an easily estimated local trunction error of 1/240 h6ypi. Since the equation is linear and homogeneous, the recurrence may be started with yo = 0 and y , an arbitrary small number. The advantage of this method is its extreme simplicity, its high order and a straightforward error estimation and interval changing capacity. It is worth noting that, for efficiency, the Numerov algorithm is simplified by writing

obtaining

In the multichannel case, calculation of the matrix inverse is the major part of the work, although it may be evaluated efficiently iteratively (Allison,

332 A . C . Allison

1970). Alternatively, it may be expanded to the same order as the formula giving

a representation ascribed to Raynal (Melkanoff et al., 1966). Note that truncating the formula after the term in h2 leads to the original method of Hartree (1957). The renormalised Numerov method of Johnson (1977) is a further rearrangement of Eq. (28).

The second method is the integral equation approach of Sams and Kouri (1969) which, as was mentioned earlier, has been extensively investigated and developed by Secrest and co-workers, see Secrest (1979) and references therein. The method relies on a reformulation of Eq. (6), viz.

{u(R’)u(R) - u(R)v(R’)} C

. V ( R’)y( R’)dR‘

where C is the Wronskian of u(R) and u(R). The key to this method is that the kernel of the first integral is zero at

R = R’, thus creating an explicit formula for the solution at R in terms of previous points which may be evaluated by a quadrature formula such as Simpson’s Rule. The second integral in Eq. (30) is a constant that may be determined at the end of the integration. In the open channel case, one may identify u(R) and u(r) with the functions kRj,(kR) and kRn,(kR), respectively. Although these special functions must be evaluated at each step, it does mean that the centrifugal term is already included, making the method efficient for weak potentials.

Both of the above methods are easy to understand and extremely easy to program. In a modern context this means that an existing code should be easy to read and easy to modify. If greater accuracy is required, it may be achieved in a reliable fashion by reducing the step size. More importantly in the multichannel case, if the actual solution is required rather than just the R matrix, then, as long as the calculated values and stabilising transformations have been stored, it is straightforward to generate the solution (McLenithan and Secrest, 1984).

All the methods mentioned above suffer from a severe disadvantage, however, in that many of the individual solutions are oscillating in the asymptotic region with a frequency that depends on the energy. Most solution following methods use a basis set of polynomials in the independent variable R and thus, to represent each half period, many points are required.


The situation gets worse as the energy increases; this is counter-physical since, for higher energy, the potential is relatively smaller and the problem ought to be simpler.

Raptis and Allison (1978) developed a new class of algorithms based on linear multistep methods of the form

Yr+ 1 - 2Yr - Yr- 1 = h2{b,y:'+ 1 + ~ I Y : ' + b2~:'- 1). (31)

Note that the Numerov formula is given by b, = b, = 1/12, and b, = 10/12. Functions 1, R, R Z . . . R5 are integrated exactly.

If the coefficients bi are allowed to depend on the interval h, then functions of the form exp(pR) can be integrated exactly (Lyche, 1972). Thus, either pure sinusoidal ( p complex) or pure exponential ( p real) solutions may be followed in a single step. In practice, much larger steps could be taken in the asymptotic region than was previously the case.

Formulae for the b,(h) for the integration with the basis set

1, R, R2, R3, exp(pr), exp(-pR)

are given in Raptis and Allison (1978), but a little algebra shows that their method reduces to the attractive form involving w, of Eq. (27)

w,, 1 - 2 c o s ( f i h)w, + w,- 1 = 0. (32)

This approach did not give any significant improvement in the nonclassical region, so Raptis and Allison (1978) proposed that the normal Numerov formula be used for small R and the exponentially fitted form be used in the outer region.

Similar formulae have been derived by Ixaru and Rizea (1980) appropriate to basis sets of the form

exp(pR), R exp(pR), R2 exp(pR). . . .

It would be a major breakthrough if one could establish a tractable formula which could fit to the Bessel functions kRj,(kR) and kRn,(kR) (for an attempt to do this, see Raptis and Cash (1986)).

Our third method has also had a long development path. Johnson (1973) showed how the phase shift of a single equation could be obtained by propagating the log-derivative y'/y rather than the solution y. Equation (25) is transformed to

2' + f + z2 = 0 (33)

where z = y'/y and integration may be started by letting z(R,) be some arbitrary large number. The features of the log-derivative may be seen easily by considering the functions exp(kR) and sin kR. In the former case, the log- derivative only increases linearly, while, for the latter, it is the cotangent

334 A. C. Allison

function that has singularities as R increases. Johnson (1973) developed an algorithm which had no difficulty propagating the solution across the singular points, both for the single channel and multichannel case which was developed in the same paper.

Recalling that the exponential growth of the solutions for small values of R is one of the major problems, the utility of the log-derivative in exhibiting smooth variation in the nonclassical region is of such importance that it has been adopted in most of the currently available software packages for the solution of molecular scattering. We shall return to this method in a later section, but point out here that if one decides to operate with the log- derivative then it is difficult to generate the actual solution.

We conclude this section with a brief mention of the iterative method developed by Thomas (1979,1982). If complex arithmetic is used then it may be possible to obtain only the required column of the S matrix. Considerable savings would be achieved if an iterative process that converged in significantly fewer steps than N could be found. Because of the errors accumulated by integration of a “guessed solution” through the nonclassical region, it has proved difficult to devise such a scheme. Thomas (1979, 1982) used the integral equation approach but had to invoke a Kato variational principle to improve convergence. Some new insight could make this approach very effective.

V. Potential Following Methods

Methods that fit this description use the special properties of the potential function that allow it to be approximated by a piecewise function simple enough that the Schrodinger equation may be solved analytically over each section.

Development of these methods owes much to the pioneering work of Gordon (1971), who used a piecewise linear fit to the potential which gives rise to analytical solutions in terms of Airy functions, and of Light (1971), who approximated the potential with a piecewise constant function.

To apply the techniques described above to multichannel scattering requires the range to be divided into sections, the potential matrix to be diagonalised at the centre point, Ri say, of each section and analytical solutions generated over each section basically as described in Eq. (11). At each step, the solutions must be transformed into the space appropriate for that section.

The potential matrix W ( R ) of Eq. (13) is expanded about the point Ri as

W ( R ) = W ( R J + ( R - Ri) ~ - Ri)2 ___


W(R,) is diagonalised by a unitary transformation Ti which, applied to Eq. (1 3), gives

[1$ f kZ - U,(R) G(R) = 0 1 (35)

where U,(R) = T,W(R)T: and G(R) = TiF(R), T: being the Hermitian conjugate of Ti. Note that UXR) is nearly diagonal in the neighbourhood of Ri.

Ui(R) is now represented as the sum of a reference potential WYf(R), for which the analytical solutions of Eq. (13) are known, plus a perturbation potential Wper*(R). The perturbation corrections may be evaluated using the integral equations to obtain an indication of the error or to predict the size of the interval for which the approximation is valid. There is enormous scope in the selection of the “best” reference potential and manner of calculation of perturbation corrections to allow the algorithms to be tuned to a wide class of problem (Rosenthal and Gordon, 1976; Light and Walker, 1976; Stechel et al., 1980; Parker et al., 1980). The latter authors go further and improve their algorithm by considering a variable number of steps within each section.

One of the advantages of the potential following methods is their ability to take large steps in the asymptotic region. For example, the above approximation may be valid over a range covering multiple periods of the solution, thus defining an interval size much larger than the solution following methods with their constraint of several points per half period. Another advantage stems from the observation that all the above transformations are largely independent of the energy. Thus, if one wishes to solve the same problem for many energies, as is frequently the case, the initial transformations can be calculated and stored for a single energy and solutions at subsequent energies can be obtained with little extra effort. A disadvantage is that if one asks for high accuracy then the number of steps required by the potential following algorithms can increase surprisingly.

Following the workshop in 1979, a number of computer codes based on both solution and potential following algorithms were compared on three test problems. The conclusion (Thomas et al., 1981) was that some of the highly developed potential following methods worked extremely well in the classical region but that all methods, with the sole exception of the log derivative method, performed poorly at small values of R. Thus, a hybrid method which used the log-derivative in the nonclassical region and a potential following approach for larger R was considered to be the best combination.

Two other factors have greatly influenced the trend of recent research. One is the desirability of solving the scattering equations in an adiabatic basis and the second is the realisation that much of the theory can be unified by appealing to the concept of propagators.

336 A. C . Allison

VI. Adiabatic and Diabatic Representations

For scattering problems which involve a change in the electronic state of the molecule, a radial coupling term appears in the adiabatic formulation of the equations (Faist and Levine, 1976). This gives rise to a term in the first derivative of the solution and the equations now take the form, identical with Eq. (9),

with the usual boundary conditions Eqs. (19) and (20). The appearance of the first derivative mitigates in favour of those methods

that can handle this feature naturally, for example the standard packages mentioned in Section IV, the method of Light (1971) from Section V, and mitigates against the methods designed solely for the case of A(R) = 0, such as Numerov (Allison, 1970), Sams-Kouri (Secrest, 1979), DeVogelaere (Lester, 1968; Coleman and Mohamed, 1978) and the original log- derivative method (Johnson, 1973). No doubt this provided the motivation for Mrugala and Secrest (1983a, 1983b) to generalise the log-derivative method to handle Eq. (36).

By substituting F(R) = Z(R)G(R), one may transform into a diabatic basis and obtain the equation

G” + Z-’(B - $A’ - A2)ZG = 0, (37)

valid only if Z satisfies the equation

22‘ + A 2 = 0 (38)

with boundary conditions Z = I at R = 03. Under certain conditions (van Dishoek et al., 1984), Eq. (37) simplifies considerably and it may be solved using any of the methods discussed previously. It is a nontrivial task to solve Eq. (38), however. For example, in curve crossing problems it is extremely difficult to integrate through the crossing point using any of the standard packages. For values of N up to three, Heil et al. (1981) have written down analytical solutions and a numerical method based on the R matrix propagation method of Light and Walker (1976) has been reported by Schwenke et al. (1987).

It does seem that, in general, it is easier to solve the adiabatic equations head on rather than transform out the first derivative, a conclusion shared by Baer et al. (1980) and Mrugala and Secrest (1983a, 1983b).


VII. Propagators

A propagator in an interval is a 2N x 2N block matrix that relates the values of the solutions of Eq. (1 1) and their derivatives at the two end points of the interval. The main features are best illustrated with a simple example, the equation y" = p z y which has exponential solutions.

The first choice is the Cauchy propagator, C, originally introduced in this field by Light (1971) and used by Mrugala and Secrest (1983a),

with C, = C , = cosh(ph), C , = p sinh(ph) and C3 = l/p sinh(ph) for our example. This formalism represents one step explicit methods for the numerical solution of first order differential equations and is closely related to stability studies for Eq. (1 1) (Lapidus and Seinfeld, 1971). An early example of an exponential propagator by Magnus (1954) has been used by Light (1971) and Garrett et a!. (1981). It is easy to extract an explicit relationship for the log-derivative at each end of the interval and this has been implemented by Alexander (1 984) in his development of the linear reference potential method.

In their generalisation of the log-derivative method, Mrugala and Secrest (1983a, 1983b) have rearranged the terms to define a propagator, L, and written down its relationship to C :

For our example, - L , = L, = l/p coth(ph) and L, = - L3 = l/p cosech(ph). The algorithm of Mrugala and Secrest (1983a, 1983b) determines the matrices Li and they report significantly improved performance over a standard Runge-Kutta routine for equations that contain first derivative terms.

A third propagator may be defined by the relation

This is just the inverse of the L propagator above and, for our example, Ri = pzLi, i = 1, 2, 3, 4, This propagator has occurred naturally in the development of the R matrix methods and Lill et al. (1983) developed a

338 A. C . Allison

formalism which subsumed the earlier work of Light and Walker (1976) and Parker et al. (1980).

Note that the latter two propagators take the form of implicit formulae and one would expect better stability properties than are obtained using the Cauchy propagator. Considering the expressions obtained for the example, it is evident that as p increases, corresponding to strongly closed channels, the Ci increase without bound while the Li and Ri are well behaved. Thus, although the computational difficulties of finding the elements of the propagation matrices is essentially unchanged, the equivalence of the generalised log-derivative propagator L and the R-matrix propagator has been established and many disparate approaches have been unified by this formalism. Alexander and Manopoulos (1987) used the log-derivative recursion derived from the L propagator and reported a major improvement in the stability characteristics of their program, particularly for a system that has several strongly closed channels.

Mrugala (1987) has exploited the new generalised log-derivative method by combining it with the R matrix code of Walker (1978) for a system that contains first derivative terms. The results are satisfactory. An interesting combination of ideas has been presented by Manopoulos (1987). Rather than diagonalise the potential, he takes a constant diagonal approximation to it and uses the L propagator, exactly as in our example above. Perturbation calculations from the residual matrix are calculated using the original log- derivative method and are combined with the propagator in an extremely simple way. Comparisons show that this algorithm is more efficient than the R matrix method and comparable with the original log derivative method.

VIII. Summary

Over the last few years, one has seen the growth of sophisticated computer packages for the determination of the S matrix for molecular scattering of different systems. The requirement has been for reliable and stable numerical algorithms at the heart of the code. Based on the theory of propagators, a stable recursion formula for the log-derivative has been developed which matches the well-known stability properties of the R matrix methods. Using a a hybrid of the original log-derivative method in the nonclassical region, together with propagation of the log-derivative matrix via a potential following technique seems to be a very effective combination. This is the approach taken in the Hibridon code of Alexander (1987).

If one needs the actual solutions, however, then they cannot be obtained by the above methods. One must wrestle with the instabilities described earlier,

SOLUTION OF THE EQUATIONS O F MOLECULAR SCATTERING 339

but these problems are well-understood. If first derivative terms are absent then the special formulae mentioned in Section VII still work extremely well. If first derivative terms are present, then one may either appeal to one of the library packages for the numerical solution of a set of first order differential equations or use some convenient personal code. More development is required to find stable efficient algorithms for the solution of Eq. (38) or, alternatively, algorithms for the solution of Eq. ( 1 1) that better exploit the special features of the Schrodinger equation.

In the future, one can see the need arising for the solution of larger sets of equations. There always will be a challenge to modify computer algorithms to increase speed, but one must be careful in that an approximation made in a close-coupling code can easily mean that one is effectively using a standard approximation such as Distorted Wave or WKB as has been pointed out by Lill et al. (1986). Of course, it may be that in many cases one should be using a valid approximation rather than the full close-coupling approach, but such thoughts are outside the scope of this paper.

One also must consider the impact of supercomputers and parallelism on the problem of molecular scattering. Although many researchers are using supercomputers very effectively and are gaining much knowledge in exploiting individual architectures (see Schwenke and Truhlar, 1985), no major breakthrough in algorithm design seems to have occurred. Application of parallelism looks to be a more fruitful area. At the simplest level, since most of the algorithms described above are written in terms of matrices, the standard parallel algorithms of linear algebra (Duff, 1987) could be applied immediately. Alternatively, one might treat the solution of each equation as an independent process that communicates with another process only when a coupling term is encountered. In any case, the numerical solution of the equations of molecular scattering will continue to be a fruitful field for research.

REFERENCES

Alexander, M. A. (1984). J . Chem. Phys. 81,4510. Alexander, M. A. and Manopoulos, D. E. (1987). J . Chem. Phys. 86, 2044. Alexander, M. A. (1987). Hibridon Scattering Code, University of Maryland. Allison, A. C. (1970). J . Comput. Phys. 6, 378. Arthurs, A. M. and Dalgarno, A. (1960). Proc. Roy. SOC. A. 256, 540. Baer, M., Drolshagen, G., and Toennies, J. P. (1980). J . Chem. Phys. 73, 1980. Bailey, P., Shampine, L. F., and Gordon, R. G. (1978). ACM Trans. Math Software. 4, 193. Brandt, M. A. and Truhlar, D. G. (1973). Chem. Phys. Lett. 23,48. Burke, P. G. and Schey, H. M. (1962). Phys. Rev. 126, 147. Coleman, J. P. and Mohamed, J. (1978). Math. Comp. 32, 751.

340 A . C . Allison

Duff, I. S. (1987). The State of the Art in Numerical Analysis (A. Iserles and M. J. D. Powell, eds.)

Faist, M. B. and Levine, R. D. (1976). J . Chem. Phys. 64, 2953. Fox, L. (1962). Numerical Solution of Ordinary and Partial Differential Equations. Pergamon,

Garrett, B. C., Redmon, M. J., Truhlar, D. G., and Melius, C. F. (1981). J . Chem. Phys. 74,412. Gordon, R. G. (1969). J . Chem. Phys. 51, 14. Gordon, R. G. (1971). Methods in Computational Physics, Vol. 10. Academic Press, New York,

Hartree, D. R. (1957) The Calculation of Atomic Structures. John Wiley, London, England. Heil, T. G., Butler, S. E., and Dalgarno, A. (1981). Phys. Rev. A. 23, 2953. Ixaru, L. Gr., and Rizea, M. (1980). Comp. Phys. Comm. 19,23. Johnson, B. R. (1973). J . Comput. Phys. 13,445. Johnson, B. R. (1977). J . Chem. Phys. 67,4086. Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations. John Wiley,

London, England. Lapidus, L. and Seinfeld, J. H. (197 1). Numerical Solution of Ordinary Differential Equations.

Academic Press, New York, New York. Lester, W. A. (1968). J. Comput. Phys. 3, 322. Light, J. C. (1971). Methods in Computational Physics, Vol. 10. Academic Press, New York, New

Light, J. C. and Walker, R. B. (1976). J . Chem. Phys. 65,4272. Lill, J. V., Schmalz, T. G., and Light, J. C. (1983). J . Chem. Phys. 78, 4456. Lill, J. V., Parker, G. A,, and Light, J. C. (1986). J . Chem. Phys. 85,900. Lyche, T. (1972). Numer. Math. 19,65. Magnus, W. (1954). Comm. Pure Appl. Math. 7, 649. Manopoulos, D. E. (1986). J . Chem. Phys. 85, 6425. Melkanoff, M. A., Sawada, T., and Raynal, J. (1966). Methods in Computational Physics.

Mott, N. F. and Massey, H. S. W. (1965). The Theory of Atomic Collisions. Clarendon, Oxford,

McLenithan, K. D. and Secrest, D. (1984). J. Chem. Phys. 80,2480. Numerov, B. V. (1924). Mont. Not. Roy. Astron. Soc. 84, 592. Mrugala, F. and Secrest, D. (1983a). J . Chem. Phys. 78, 5954. Mrugala, F. and Secrest, D. (1983b). J. Chem. Phys. 79, 5960. Mrugala, F. (1987). J. Comput. Phys. 68, 393. Parker, G. A,, Schmalz, J. G., and Light, J. C. (1980). J. Chem. Phys. 73, 1757. Press, W. H., Flannery, B. P., Teukolsky, S. A,, and Vettering, W. T. (1986). Numerical Recipes:

The Art of Scientijc Computing. Cambridge University Press, Cambridge, England. Raptis, A. D. and Allison, A. C. (1978). Comp. Phys. Comm. 14, 1. Raptis, A. D. and Cash, J. R. (1987). Comp. Phys. Comm. 44,95. Rosenthal, A. and Gordon, R. G. (1976). J . Chem. Phys. 64, 1621. Sam, W. M. and Kouri, D. J. (1969). J . Chem. Phys. 51,4809. Schwenke, D. W. and Truhlar, D. G. (1985). Supercomputer Applications. Plenum, New York,

Schwenke, D. W., Truhlar, D. G., and Kouri, D. J. (1987). J . Chem. Phys. 86,2772. Secrest, D. (1979). Atom-Molecule Collision Theory: A Guide for the Experimentalist. Plenum,

Shampine, L. F. and Gordon, R. G. (1975). Computer Solution of Ordinary Differential Equations.

Clarendon, Oxford, England.

Oxford, England, Ch. 1.

New York, p. 81.

York, p. 1 1 1 .

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England, p. 71.

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Stechel, E. B., Walker, R. B., and Light, J. C. (1980). J. Chem. Phys. 68, 3518. Thomas, L. D. (1979). J . Chem. Phys. 70,2979. Thomas, L. D., Alexander, M. H., Johnson, B. R., Lester, W. A., Light, J. C., McLenithan, K. D.,

Parker, G. A,, Redmon, M. J., Schmalz, T. G.. Secrest, D., and Walker, R. B. (1981). J . Comput. Phys. 41,407.

Thomas, L. D. (1982). J . Chem. Phys. 76,4925. van Dishoek, E. F., van Hemert, M. C., Allison, A. C., and Dalgarno, A. (1984). J. Chem. Phys. 81,

Walker, R. B. (1978). Quantum Chemistry Program Exchange, Program No. 352. 5709.



I 1 HIGH ENERGY CHARGE TRANSFER B. H . BRANSDEN and D . P . D E WANGAN? Department of Physics Universitj) of’ Durham Durhum, Enylund

I. Introduction . . . . . . . . . . . . . . , . . . . . . . . 11. Transition Amplitudes . . . . . . . . . . . . . . . . . . .

A. The Role of the Coulomb Phase. . . . . . . . . . . . . . . 111. Distorted Wave Series . . . . . . . . . . , . . , . . . . ,

A. Post-prior Discrepancy . . . . . . . . . . . . . . . . . . IV. First Order Models . . . . . . . . . , . . . . . , . . . .

A. Multi-Channel Distorted Wave (MCDW) Treatment . . . . . . . B. The Boundary Corrected First Born Approximation . . . . . . . C. The Bates Distorted Wave Model and Its Generalisation . . . . . ,

D. The Single-Centre Expansion Approximation . . . . . . . . . .

Sobelman, Glauber and Symmetrical Eikonal Models . . . . . . . . A. The Continuum Distorted Wave (CDW) Approximation . . . . . . B. The VPS Approximation . . . . . . . . , . . . . . . . . C. The Glauber Eikonal (GE) Approximation . . . . . . . . . . . D. The Symmetrical Eikonal Approximation . . . . . . . . . . .

V1. Second Order Theories . . . . . . . . . . . . . . . . . . . A. Intermediate Energy . . . . . . . . . . . . . . . . . . . B. High Energy Behaviour. . . . . . . . . . . . . . . . . .

VII. Relativistic Electron Capture . . . . . . . , . . . . . , . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . .

V. The Continuum Distorted Wave, Vainshtein Presnyakov and

343 345 347 348 349 349 350 35 1 354 357

357 359 360 361 362 363 363 366 369 37 I 37 1

I. Introduction

Collisions between ions and atoms can be considered to be “fast,” when the relative collision velocity, u, exceeds the Bohr velocity, u,, of the target electron or electrons taking part in charge exchange or some other process. Both coupled channel and perturbative models for fast collisions can be developed, the coupled channel model being particularly useful for intermediate energies ( 1 1 - uo). In this article we shall discuss in some detail the

‘t On leave from Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India.

343 Copyright 1988 by Academic Press, Inc

All rights of reproduction in any form reserved I C R N n.i?. i?m~xn

344 B. H . Bransden and D. P . Dewangan

perturbative approach to the nonrelativistic theory, which although introduced by Oppenheimer (1928) and Brinkman and Kramers (1930) some sixty years ago, has been properly clarified only recently. This is a particularly appropriate topic for this volume, since much of Alex Dalgarno's earlier work on charge exchange, in collaboration with D. R. Bates and also with N. M. King and one of the authors (B.H.B.), was concerned with the use of the Born approximation to describe electron capture from neutral atoms by fast protons (Bates and Dalgarno 1952, 1953; Bransden et al., 1954). A brief account of the theory at relativistic energies will be given in Section VII.

The theory will be discussed for a system in which a fully stripped ion ( P ) of charge Z p and mass M , is incident on a hydrogenic atom with a nucleus ( T ) of charge Z , and mass M , (atomic units are used here and throughout). For fast collisions, the relative motion of the target and projectile nuclei can be described by a classical straight line trajectory at constant velocity v and impact parameter b. Thus, if R(t) is the internuclear separation

R(t) = b + ~t b . v = 0. (1)

In this impact parameter treatment, which is entirely equivalent to a completely quanta1 description for fast collisions provided we are not concerned with large angle scattering, the electronic wave function tj *(rT, t ) satisfies the time-dependent Schrodinger equation

where rT and rp are the position vectors of the electron with respect to the target and projectile nuclei, respectively. The target nucleus T has been taken to be the origin, so that rdt) = -R(t) + IT, rT and t being independent variables so that the differential d / d t is taken with r T fixed. Any other choice of origin along the internuclear line yields the same cross section, since the Schrodinger Eq. (2) is translationally invariant (see Bransden, 1972, 1983).

Inspection of Eq. (2) shows that the internuclear potential (ZTZ,/R) can be removed bv the transformation

Since Eq. (3) is a phase transformation, it follows that the internuclear potential has no effect on transition probabilities, which can be determined from Eq. (4). In contrast, the calculation of differential cross sections requires the restoration of the correct phase, if the internuclear potential is omitted (see Bransden, 1972, BelkiC et al., 1979).

HIGH ENERGY CHARGE TRANSFER 345

Oppenheimer (1928), Brinkman and Kramers (1930) and Takayanagi (1952), realising that the internuclear potential should not contribute to transition probabilities, developed a first-order perturbation theory known as the OBK approximation in both wave and impact parameter versions, by taking either V, = - Z p / r p or V, = -ZT/rT as the perturbation corresponding to the initial arrangement P + (T + e - ) or the final arrangement (P + e - ) + T, respectively. The fact that the overall interaction in both the initial and final arrangements is long-range was ignored, so that in the wave treatment, for example, the relative motion of P and T was described by plane waves. As we shall see below, this accounts for the fact that the OBK cross sections are in very poor agreement with experiment. Noting this disagreement, Bates and Dalgarno (1952, 1953) and, independently, Jackson and Schiff (1953) argued that in the case of H + + H for which Z , = Z , = 1, the perturbation theory should be based on Eq. (2), which retains the internuclear potential. It was suggested that the partial cancellation between the matrix elements of the electron-nuclear term and the internuclear term should account in some degree for the error due to the nonorthogonality of the initial and the final state wave functions, and should lead to improved values of cross sections. This prescription is in fact very successful for protons incident on a neutral atom, but subsequent work shows that if extended to other systems in which an overall Coulomb interaction exists in either the initial or final arrangement, agreement with experiment becomes very poor.

In the following sections of this review, we shall show how it has been established that a correct perturbation theory must take into account the overall long-range interaction in each channel. In retrospect, we can see why the Bates-Dalgarno (Jackson-Schiff) Born approximation, commonly known as the JS approximation, is successful for reactions like H + + H(1s) -+ H(n,lJm,) + H + . It is because, in this case, the chosen perturbations (Z ,Z, /R - Z p / r p ) or ( Z , Z p / R - ZT/rT) are of short-range, and the assumption of plane wave relative motion is then correct.

We refer the reader to Bransden (1988) for a discussion of advances in the coupled channel approach and of correlation effects at intermediate energies.

11. Transition Amplitudes

Let us define the unperturbed wave functions in the initial and final arrangement by @T(t) and @F(t), respectively. These functions satisfy the Schrodinger equations

346

and

B. H . Bransden and D. P . Dewangan

[Hf - i ;-J@(t) = [ - ;(t) = 0. (5b)

Taking into account the translational motion of the electron when attached to the projectile, P, we have

and

where 4; and 4; are hydrogenic eigenfunctions centered on T and P, respectively.

It is clear from Eq. (4) that the electron near one nucleus experiences the field due to the other nucleus, even at asymptotically large separations. Cheshire (1964) and BelkiC et al. (1979) have shown how this residual Coulomb interaction can be taken into account by specifying the Coulomb boundary conditions

~ + ( t ) ___+ t + - m @T(t)e-iu{(t) (7a)

‘Y; ( t ) x OF( t)eiar(*) (7b)

where oi = up ln(Ru - u2t) and of = uT ln(Ru + u2t) with up = Zp/u and uT = ZT/u are the familiar logarithmic Coulomb phases. It should be emphasized that in the asymptotic limits t -+ 00, the wavefunctions Y * ( t ) satisfy the Schrodinger Eq. (4) only when the Coulomb phases are properly retained.

In general, distorted waves t+(t) and q;(t) can be introduced that are the solutions of the following time-dependent Schrodinger equations:

where Ui and U , are arbitrary short-range potentials. Clearly, <+(t) and q;(t) satisfy the Coulomb boundary conditions

t + ( t ) @T(t)e-iul ( 9 4


The post form of the transition amplitude, A+(b), is given by lim,++ m(q;(t)l'€'T), which can be manipulated into the following form

The second term vanishes since limt+ - ,(v; IY,?) = limz-.-m(q; I@Teiai). We then have from Eq. (8b) and (4) that

A + @ ) = - i j-,(q;l -H,DW + HlYZ) dt = -i

where we have introduced

m m

( r , q l ~ l Y ~ ) dt (11) I-, Lq=q-ui ; w,=V,-u,

where

The expression (Eq. (1 1)) for the transition amplitude is the impact parameter version of the well-known quanta1 formula for the distorted wave transition amplitude (Bransden, 1983). Similarly, the prior form of the transition amplitude is given by

A. THE ROLE OF THE COULOMB PHASE

The crucial role of the Coulomb phases and, therefore, the long range nature of the Coulomb interaction in perturbative approaches, can be demonstrated easily in a simple treatment given by Dewangan (1984) and Dewangan and Eichler (1987b). It often is stated in the literature that the exact transition amplitudes are given by (e.g., Macek and Alston, 1982, Dub4 1986)

ZP m

d-(b) = -i dt(Y; 1 - -l@T(t)). 5- m rP

Equations (14a and b) are obtained from Eqs. (11) and (13) by setting Ui = U , = 0 and assuming that the interactions involved in the problem are

348 B. H . Bransden and D. P. Dewangan

of short-range so that there are no asymptotic Coulomb phases. The conditions of Eqs. (9a and b) immediately show that

(1 5a)

A - ( b ) = [li~I,-me-'"~l"(R"-" "]d-(b) . (15b)

A+(b) = [lirn,, + , - i u ~ l n ( R v + v I ' ] d f ( b )

Since the phase factors in square brackets are singular, it follows that d *(b ) must contain a compensating singularity, since A*(b) is itself well defined. In order to develop a perturbation series for d*(b), the effect of this singularity must be identified in each term of the series and removed systematically. This has been studied by Macek (1988), but here we shall avoid this complication by working throughout with the appropriate Coulomb phases. The related question of how the boundary conditions are satisfied in coupled channel formulations have been studied by Becker et al. (1980) and will not be discussed here.

111. Distorted Wave Series

An integral equation can be obtained for the exact wave function 'PT(t), satisfying the boundary condition in the form

vT(rT, t ) = (T(rT, t ) + dt' dr;G;(rT, t ; rkt')w(rk, Z')y(rk, 2 ' ) (16) SP, s where

G,'(r,, t ; r;, t ' ) = -iO(t - t ' ) 1 tn-(rT, t)&-*(rk, t ') (17) n

in which O ( t - t ' ) is the step function. An expansion for 'Y: is obtained by iteration of Eq. (16) leading to a series of time-ordered products. Inserting this series into Eq. (1 l), a corresponding expansion of the amplitude A + ( b ) is obtained, which can be written symbolically as

A + ( b ) = -i 1 dt(q;l W, + W,C,' W; + W,G,f WG,' + ...I&+). (18)

A corresponding series can be obtained in terms of the Green's function G;, associated with the rearranged channel,

A - ( b ) = - i 1 dt(q; Iw + W,G; + W,G: W,G; WT + ...I&+), (19)

where

m

- m

m

- m


The idea behind the distorted wave method is that by a suitable choice of Ui and U , the series (Eqs. (18) and (19)) can be made to converge rapidly. In particular, the first order approximations

m

Agw,(b) = -i j dt(q; -41

or m

may be sufficiently accurate over a particular energy interval.

A. POST-PRIOR DISCREPANCY

The initial and the final channel distorted wave functions, defined as solutions of Eqs. (8a) and (8b) are asymptotically orthogonal,

lim, + * m (?; (9 I ti+ (t)> = 0.

m a

(22)

Writing this condition in the form

(23) j dt (rl;(t)Iti+(t)> = 0,

- i J d t ( q ; I w / t + ) = - i s d t < v ; ~ ~ ~ t + ) ,

- m

and using Eqs. (8), we easily find that m m

(24) - m - m

so that, even for an arbitrary choice of Ui and U,, there is no post-prior discrepancy in the first-order distorted amplitudes of Eqs. (21a) and (21b). In contrast, the post and prior forms of the second and higher order terms of the theory are not usually equivalent.

IV. First Order Models

Corresponding to various “physical” choices of Ui and U,, there exist in the literature a variety of first-order distorted wave approximations. It is not possible to discuss here all first-order models in detail, instead we shall discuss the physical contents of some frequently employed first-order approximations.

350 B. H . Bransden and D . P . Dewangan

A. MULTI-CHANNEL DISTORTED WAVE (MCDW) TREATMENT

In order to discuss systematically several first-order models from a unified point of view, we consider a MCDW treatment following Dewangan and Eichler (1987b). We introduce the following projection operators:

and define the distorting potentials

ui = Pi yPi (264

u, = P,'v,P,. (26b)

With this choice, the time-dependent Schrodiner Eqs. ( 5 ) become the usual set of single-centre coupled channel equations in which couplings amongst the N target levels in Eq. (8a) and the couplings amongst the M projectile levels in Eq. (8b) are treated exactly. These equations can be conveniently solved by standard techniques by using single-centre expansion (SCE) basis to represent C;' and q;, viz.

N

and

Explicit inclusion of the phases makes the amplitudes a,([) and b,(t) become independent of time as t + k 00. In first order, the MCDW amplitude is given by Eq. (21) using C;: and q;, defined by Eq. (27). The physical content of the model is that the target interacts strongly with the projectile up to a time t during which it undergoes any sequence of transitions within the group of N preselected target levels. Then, at time t , under the influence of the reduced interaction (6 - Uf), the target electron is captured by the projectile into any of the group of M preselected levels. Following capture, the projectile interacts with the target nucleus and is excited and deexcited within the group of M levels any number of times before entering the final state.


B. THE BOUNDARY CORRECTED FIRST BORN APPROXIMATION

Probably the simplest choice of distorting potentials is Ui = 0 and Uf = 0. This leads to the "boundary corrected first Born" or the B1B approximation (BelkiC et al., 1986a, Dewangan and Eichler, 1985)

and

These amplitudes may be interpreted as representing single collisions in which the projectile interacts only once with the target via the potential V, (in the post form), or (in the prior form).

1, The Oppenheimer-Brinkman-Kramers Approximation

Section I) (for simplicity we consider only the post form) The B1B amplitudes can be contrasted with the OBK amplitude (see

This expression does not include the residual electron-nucleus term ZT/R and omits the Coulomb phase functions from the initial and the final channel functions. The expression with the perturbative expansion (Eq. (18)) of Eq. (14a), which is obtained by setting C: = G:, the free electron Green's function corresponding to the choice tq-(rT, t ) 3 ( 2 ~ ) - ~ ' ~ exp(ik-rT - ik2/2t), the undistorted plane wave states, in Eq. (17). Such an expansion is strictly valid only for short-range interactions. Although this expansion is believed to provide the correct high velocity form of the capture amplitude, it cannot be expected to be useful at finite impact velocities. Indeed, Miraglia et al. (1981), Simony and McGuire (1981) and Simony et al. (1982) have shown that the second-order version of the model, to be referred to as the OBK2 approximation, provides an even poorer description of the experimental data than the first-order OBK approximation, over the experimentally accessible region (see Table I and Section VI).

2. The Bates- Dalgarno (Jackson-Schir) Approximation

As was noted in Section I, the earliest attempts to improve the OBK approximation were made independently by Bates and Dalgarno (1952) and


Jackson and Schiff (1953) who introduced a first order model, in which the post form of the capture amplitude is expressed as

m

A,,(b) = -i dt(Cf(t)I - 2 J(DT(t)). s, The validity of this (JS) approximation often has been questioned for the

following reason. This approximation may be viewed as a first-order term of a perturbative expansion on a plane wave basis of the capture amplitude

in which Y+ is a solution of the Schrodinger Eq. (2). Since the phase transformation (Eq. (3)) removes the internuclear potential from the Schrod- inger equation, it follows that in an exact treatment, this potential does not contribute to total cross sections. Based on this fact, it has been argued that the internuclear potential should be dropped from Eq. (31) at the outset to avoid any spurious contributions.

Many discussions have been presented in the literature as to which of the two approximations, the OBK approximation or the JS approximation, is an acceptable form of first order theory (see Bransden, 1965, 1972; McDowell and Coleman, 1970; Mapleton, 1972). In order to resolve the problem, both these approximations were applied to study the asymmetrical collision (2, > 2,) of electron capture by protons from the K-shell of argon (Halpern and Law, 1975; Band, 1976). These studies revealed that, in an asymmetrical collision, the JS approximation can overestimate the experimental differential cross sections (Cocke et al., 1976) as well as the total cross sections (MacDonald et al., 1974) by two to three orders of magnitude. In addition, the predictions of the OBK approximation also are not satisfactory, since they overestimate the experimental data by three to four times.

The poor performance of the OBK and the JS approximations can be related to the fact that neither of these preserve the correct boundary conditions. Although the role of the proper Coulomb boundary conditions was discussed in detail by Cheshire (1964) and later by BelkiC: et al. (1979), (see also Omidvar et al., 1976; Lin et al., 1978), the Coulomb interactions involved in the problems often have been treated incorrectly as if they were of short-range. The crucial fact is that the amplitude d i given by Eq. (31) is related to the exact amplitude A + of Eq. (1 1) by a relationship similar to Eq. (M), showing that d; contains an ill-defined phase factor. From the discussions following Eq. (15), it is clear that neither Eq. (14) nor Eq. (31) form a suitable basis for perturbative models. A particular example is a perturbative expansion of Eq. (14), which employs either Eqs. (17) and (18), or Eqs. (19) and (20), modified by omission of all Coulomb phase factors and with W;, set


equal to (- Zp/rp) and W, set equal to (- ZT/rT). In this model, Dewangan and Eichler (1 975) have shown that the second-order approximation (called the Strong Potential Born approximation, Macek and Alston, 1982) is singular, a fact that had been noticed previously by Mapleton (1967, 1972) in connection with a related second-order amplitude for p-H collisions. The importance of using an expression such as Eq. ( l l ) , which contains the correct Coulomb phases, to obtain divergence free approximations must be emphasised.

3. Application of the B1 B Approximation

The B 1 B approximation appears to be the simplest first-order approximation consistent with the boundary conditions. Attention has been focused on this approximation by the work of Dewangan and Eichler (1989, though Eqs. (28) were written several years ago (Belkik et al., 1979). Following the initial demonstration (Dewangan and Eichler, 1986; Belkik et al., 1986a) that this approximation gives results in reasonable agreement with the experimental data, even for asymmetric charge transfer collisions, several calculations have been reported (e.g., Belkii. et al. 1986b, 1987a,b; Dewangan and Eichler, 1987a; Deco et al. 1986a). Programmes for computing nilirni + nrlfrnf cross sections have been developed (Belkii. et al. 1987a; Roy et al. 1980). It is found that the predictions of the BIB approximation in the intermediate energy regions are in reasonable agreement with available experimental data. An illustrative example of the application of the asymmetric model to the p-Ar collision is shown in Fig. 1.

Even more telling is a series of calculations for capture by the fully stripped ions H + , He2+, Li3+, Be4+, B5+, C6+ from H(1s) by Belkik et al. (1987b). Over an impact energy range of at least from v = vo to u = lo,, good agreement is found with the experimental data (where this exists). The result of the JS approximation, which is identical with that of the B1B approximation for H + + H(ls), gradually diverges from the B1B cross section and the data, as the charge on the incident ion increases (see Figs 2 and 3). Further results for H + + H(1s) have been reported by Omidvar (1975) and Roy et al. (1 980).

It is interesting to note that for electron capture by protons from neutral multielectron atoms, the JS approximation again becomes rigorously equivalent to the B1 B approximation. Many years ago, Bransden et al. (1954) and subsequently, Mapleton (1961, 1968, 1972), computed the JS cross section for proton-helium charge transfer. The remarkably good agreement which the JS cross sections show with experiment for p-He charge transfer (see Mapleton, 1972) suggests that the B1B approximations can be expected to provide reasonable results for multielectron systems in the intermediate energy

354 B. H, Bransden and D. P. Dewangan

100

- (Y

E 4"

7 0,

.!? 10

- c

c U al v)

v) v) 0 L U

I I I I l l l l l

1 2 5 10 Energy (MeV I

3

FIG. 1. Electron capture by protons from the K shell of argon: - B1B: the first order boundary corrected total cross section (Dewangan and Eichler, 1986). Experimental data points from Horsdal-Pedersen et al. (1983a).

region. Further work will help assess the situation. It should be noted that in the case of proton impact, and only in this case, the Coulomb-Born approximation (see, for example, Mandal et al., 1981 and references therein) is equivalent to the B1B approximation, and this accounts for its success.

C. THE BATES DISTORTED WAVE MODEL AND ITS GENERALIZATION

Almost thirty years ago, Bates (1958) derived a first-order distorted wave model as an approximation to the two-state coupled channeI approach, in which only the initial state around the target and the final state around the

355

t 10Z2L

-21 t_ 10 1 I I l l 1 1 I I I I 1 1 I l l V

0.05 0.1 0.5 1.0 2.0 Laboratory Energy (MeV)

FIG. 2. Electron capture by protons from atomic hydrogen: the calculated total cross section in the BIB approximation (Belkii. et al., 1987b). This coincides with the Jackson-Schiff Born approximation for this system. + Second order Jackson-Schiff approximation for capture into the Is level only (Kramer, 1972). Experimental data: (a) H target; Bayfield (1969), Fite et al. (1960), 0 Gilbody and Ryding (1966), A McClure (1966), + Wittkower et al. (1966). (b) derived from experiments with an H, target; 0 Stier and Barnett (1956), A Barnett and Reynolds (1958), 0 Welsh et al. (1967), V Williams (1967), A Schryber (1967), Toburen et al. (1968).


9 0.1, 0.6 0.8 1 2 3

10'' o,2

Laboratory Energy (MeV)

FIG. 3. Electron capture by C 6 + from atomic hydrogen: total cross sections calculated by Belkii: et ul. (1987b). - BIB approximation, - ~ - JS approximation. Experimental data: 0 Goffe et ul. (1979).

projectile are retained. This approximation is closely related to a special case of the MCDW treatment corresponding to the choice

(324

(324

ui = Ki(R) = (4T(rT) I V I +T(r,>>

Uf = V f W = <4,'(r,)I v,l +,'@,)>,

and the amplitude in this approximation is given by

ABDw = - 1 dt(q$'e+iof+ilYo,Yrrdt 1~ - v.1 @ T e - i a i - l t - , V i i d t >. 133) *

An important feature of this model is that it allows for elastic scattering by the initial and the final state charge distributions. Only a few detailed calculations have been performed with this interesting approximation', however (Toshima et al., 1987). A unitarised version of the Bates distorted wave model has been developed by Ryufuko and Watanabe (1978) and

' A related approximation introduced by Bassel and Gerjuoy (1960) is rigorously equivalent to the approximation of Eq. (33) in the case of p-H resonant capture. Numerical results have been given by Roy et ul. (1980).


applied to a large number of charge transfer reactions (see Bransden and Janev, 1983).

D. THE SINGLE-CENTRE EXPANSION APPROXIMATION

Within the framework of the MCDW treatment, a simple generalization of the Bates distorted wave approximation is obtained by choosing

U , = V,, and Ui = PivPi

where the projection operator Pi is given by Eq. (25a) and where the post form of the amplitude, Eq. (21a), is used. This model was originally proposed by Reading et al. (1976) and is called the single-centre expansion (SCE) approximation, since the distorted wave (: in the entrance channel given by Eq. (27a) is obtained by solving a set of coupled channel equations in which the atomic basis set around the target is used. The basic idea of this model is to obtain an accurate approximation to the exact total wavefunction, Y +, of the system by expanding it in a large but manageable basis set, which may also contain pseudo-states to improve the convergence properties of the solution. This model has been applied to many systems in studies of inner shell vacancy production, ionization and charge transfer (Ford et al., 1979; Reading et al., 1979; Becker et al., 1980) are found to give a good description of the experimental data at intermediate energies.

V. The Continuum Distorted Wave, Vainshtein Presnyakov and Sobelman, Glauber and Symmetrical Eikonal Models

We shall now discuss a group of models that attempt to allow, as completely as possible, for the time-dependent interaction of the projectile with the target electron, at the expense, of course, of approximations elsewhere-for example, in the electronic binding energy. We can start by treating the target electron as a “free” electron, in which case its wavefunction, S?:, in the field of the projectile satisfies the Schrodinger equation

i - 9 : = 0 . ITT - R I at ”1

a a at 82,

Since ~ = - u - = - v . V,,, we immediately find

9: = N+,F,[io,, l,iuip + iv.r,]

(34)

(35)


where N + = r(l - iup)e+l’Z””P. Since

9: exp(-iaJ,

the distorted wave function defined by the factorable form (:Dwi = Oi(t)9; is consistent with the boundary conditions in Eq. (9). Note that the function 9: does not contain any information about the atomic state of the system and is the same for all possible initial states. The distorting potential Ui = UdDwi is readily computed by substituting (GDwi in Eq. (8a). The result is

uic: = u:DWi{&Wi = q t & W j + v@T *v6R: f (36)

We now define an eikonal function 9& by

(37) 9: , 9ii = e-iupln(urp+v.rp) zp’ di

Clearly, 9& satisfies the following differential equation

It is easily verified that 9: is the solution of the following integral equation:

where (rk)z = zb; (rb)x,y = (rp)x,y and the Green’s function is defined by

(40) 1

U G2(rp, rk) = - - 9&(rp)9L;(rb)@(z;l - zp).

It should be noted that the conversion of the differential Eq. (33) into integral Eq. (39) is possible because the distorted wave basis 9 Z i has been used. Indeed, it is not possible to obtain an integral equation for 9:(rP) starting with the boundary condition

=%+(TP) Zp’m. 1,

reflecting the fact that the Lippman-Schwinger equation for a Coulomb potential in a plane wave basis does not exist (McDowell and Coleman, 1970).

Proceeding in a similar way, we can obtain the continuum wavefunction 9; of an electron in the field of the target nucleus as

9 F ( r T ) = N;F,(-iu,, 1, -iurT - iv .rT) (41)


where N - ( o , ) = exp(1/2~uT)r(l + iuT). Clearly, in the exit channel the factorable form uCDwf = @,fP(rp)9;(rT) is a valid distorted wave function which satisfies the Schrodinger Eq. (8b) provided

u f q t = UCDWf‘l&Wf = 6 q G D W f + @v In 4,fP’vyL(rT). (42)

(43)

As before, we can introduce an eikonal function 9&rT) defined by

2; + yif(rT) = eivTln(urT+v.rr)

ZT-m

which satisfies

(iv . v + ?)9&) = 0. (44)

It is readily verified that 9; (rT) satisfies the following integral equation

in which

(46) i

Gif(rT, r;) = - 2’&(rT)9EF (r:)@(zk - zT) U

and ( G ) Z = 2;; (rk)x. y = (TTIX. Y .

I t is clear that the eikonal distorted wave functions defiened by Czi = @:( t ) 9 i i ( r p ) and qEf = @‘(t)9if(rT) satisfy the Coulomb boundary conditions. It is readily verified that tii and qEf are solutions of Schrodinger Eqs. (8a) and (8b), respectively, provided distorting potentials Ui = UEi and Uf = UEf are chosen according to

U E i l & = @T()V2 + V In 4:. V)LZG~ + vtGi u E f & = @r(&v’ + V In 4:. V ) y i f + Kq&.

(47)

(48)

We shall now use these results to derive a number of specific approximations.

A. THE CONTINUUM DISTORTED WAVE (CDW) APPROXIMATION

In 1964, Cheshire proposed a model in which distortions are included symmetrically in both the channels. In this case, Ui and U , are given by Eqs. (36) and (42), respectively, so that, from Eq. (21) the post and prior forms of the CDW amplitudes are given by

m

AGDw(b) = i J cit(~,f~v In +~P.v-Y; I <bDwi> (494 - m

360 B. H. Bransden and D. P . Dewangan

and m

A&w(b) = i 1 dt(tj'&wfl(DTV In 4T.vgy). (49b) - 0 0

Note that there is no post-prior discrepancy in this model. We see that this derivation of the CDW amplitude is based essentially on the conventional distorted wave formalism of the transition amplitude outlined in Section I11 (Cheshire, 1964; McCarroll and Salin, 1967; McDowall and Coleman, 1970).

It has been shown by Gayet (1972) that the CDW amplitude can also be derived as a first-order term of the form of the perturbation series introduced by the Dodd and Greider formulation (1966). From a theoretical point of view, the Dodd and Greider theory is particularly suited for rearrangement collisions (McDowell and Coleman, 1970), since the higher order terms of this series, unlike the usual distorted wave series, do not correspond to disconnected diagrams and the series converges at sufficiently high energy.

Since the CDW approximation can be viewed as a first-order term of the Dodd and Greider formulation and satisfies the basic conditions of a well- founded theory, it has received a great deal of theoretical attention. During the last few years, general features of this approximation have been analysed and closed form expressions have been obtained for arbitrary nilimi -+ n, l,m, capture (Crothers, 1981, 1985a; DubC, 1984; Saha et al., 1986). In the intermediate and high energy region, it is found that the CDW approximation gives satisfactory agreement with experimental data, both for the total and the differential cross sections, and for a wide range of 2, and 2,.

An interesting coupled channel formulation employing the CDW wave functions in the initial and the final states has been developed and applied to p-H charge transfer (see Crothers and McCann, 1985).

B. THE VPS APPROXIMATION

In order to connect the CDW amplitude with first-order theories in which distortion is retained asymmetrically in the initial channel, ie., Uf = 0, Ui # 0, we replace t j ' & , f in Eq. (49b) by its asymptotic form @;g&, giving

J - a ,

The phase factor 3if can be expanded in powers of V, according to 9- - eiu.rln(ur.r+v.rT)

Ef -

HIGH ENERGY CHARGE TRANSFER 36 1

where v(t') is defined by Eq. (12) with rT(t') = rp + R(t'). It should be noted that the potential vf(t') has no Coulomb tail and, as a consequence, the integral over t' converges. Substituting Eq. (51) in Eq. (50) and retaining only the term corresponding to n = 0, we obtain

m

1- m

s_,

A,,,(b) = i dt(ei"'@FIV@,T.V2;). (524

A straightforward application of Eq. (24) with Ui = UGDwi and U , = 0 shows that

m

Avp,(b) = -i dt(eiuf@FI V,( tCDwi). (52b)

This approximation was first discussed for charge transfer McCarroll and Salin (1967) and, due to its resemblance to a corresponding approximation for direct excitation, may be called the VPS (Vainshtein, Presnyakov and Sobelman, 1964) approximation'. The VPS cross sections for K - K capture in collisions of protons with C, N, 0, Ne and Ar have been computed by Ghosh et al. (1987) and are in good agreement with the experimental data.

C. THE GLAUBER EIKONAL (GE) APPROXIMATION

A simple first-order distorted wave theory consistent with the Coulomb boundary condition is the Glauber eikonal (GE) approximation first discussed by Dewangan (1982) and applied to p-H charge transfer by Sinha et al. (1986). The post form of this approximation is found by substituting @F(t)2'& for <GDwi in Eq. (52b). We obtain

It is instructive to obtain a Born-type series by expanding the G E amplitude in powers of y . Using an expansion similar to Eq. (51) for 9&, it is straightforward to show that

Each term of the series is consistent with the Coulomb boundary conditions and is well defined. The first term in this expansion is identical with the B1B

An analysis of the VPS approximation that can be extended to the case of charge exhange has been given by Dewangan and Bransden (1982).


amplitude and therefore, if only the first term in Eq. (54) is important, the GE approximation reduces to the B 1B approximation.

by Had we replaced e - i u ~ l n ( U r P + v . r p )

in Eq. (53) and expanded it in powers of the integral

we would have found that all terms of Eq. (54) containing this integral were divergent. Further, none of the terms of such an expansion would have been consistent with the Coulomb boundary conditions and truncation of this series for any arbitrary value of n 2 2 would lead to a divergent result. This example illustrates how one can get into difficulties for a well-defined problem containing a Coulomb potential if the long-range nature of the Coulomb interaction is disregarded.

In order to make the nature of the higher order terms clear, we consider the second term of Eq. (54) which, after introducing a complete set of target states, may be written as

r m

x 1; a, dt'(4Te-ieit'l K(t')I @T(t')). ( 5 5 )

Thus, the intermediate states in A& all have the same energy ci as the initial state. The same result clearly is true for all higher order terms. An immediate consequence of this result is that only the interaction vi is treated exactly to all orders in this approximation.

D. THE SYMMETRICAL EIKONAL APPROXIMATION

Let us consider the amplitude A defined by Eq. (50). Noting that q;f(=@:9;f) satisfies the distorted wave Schrodinger Eq. (8b), with U , = U,, given by Eq. (48), we immediately obtain, by using Eq. (24),

m

A = i S_ d t ( ( ~ ~ ( + ~ 2 + v In dF.v)Tif 1 tzDwi). (56)


Further approximating tcDwi by t& 3 @T(t)5?&(rp), we find

00

= i j dt(q;fImT(iV2 + v In ~T.V)Y;J (57b) -a

where Eq. (24) has been used to obtain the last expression. The approximation of Eqs. (57), called the symmetrical eikonal (SE) approximation, was first discussed by Maidagan and Rivarola (1984) to symmetrise the eikonal approximation. Closed form expression for arbitrary nilimi -+ n,Zf mf capture amplitude has been evaluated by Deco et al. (1986b). This approximation gives a good description of experimental total cross sections.

It is interesting to compare the SE approximation with the G E approximation. We easily see that substituting eiaf for YpEf (see Eq. (51)) in Eq. (57b) and using Eq. (24), we obtain the G E approximation (Eq. (53)). This shows that, if the SE amplitude is expanded in a particular way, it contains as its first-order term the GE amplitude, but also includes contributions from higher order terms in the exit channel.

We already have seen that the post form of the GE approximation takes full account of the elastic scattering by the static potential, Fi. It also can be shown that the SE approximation includes, in addition, the corresponding contributions to elastic scattering by the static potential in the final channel. In fact, it contains the Bates distorted wave approximation as the first term in an expansion.

From a theoretical point of view, our analysis reveals that the CDW approximation has many desirable features. In particular, it is possible to obtain several other useful approximations (e.g., the BlB, the VPS, the SE, the Bates approximations) by expanding the CDW amplitude in certain ways. It should be noted that the CDW, SE, Bates and B1B approximations all treat the initial and final configurations symmetrically, so that there is no post-prior discrepancy. In contrast, the GE and VPS approximations are unsymmetrical, only the description of either the initial or the final channel wave function being improved.

VI. Second Order Theories

A. INTERMEDIATE ENERGY

Although the first-order models predict total cross sections in reasonable agreement with experimental observations at intermediate energies, they do


not provide adequate desriptions of all the experimental data. For example, the B1 B approximation predicts an unphysical zero in the differential cross sections. Similarly, measurements of alignment and orientation parameters in p-He collisions (Hippler et al., 1986) show that for 100 < E < 300 keV, the CDW calculations do not give satisfactory agreement with this data, see Bransden (1988) for a further discussion. Such measurements of angular distribution provide a much more sensitive test of the theoretical models than measurements of total cross sections. The unsatisfactory agreement obtained between first-order approximations and the experimental angular distribution underlines the need for further refinement of the models. Current research is being directed to testing the adequacy of higher order models.

As was pointed out in Section IV B, in the past, the free electron Green’s function, Go, has frequently been employed to discuss the second- and higher order terms of charge transfer theory (Drisko, 1955; Bransden, 1965, 1983; McDowell and Coleman, 1970) without taking into account the Coulomb boundary conditions. The corresponding perturbation series in the boundary corrected approach is obtained from Eqs. (16) to (20) by choosing

T -ioi &+ = Qi e

and

qf- = @fPeiar, & = 6 and W, = vf and by setting G; and Gp’ equal to the corresponding Coulomb distorted free electron Green’s functions obtained by taking

!&(rT, t ) = exp

and

qn-(rT, t ) = exp

in Eqs. (17) and (20), respectively. Clearly, the first term of this series is the B1B approximation discussed in Section IV B.

It turns out that the second-order theory of this approach, to be referred to as the B2B, approximation, is rigorously equivalent, for proton-hydrogen collisions only, to the one considered by Drisko (1955) in his analysis of the high energy behaviour of proton-hydrogen ground state to ground state charge transfer cross sections. Kramer (1972) computed proton-hydrogen Is-1s cross sections obtained in the B1B approximation and the OBK2 approximation in Table I. It is interesting to note that in the energy region, 100 keV I E I 2.5 MeV, the B1B cross sections have similar magnitudes to the B2B, cross sections. Indeed, both the B1B and the B2B, cross sections are

HIGH ENERGY CHARGE TRANSFER

TABLE I CROSS SECTIONS IN ATOMIC UNITS FOR p + H(1s) + H(1.s) + p

E (keV) OBK OBK2" B1B B2BOb

100 1.26 5.0 2.31( - 1)* 3.60(- 1) 500 1.03(-3) - 3.16(-4) 2.96(-4)

1 OOO 2.50( - 5) 4.4( - 5) 9.23( - 6) 6.58( - 6) 2500 1.35 (-7) - 6.09( - 8) 4.20( - 8)

365

*The number in the parenthesis indicates the power of 10

a Miraglia et al. (1981). Simony and McGuire (1981). by which the corresponding entry is to be multiplied.

Kramer (1972).

in good agreement with the experimental cross sections (see Fig. 2 and also Schwab et al. (1987) for measurements of cross sections in atomic hydrogen at high energies). The OBK2 cross sections, being larger than the OBK cross sections in the energy region 100 < E < 3 MeV, fail to provide any improvement in the agreement over the OBK cross section (Miraglia et al., 1981; Simony and McGuire, 1981).

Recently, Belkik (1987) has computed differential cross sections for p-H ground state charge transfer at 125 keV in the B2B, approximation and, as expected, they do not possess a zero at this energy and are in reasonable agreement with the measurements of Martin et al. (1981).

A tractable and useful second-order theory improving on the B2B0 approximation is the boundary corrected second Born or B2B approximation (Dewangan and Eichler, 1985) which is obtained by replacing the Coulomb distorted free electron Green's functions by the target Green's function GTf, given by Eq. (17) with cn-(rT, t ) = (D;f(t)efiUi(-') or the projectile Green's function Gp' of Eqs. (19) with qn-(rT, t) E cD:(t)eiuf(').

In order to assess the importance of the second-order term, Dewangan and Bransden (1988) have performed a numerical calculation of the B2B amplitude for proton-hydrogen ground state to ground state charge transfer at E = 125 keV by retaining essentially all s, p, d and f target intermediate states. The results of the calculations are summarised in Table 11. Note that, at these energies, in a similar second Born calculation for direct excitation in proton-hydrogen collisions, the higher order terms were found to be important (Bransden and Dewangan, 1979). It is encouraging to note, however, that the B2B cross section is not very different from the B1B or the Bates approximation cross sections. The OBK cross section at this energy is larger by a factor of 4 than the B2B cross section.

Progress has been made towards the computation of the second order terms of the CDW (Crothers, 1985b) and the SE (Deco and Rivarola, 1985)


TABLE I1 CROSS SECTIONS IN ATOMIC UNlTS FOR

p + H(1s) + H(1s) + p AT 125 keV

B2B (bound intermediate)

OBK BIB state only) B2B

0.557 0.109 0.126 0.14

perturbation series. Much work is needed to assess the merits of the various second order approximations at intermediate energies.

B. HIGH ENERGY BEHAVIOUR

The OBK cross section for arbitrary initial (nilImi) and final (n f l fmf ) states has been evaluated in a closed form by Omdivar (1967). At asymptotic velocities, the OBK cross section varies as v - 12-21i-2'r (McDowell and Coleman, 1970; Briggs and DubC, 1980; Shakeshaft and Spruch, 1979). The B1B cross section at asymptotic velocities also varies as u-12-21 i -21r but with a different constant of proportionality (Omdivar, 1975; Karnakov, 1982). For high energy behaviour of the cross sections of several first-order models, we refer to Belkii- et al. (1979). Dewangan (1982), and Maidagan and Rivarola (1984). The fact that different first order models differ at asymptotically high energies indicates that higher order terms must be important even at arbitrarily high energies. Indeed, more than thirty years ago Drisko (1955) employed the second order Born (B2B0) approximation discussed above to analyse proton-hydrogen ground state to ground state charge transfer and showed that the high energy behaviour of the cross section is modified from a

to a 0-l' dependence. He also showed that while the third-order term v - 12

altered the coefficient of the v - 1 2 term, it does not alter the asymptotically dominant Y - " term (see also Shakeshaft, 1978).

The fact that the second Born term changes the asymptotic velocity dependence of the cross section to that computed in the first Born approximation has focused considerable attention on the behaviour of the Born series for rearrangement collisions at high energies. It has been shown that for a wide class of short range potentials, the second Born term provides the leading contributions to the cross section whether or not the Born series converges (Dettman and Liebfried, 1968, 1969; Dettman, 1971; Shakeshaft and Spruch, 1973, 1979). For charge transfer that involves long-range Coulomb interactions, it is widely accepted that the same holds true and that


the OBK2 approximation gives the correct leading up' ' term of the cross section.

Using the OBK2 approximation, Shakeshaft (1974b) (see also Shakeshaft and Spruch, 1979) has derived an elegant expression for the asymptotic form for arbitrary ni Iimi -, n, I , m, capture cross section

with

in which &(r) and q,,,(?) are the radial and the angular part of the bound state hydrogenic wavefunction 4nlm(r). This result was also obtained using a different method by Karnakov (1982). Averaging over mi and summing over m, immediately gives

showing that the nl distribution of the final state is related to the atomic radial wavefunctions through a simple relation (Eq. (59)). For further explicit results on u - l 1 behaviour, see Briggs and Dub6 (1980), Crothers (1985b). Dube and Briggs (1981).

Some first-order distorted wave theories like the VPS and the CDW approximations also provide a u - l 1 behaviour of the cross section at high energies (Cheshire, 1964; BelkiC: et al., 1979; Dube, 1984; Crothers, 1985a; McCarroll and Salin, 1967), but usually with an incorrect coefficient of this term. Other distorted wave theories like the Bates approximation, the GE approximation and the SE approximation, however, do not contain the u-" term.

The scaling properties of the B1B cross section for p-H collisions with respect to principal quantum numbers ni and n, at high energies have been considered by Karnakov (1982); see also Omidvar (1975), Roy et af. (1980). Crothers (1985a) has analysed the scaling behaviour of the CDW cross sections for 1s -, n, capture for (Zp, Z,) collisions. These analyses show that at high energies the cross sections of all these approximations follow the l/(n?n;) law, as do the OBK (Oppenheimer, 1928; McDowell and Coleman, 1970) and the second Born cross sections (Dube and Briggs, 1981; Shakeshaft, 1974b; Karnakov, 1982).

The 0 - l1 dependence of the charge transfer cross section was first obtained by Thomas (1927) using a classical model of double scattering. In this description, the target electron assumed to be initially at rest, in the


laboratory frame, is deflected with velocity u through an angle of about 60" with respect to the beam direction in the first collision with the projectile. Subsequently, the electron is scattered elastically by the target nucleus through an angle of about 60" so that the electron emerges with almost zero speed relative to the projectile and hence, can be captured. The conservation of energy and momentum requires that the projectile itself be deflected through an angle called the Thomas angle

(~0.054" for proton incidence) in the laboratory frame, and the angle of deflection depends only on the ratio of the electronic mass (me) to the projectile mass (Mp). Classical kinematics indicate that a deflection of the projectile through the Thomas angle signals a capture and one may expect a peak in the differential cross section, called the Thomas peak. (See Shakeshaft and Spruch, 1979 and Briggs, 1986 for discussions on a second peak at larger angles.)

The connection between the classical double scattering mechanism and the quanta1 second Born description has been considered by Shakeshaft (1974a). Spruch (1978) has shown that for capture involving atomic Rydberg states,

Lc 0 v) c c

Angle of Scattering (Laboratory System) mrad.

FIG. 4. The measured angular distribution for electron capture by 5.0 MeV protons from atomic hydrogen showing the Thomas peak (Vogt et al., 1986).


the classical result not only becomes accurate but also identical to the quanta1 result as nl increases.

The possibilities of observing signatures of the double scattering mechanism have been discussed (Shakeshaft and Spruch, 1978, 1979; Spruch, 1978; Dube and Briggs, 1981; Briggs, 1985; Dub& 1986). The Thomas peak has been observed in the differential cross sections in p-He and p-H collisions (Horsdal-Pedersen et al., 1983b; McGuire et al., 1984; Vogt et al., 1986). Fig. 4 shows an example of the measured Thomas peak in atomic hydrogen (Vogt et al., 1986). The direct observation of the u - l l variation of the cross section is not possible in the 1s-1s capture, however, because the u - l l term only dominates at energies for which relativistic corrections become important (E 2 100 MeV). In contrast, because of the rapid decrease of the first Born contribution - v - 1 2 - 2 1 1 - 2 1 f to the cross section in comparison with the second Born contribution N u - " , the capture involving changes in angular momentum quantum numbers offer possibilities of observing the u - l l tail of the cross section. A recent analysis based on the B2B, approximation suggests that the u - l 1 behaviour may be observable in proton-hydrogen 1s -+ 2p cross sections (Dewangan, 1988) at comparatively low energies (- 10 MeV).

VII. Relativistic Electron Capture

The theory outlined in previous sections of this article is nonrelativistic and must be modified when the impact velocity u becomes comparable with the velocity of light, and also when the target or projectile ions are of high Z . At relativistic impact velocities, the nonradiative electron capture process competes with radiative capture, typified by the reaction (see Briggs and Dettmann, 1977; Anholt and Gould, 1986)

P + (T + e-) --* (P + e-) + T + hu. (61)

At sufficiently high energies, for which the binding of the electron with the target can be ignored, this process is just the inverse of the photo-electric effect, which has been much studied. The nonradiative capture process at high velocities can also be identified experimentally and has been the subject of investigation starting with the relativistic version of the PBK approximation (Mittleman, 1964; Shakeshaft, 1979; Moiseiwitsch and Stockman, 1980; Moiseiwitsch, 1985).

Although the OBK approximation is not adequate, as it does not satisfy the Coulomb boundary is, it is interesting to consider the relativistic


modifications to the nonrelativistic formula (Eq. (29)), based on the unperturbed initial and final state functions, Eqs. (6a) and (6b). The relativistic unperturbed wave functions for the initial and final states are

@: = 4T(rT)e - (624

(62b)

where 4: and 4; are Dirac spinor functions and Ei, E , are initial and final electronic energies including the rest mass energy. The primed quantities are defined in the rest frame of the projectile and the unprimed quantities in the rest frame of the target. Any spinor function defined in the rest frame of the target can be transformed into the rest frame of the projectile by S where

@;P = 4 P ( f le - i E d f P

with y = (1 - u ’ / c ~ ) - ~ / ’ while u and /3 are the usual Dirac matrices. The 2 component of u appears because the transformation is parallel to the relative motion of the projectile and target.

The perturbation in the target frame for the “post” version of the theory is (zT/rT), so that the relativistic OBK approximation can be written as

where S - is the inverse transformation to S, obtained by replacing p by -p. By noting that t‘ = y [ t - fl(v-r,)/(uc)], it is easy to see that Eq. (64) reduces

to the nonrelativistic form of Eq. (29) in the low energy limit, (y -P 1, S = 1). In the opposite limit of increasing energy, v approaches c, which is a constant, with the consequence that the momentum transfer also approaches a constant. This profoundly modifies the high energy behaviour of the cross section which varies like y - l (i.e., E - ’ ) at high energies, compared to nonrelativistic E - 6 behaviour. For 1s capture from a 1s level, the relativistic second order OBK2 cross section in a peaking approximation has been shown by Humphries and Moiseiwitsch (1984, 1985) to vary like y - l ; Jakubassa-Amundsen and Amundsen (1980, 1985), without using a peaking approximation, have found a variation like (log y)’ /y, in contrast to the E-11/2

Both the first and second-order OBK models fail to agree with the experimental data of Crawford (1979) and Gould et al. (1985) who measured K shell capture of electrons by 1050 Mev/amu Ne ions from targets with 2, in the range Z , = 13 to 2, = 92. The rsion of the BlB, the boundary corrected model has been dei Eichler (1987). This

variation of the nonrelativistic second order theory.

HIGH ENERGY CHARGE TRANSFER 37 1

formulation deals properly with the Coulomb boundary conditions, exhibits no post-prior discrepancy and is in satisfactory accord with the data.

Finally, we should mention that a relativistic version of the symmetrical eikonal model has been developed by Moiseiwitsch (1986) and by Deco and Rivarola (1987) which provides as good a description of experimental cross section as the relativistic B1B model, and progress has also been made with the relativistic CDW approach (Deco and Rivarola, 1986; McCann, 1985) that should be equally satisfactory.

ACKNOWLEDGMENT

We are greatly indebted to our colleagues in many institutions for sending us details of their published and unpublished work. We should also like to record illuminating conversations on the topic of this review with Professors J. H. McGuire and J. Eichler. One of us (DPD) was supported by a Visiting Fellowship from the U.K. Science and Engineering Research Council during the period in which this review was written.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 25 l l R ELA TIVISTIC

APPRO XIMA TION RANDOM- PHASE

W. R. JOHNSON Department of Physics University of Norre Dame Norre Dame, lndiana

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . 375 376

111. Reduction to Radial Equations. . . . . . . . . . . . . . . . . 379 A. Hartree-Fock Equations. . . . . . . . . . . . . . . . . . 379 B. Homogeneous RRPA Equations . . . . . . . . . . . . . . . 381 C. Transition Amplitudes . . . . . . . . . . . . . . . . . . 383

IV. Basis Set Expansion of the Radial RRPA Equations . . . . . . . . . 386 A. Basic Formulas. . . . . . . . . . . . . . . . . . . . . 386 B. Applications to Dipole Excitations in Helium-Like Ions. . . . . . . 388 Acknowledgment . . . . . . . . . . . . . . . . . . . . . 390 References . . . . . . . . . . . . . . . . . . . . . . . . 390

11. Derivation of the RRPA Equations . . . . . . . . . . . . . . .

I. Introduction

The nonrelativistic random-phase approximation (RPA) is a simple but elegant method for including correlation corrections in studies of atomic transitions. Starting from a Hartree-Fock description of the atomic ground state, the RPA gives a prescription for determining frequencies of transitions to single-particle excited states, together with rules for determining amplitudes of transitions from the ground state to such excited states. From the point of view of many-body perturbation theory, the RPA amplitudes include all first-order correlation corrections together with those corrections of second and higher order that are obtained by iterating the first-order corrections. The RPA leads to a fully coupled, multichannel description of the final states. Thus, for example, the dipole excited states of neon obtained from RPA include contributions from 2p -+ s, 2p + d, 2s + p and 1s -+ p excitation channels. Finally, the R P A amplitudes are gauge independent so that, for example, dipole amplitudes calculated using the “length” form of the dipole operator are identical to those calculated using the “velocity” form. Early applications of the RPA to study correlation effects in atoms were made by

375

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376 W, R . Johnson

Dalgarno and his coworkers (Dalgarno and Victor, 1966; Stewart, Watson and Dalgarno, 1975; Watson, Stewart and Dalgarno, 1976a, 1976b). These pioneering studies demonstrated how the RPA could be applied to obtain accurate values for excitation energies and transition amplitudes in few- electron atoms and molecules. Theoretical studies of photoionization, particularly the extensive studies by Amusia and Cherepkov (1975), have in a similar way illustrated the predictive power of the RPA for heavier many- electron systems.

Relativistic effects become important for heavy atoms and for ions with high nuclear charge, leading to an enhanced spin-orbit splitting of the atomic shells. It is therefore of interest to include relativistic effects along with correlation corrections to describe transitions in such systems. The extension of the RPA to include relativistic effects is the relativistic random-phase approximation (RRPA). The RRPA was developed by the author in collaboration with C.D. Lin and A. Dalgarno (Johnson and Lin, 1976; Johnson, Lin and Dalgarno, 1976; Lin, Johnson and Dalgarno, 1977) and has been applied extensively to the study of transition energies and amplitudes in highly charged ions and heavy atoms. In Section I1 we derive the RRPA equations as linearized time-dependent Hartree-Fock (TDHF) equations. Following this derivation, in Section I11 we reduce the RRPA equations to a set of coupled radial differential equations. The RRPA equations are solved using finite-basis techniques in Section IV with attention to the treatment of negative energy states. An application of the finite-basis method to dipole excitations of He-like ions is also given in Section IV and comparisons are made with other calculations and with experiment.

11. Derivation of the RRPA Equations

Our point of departure for this discussion of the RRPA is the Dirac-Breit many-electron Hamiltonian (Bethe and Salpeter, 1957),

e2 H = C h , + C - - ,

i i > j rij

Ze2 hi = cai. pi + (pi - 1)rnc2 - -,

Ti

where the one-electron Hamiltonian, hi, includes the electron’s kinetic energy and its interaction with the nucleus but omits the electron’s rest energy to make comparisons with nonrelativistic theory simpler. The quantities a and 0 in Eq. (2) are the usual Dirac matrices. The configuration space Hamiltonian in Eq. ( I ) , with certain restrictions on the treatment of negative energy states,

RELATIVISTIC RANDOM-PHASE APPROXIMATION 377

can be obtained from the underlying field equations of quantum electrodynamics. (Sucher, 1980; Mittleman, 198 1). Although the restrictions just mentioned are ignored in the following development, they will be taken into account in the application given in Section IV. The Breit interaction (Bethe and Salpeter, 1957) should be considered along with the Coulomb interaction in Eq. (l), but contributions from the Breit interaction are typically smaller by a factor of (Za)’ than those from the Coulomb interaction, so the Breit interaction will be ignored also in the discussion to follow.

Among the many approaches that lead to the RRPA equations, perhaps the simplest method to obtain the equations in a form convenient for applications is that used by Dalgarno and Victor (1966), the TDHF method. One considers a closed-shell many-electron system such as a noble gas atom that can be described in the independent-particle approximation by a wave function consisting of an antisymmetric product of one-electron orbitals. A weak time-dependent interaction &(t) of the form,

h(t) = 1 [v(r,)e-’O‘ + v+(ri)eiwz], (3) i

is added to the many-body Hamiltonian of Eq. (l), and the influence of this interaction on the atomic wave function is examined. The time-dependent atomic wave function, Y(t), is taken to be a Slater determinant of time- dependent orbitals u,(r, t ) and the time-dependent variational principle (Kramer and Saraceno, 1981),

(4)

is invoked to obtain equations for the orbitals. This variational principle leads to a set of nonlinear coupled TDHF equations for the one-electron orbitals:

( 5 ) a at

i ~ u,(r, t ) = [ h + VHF(t) + v(r)e-iw‘ + o+(r)eiw‘]u,(r, t ) ,

- ud(r’, t)u(r’, t)u,(r, t ) ] . (6)

Equation ( 5 ) is satisfied for each occupied orbital, a, in the Slater determinant, and the sum in Eq. (6) for the TDHF potential, VHF(t), extends over all occupied orbitals, b. To obtain the RRPA equations from the TDHF Eqs. (5-6), the time-dependent orbitals u,(r, t ) are expanded about the time- independent orbitals u,(r) in powers of the external potential:

(7) u,(r, t ) = e-”J[u,(r) + ~ , + ( r ) e - ’ ~ ~ + w,_(r)e+’Ot + a*. ] ,

378 W. R. Johnson

where w, rt (r) are terms linear in the external field and where the ellipsis represents terms of second and higher order. Substituting the expansion given in Eq. (7) into the TDHF Eq. (6), equating terms with common phases and discarding terms of higher order in the external field, leads to the following set of linear coupled differential equations for the functions u,(r) and wa*(r):

(8) C h + VHFIUa(r) = Eaua(r)9

Ch + VHF - Ea - wlwa+(r) = + v c k + l u O ( r ) - 1 &b+ ub(r), (9a> b

[ h + VHF - E, + W]W,-(r) = - [Ut(r) -k V&-]U,(T) - 1 &b-Ub(r). (9b) b

Equation (8) is the time-independent HF equation for orbital a and VHF is the time-independent HF potential obtained by replacing time-dependent orbitals with time-independent orbitals in Eq. (6). The inhomogeneous Eqs. (9a,b), which describe the linear response of the atom to the perturbation V,, are the RRPA equations. We denote the first-order corrections to the HF potential by V#* in Eqs. (9a,b) and find:

+ wi-(r’)ub(r’)u,(r) - w~-(r’)ua(r’)ub(r)]. (10)

The perturbed potential VgA- in Eq. (9b) is given by Eq. (10) with the + and - signs on the right-hand side interchanged. The parameters in Eqs. (9a,b) are Lagrange multipliers introduced to insure that the perturbed orbitals, w, &), are orthogonal to all occupied orbitals, @), in the ground state Slater determinant. There is no loss of generality in requiring that the perturbed orbitals be orthogonal to the occupied orbitals, since contributions from the occupied orbitals cancel in the antisymmetric many-electron wave function.

It is particularly interesting to study the eigenvalue equations that follow from Eqs. (9a,b) on discarding the driving terms, u and ut, on the right-hand side:

Ch + VHF - EalW, + (r) + V#J + ua(r) = Ow, + (r) - 1 Aob + ub(r), ( a) b

[ h + VHF - E,]W,-(r) + V#-U,(r) = - w W , - ( r ) - &b+Ub(r). ( l lb ) b

We refer to Eqs. (Ila,b) as homogeneous RRPA equations. The positive frequency eigenvalues, wp, of Eqs. (1 la,b) give an approximate single-particle excitation spectrum of atom. From the symmetry of the homogeneous equations, one can show that if wlr is an eigenvalue belonging to w:*(r), then -0, is an eigenvalue belonging to wz r(r). The eigenfunctions, wif(r) and


wf+(r) belonging to eigenvalues o, and oA satisfy the basic orthonormality relations

j? [w:l(r)wB+(r> ~ w;~(r)w:-(r)]d3r = sign(oA)dA,. (12)

Solutions to the inhomogeneous RRPA Eqs. (9a,b) can be expanded in terms of the eigenfunctions in Eqs. (1 la,b) to give

m

'p

with

T, = C [wii(r)u(r)ub(r) + u&)u(r)wi-(r)]d3r. (14) b s

It follows from Eqs. (13a,b) that the many-electron TDHF wave function, "( t ) , has poles at the frequencies o, with residues given by the quantities T, in Eq. (14). By comparing the many-electron TDHF wave function with that obtained directly from first-order perturbation theory, one is led to interpret the positive frequency eigenvalue, o,, as the excitation frequency to the pth excitated state and the residue, T,, as the amplitude of the transition from the ground state to the pth excited state induced by the perturbing potential, VI . If V, is an electromagnetic multipole potential, then 5 is the associated multipole transition amplitude. The problem of determining transition amplitudes in the RRPA is thus reduced to the problem of solving the homogeneous RRPA Eqs. (1 la,b) and calculating the transition amplitudes from Eq. (14). In Section 111 B we reduce Eqs. (lla,b) to radial differential equations suitable for numerical integration and in Section I11 C we obtain corresponding expressions for electromagnetic transition amplitudes from Eq. (14).

111. Reduction to Radial Equations

A. HARTREE-FOCK EQUATIONS

Before discussing the reduction of the RRPA equations to radial differential equations, let us first review briefly the radial reduction of the relativistic H F Eq. (8). For a closed-shell atom, the H F equations are separable in a

380 W . R . Johnson

spherical basis. An orbital unKm(r), describing a Dirac electron with quantum numbers n, K ( K = t for j = L' - 1/2 and K = -t = 1 for j = t + 1/2), and m, is decomposed into a product of radial and angular factors as follows:

where x,,(O, p) is an L' - s coupled spinor. With the aid of Eq. (15), the H F equations given in Eq. ( 8 ) can be rewritten as a set of coupled radial differential equations. We introduce the two-component radial orbital

and the 2 x 2 radial HF Hamiltonian

where

Z r

V ( r ) = - - + VHF, (18)

and

vHFpo(r) = [jb1[.db9 b, r)pa(r) - 1 AL(b, a)uL(b, a, r)Pb(r) . (19) b L 1

In Eqs. (1 7- 19) we have adopted atomic units ( h = e = m = 1). The indices a and b in these equations designate the quantum numbers (n,, K , ) and (nb, K ~ )

of a closed atomic subshells. The quantity [jJ = 2j, + 1 in Eq. (19) is the degeneracy of subshell a; the functions uL(a, b, r ) are Hartree screening potentials defined by

uL(a, b, r ) = r - L - l dr'r'LPi(r')Pb(rr) + rL d r r r ' - L - l ~ ~ ( r ' ) P b ( r r ) , (20)

and the angular coupling factor in the exchange term of Eq. (19), &(a, b), is given in terms of Wigner 3-j coefficients by

I: Jrn

RELATIVISTIC RANDOM-PHASE APPROXIMATION 38 1

where

1 for 8, + / b + L even 0 for 8, + /,, + L odd. n(/b 3 / b 3 L) =

Using this notation, the radial H F can be written simply as

H"vf'a(r) = ~of'a(r1. (22)

Numerical solutions to these equations, which are also referred to as Dirac-Hartree-Fock(DHF) equations, can be obtained quickly and easily. As an example, for xenon, where the index a ranges over 17 subshells, ls1/,, 2s,/,, . . . , 5p,,,, 5p,,,, the radial D H F equations can be solved self-consistently for the radial orbitals, P,(r), and the corresponding eigenvalues, E,, in about one-half hour on a minicomputer or in a few seconds on a supercomputer.

B. HOMOGENEOUS RRPA EQUATIONS

Now we turn to the homogeneous RRPA Eqs. (lla,b). To construct a solution to these equations describing an excited state of the atom with angular momentum components J and M , we expand the perturbed orbitals w,*(r) in terms of a set of orbitals w;$(r) that can be separated in a spherical basis:

w,+(r) = 2 (-IF-"', < j , - ma j , m , ) J M > w;I,M+(r), (234 n

w,-(r) = C (- l ) j - - m a + M < j , - ma j,rn,lJM > wiE(r). (23b)

The perturbed orbitals w:$(r) describe the excitation channels, a n, contributing to an atomic excited state with angular momentum J , M . The perturbed orbitals w i s ( r ) have angular momentum components, j,, m,, and parity, (- 1)'". The parity of the atomic state described by Eqs. (23a,b) is ( - l)'n+eQ, and only those terms, n, leading to states having one fixed value of parity are included in the expansion. The orbitals wi$(r) can be decomposed into radial and angular components following the pattern of Eq. (13,

n

The two-component radial orbital associated with wi$(r) is designated by

382 W. R . Johnson

With the aid of the expansion in Eqs. (23a,b), the homogeneous RRPA equations can be rewritten as a family of coupled radial equations for the channel functions Ri,, *(r). These radial functions are independent of the magnetic quantum numbers, ma and m,, of the states a and n. To simplify the form of the radial RRPA equations, we introduce two auxiliary angular momentum coefficients

The radial Hamiltonian H H F in Eq. (28) is that written out in Eq. (17) with K = K , . The arguments bm -t appearing in the Hartree screening functions in Eq. (28) signify that the radial RRPA orbitals R&,+(r) are to be used in place of the corresponding H F orbitals in calculating the screening functions.

For the “magnetic case,” where the angular momentum of the excited state is J and the parity is (- l)J+ ’, the RRPA equations are

CHH, - &a T wIR;[,* = 1 CA(-a, -6, n, m, L J)u,,(b, a, rWim*(r) b . m , L

+(-l)’b--jnA(-u, - m , n , b , L , J )


The - signs in the arguments of the A-functions signify that the sign of the corresponding angular momentum quantum numbers, K, are to be reversed. This reversal modifies only the parity selection rule in Eq. (27) but leaves the angular momentum selection rules unchanged.

To describe excited bound states, Eqs. (28) or (29) are solved as eigenvalue equations for w, while for continuum states o is chosen arbitrarily and solutions to the equations that satisfy “incoming wave” boundary conditions are sought. Before we discuss specific solutions to the RRPA equations, let us first consider the reduction of the expressions for transition amplitudes to radial form.

C . TRANSITION AMPLITUDES

States described by Eqs. (28) are excited from the IS, atomic ground state by electric multipoles, while those described by Eq. (29) are excited from the ground state by magnetic multipoles. We designate the vector potential for a multipole field of angular momentum J , A4 by a$%(r), where I = 1 for electric multipoles and I = 0 for magnetic multipoles; the corresponding scalar potential is designated by cpS%(r). To describe the excitation of an atom by a multipole field, we replace the perturbing potential u(r) in Eq. (3) by a multipole potential

u(r) + u$%(r) = a * aS%(r) - rp$%(r). (30)

In the Coulomb gauge, the scalar potential vanishes and the vector potential is given by

af&(r) = j,(kr)Y$%(P), magnetic case, ( 3 W

electric case,

where k = w/c, w being the frequency of t..e multipole fit

(3 1 b)

The vector spherical harmonics Y$%(P) in Eqs. (31a,b) are those defined in Akhiezer and Berestetskii (1969, the quantity j J ( k r ) is a spherical Bessel function of order J , and j J ( k r ) is its derivative. Electric dipole transition amplitudes calculated in the Coulomb gauge reduce to “velocity” form amplitudes in the nonrelativistic limit. Alternative expressions for electric multipole fields that lead to

384 W . R . Johnson

“length” form amplitudes nonrelativistically are obtained from Eq. (3 lb) by a gauge transformation and are given by

The RRPA transition amplitude, T$%, from the ground state to an excited state with angular momentum J , M and parity (- l )J+’-* induced by the multipole field u$%(r) in Eq. (30) is given, according to Eq. (14), by

~ 5 % = [ ~ w f +(r)u$%(r)uo(rl + u;(r)u$%(r>w, - (r)ld’r. (33)

The corresponding absorption probability per unit time is

B$% = 8 ~ ~ 1 Ti%[’. (34)

Before carrying out the angular integrations in Eq. (33), it is useful to express the perturbing potential, u$%(r), in terms of a multipole moment operator, q$$,?,(r), related to @,(r) by

The following expressions for one-electron reduced matrix elements of the multipole moments can then be used to simplify Eq. (33).

Coulomb Gauge

( K 2 114$o~11 K1)


Length Gauge

(25 + l)!! kJ = C J ( K 2 , K 1 )

where C J ( ~ ’ , K ~ ) are the angular momentum coefficients introduced in Eq. (26). Substituting the expansion of Eq. (23a,b) into Eq. (33) and carrying out the sums over magnetic quantum numbers ma and m,, one obtains the following result for the transition amplitude:

J + 1 u2 kJ ~ $ 2 = i ~ QS”’, [ 4 n J ] (25 + l)!!

where

Q$’) = (- 1)J- 1 C [(- l)jn-ja(an+ 11qS’)Ila) + <allq$’)IIan-)]. (39)

The symbols anf in Eq. (39) indicate that the radial RRPA functions, Rin*(r) from Eqs. (28 and 29) are to be used in evaluating the single-particle reduced matrix elements. The absorption probability per unit time to a state of angular momentum J and parity (- l )J+ -’ is found by summing Eq. (34) over M substates to give

a. n

[Q$’)]’ccR,. 4n(2J + 2)k2J+1

(.I) - BJ - J(2J + 1)[(2J - l)!!]’

Electric dipole excitations are of particular interest. The expressions for the reduced matrix elements of dipole operator q\’) take an especially simple form if the Bessel functions in Eq. (34) are expanded in a power series and terms of relative order (kr )2 % (Za)2 are neglected. The resulting expressions are

Velocity form

386 W. R. Johnson

Length form

(~zlldil’ll~~) = Cl(rcz, rcl) drrCG2(r)Gl(r) + Fz(r)F1(r)l . (42) s The corresponding dipole absorption probability per unit time is

In the example given in Section IV B, dipole excitations of He-like ions are treated using the simplifications given in Eqs. (41-43).

IV. Basis Set Expansion of the Radial RRPA Equations

A. BASIC FORMULAS

One approach that has been applied successfully to solve the radial RRPA equations is direct point-by-point numerical integration of Eqs. (28) and (29). Applications of the RRPA using this direct approach have been given, for example, by Johnson and Lin (1976). An alternative that has certain attractive features is to expand the perturbed radial orbitals, Rin*(r), of Eqs. (28) and (29) in a set of basis functions and thereby reduce the differential equations to algebraic equations for the expansion coefficients.

The basis expansion approach is particularly attractive in the relativistic case since it permits one to deal with the modifications required by QED of the many-body Hamiltonian written down in Eq. (1). In the no-pair approximation to QED (Sucher, 1980; Mittleman, 1981), in which all effects of virtual electron-positron pairs are ignored, the Coulomb potential in Eq. (1) is replaced by a modified Coulomb potential,

where A + is a positive-energy projection operator. The appearance of A+ in the potential (Eq. (44)) reflects the fact that the Dirac negative-energy sea is filled. For one electron moving in a potential V(r), the positive-energy projection operator is given by

where u,(r) is a solution to the one-electron Dirac equation in the potential V(r ) and where the sum is carried out over all positive energy states. It has


been suggested by Mittleman (1981) that an optimal choice for the atomic projection operator, A + , in Eq. (44) is a direct product of one-electron positive-energy Hartree-Fock projection operators. This choice leaves the HF equations in Eq. (8) unchanged but modifies the RRPA equations written down in Eqs. (9a and b). The required modifications have the effect of projecting out all negative-energy H F components from the solutions to the homogeneous RRPA Eqs. (1 l a and b). An obvious way to implement the H F projection operators is to expand the solutions to the RRPA equations in a basis consisting of positive-energy H F orbitals only. We describe such a basis expansion technique in the paragraphs below and apply it to a specific example in Section IV B. It should be mentioned that numerical results obtained using the positive energy H F basis are very close to those obtained by direct numerical integration of the radial RRPA equations, which contain contributions from both positive and negative energy states. This is a result of the fact that the negative energy terms in the expansion are strongly suppressed by energy denominators of the order cz au, while the matrix elements in the numerators of the negative-energy terms are suppressed by an additional factor of order uz.

We suppose that the radial H F Eq. (22) has been solved self-consistently for the ground state orbitals. An excited H F orbital, Pkn(r), with principal quantum number k and angular momentum quantum number K = K , , can be found by solving the linear H F equation

(46) HHF Pkn(Y) = &kn Pkn(r) ,

where the Hamiltonian H,, is that given in Eq. (17). Equation (46) has a spectrum, cknr consisting of possible discrete states, a positive energy continuum and a negative energy continuum. The radial RRPA orbitals Ri,*(r) are expanded in terms of the H F eigenfunctions with K = K , , excluding the negative energy continuum, according to

R&-(r) = 1 y & t ) k p k n ( r ) . (47b) k

Substituting the expansions (47a,b) into Eq. (28) or (29) leads to the following system of linear equations for the expansion coefficients X ( a n ) k and Fan),(

CP(an)k , ( b m ) d X ( b m ) C + Q(an)k , (bm)d q b m ) d l = ox(an)k , (484

2 CQfan)k , (bm)dX(brn)8 + ‘ (an)k , (bm)d t 5 b m ) d l = - W F a n ) k * (48b)

bmC

bmC

Equations (48a and b) describe a matrix eigenvalue problem that can be treated using standard numerical techniques. From the structure of these

388 W. R. Johnson

equations, it follows that there are both positive and negative frequency solutions. Moreover, the negative frequency solutions can be obtained from the positive frequency solutions by simply replacing the pair (X(an)k, Tan)k) by - ( Y & ) k , x&,)k). It should be noted that the negative-frequency solutions to the RRPA equations are unrelated to the negative-energy solutions of the Dirac equations and also occur in the nonrelativistic treatment of the RPA. Explicit formulas for the matrices P and Q in Eqs. (48a and b) can be obtained easily from Eqs. (28 and 29). Once Eqs. (48a and b) are solved, the radial RRPA orbitals associated with the various eigenvalues o,, can be reconstructed and the corresponding transition amplitudes can be calculated.

In our applications of the basis set expansion, we artificially confine the atom to a cavity of finite radius in order to discretize the continuum states. By choosing the cavity radius sufficiently large, the low-lying bound states in the cavity can be brought arbitrarily close to the actual H F bound states. Care must be taken with the cavity boundary conditions in the relativistic case to avoid the Klein paradox (Sakurai, 1967). To this end, we adopt MIT bag model boundary conditions (Chodos et al., 1974) at the cavity wall.

Although the cavity spectrum is discrete, it is infinite. To avoid dealing with infinite matrices, we introduce a finite H F pseudospectrum obtained by expanding the HF orbitals in terms of a finite number of B-splines. (de Boor, 1978) The low-lying states in this pseudospectrum can be made to agree to any desired level of accuracy with the HF states by exploiting the freedom available in choosing the number and order of the B-splines used to approximate the HF orbitals. Moreover, the spectrum obtained by using B- splines of order k is complete in the space of piecewise polynomials of degree k - 1. A detailed description of the construction of a B-spline pseudospectrum is given by Johnson, Blundell, and Sapirstein (1987), together with a number of tests that have been made to ensure the quality of the approximation. It is found that half of the terms in the pseudospectrum of the radial DHF equation are in the positive energy branch of the spectrum and the remaining half are in the negative energy branch. We include only the positive energy terms from the DHF pseudospectrum in the expansion of the RRPA equations in Eqs. (48a and b).

B. APPLICATION TO DIPOLE EXCITATIONS IN HELIUM-LIKE IONS

Helium-like ions are particularly simple since there is only a single radial DHF orbital, Pls(i-), to consider. An external electric dipole field gives rise to two excitation channels 1s -, pl,2 and 1s -, p3/2. Associated with each of these channels are two radial RRPA orbitals, R1,,l *(I - ) for the pli2 channel ( K = 1) and Rls,-2*(r) for the ~ 3 1 2 channel (K = -2). These radial RRPA


functions are expanded in terms of the positive energy HF pseudostates for K = 1 and K = -2, respectively. Typically, we use 40 B-splines to obtain the DHF pseudospectrum for a given value of K and obtain a spectrum containing 40 positive energy states. The dimension of the resulting eigenvalue problem in Eqs. (48a and b) is then 160. The atomic excitation spectrum contains 80 positive frequencies, q,, and 80 negative frequencies, - m F . The RRPA eigenvalues appear in closely spaced pairs, the lower member of each pair being the excitation frequency of an n3P, state of the atom and the upper member being the frequency of an n'P, state. The corresponding spectrum of the nonrelativistic RPA equations contains only the n lP , states unless a spin-dependent interaction is added to the nonrelativistic many-body Hamiltonian.

In Table I we compare the frequencies of the first four terms in the RRPA spectra determined by solving the RRPA Eqs. (48 and b) for three different helium-like ions with measurements (Moore, 1971). The agreement with the measured spectra is found to be remarkably good in view of the fact that the

TABLE I RRPA EXCITATION ENERGIES(AU) AND OSCILLATOR STRENGTHS FOR THE FIRST

FOUR SINGLET AND TRIPLET J = 1 ODD PARITY STATES OF He I, Li 11, AND

Ne Ix COMPARED WITH OTHER VALUES

He I

2 0.797 0.780 0.252 0.276 0.780 0.770 1.56(-9) 3 0.864 0.848 0.070 0.073 0.858 0.846 3.76(- 10) 4 0.887 0.873 0.030 0.030 0.885 0.871 1.57(-10) 5 0.900 0.884 0.024 0.015 0.899 0.883 1.40(-10)

Li I1

2 2.305 2.286 0.444 0.457 2.258 2.252 5.90(-8) 3 2.574 2.560 0.111 0.111 2.560 2.549 1.40( -8) 4 2.669 2.656 0.044 0.044 2.663 2.651 5.77(-9) 5 2.713 2.700 0.022 0.022 2.710 2.698 3.57(-9)

Ne IX

2 33.92 33.88 0.720 0.723 33.63 33.63 2.76(-4) 3 39.49 39.47 0.149 0.149 39.41 39.37 6.47(-5) 4 41.46 41.42 0.059 0.056 41.43 41.36 2.64(-5) 5 42.67 42.33 0.054 0.027 42.64 42.30 2.44-5)

a Moore (1971). Schiff et a/., (1971).

390 W. R. Johnson

RRPA treatment of correlation is incomplete in the sense that some of the higher-order terms from many-body perturbation theory are ignored. Oscil- lator strengths for the transitions to the ground state, defined in terms of the reduced matrix element of the dipole operator by

f = $ W [ Q \ ” ] ~ ,

are also presented in Table I. Comparisons of the singlet-state oscillator strengths with values obtained by Schiff, Pekeris, and Accad (1971) using accurate variational wave functions are also given in the table. Again, the agreement between the RRPA values and the precise variational calculations is good. The oscillator strengths for the triplet states are very sensitive to the singlet-triplet splitting, and the values for the triplet transitions given in Table I, although qualitatively correct, are modified significantly by the Breit interaction (Johnson and Lin, 1976).

The numerical values obtained here using a finite basis expansion are found to be in excellent agreement with previous solutions obtained by direct numerical solution of the RRPA equations, even though the direct numerical solutions include contributions from the negative energy states. This illustrates the observation made previously that the negative-energy state contributions to RRPA are small. Even though there are no significant numerical differences for the cases studied between the basis set solutions and those obtained by direct integration of the radial RRPA equations, the basis set technique is still of interest since it permits one to isolate and eliminate the negative energy contributions as required in the no-pair approximation to QED.

In summary, the RRPA provides a simple and effective approach to the problem of including correlation corrections in a class of relativistic atomic structure calculations. With the use of finite-basis techniques, the RRPA equations can be solved within the framework of the no-pair approximation to QED. It is expected, and indeed, found by explicit calculation, that the contributions from the negative energy states is insignificant.

ACKNOWLEDGMENTS

This work was supported in part by NSF grant No. PHY 86-08101.

REFERENCES

Akhiezer, A. I. and Berestetskii, V. B. (1965). Quantum Electrodynamics. Interscience, New York,

Amusia, M. Ya. and Cherepkov, N. A. (1975). Case Studies in Atomic Physics 5, 49-179. Bethe, H. A. and Salpeter, E. E. (1957). Quantum Mechanics of One and 7bo-Electron Atoms.

New York, p. 28.

Springer, Berlin, West Germany, p. 170-205.

RELATIVISTIC RANDOM-PHASE APPROXIMATION 39 1

Chodos, A., Jaffe, R. L., Johnson, K., Thorn, C. B., and Weiskopf, V. W. (1974). Phys. Rev. D 9,

Dalgarno, A. and Victor, G. A. (1966). Proc. Roy. Soc. A 291, 291-5. de Boor, Carl ( 1 978). A Practical Guide to Splines. Springer, New York, New York. Johnson, W. R. and Lin, C. D. (1976). Phys. Rev. A 14, 565-575. Johnson, W. R., Lin, C. D., and Dalgarno, A. (1976). J . Phys. B 9, L303-6. Johnson, W. R., Blundell, S. A., and Sapirstein, J. (1987). Phys. Rev. A. To be published. Kramer, P. and Saraceno, M. (1981). Geometry oi the Time-Dependent Variational Principle in

Qitantum Mechanics, Lecture Notes in Physics 140, Chap. 2, Springer, Berlin, West Germany.

3471-95.

Lin, C. D., Johnson, W. R., and Dalgarno, A. (1977). Phys. Rev. A 15, 154-61. Mittleman, M. H. (1981). Phys. Rev. A 24, 1167-75. Moore, C. E. (1971). Atomic Energy Levels. Natl. Bur. Stand. Ref. Data Ser., Natl. Bur. Stand.

Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison-Wesley, Reading, Massachusetts,

Schiff, B., Pekeris, C. L., and Accad, Y. (1971). Phys. Rev. A 4, 885-93. Stewart, R. F., Watson, D. K. and Dalgarno, A. (1975). J . Chem. Phys. 63, 3222-7. Sucher, J. (1980). Phys. Rev. A 33, 348-62. Watson, D. K., Stewart, R. F., and Dalgarno, A. (1976a). J . Chem. Phys. 64,4995-9. Watson, D. K., Stewart, R. F., and Dalgarno, A. (1976b). Mol. Phys. 32, 1661-70.

(U.S.) Circ. No. 35, Vols. I and 11, U.S.GP0, Washington, D.C.

p. 120.



RELATIVISTIC STURMIAN AND FINITE BASIS SET METHODS IN ATOMIC PHYSICS G. W. F. DRAKE Deparlment of Physics University of Windsor Windsor. Canada

S. P . GOLDMAN Department of Physics University of Western Ontario London, Canada

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . 393 11. Variational Representation of the Dirac Equation . . . . . . . . . . 396

111. Relativistic Sturmian Basis Sets . . . . . . . . . . . . . . . . 402 IV. Test Calculations with Relativistic Sturmian Basis Sets and Comparisons with

B-Spline Methods . . . . . . . . . . . . . . . . . . . . . 404 V. Variational Dirac-Hartree-Fock Calculations. . . . . . . . . . . . 410

VI. Suggestions for Future Work . . . . . . . . . . . . . . . . . 414 Acknowledgments . . . . . . . . . . . . . . . . . . . . . 414 References . . . . . . . . . . . . . . . . . . . . . . . . 414

I. Introduction

Finite basis set variational techniques have been used widely in nonrelativistic atomic physics for the calculation of energy levels, transition rates, processes involving the absorption and emission of radiation, and scattering cross sections. To review briefly, given a Hamiltonian H and any trial function Y t , normalized so that

if E l is the lowest energy eigenvalue of the exact spectrum of solutions of H , then

Therefore, the best choice of a trial function to approximate the ground state in the spectrum of H is the one that minimizes the expectation value of H. The variational method then relies on a systematic variation of Y t in order to find the best representation of the eigenstate with energy E , . The important

393 Copyright Q 1988 by Academic Press, Inc

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394 G. W . F . Drake and S . P . Goldman

characteristic that makes this procedure useful is that the minimum of (YtlHIYf) is always an upper bound to the exact value E l .

Consider an expansion of Y , in terms of an orthonormal set of functions { A } , i = l , 2 ,..., N

N

Yt = Caif i . i = 1

(3)

A minimization with respect to the linear coefficients {a,} is equivalent to the diagonalization of the Hamiltonian matrix H , = (fi I H I A). This diagonalization yields N variational eigenvectors and N variational eigenvalues that, according to the Hylleraas-Undheim theorem (Hylleraas and Undheim, 1930), are upper bounds to the first N exact eigenvalues of H . Each of these variational eigenvalues can, in turn, be further minimized by the variation of all the nonlinear parameters contained in the basis set {A} . A typical optimization process then consists of a sequence of diagonalizations for different sets of nonlinear parameters while searching for the set that minimizes the energy of the state under consideration.

Variational procedures are applied widely in this way to find accurate eigenstates using correlated basis sets for two- and three-electron systems and using hydrogenic basis sets in the Hartree-Fock scheme for many-electron atomic and molecular systems.

There is, however, a second very powerful use of this method that can be applied to the systems mentioned above but that is also advantageous for hydrogenic systems for which the exact solutions are well known. The set of N variational eigenstates,

N

Y i = c a i j f j , i = 1 , 2 ,..., N , (4) j = 1

forms a discrete variational representation of the complete spectrum of solutions of H , including both bound and continuum energy states. If the basis functions can be expressed as finite linear combinations of N Sturmian functions (see Section 111 and Rotenberg, 1962), then the basis set will be referred to as a Sturmian-type basisset. In the limit N + a, a Sturmian-type basis set satisfies the closure relation

The limiting condition in Eq. ( 5 ) defines the completeness of the basis set used. The convergence and completeness properties of Sturmian basis sets are discussed in detail by Klahn and Bingel (1977). The approximation (Eq. ( 5 ) ) can be used to perform calculations involving sums over intermediate states by adding a finite number of terms involving only normalizable eigenfunc-

RELATIVISTIC STURMIAN AND FINITE BASIS SET METHODS 395

tions (see, for example, Drake et al. 1969; Drake, 1986). For expressions involving single sums over intermediate states, the result is equivalent to performing integrals over the solutions to inhomogeneous differential equations using the methods developed by Dalgarno and Lewis (1955) and Sternheimer (1950).

For both eigenvalue and intermediate state summation calculations, as the size of the basis set is increased, i.e., as the basis set tends toward completeness, the variational results converge towards the exact values. At the same time, the results become less susceptible to changes in the nonlinear parameters. These two characteristics of completeness can then be used to estimate the accuracy of the variational results obtained. Sturmian representations of the Coulomb Green’s function have recently been discussed by Shakeshaft (1985) and by Engelmann and Natiello (1987).

The properties of bounds presented above cannot be extended readily to the relativistic case. There are two fundamental difficulties to overcome: variational collapse and, for systems with more than one electron, continuum dissolution as discussed below. Both of these stumbling blocks, although very different in nature, are related to the fact that the relativistic description of the electron contains a positive-energy spectrum of eigenvalues E ,, with 0 < E, < mc2 for bound states and E, 2 mc2 for the scattering continuum, and a negative-energy spectrum of eigenvalues E , , with E, I -mc2.

The lowest possible eigenvalue is now - co, and equation (2) is rendered meaningless. An arbitrary trial function no longer yields an upper bound to the ground state. Moreover, a minimization process in principle forces the expectation value of the relativistic Hamiltonian to fall into the bottomless negative-energy continuum. This effect is known as variational collapse.

The effect of continuum dissolution was first discussed by Brown and Ravenhall (1951) who pointed out that, for systems with more than one electron, any accessible positive-energy state for an electron is degenerate with an infinite number of negative-energy states of any of the other electrons. The total energy spectrum for all the accessible electronic states thus consists of a continuum of all positive and negative real values. It follows that a variational eigenvalue can have any value whatever.

Provided that the above problems can be overcome, relativistic variational calculations are useful in many circumstances, such as

(i) The calculation of higher order relativistic corrections to atomic properties. For small values of the nuclear charge 2, relativistic corrections can be expanded in powers of (az)’, where a is the fine structure constant. A relativistic variational calculation, in effect, sums the series to infinity. This is especially important for highly ionized atoms with large nuclear charge where the (IxZ)’ expansion may be slowly convergent or divergent.

(ii) QED calculations, in which virtual transitions to all positive and

396 G. W. F . Drake and S . P . Goldman

negative energy states must be included. In this case, the expansion (Eq. ( 5 ) ) is extended to include the variational representation of the negative-energy spectrum. This allows one to approximate the one-electron Dirac-Green's operator by the expansion

here H , is the Dirac Hamiltonian and the sum is taken over all the positive- and negative-energy eigenstates.

In the following sections, different approaches to the relativistic variational representation of atomic systems will be presented, together with different strategies to avoid the problems inherent in a relativistic variational formulation and permit its application to the rigorous relativistic calculation of atomic properties.

11. Variational Representation of the Dirac Equation

As suggested in the Introduction, due to the characteristics of the Dirac spectrum, care must be taken with the choice of trial functions. An arbitrary choice leads, in general, to a variational spectrum that does not yield upper bounds to the states represented, and results in variational collapse or the appearance of spurious roots, i.e., eigenvalues in the forbidden gap or more than one variational eigenvalue between two exact solutions (Mark and Rosicky, 1980; Wallmeier and Kutzelnigg, 1983; Ishikawa et al., 1985).

We shall concentrate in this section on the one-electron Coulomb case. Relativistic units with h = c = 1 will be used throughout unless otherwise noted. For any central potential, one can separate the radial part of the Dirac equation from the angular part. The resulting radial eigenfunctions can be expressed in terms of a real two-component radial spinor

that satisfies the radial Dirac equation,

where the radial Dirac Hamiltonian is given by


where K is the Dirac quantum number: K = + ( j + $) for 8 = j f 3. g(r) and f ( r ) are called the large and small components, respectively.

The exact bound state solutions to Eq. (8) are of the form

where

= [rc2 - ( a ~ ) ’ ] ’ ’ *

and

J, = (m’ - E’ )I/’. n, K n, K

The exact positive energy eigenvalues are given by

E n , K = m ( 1---

with

N = (ic2 + 2ny + n2) l / ’

Drake and Goldman (198 1) constructed a successful finite basis set representation of the solutions (Eq. (10)) analagous to the nonrelativistic Sturmian functions. In this method, independent sets of basis functions are used to expand the large and small components in order to obtain the most general representation of the solutions of the form of Eq. (7). The basis set used is

where Ai are normalization constants, y is defined in Eq. ( l l a ) and A is a nonlinear variational parameter. The ai and BN + are orthogonal by spinor orthogonality. As discussed in Section 111, the above basis set is equivalent to a relativistic Sturmian basis set. Upon orthonormalization of the 2N-dimensional basis set (Eq. (1 2a and b)) and diagonalization of the Hamiltonian (Eq. (9)), the following results are obtained:

(i) Of the 2N variational eigenvalues, N are positive and N are negative. (ii) Except as noted in (iv) below, the eigenvalues satisfy a generalized

Hylleraas-Undheim theorem in that the positive-energy eigenvalues all lie above the corresponding exact eigenvalues as if the Dirac Hamiltonian were a positive definite operator.


(iii) The negative-energy eigenvalues all lie below -mc2 and move progressively towards -mc2 as the dimension of the basis set is increased.

(iv) There is one spurious root in the case IC > 0. This spurious energy eigenvalue is degenerate with that of the lowest variational state of the same ( K I , but with K < 0 (e.g. a lpl,, state degenerate with the variational ls,,, state).

(v) The basis set (Eq. (12a and b)) is complete in the sense that

As a typical example, Fig. 1 illustrates how the variational ns,,, eigenvalues for Z = 92 progressively move as the size of the basis set increases.

Drake and Goldman performed extensive numerical testing and discussion of this procedure for different sizes of the basis set, for different values of the nuclear charge and for different nonlinear parameters. They found that, in all cases, the set (Eqs. (12a and b)) provides the necessary conditions of boundness and completeness. Also, by means of relativistic sum rules, they showed that the spurious root does not affect calculations involving summations over the complete spectrum (Drake and Goldman, 1981; Goldman and Drake, 1982).

Since its introduction, this method has been successfully applied to several problems of which we mention a few

(a) The calculation of two-photon decay rates in hydrogenic ions (Gold- man and Drake, 1981) and the perturbative calculation of the El-M1 decay rates in He-like ions (Drake, 1985). To account for all virtual transitions in these calculations, the variational representation of the positive- and negative-energy spectra is used to expand the Dirac-Green’s operator Eq. (6).

(b) The calculation of photoionization cross-sections in hydrogenic ions (Goldman and Drake, 1983; see also Section IV) in which the variational projection over positive-energy states only is used to account for all real transitions, and Stieltjes imaging techniques are applied to approximate the oscillator strength densities in the continuum.

(c) Relativistic energy eigenstate calculations of screened Coulomb potentials such as the Yukawa potential (Gazdy and Ladanyi, 1984) and the Dirac-Fock-Slater potential (Mukoyama and Lin, 1987).

Grant (1982, 1986) has analyzed the results (i-iii) above by identifying conditions that the basis set must satisfy in order for a separation theorem to hold. Grant and Quiney (1988) review many aspects of relativistic finite basis sets, and present an extensive comparison of methods for constructing them.

A rigorous proof of bounds and completeness and the removal a priori of the spurious root in the spectrum was given by Goldman (1985a) for the


1.5-

1.4

?

0 3

b 1.391

“ O t

1 -

-

L

‘ I I I I I 1 I 1 I I I I I I

3 -.

-2.0 -

-3.0-

-10.0-

-eo.+ , I 1 I I I 1 1 I I I I I I 1 2 3 4 5 6 7 8 9 l0 11 12 13 14

N FIG. 1. Distribution of the ns,,* variational eigenvalues for Z = 92 and 1 = 65.2/a0 as a

function of the size of the basis set. Each basis set of size 2N has N positive eigenvalues in the upper half of the diagram and N negative eigenvalues in the lower half of the diagram. The vertical scale is logarithmic. (From Drake and Goldman, 1981.)

400 G. W . F. Drake and S. P . Goldman

Coulomb case. The results of this work are obtained by means of the unitary transformation

where q is the ratio at the origin of the large and small components of all the eigenstates for a given value of IC

It is shown in this work that all the necessary conditions of bounds and completeness are satisfied if the new bispinor

0 = (g) is expanded in terms of the variational basis set

with the additional vector

for states with IC < 0. Some of the rigorously proven results presented are

(i) The exact eigenvalue for the ground state in the case IC < 0 (call this eigenvalue e l ) is always present in the energy spectrum. This corresponds to the eigenstate with O(r) = 0.

(ii) The variational eigenvalues for K = I IC I and IC = - 1 K I , excluding e l , are degenerate.

(iii) Every positive eigenvalue is an upper bound to the corresponding exact eigenvalue and every negative eigenvalue is a lower bound to -mc2 (generalized Hylleraas-Undheim theorem).

(iv) The number of positive eigenvalues (excluding e l ) and negative eigenvalues is the same.

(v) There is no spurious root in the spectrum.

Other approaches to the variational representation of the Dirac spectrum have been proposed that differ in the way in which the optimization process is


carried out, the choice of boundary conditions for the basis set, or the way in which the Dirac Hamiltonian is implemented. We now review briefly some of these methods.

I . Correct Nonrelativistic Limit

In this approach, the bais set is forced to have the correct nonrelativistic limit by imposing certain constraints relating the large and small components. One method proposed (Goldman, 1985a,b) is to use the basis set (Eqs. (12a and b)) with or without the zeroth order constraint (Eq. (15)) and to eliminate the spurious roots for ic > 0 by means of an exact energy-level independent first-order differential condition at the origin. This condition, relating the large and small components, is given by (Goldman, 1985a,b)

where q is defined in Eq. (15) and

The spectrum obtained in this way does not contain spurious roots and satisfies all the required conditions of bounds and completeness mentioned earlier. This method has been successfully applied to yield accurate results in variational Dirac-Hartree-Fock calculations (Goldman and Dalgarno, 1986; Goldman, 1988).

A variation of the above is the “balanced basis set” method in which the large and small components are related by the (nonrelativistic) differential condition at c -+ 00 (Ishikawa et a!., 1983, 1984, 1985; Dyall et al., 1984; Grant, 1986). Variational collapse has been observed in some applications of this method to states with K > 0 (Ishikawa et ul., 1985).

2. The Minimax Approach

In this class of methods, the problem of variational collapse is avoided by abandoning the usual energy minimization process. The underlying idea is to maximize with respect to the negative-energy portion of the variational spectrum and to minimize with respect to the positive-energy portion of the spectrum. The respective projection operators (and therefore their dependence on the nonlinear parameters) are not known a priori, however, and one must use an approximation. One approach (Talman, 1986) is to maximize the energy functional with respect to the small component and then minimize it


with respect to the large component. Another approach is to use the Casimir positive- and negative-energy projection operators for a free electron. In this method (Rosenberg and Spruch, 1986) a maximum principle is employed to construct an effective potential that accounts for the negative-energy continuum. This potential is later used to construct an effective Hamiltonian with a positive-energy spectrum of solutions, and here a minimum principle is used to find the variational eigenvalue. These methods provide approximations to the actual eigenvalues that may converge for large basis sets but do not yet provide variational bounds.

3. Squared Dirac Hamiltonian

In this method (Baylis and Peel, 1983; Wallmeier and Kutzelnigg, 1981) the problem of variational collapse is avoided by working with the square of the Dirac Hamiltonian Hh rather than with H,. Hi is a positive-definite operator and, hence, all its eigenvalues are bounded from below. A variational procedure then provides upper bounds to the exact eigenvalues. This positive spectrum of solutions contains eigenvalues belonging to both the positive- and negative-energy spectrum of Dirac energies, each of which can be identified by the relative norms of the large and small components. Little work has been done, however, on the application of this method to more complex problems, or on the use of the resulting set of eigenvectors for calculations involving sums over the complete spectrum.

A review of several other approaches involving different manipulations or approximations of the Dirac Hamiltonian is given in a detail by Kutzelnigg (1984).

111. Relativistic Sturmian Basis Sets

A systematic way of generating a finite basis set of polynomials that is functionally equivalent to Eqs. (12a and b) is provided by a relativistic generalization of the Sturmian basis sets widely used in nonrelativistic calculations (Rotenberg, 1962). In the nonrelativistic case, one keeps the energy fixed at an arbitrary value, E , and varies the nuclear charge to satisfy the Sturmian eigenvalue problem

H(Z,) = EYn. (19)

The eigenvalues are 2, = Z(E/E, ) ' /2 , where the En are the energy eigenvalues of H ( 2 ) with Z fixed at the physical nuclear charge. The radial eigenvectors expressed in terms of the confluent hypergeometric functions

a a(a + 1) z2 + ... F ( a , b ; z ) = 1 f - z + b l ! b(b + 1)2!


are, for angular momentum i,

x (2Ar)’F( - n + 1 + 1,21+ 2; 21r) (21) with A = (-2E)’”. These form a complete set of finite polynomials for n 2 I + 1. They are orthogonal with respect to the potential as a weight function.

The above cannot be generalized to the Dirac equation in a straightforward way. Here, the problem is to find a Dirac-like equation that generates a sequence of polynomials consisting of linear combinations of the Qi and defined by Eqs. (12a and b) with fixed y and A. The asymptotic form of the Dirac equation determines A = (mZ - E 2 ) l / ’ , and hence, E is fixed. One cannot now simply vary 2 to satisfy the relativistic analogue of Eq. (19) because y simultaneously changes (see Eqs. (11)) and eventually becomes imaginary. The problem of constructing relativistic Sturmian functions is discussed by Hostler (1987) within the context of the second order Dirac equation. We present here an alternative way of systematically generating a complete set of discrete functions. One begins by introducing two parameters in place of Z so that radial Dirac Hamiltonian becomes

H D . , =

CI

The condition that the radial functions behave as ry as r -+ 0 results in

2 ( K : - K ; ) = KZ - yz . (23)

If y is held fixed at the value (K’ - c12Z2)112, then

K , = ( K t + Z2) l / ’ .

K , can now be varied to satisfy the Sturmian eigenvalue problem

H D , r ( K p ) Y ; = EY,+ (25)

as many times as desired, and K , is determined by Eq. (24). The Sturmian eigenvalues are

(26) Z m

K(2n) = - ( E - En) 24

n = 0, 1,2, .

404 G. W. F. Drake and S. P. Goldman

where A,, = (m2 - E;)l i2 and En is the nth bound state eigenvalue of the Dirac spectrum given by Eqs. (12). If E = En, then ICY) = Z , KF) = 0 and Eq. (22) reduces to the standard Dirac Hamiltonian. For fixed basis set size N , the original eigenvalue probelm is satisfied exactly N times as E varies between 0 and mc2.

The functions that span the negative energy spectrum are generated by the complementary equation

HD,JKY'-)Y,,- = - EY,- (E > 0). (28)

The Sturmian eigenvalues are solution for the two cases yields the radial eigenfunctions

= KF) and KY)- = -KY). A detailed

x [TnF(-n + 1,2y + 1; 2 h ) + ( N f K)F(--n, 2y + 1; 2h)] (29)

x [TnF(-n+ 1 ,2y+ 1 ; 2 1 r ) - ( N T ~ ) F ( - n , 2 y + 1;21r)] (30)

where E = Elm and

Since N = ( K ~ + 2ny + n2)'I2, there is no n = 0 solution for 'u: if K > 0 and for Yo- if ic < 0. The g,' and f,' clearly become the well-known eigenfunctions for the Coulomb-Dirac Hamiltonian if 1 = 1,.

The above functions constitute a complete set of finite polynomials of the Sturmian type for the Dirac equation. Since the first 2N - 1 Sturmians Yo', . . , , Y;, . . . , Y i - l (with either 'Yo* or YO omitted, depending of the sign of K) can be written as linear combinations of the mi and mN+ in Eqs. (12), the two basis sets are equivalent and yield identical eigenvalues when HD,r is diagonalized (with the possible exception of spurious roots corresponding to the nonexistent n = 0 solutions). The Oi, mN+ basis set therefore will be referred to as a relativistic Sturmian basis set.

IV. Test Calculations with Relativistic Sturmian Basis Sets and Comparisons with B-Spline Methods

The basis-spline (B-spline) method for constructing solutions to relativistic field equations in elementary particle and nuclear physics (Drouffe and Itzykson, 1978; Rabin 1982; Bender et a/., 1985) has been adapted to


problems in atomic physics by Bottcher and Strayer (1987), and Johnson et al. (1988). For a one-dimensional problem involving, say, a radial coordinate r, the basis idea is to approximate the solution to a differential equation over a finite interval from r = 0 to r = R by first dividing the interval int0.a radial grid. The radial grid is then partitioned into overlapping segments, each containing several grid points, and a family of polynomials of fixed degree is introduced on each segment. Finally, the coefficients of the piecewise polynomials are adjusted to fit the functions of interest as well as possible. The B-spline method provides a systematic way of carrying through the above procedure (deBoor, 1978).

Solving the Dirac equation by the B-spline method with N segments gives 2N discrete eigenvalues that form a representation of the complete Dirac spectrum, just as is the case for the relativistic Sturmian basis sets. The B-spline eigenvalues, however, in some cases fall below the exact eigenvalues of H , for R = co (Johnson et al. 1988), and they are therefore not upper bounds for the true spectrum. Since a full discussion of the B-spline method is given by Johnson et al., (1988) the reader is referred to their article for further details. Here, we compare various tests of the quality of the two methods for performing summations over intermediate states.

A comparison with the exact values of the energy-weighted dipole matrix element sum rules provides one such test. The sum rules are of the form

S, = C (En - Eo)kI(yoIrIYn>12 (31) n

where En denotes summation over the discrete states of the Dirac spectrum and integration over both positive and negative energy continua. The exact values of S, are known for 0 I k I 5 (Goldman and Drake, 1982). The results for an arbitrary central potential V ( r ) are (in atomic units with a-' = c)

so = W o l r 2 I ~ o ) (32)

s, = 0 (33)

s, = 3/a2 (34)

S, is the relativistic generalization of the well-known Thomas-Reiche-Kuhn sum rule first obtained by Levinger et al., (1957). The sum is zero because the

406 G. W. F. Drake and S. P . Goldman

TABLE I COMPARISON OF EXACT VALUES OF THE DIPOLE OSCILLATOR STRENGTH SUM RULES WITH THE

RESULTS OBTAINED FROM FINITE BASIS SETS

ASiJSi AS,/S, Sturmian basis set' B-spline methodb

-

Sum Exact S, z=1 Z = 50 z = 2 SO ( Y + 1)(2Y + 1)/2Z2 - 5 x 10-1° 7 x Sl 0 6 x lo-' 5 x lo-' 7 x lo-' 1 x s 2 3/at 1 x lo-' 6 x 1 x lo-' I x

In each case, ASi is the deviation from the exact value S,. For i = 1, the number tabulated is ASl rather than AS1/Sl.

From Drake and Goldman (1981), using a relativistic Sturmian basis set with N = 14. * From Johnson et al. (1988). The first number of each pair is obtained with N = 40, k = 7,

and the second number with N = SO, k = 9.

contributions from positive and negative energy states cancel exactly. S3 simplifies in the case of a Coulomb potential to

S , diverges for ns1,2 and 2p,,, states, but not for states of higher 1x1. Exact values of the sum rules for the case where Y o is the wave function for the ground state are compared in Table I with values obtained by performing explicitly the sum in Eq. (31) by means of finite basis sets. For cases where comparisons can be made, the relativistic Sturmian basis set results with N = 14 are comparable in accuracy to the B-spline results with N = 40 and k = 7 (k is the order of the polynomial in each segment). The Sturmian basis set therefore appears to offer a somewhat more compact representation of the Dirac-Coulomb Green's function. A disadvantage of the method is that N cannot be made much larger than 14 with ordinary double precision (16 digit) arithmetic due to a progressive loss of significant figures. This problem can be overcome by using multiple precision arithmetic.

Another interesting test quantity is the dipole polarizability ocD. It is given by the sum rule

tlD = $-, (39)


and is known analytically up to order (aZ)’ from the expansion (Kaneko, 1977)

= 0.281 18783 . . . u: for 2 = 2.

This case is particularly advantageous for finite basis set expansions because the energy difference appears in the denominator. The Sturmian basis set results up to N = 14 converge to

aD = 0.28 1 18787499375a&

while the B-spline calculation with N = 50 and k = 9 gives (Johnson et al., 1988)

CID = 0.28 1 187877~:.

In this case, the Sturmian basis set is clearly superior. The accuracy of the results for other values of Z in Table I1 is sufficient to determine the expansion

a D = [i - y’ + 0.5285344(3)(~rZ)~ + 0.0336(1)(aZ)6 + ---I (41)

where numbers in brackets denote the uncertainties in the final figure quoted.

TABLE I1

DIPOLE POLARIZABILITIES FOR HYDROGENIC IONS OBTAINED WITH A RELATIVISTIC STURMIAN 2 X 14 TERM BASIS SET

Z Z

4.49975 1495 143 2.811878749096 x lo-’ 5.552794523705 x lo-* 1.756259485161 x lo-’ 7.19006124466 x 3.46532076684 x 1.86914901612 x 1.09475139978 x lo-’ 6.82804575717 x

10 20 30 40 50 60 70 80 90

4.4751643572 x 2.750523490 x lo-’ 5.28094069 x 1.60400282 x 6.2210863 x lo - ’ 2.7970904 x lo-’ 1.3822846 x lo-’ 7.256230 x lo-* 3.944093 x lo-*

The results have converged to the number of figures quoted, using E-’ = 137.0359895 for the fine structure constant.


A final test of finite basis sets is the calculation of photoionization cross sections given by (in atomic units)

(Akhiezer and Berestetskii, 1965), where d f / d E is the oscillator strength density

C2 - C 1 (Yo la. e*ie-ik'r( Y E ) 12.

dE - 2n(2j0 + 1) o df 1 -- (43)

Here, gi is the photon polarization vector, Y8 is a positive continuum wave function normalized to 6(E' - E), and o = ( E - Eo)h. The usual expansion of the plane wave in Eq. (43) into electric and magnetic multipoles results in

where

with A = 1 for electric multipoles and A = 0 for magnetic multipoles. In the above, j, is the angular momentum of the continuum electronic state and the radial integrals are (Grant, 1974)

and j,(or/c> is a spherical Bessel function. The above integrals can be calculated directly as a function of E using the

exact continuum solutions YE to the Coulomb-Dirac equation (Goldman and Drake, 1983). The finite basis sets, however, also represent the continuum by providing eigenvalues at a discrete distribution of points E,. The corresponding points on the continuous d f / d E curve can be approximated from the calculated oscillator strengths f k at the points E , by the method of


10-‘ A

3

0

6

Y

-0 \

0 .c

10-2

Stieltjes imaging (Langhoff and Corcoran, 1974). The optimal representation is

i-

-

-

with Ek = (Ek+ I - &)/2 Figures. 2 and 3 from Goldman and Drake (1983) compare the exact electric dipole ( L = 1 ,3 , = 1) contribution to df/dE for the ground state with the N = 13 Sturmian basis set results obtained from Eq. (49). For both Z = 1 and Z = 82, the agreement is excellent. Only for the highest eigenvalue, which “represents” the entire continuum lying above it, is there a significant deviation from the exact calculation. This demonstrates that Sturmian basis sets provide an accurate representation of relativistic continuum states in the region of space near the nucleus.

I I I \ o I 1 1 , 10-31 1 2 3 5 7

7)

FIG. 2. Comparison between the exact and variational results for the electric dipole contribution to the oscillator strength density for Z = 1. q is the energy measured in units of the ionization energy. The circles denote the variational values for df/dq, with (-) being the p S i z contribution and ( - ~ -) being the P , , ~ contribution obtained by exact calculation. (From Goldman and Drake 1983.)

410 G . W. F . Drake and S. P . Goldman

lo-' ' h

3

0 Y

F U \

-0 rc

10-2.

1 I I I I l l 1 2 3 5 7

T FIG. 3. Comparison between the exact and variational results for the electric dipole

contribution to the oscillator strength density for Z = 82. The symbols are as defined in Fig. 2. (From Goldman and Drake, 1983.)

V. Variational Dirac-Hartree-Fock Calculations

In recent years there has been an increasing interest in variational relativistic self-consistent field calculations. The first attempt at a relativistic extension of the nonrelativistic SCF method was made by Swirles (1935, 1936). Later on, using the more powerful group-theoretical techniques by then available, Grant (1961, 1965) derived the relativistic Dirac-Hartree- Fock (DHF) equations for closed shell atoms. Since then, numerical techniques have been developed to integrate the D H F equations that have proven to be very successful in relativistic atomic physics calculations through the implementation of DHF and MCDHF routines (Desclaux, 1969,1975; Grant et al., 1980; Lindgren and Rose, 1973) and the relativistic random-phase approximation method (Johnson et al. 1976, Lin et al., 1977). On the other


hand, the implementation of an analytic DHF routine, of the type introduced by Roothaan (1985) for the nonrelativistic case, has until very recently been unsuccessful. Such an approach is of interest for several reasons. It simplifies complicated atomic physics calculations, and it provides a representation of the complete spectrum (in the sense of Eqs. (13) and (6) of states of a given angular symmetry type (including continuum states) for use in MCDHF calculations, and in the calculation of quantities involving infinite summations over intermediate states. Finally, straightforward extensions to relativistic calculations in molecules are possible.

The first attempt at an analytical DHF procedure was made by Kim (1967). In his work, a variation of the basis set parameters does not yield upper bounds to the total energy. In order to find the variational eigenstates, Kim used the relativistic virial theorem

that is satisfied at the variational minimum (Lindgren and Rosen, 1973; Kim, 1967). The same method was later applied by Kagawa (1975, 1980) to open shell atomic systems. Different analytic D H F procedures have been tried since then (Mark and Rosicky, 1980; Mark et al., 1980; Wallmeier and Kutzelnigg, 1983; Ishikawa et al., 1983, 1984, 1985; Dyall et al., 1984; Kutzelnigg, 1984) but the problem of collapse or spurious roots hindered their usefulness and reliability.

The idea of avoiding these problems by the use of positive-energy projection operators has been explored (Mittleman, 1971,1981; Sucher, 1980, 1985; Datta, 1980; Schwarz and Wechsel-Trakowski, 1981). Although such a projection would avoid the problem of continuum dissolution, its application to the finite basis set method is not straightforward. The reason is that the exact projection operators for the DHF case are not known. The use of any other projection scheme to eliminate the negative-energy states a priori would result in an incomplete projection and therefore, as shown in detail for the one-electron case (Drake and Goldman, 198 1 ; Goldman, 1985a), in variational collapse.

A different approach was suggested by Goldman and Dalgarno (1986). In their work they propose the use of constraints in the basis set in order to ensure stability. It is interesting that, unlike the nonrelativistic case, the set of eigenstates of the Dirac-Coulomb Hamiltonian does not provide a complete representation for the DHF case. This can be seen using the unitary transformation (Eq. (14)). It is only for the Coulomb Hamiltonian that the

412 G. W . F . Drake and S. P . Goldman

solution 0 = 0 exists (see Eq. (17~)); i.e., there is one degree of freedom missing in an expansion of the hydrogenic wavefunctions, with the upper component q ( r ) behaving at the origin as r y + I . In other words, an expansion in terms of hydrogenic wavefunctions fails to represent the zy terms present in the lower component in the DHF case (or in any other case but the Coulomb potential) (Goldman and Dalgarno, 1986).

On the other hand, it can be shown that condition (18) still holds in the DHF case (Goldman, 1988) and then can be used to ensure that the variational solutions have the correct nonrelativistic limit, while satisfying the relativistic boundary conditions at the origin. It then was proposed (Goldman and Dalgarno, 1986) to use a basis set of the form of Eqs. (12) with multi-exponential parameters constrained by the boundary conditions (Eq. (18)) to represent the one-electron D H F states. In this approach, the radial variational eigenfunctions of the form of Eq. (7) are written as the linear combination

In the case K < 0, the bispinors ‘p i , are given by

with j = 1,2 ,..., M ; i = O , 1 ,..., N - 1.

In the case K > 0, for i = 1, c p l , j and c p N f l S j are given by (in a.u.)

for all other values of i, the bispinors qi, are given by


with

j = l , 2 , . . . , M . (53c)

This approach has proven to be successful and has been applied to obtain very accurate results for the He, Be, C and Ne isoelectronic sequences (Goldman, 1988). These were obtained with basis sets including two exponential parameters and six powers for each of the electronic configurations represented, and include the Breit interaction corrections.

A crucial point for the accuracy of the variational DHF results is the minimization of the total energy with respect to the nonlinear parameters. This is important not only to produce the best representation with a given size of the basis set, but also to guarantee that the one-electron energies are upper bounds to exact results (Goldman and Dalgarno, 1986; Goldman, 1988). While the total energy is an upper bound to the exact value, the one- electron energies are upper bounds at the variational minimum only. The lack of bounds on the one-electron energies is a consequence of the inadequate screening provided by the nonoptimized wavefunctions.

The process of nonlinear optimization is, in general, long and costly even for small sets of nonlinear parameters. There are standard strategies to pursue the energy minimization (Press et al., 1986), but the convergence is slow when the gradient of the function to be minimized cannot be written explicitly in terms of the nonlinear parameters, as is the case here. An advantage of the Sturmian-type basis sets is that a fast converging nonlinear optimization method based on the relativistic virial theorem (Eq. (19)) can be used. In this method, one follows the directions defined by the gradients of E and ( B ) to move towards the region where Eq. (19) is satisfied and E is a variational minimum (Goldman, 1987). If one defines

dB B E ( p ) and B =-

- ax, (54)

where xk is the kth nonlinear parameter, then starting from a set of nonlinear parameters x*, the next set of parameters predicted by the virial theorem iteration method is given by

x , = x f + & ( 5 5 )

where E is the average error E - ( p ) in the previous iteration. The minimization concludes when either a convergence criterion has been met, or the basis set is not able to better satisfy the virial theorem using the values where ( p ) and E are stable with respect to the nonlinear parameters. In other words, the routine provides a natural end to the iteration process. As an example of the


efficiency, a virial-DHF calculation for the ground state of carbon using a basis set with four nonlinear parameters needed five and 15 iterations to achieve relative errors of and 2 x lo-’ in the total energy, respectively. For comparison, the downhill simplex method needed about 50 and 140 iterations to achieve the same accuracy (Goldman, 1987). The number of iterations remains approximately the same as the number of nonlinear parameter changes in order to achieve a given relative error for different atoms. In summary, the virial theorem method provides a very efficient optimization technique with a natural stopping point for the iteration process.

VI. Suggestions for Future Work

There is a need for rigorous proofs of bounds and completeness for the variational representation of general screened-Coulomb, finite nuclear size and DHF potentials. For the first two cases, it might be possible to extend the unitary transformation of Eq. (14) or the Sturmian basis set methods of Section I11 to more general cases. Open shell DHF and multiconfiguration DHF calculations are already under way. The extension of this method to time-dependent DHF calculations and to multicentered systems is desirable. This would allow, for example, the study of forbidden transitions in H, that are of importance in astrophysics. Further studies should be made of the potential applications for alternative methods, such as the different “minimax” strategies discussed in Section 11.

ACKNOWLEDGMENTS

The authors wish to express their gratitude to Professor Alex Dalgarno for his inspiration and encouragement over many years.

REFERENCES

Akhiezer, A. I. and Berestetskii, V. B. (1965). Quantum Efectrodynamics, Interscience, New York,

Baylis, W. E. and Peel, S. J. (1983). Phys. Rev. A 28, 2552. New York.


Bender, C. M., Milton, K. A., Sharp, D. H., and Strong, R. (1985). Phys. Rev. D 32, 1476. Bottcher. C. and Strayer, M. R. (1987). Ann. Phys. (N.Y.) 175, 64. Brown, G. E. and Ravenhall, D. G. (1951). Proc. Roy . SOC. (Lond.) A 208, 552. Dalgarno, A. and Lewis, J. T. (1955). Proc. Roy. SOC. (Lond.) A 233, 70. Datta, S. N. (1980). Chem. Phys. Lett. 74, 568. deBoor, C. (1978). A Practical Guide to Splines. Springer, New York, New York. Desclaux, J. P. (1969). Comp. Phys. Commun. 1, 216. Desclaux, J. P. (1975). Comp. Phys. Commun. 9, 31. Drake, G. W. F. (1985). Nucl. Instrum. Meth. B 9,465. Drake, G. W. F. (1986). Phys. Rev. A 34, 2871. Drake, G. W. F. and Goldman, S. P. (1981). Phys. Rev. A 23, 2093. Drake, G. W. F., Victor, G. A,, and Dalgarno, A. (1969). Phys. Rev. 180,25. Drouffe, J. M. and Itzykson, C. (1978). Phys. Rep. C 38, 133. Dyall, K. G., Grant, 1. P., and Wilson, S. (1984). J. Phys. B 17, 1201. Engelmann, A. R. and Natiello, M. A. (1987). Int. J. Quant. Chem. 32,457. Gazdy. B. and Ladanyi, K. (1984). J . Chem. Phys. 80,4333. Goldman, S . P. (1985a). Phys. Rev. A 31, 3541. Goldman, S. P. (1985b). Nucl. Instrum. and Meth. B 9. 493. Goldman, S. P. ( 1 987). Phys. Rev. A. 36, 3054. Goldman, S. P. (1988). Phys. Rev. A. 37, 16. Goldman, S. P. and Dalgarno, A. (1986). Phys. Rev. Lett. 57, 408. Goldman, S. P. and Drake, G. W. F. (1981). Phys. Rev. A 24, 183. Goldman, S. P. and Drake, G. W. F. (1982). Phys. Rev. A 25, 2877. Goldman, S. P. and Drake, G. W. F. (1983). Can. J. Phys. 61, 198. Grant, I. P. (1961). Proc. Roy . SOC. (Lond.) A 262, 555. Grant, I. P. (1965). Proc. Phys. SOC. (Lond.) 86, 523. Grant, I. P. (1974). J . Phys. B 7, 1458. Grant, I. P. (1982). Phys. Reo. A 25, 1230. Grant, I. P. (1986). J. Phys. B 19, 3187. Grant, I. P. and Quiney, H. M. (1988). Adv. At. Mol. Phys. 23, 37. Grant, I. P., McKenzie, B. J., Norrington, P. J., Mayers, D. F., and Pyper, N. C. (1980). Comp.

Hostler, L. (1987). J. Math. Phys. 28, 2984. Hylleraas, E. A. and Undheim, B. (1930). 2. Phys. 65, 759. Ishikawa, Y., Binning, R. C., and Sando, K. M. (1983). Chem. Phys. Lett. 101, 111. Ishikawa, Y., Binning, R. C., and Sando, K. M. (1984). Chem. Phys. Lett. 105, 189. Ishikawa, Y., Baretty, R., and Sando, K. M. (1985). Chem. Phys. Lett. 117, 444. Johnson, W. R., Lin, C. D., and Dalgarno, A. (1976). J . Phys. B 9, L303. Johnson, W. R., Blundell, S. A., and Sapirstein, J. (1988). Phys. Rev. A . 37, 307. Kagawa, T. (1975). Phys. Rev. A 12,2245. Kagawa, T. (1980). Phys. Rev. A 22, 2340. Kaneko, S. (1977). J. Phys. B 10, 3347. Kim, Y. K. (1967). Phys. Rev. 154, 17. Klahn, B. and Bingel, W. A. (1977). Theo. Chem. Acta (Bed . ) 44,9 and 27. Kutzelnigg, W. (1984). Int. J. Quantum Chem. 25, 107. Langhoff, P. W. and Corcoran, C. T. (1974). J. Chem. Phys. 61, 146. Levinger, J. S., Rustgi, M. L. and Okamoto, K. (1957). Phys. Rev. 106, 1191. Lin, C. D., Johnson, W. R. and Dalgarno, A. (1977). Phys. Rev. A 15, 154. Lindgren, I. and Rosen, A. (1973). Case Stud. At. Phys. 4,93. Mark, F . and Rosicky, F. (1980). Chem. Phys. Lett. 74, 562. Mark, F., Lischka, H., and Rosicky, F. (1980). Chem. Phys. Lett. 71, 507.

Phys. Commun. 21, 207.

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Mittleman, M. H. (1971). Phys. Rev. A 4, 893. Mittleman, M. H. (1981). Phys. Rev. A 24, 1167. Mukoyama, T. and Lin, C. D. (1987). Phys. Rev. A 35,4942. Press, W. H., Flannery, B. P., Teukolsky, S. A,, and Veterling, W. T. (1986). Numerical Recipes.

Cambridge University Press, Cambridge, England. Rabin, J. M. (1982). Nucl. Phys. B 201, 315. Roothaan, C. C. J. (1985). Rev. Mod. Phys. 55, 1033. Rosenberg, L. and Spruch, L. (1986). Phys. Rev. A 34, 1720. Rotenberg, M. (1962). Ann. Phys. (N .Y . ) 19,262. Schwarz, W. H. E. and Wechsel-Trakowski, E. (1981). Chem. Phys. Lett. 85,94. Shakeshaft, R. (1985). J . Phys. B 18, L611. Sternheimer, R. M. (1950). Phys. Rev. 80, 102. Sucher, J. (1980). Phys. Rev. A 22, 348. Sucher, J. (1985). Phys. Rev. Lett. 55, 1033. Swirles, B. (1935). Proc. Roy. Soc. A 152, 625. Swirles, B. (1936). Proc. Roy. Soc. A 157, 680. Talman, J. D. (1986). Phys. Rev. Left. 57, 1091. Wallmeier, H. and Kutzelnigg, W. (1981). Chem. Phys. Lett. 74, 341. Wallmeier, H. and Kutzelnigg, W. (1983). Phys. Rev. A 28, 3092.


l l DISSOCIATION DYNAMICS OF POLYATOMIC MOLECULES T. Uzer School of' Physics Georgia Instituie of Technology Atlanta, Georgia

1. Introduction . . . . . . . . . . . . . . . . . . . . . . 11. Unimolecular Reaction Rate Theories . . . . . . . . . . . . .

111. Semiclassical and Quasiclassical Trajectory Methods . . . . . . . . IV. Unimolecular Dissociation Through State Selection. . . . . . . . . V. Overtone-Excited Processes . . . . . . . . . . . . . . . . .

VI. Case Study: Overtone-Induced Dissociation of Hydrogen Peroxide-Experi- ment and Theory. . . . . . . . . . . . . . . . . . . . .

Photodissociation and IVR . . . . . . . . . . . . . . . . .

XI. Concluding Remarks . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

VII. Dissociation through Electronically Excited States-Interface between

VIII. Unimolecular Quantum Dynamics and Molecular Dissociation . . . .

417 418 422 424 425

426

428 43 1 431 432 433

I. Introduction

Unraveling the mechanisms by which many-electron systems evolve from one state or another has been identified as the central challenge of low-energy physics (Fano and Rao, 1986). To review recent advances in our understanding of one of these transformations, molecular dissociation, is a formidable task. Doing i t any justice would require the survey of much that is key to molecular physics and modern-day chemical physics: unimolecular reactions, which form the basis for understanding chemical reactivity and chemical reaction dynamics; quantized state structure in molecules and the coupling of their internal and external degrees of freedom; the interaction of radiation with molecules; and last, but by no means least, intramolecular energy sharing. Since limitations of space preclude an exhaustive review, I have instead selected a number of topics that display the intimate connections between these subjects. This selection is necessarily limited and largely arbitrary. It aims to give an impression of the subject, and by emphasizing recent developments in both experiment and theory, to point out the directions in which chemical physicists' understanding of molecular dissociation is developing.

417 Copynght 0 1988 by Academic Press, Inc.


418 T. Uzer

Any discussion of molecular dissociation inevitably leads to questions about the way energy flows inside a molecule, and in particular, how vibrational motions are energized. The theory of intramolecular energy redistribution (IVR) are reviewed in another publication (Uzer, 1989). Other subjects excluded from this review are dissociation of van der Waals molecules (Miller, 1986), bimolecular reactions (Leone, 1989, molecular- beam investigations of reactive scattering (Lee, 1987), and most polyatomic photodissociation processes (see Simons, 1984 for a critical review), the dissociation of ions (Baer, 1986), and the photodissociation of diatomic molecules, reviewed by Kirby and van Dishoeck in this volume. Laser- induced phenomena (Reisler and Wittig, 1986) will, however, figure promin- ently, because they represent the most state-specific way of depositing energy into molecules.

The last decade has seen the blurring of the distinction between the traditionally separate fields of molecular spectroscopy and reactive scattering. Traditional spectroscopy was concerned with long-lived states residing in low-energy regions of the potential surface, whereas high-energy processes involving mostly unbound states were the province of scattering. Now it is possible to perform quite accurate spectroscopy on very short-lived species (e.g., Imre et al., 1984) and infer much about the transition state and the progress towards dissociation from attributes of products. Thereby, the unity between the spectroscopic phenomenon of predissociation and unimolecular reactions, which had been recognized earlier (Rice, 1930) theoretically has finally been reached experimentally as well. At present, photodissociation and unimolecular reactions of transient species are being described as “half- collision events” to emphasize their connection with scattering phenomena. Theoretical aspects of photodissociation and intramolecular dynamics have been described thoroughly by Brumer and Shapiro (1985).

The plan of this review is as follows. After a brief statement about traditional unimolecular theories and citation of the most prominent reviews, contemporary methods of producing highly excited molecules will be mentioned in their outlines. Then, various methods for calculating the aftermath of the excitation will be reviewed. Special emphasis will be given to the overtone-induced dissociation technique (Crim, 1984) and the dissociation of hydrogen peroxide. We will conclude by mentioning a number of experiments that incisively probe the validity of traditional views of molecular dissociation.

11. Unimolecular Reaction Rate Theories

Several extensive monographs and reviews on the subject of unimolecular reactions are available (the books by Slater, 1959; Robinson and Holbrook, 1972; Forst 1973; and the thorough reviews of Hase, 1976, 1981) which treat

DISSOCIATION DYNAMICS OF POLYATOMIC MOLECULES 419

the classical theory, and one that treats the quantum theory (Pritchard, 1984). Several new developments have been presented in the R. A. Marcus Commemorative Issue of the Journal of Physical Chemistry (El-Sayed and Zewail, 1986), as well as by Wardlaw and Marcus (1987).

The currently accepted varieties of unimolecular reaction rate theory arose through the testing of Slater’s dynamical theory (Slater 1959), and the statistical Rice-Ramsperger-Kassel-Marcus (RRKM) theories (Marcus, 1952) in the 1950s and 1960s. Slater’s theory pictured the molecule as an assembly of harmonic oscillators. Within this framework, the vibrational energy relaxation between the normal modes is forbidden, and reaction occurs only if a reaction progress variable, the “reaction coordinate,” reaches a critical extension by the superposition of various normal mode displace- ments. In contrast, the RRKM theory assumes rapid relaxation of vibrational energy. The statistical theory has been very successful in accounting for a large body of experimental data in a variety of systems. For a review of statistical unimolecular reaction rate theories, see Wardlaw and Marcus (1 987).

Inherent in the statistical theory is the assumption that energy, once put into the molecule, randomizes rapidly compared to the reaction rate, so that it is distributed statistically prior to the reaction. This hypothesis was introduced very early in the theoretical development because not much was known about energy-sharing inside molecules (the problem of intramolecular energy sharing needs to be solved before a truly dynamical unimolecular reaction rate theory can be developed). The assumption of randomization, however, turns out to be a widely valid one. Much detail about the progress of the reaction is lost due to the use of statistics, but the essence of the behavior of real molecules is preserved, as it turns out, very successfully.

Strong coupling among the modes is introduced by ignoring all the symmetry relations and by evaluating the “microcanonical density of states” by counting the number of ways the given energy can be shared among the pertinent degrees of freedom. In this process, it is assumed that any one way of distributing this energy is just as likely as another (for a treatment of densities of rovibronic states, see Quack, 1985).

The statistical approach also reduces the (usually very high) dimensionality of the problem in a manner similar to traditional “Transition State Theory” (Glasstone et al., 1941): The motion of the system along the reaction coordinate is assumed to be separable from the motion of the system in all possible other modes, at least in the vicinity of a multidimensional surface in phase space that separates reagents from products. This latter construct is called the “dividing surface,” and the reaction rate calculation is then reduced to calculating the rate of passage of systems across the dividing surface in one dimension only. This is a remarkable simplification of the multidimensional problem. Transition State Theory makes do with information about the

420 T. Uzer

potential energy surface that is limited to the knowledge of the change in effective potential energy associated with the degree of freedom involved in the transformation. The other degrees of freedom are of course not discarded, but rather act as a reservoir and contribute to the rate through their densities of states.

The RRKM theory is basically a statistical rate theory couched in the transition state formulation (Truhler et al., 1983). The success of the theory derives from two factors: the use of statistics avoids the spelling out of detailed molecular dynamics, utilizing very limited information about molecular potentials; and various theoretical parameters of the theory can be fairly directly related to experimental observables. A similar formalism, albeit for the decay of excited nuclei, had been conceived of by Bohr and Wheeler (1 939) prior to its independent development in chemical physics.

For application to modern laser-induced processes, which are many times state-selected, state-to-state rate constants are needed. The detailed information necessary for such an accurate calculation is, in many cases, not available even today. On the other hand, the statistical theory assumes that the decay process can be described by a single time-independent rate constant, and that the total energy (and, in some versions, the angular momentum) suffices for the description of the unimolecular process. It is known that every state may have a different unimolecular reaction rate. Indeed, the variation in these rates can be quite dramatic, as is seen in, say, the accidental predissociation process (Uzer and Dalgarno, 1980; Cooper et al., 1982; Baumgartner et al., 1984; Preuss and Baumgartner, 1985). Experimental evidence includes Dai et al. (1985), Knee et al. (1985), Guyer et al. (1986), and Khundkar et al. (1987). Such detail is given up in the statistical description, and a rate constant averaged over the initial states is sought instead. It is assumed that the individual initial states comprising the collection must communicate, and must do so more rapidly than the time it would take any one of them to decay into products. Clearly, it makes sense to speak of such a rate constant that is the same for all members of the collection of initial states only if, on the relevant time scale, the initial states have no dearly definable individuality with respect to the decay process. Of course, it is precisely this assumption that is probed increasingly incisively by modern-day state-to-state experiments.

Given the approximations inherent in the transition state, there have been a number of attempts to supplant it, or provide new definitions of it that correspond more closely to our understanding of the phase space of highly excited molecules (e.g., Kato, 1985). With the recent developments in nonlinear mechanics, and its application to coupled oscillator systems in chemical physics, a new treatment of intramolecular relaxation rates has been formulated. This theory, referred to as “intramolecular bottlenecks” (Carter

DISSOCIATION DYNAMICS OF POLYATOMIC MOLECULES 42 1

and Brumer, 1982; Davis 1985; Gray et al., 1985; Davis and Gray, 1986; Gray and Rice, 1987) attempts to correct the shortcomings of the traditional RRKM theory by improving the definition of the transition state.

Other statistical theories that have found wide use because of the agreement they provide with experiment are Phase Space Theory (PST) (Pechukas and Light, 1965), the Statistical Adiabatic Channel Method (SACM) (Quack and Troe, 1974,1975; Quack, 1979, Troe, 1981,1983; Cobos and Troe, 1985), and, to a lesser extent, the “Separate Statistical Ensembles” (SSE ) method (Wittig et al., 1985). These methods have been reviewed (Wardlaw and Marcus, 1987) and critically compared by Troe (1986) and Wittig et al. (1986). Attempts to incorporate the detailed dynamical information by lifetime probability densities at various levels of complexity to yield predictions about the reaction process have yielded the Separable Unimolecular Rate Theory (SURT), which includes other statistical rate theories as subsets (Nordholm, 1975, 1976).

In the meantime, corrections and improvements to the traditional RRKM theory are being performed. The highly coupled bending vibrational-molecular rotational motion, which determines the transition state of a number of reactions, has been included in the theory (Wardlaw and Marcus, 1984, 1985, 1986), and the resultant flexible transition theory has been applied to the recombination of methyl radicals (Wagner and Wardlaw, 1988). Another development, the Reaction Path Hamiltonian (Miller et al., 1980), has been applied to the dynamics of polyatomic molecules (Cerjan et al., 1982) and a number of unimolecular reactions (e.g., formaldehyde dissociation, Waite et al., 1983).

Until the early 1970s, it was thought (Forst, 1973) that it is not feasible to observe the decay of an undisturbed isolated molecule of specified energy and angular momentum, because molecules cannot be prepared within a narrow range of energies and angular momenta, and because the decay of interest always occurs in competition with some other processes. The unimolecular theory provided the researcher with many averaged quantities, and it is due to the development of laser techniques that state-to-state rate constants which involve very little averaging are being measured.

The recognition that individual states have different rates of decay, and the feasibility of actually measuring such rates, has encouraged some researchers to abandon the Transition State Theory and classical phase-space arguments in favor of fully quanta1 theories, like the Pauli equation formalism (Quack, 1981) and the Master Equation formalism (Pritchard, 1984). This approach has been applied to the unimolecular dissociation of N,O to 0 and N, (Yau and Pritchard, 1979a,b), showing that the calculated state-to-state rate constants are highly oscillatory. The feasibility of performing these calculations on molecules larger than triatomics remains to be established.

422 T. Uzer

111. Semiclassical and Quasiclassical Trajectory Methods

Molecular dissociation is a quantum mechanical process and, ideally, its calculation should rely on detailed state-to-state transition (or coupling) matrix elements and their influence on the propagation of the wavefunction. In some cases, this fully quantum mechanical scheme has indeed been implemented (Pritchard, 1984). But the realization that great amounts of vibrational energy are being exchanged indicates that an overwhelming number of quantum states are involved in the process, thereby removing such a calculation from the realm of practicality in most instances. In addition to the arguments for feasibility, there is another strong argument in favor of intuitive appeal: state-to-state transition matrix elements are not a pictorial way of understanding the variety of processes that lead to molecular dissociation. Thus, one is faced with a situation similar to that in heavy- particle scattering, and one resorts to similar approximation methods, semiclassical or classical (Miller, 1974, 1976, 1986).

The most current semiclassical methods used for molecular dissociation, especially through photodissociation, are the wave packet methods developed by Heller and his coworkers (Heller, 1981a,b). The calculational procedure for wave packets remains to be implemented for larger molecules. In contrast, quasiclassical trajectory methods have been applied from the beginning to large molecules, and generally give results in harmony with experiment if the parameters that are involved (or, in ideal cases, the potential energy surface) are known accurately. (For a critical study of classical trajectory methods and the connection between their results and potential surface features, see Hase and Buckowski, 1982 and Hase, 1986). These methods have been reviewed in the context of reactive scattering by Raff and Thompson (1984).

For unimolecular reaction studies, if the potential surface is known accurately and with all its dissociation channels, the motion of the phase point on this multidimensional potential surface can be monitored using Hamilton’s equations, and resulting trajectories analyzed for the outcome of motion. But it is comparatively rare to have an accurate potential surface for all possible configurations of a polyatomic molecule (where the stretching, bending, and torsional degrees of freedom are taken into account in detail). It can be said safely that the major problem of applying trajectory calculations to unimolecular reactions is obtaining accurate potential energy surfaces, especially for large-amplitude motions which are precursors to dissociation. Remarkable advances have been made in the procedures of obtaining accurate force constants and anharmonicities for molecules of interest (Fogarasi and Pulay, 1984).

In the absence of detailed information, the molecule is modelled as a collection of anharmonic, nonlinear oscillators. Then, quasiclassical trajec-


tories are calculated using Hamilton's equations, preferably in Cartesian coordinates, which makes the treatment of torsional and rotational motions easier. The numerical procedures for propagating trajectories are well developed, and indeed, there are general computer programs to calculate trajectories in many systems. The applicability of classical mechanics to unimolecular reaction calculations has been critically examined, including the difference between classical and quanta1 unimolecular dynamics (Hase and Buckowski, 1982).

The selection of initial conditions requires some thought and ingenuity if there is to be a close connection between the experimental results and the calculation. Many times, one would like to have a microcanonical ensemble of trajectories, all at the same total energy, with this energy distributed among the various motions as outlined by the initial conditions of the experiment. Especially when several modes are excited, or the sample is excited thermally, refined procedures are needed. These have been reviewed by Raff and Thompson (1984), and the task is made easier if good action- angle variables exist for the given molecule. During the apportioning of the total energy into various normal modes, molecular rotation also can be included by first specifying the angular momentum along each principal axis. This is equivalent to choosing the overall direction of the molecular angular momentum vector. From these values and the geometry of the molecule, the linear momentum magnitude of each atom is computed. The initial Cartesian momentum components for each atom due to rotation are then given by the multiplication of these magnitudes by appropriate direction cosines.

These statistical methods of selecting initial conditions assume that probability distributions from which the initial state selection is made are those characteristic of classical systems. These probability distributions will approach the correct quantum mechanical distribution only in the correspondence-principle limit. Moreover, averaging procedures that employ classical distribution functions assume precise knowledge of the coordinate and momentum states. The fact that the uncertainty principle is not incorporated in these calculations, and the absence of any tunneling motion makes these calculations unrealistic from the quantum mechanical point of view. Better accuracy can be expected if the initial state selection is made based on quantum mechanical considerations. Therefore, it is convenient to use probability density functions derived from appropriate initial state quantum mechanical wavefunctions. The Wigner Distribution Function is ideally suited to this purpose (Wigner, 1932). So far, this formalism has been used for the dissociation of negative ions (Goursand et al., 1976, 1978), and for the photodissociation of ICN (Brown and Heller, 1981), and methyl 'iodide (Hendricksen, 1985).

The various unimolecular reactions that have been treated using classical trajectories are enumerated in the cited review (Raff and Thompson, 1984) as

424 T. Uzer

well as in earlier ones (Hase, 1976, 1981). In this context, Bunker initiated the earliest studies, and he and his coworkers treated many dissociating systems, model and actual; references to this large body of work can be found in the reviews by Hase (1976, 1981). Among applications to specific molecules are the unimolecular dissociation of methane (Raff et al., 1984; Viswanathan et. al., 1984a, 1985), silane (Viswanathan et. al., 1984b), silylene (NoorBatcha et. al., 1986), and ammonia (Rice et. al., 1986). The dependence of lifetimes of triatomic collision complexes on a variety of factors has been examined by Schlier and Vix (1985). Energy transfer and dissociation processes in chains of nonlinear oscillators are still of interest (Hamilton and Brumer, 1985), especially in connection with the “heavy atom problem” (Lopez and Marcus, 1982; Swamy and Hase, 1985). The fundamental assumptions of RRKM theory about the internal dynamics of one-dimensional chain “molecules” have been examined by Schranz et. al. (1986a,b). They find that nonuniformi- ties in the chain can indeed affect the validity of RRKM theory (Schranz et. al., 1986c), though as Swamy and Hase (1985) point out, in a true molecule, the intricacies of the potential surface will alter, in a very significant way, any clear-cut conclusions derived from models.

The modeling of an overtone-induced reaction in a supersonic beam, where all vibrations other than the excited overtone are zero-point vibrational motions, and most of the molecular rotations are frozen out, is straightforward, and the mode and site-specificity of these reactions makes them readily amenable to comparison with experiment. The sensitivity of vibrational overtone excitation experiments to non-RRKM behavior has been examined by Hase (1 985).

IV. Unimolecular Dissociation through State Selection

The experimental situation in mode-selective unimolecular reactions has been reviewed by Crim (1984). An ideal experimental technique for studying unimolecular reactions would prepare the reactant molecule in a well characterized and highly selective manner while detecting the rate of product formation in individual quantum states. At the least, the excitation scheme should be energy selective in order to create reactants with a narrow distribution of total energy. Since unimolecular reaction rates are very strong functions of energy content, an experiment that averages over a large spread of energies is difficult to compare with theory. A monoenergetic site-selective deposition scheme is even more informative, but true state-selective preparation, in which only one or very few quantum states are initially populated, is


ideal. The product detection scheme should yield the reaction rate of the selectively excited molecules, as well as measure the energy content of products. Information about energy disposal provides additional insight into the reaction even if statistical behavior completely controls the rate.

Of course, highly vibrationally excited species are not solely of interest for photophysics and photodissociation. They are also reactive species in most of thermal chemistry, as well as being part of atmospheric chemistry and combustion processes. The variety of techniques that approach the description above have been explained in the review cited above with relevant references. Briefly, these are

(i) Chemical activation, which makes use of exothermic reactions such as free radical addition to create highly excited molecules;

(ii) Infrared multiphoton excitation, in which the molecule absorbs a large number of infrared photons from an intense laser pulse;

(iii) Internal conversion, which creates highly vibrationally excited molecules by the isoenergetic crossing of electronically excited molecules into high vibrational states of the ground state;

Stimulated emission pumping, which produces highly vibrationally excited molecules by a two-photon process in which the first photon excites the molecule to an electronically excited state, and the second transfers it to a high vibrational state of the ground electronic state by stimulated emission; and

(v) Overtone vibration excitation, which generates vibrationally excited molecules by single-photon preparation of high vibrational states. This is a mode-selective technique that retains the initial distribution of energy in the degrees of freedom that do not interact with the excitation photon. For molecules with sparse rotational structure, excitation of an individual angular momentum state is possible.

(iv)

V. Overtone-Excited Processes

The technique of direct overtone excitation was introduced by Reddy and Berry ( 1979) who studied, among other reactions, the isomerization of methyl cyanide. Prominent in subsequent dissociation work is the study of unimolecular decomposition of t-butyl hydroperoxide by overtone excitation (Rizzo and Crim, 1982; Chandler et al., 1982; Chuang et al., 1983). Overtone- induced dissociation accomplishes two desirable aims of unimolecular reactions at once: a high degree of mode specificity and site specificity. In this context, most of the overtone work has been performed on molecules that

426 T. Uzer

have local modes, e.g., hydrogen atom bound to oxygen or carbon. It turns out (Child and Halonen, 1984) that such vibrations are, to a large extent, decoupled from the vibrations of the rest of the molecule. The notion that the initially excited state is mixed with background states implies that excitation by a nanosecond pulsed laser or a continuous laser cannot prepare a pure local mode state, but rather that a coherent excitation of the entire linewidth by a transform-limited laser pulse is required (Jasinski et al., 1983). Vibra- tional overtone excitation prepares an initial state having far from statisitical stretching excitation in a molecule. Knowing the exact nature of this state, however, requires detailed information about the molecular Hamiltonian.

The disadvantage of the method is that with a single photon, transitions are excited that are forbidden in harmonic systems, but only take place in anharmonic systems with a small probability. It is often necessary to excite high overtones to deposit enough energy for a reaction, and therefore the transition probability is very small. Therefore, extremely sensitive techniques need to be used for the detection.

With its site- and state-specificity, overtone-induced unimolecular reactions are among the most detailed investigations of unimolecular reaction dynamics so far (Crim, 1984), and constitute a window on the intramolecular dynamics, and on the reaction rates of highly vibrationally excited molecules. For a survey of theoretical investigations of overtone-induced reactions, see Uzer and Hynes (1987). We will single out the overtone-induced dissociation of hydrogen peroxide for further scrutiny as a case study.

VI. Case Study: Overtone-Induced Dissociation of Hydrogen Peroxide-Experiment and Theory

Unimolecular reaction rate constant measurements, on systems with well- known total energy, permit detailed comparisons with theoretical predictions. Vibrational overtone excitation is a technique that adds a precise energy increment to the reacting molecules and, when applied to a sample cooled in a free jet expansion, produces an ensemble with a known energy content. Observation of the width of individual features in a well-resolved vibrational overtone excitation spectrum establishes an upper limit to the unimolecular reaction rate constant for these molecules in their lowest few rotational states.

Small molecules offer several advantages in studies of unimolecular dynamics. Because their density of states is less than that in larger molecules, the uncertainty in the initial energy content following the excitation is


reduced. The accompanying reduction in spectral congestion increases the selectivity also, and theoretical studies using realistic potential energy surfaces are certainly more likely to be practicable on a molecule containing only a few atoms.

The disadvantage of small molecules is that direct measurement of their shorter unimolecular lifetimes requires greater time resolution. Crim's research group in Wisconsin performed overtone-induced dissociation studies on hydrogen peroxide (Rizzo et al., 1983, 1984). Hydrogen peroxide has a readily identifiable reaction coordinate in its weak 0-0 bond. Moreover, it has two local mode oscillators that can be excited selectively with enough energy to break the 0-0 bond on the ground electronic surface. With the variety of experimental and theoretical information that is becoming available, the simple 0-0 bond breaking in hydrogen peroxide is becoming a prototype case for unimolecular bond fission processes (for an account of these studies, see Brouwer et al., 1987).

The u = 6 (fifth overtone) stretching vibration of the OH group has enough energy to cause the 0-0 bond to break. This particular overtone was excited using a high energy 10 nanosecond pulsed laser and the emerging OH radicals were analysed using laser-induced fluorescence on a 20 nanosecond time scale. Since this particular excitation is very mode- and site-specific, it is well known from where the dissociation energy is coming. The early room temperature experiments produced a FWHM for the u = 6 excitation of about 86 cm- '. The statistical reaction rate theories suggest a lifetime for the excited molecule between 5 and 50 picoseconds (Rizzo et al., 1983), which, of course, correspond to a far smaller linewidth than observed experimentally.

To understand the processes that lead to the dissociation of this simple molecule, the dissociation of the rotationless molecule was simulated by a classical trajectory calculation (Sumpter and Thompson, 1985; Uzer et al., 1985, 1986). Picturing the dissociating molecule as a collection of nonlinear oscillators, the frequencies of which change dynamically as energy is transferred, was very helpful in rationalizing the results of the trajectory calculation. We saw that it would take more than half a picosecond incubation time for the molecule to dissociate, and we extracted a l/e lifetime of about 6 picoseconds (Uzer et al., 1985). We predicted that the linewidth for u = 6 would shrink dramatically to about 1 cm- ' in a supersonic beam. Later, that free-jet experiment was performed, and resulted in a lower limit of the lifetime of 3.5 picoseconds (Butler et al., 1986a). The experimental line still contains small amounts of rotational congestion, the elimination of which should bring the theoretical and experimental results into even closer agreement.

The lifetime from trajectory calculations agrees with the lower limit of the statistical lifetime. On the other hand, the vibrational energy distribution in the molecule prior to dissociation is not statistical-the initially unexcited

428 T. Uzer

OH stretch remains unenergized during the progress of the energy transfer. Yet, a statistical adiabatic channel calculation (Brouwer et al., 1987) gives a lifetime that is in good agreement with both the experiment and the dynamical trajectory calculation. The applicability of a statistical theory to such a small molecule is by no means obvious, but the trajectory calculation gives a clue to its success. The unexcited, unenergized, high-frequency O H stretching vibration contributes little to the sum of the states and hence, the statistically calculated unimolecular rate constant (Brouwer et al., 1987). Thus, the statistical calculation and the rate constant measurement are both blind to the participation of that degree of freedom.

There is still ongoing interest in the dissociation of hydrogen peroxide. The v = 5 excitation by itself falls short of the dissociation energy by about 1100 cm- ', but this difference can be made up thermally from combination vibrations involving the low-frequency modes (especially the torsion) and the rotational energy of the molecule (Ticich et al., 1986). Therefore, many factors come together in this dissociation, making it somewhat more involved than the v = 6 excitation dissociation. Exciting the same overtone with a picosecond laser, Zewail's group at Caltech measured the thermally assisted dissociation rate of hydrogen peroxide (Scherer et al., 1986; Scherer and Zewail, 1987) and have found lifetimes that are much longer than those from the v = 6 excitation.

VII. Dissociation Through Electronically Excited States- Interface between Photodissociation and IVR

As mentioned in the introduction, because laser excitation is the most effective and informative method for investigating state-by-state unimolecular dissociations, it is at times impossible to draw a clear demarcation between photodissociation and unimolecular dissociation, except possibly (arbitrarily) through the involvement of excited electronic states. Let us mention some experiments and theoretical calculations in this context because they involve the intramolecular energy sharing process significantly. Much of this material has been reviewed before (Heller, 1981b; Leone, 1982; Moore and Weisshaar, 1983; Lawrance et al., 1985), and therefore we will discuss only a few experiments.

When a diatomic molecule is photoexcited to a repulsive electronic state, it dissociates in one-half of a molecular vibration to atomic fragments. Excita- tion to a bound excited electronic state results in fluorescence back to the

DISSOCIATION DYNAMICS OF POLY-ATOMIC MOLECULES 429

vibrational levels of the ground electronic state or predissociation to a repulsive electronic state, or both. The potential energy surfaces of polyatomic molecules are usually very complex. This complexity opens many pathways for energy flow from the initial excitation to reaction products. Crossing from a completely bound potential surface to one with an energetically accessible exit valley may occur. A molecule with sufficient energy to dissociate may execute many vibrations before passing through an exit valley of the surface to fragments. Molecular fragments may be vibrationally, rotationally, and electronically excited. How dynamical information may be derived from high-resolution photofragment spectroscopy has been reviewed (Leone, 1982; Simons, 1984). One of the best documented cases of unimolecular dissociation through an intermediate electronic state has been the photofragmentation of formaldehyde, H,CO to H, and CO (Moore and Weisshaar, 1983). In these experiments, the formaldehyde molecule is excited to a bound singlet state, which then internally converts to a highly vibrationally excited ground singlet state by the intramolecular conversion of electronic to vibrational energy. This highly excited molecule then vibrates until nearly all of its energy appears in the dissociation coordinate to bring the molecule to a transition state. As the fragments push off from each other, energy is released; this energy then appears in the fragments. An experiment on the order-of-magnitude variation in the unimolecular dissociation rates of rovibrational states within 0.2 cm- has been reported, demonstrating significant differences among neighbouring states (Guyer et al., 1986).

A similar intramolecular electronic to vibrational energy conversion is implicated in the bond-selective photochemistry of CH,BrI (Butler et al., 1986b, 1987). When the molecule is excited with 210 nm photons, selective breaking of the stronger C-Br bond results without any fission of the weaker C-I bond. In contrast to vibrational states, pathways following the electronic excitation of a molecule are critically sensitive to the nature of the initial excitation (Levy and Simons, 1975). It is perhaps not surprising that such reaction specificity has been achieved with electronic excitation. It is again mostly in the context of electronic excitation that experimental control of reaction products has been proposed (Shapiro and Brumer, 1986; Tannor et al., 1986), though some theoretical schemes for vibrational mode-specificity have also been reported (Tannor et al., 1984) and examined (Lami, 1987).

A study of vibration-rotation excitation of CN from the photofragmentation of nitrosyl cyanide, NCNO, has been carried out by Wittig and colleagues (Nadler et al., 1985; Qian et al., 1985). Statistical theories are able to match the experimental results up to a certain photon energy; however, when it becomes possible to produce vibrationally excited CN, phase space theory does not reproduce the experimental distributions. The vibrational

430 T. Uzer

excitation of CN is matched by a statistical calculation that distributes the excess energy among the six vibrations of a very loose NCNO transition state, instead of all the nine vibrational-rotational degrees of freedom.

A striking visualization of the dissociation process has been achieved by Imre et al. (1984), who infer the nuclear motion on the dissociative surface from the product energy distribution: they observe small molecules spectro- scopically during the half-vibrational period that it takes the molecule to dissociate. (Their article is also a prominent example of the application of Heller’s wave packet methods to molecular dissociation.) They excite the molecule to a dissociative electronic state. The multidimensional potential surface for such a molecule will have bound motion for all vibrations except the reaction coordinate. Although the molecule dissociates in half a vibrational period (10- l 4 seconds), there is a very small but observable probability of emission. According to the Franck-Condon principle, the emission takes place vertically to increasingly highly excited vibrational levels of the ground electronic state, thereby reflecting the change in the nuclear geometry as the excited state evolves into products. The utility and elegance of this technique has been demonstrated with spectra of methyl iodide and ozone excited to a dissociative continuum. Along with the information on the excited state potential, this technique provides spectra of very highly vibrationally excited levels of the ground electronic state that should be of great value in studying IVR.

Time-resolved studies of the photofragmentation dynamics offer new opportunities for the direct viewing of the bond fission process and its dependence on the transition state, and the internal states of the products. Prominent among these are the femtosecond photofragmentation spectroscopy experiments of the Zewail group on ICN (Scherer et al., 1985) and the measurement of state-to-state reaction rates for ICN and NCNO, nitrosyl cyanide (Knee et al., 1985), and nitrosyl cyanide (Knee et al., 1985; Khundkar et al., 1987). High-resolution spectroscopic studies of molecular dynamics have been reviewed by Hirota and Kawaguchi (1985).

One of the reasons for the intense study of state-to-state unimolecular reactions was the hope for mode-specific laser chemistry; that is, to be able to influence chemical reactions by experimental means (Bloembergen and Zewail, 1984). This prospect still looms far on the horizon, simply because numerous and intricate couplings redistribute vibrational energy efficiently even in molecules as seemingly uncomplicated as hydrogen peroxide. Thus, most of the assumptions of statistical theories hold, and it is very difficult to hamper or channel energy flow inside molecules. There is some reason for optimism when electronic excitation is involved. Barring that, most of the studies today are done with a view to understanding the dissociation process more thoroughly.

DISSOCIATION DYNAMICS OF POLYATOMIC MOLECULES 43 1

VIII. Unimolecular Quantum Dynamics and Molecular Dissociation

Unimolecular dynamics is a field distinct from unimolecular reaction rate theory because it assumes that the intramolecular dynamics of the molecule and its state structure is reflected, in a measurable way, in the state-to-state information experimenters are generating. The tremendous interest in intramolecular energy redistribution (IVR) has spurred efforts to understand the connections between the classical dynamics of coupled oscillator systems, the wavefunctions of metastable states representing highly vibrational states of molecules, and their decay properties. The structure of phase space is being examined in increasing detail for both model systems and systems that closely correspond to molecules (e.g., Kato, 1985). The classical, quantal, and statistical behavior of dissociating model triatomics has been examined by Rai and Kay (1984). The unimolecular decay corresponding to a “regular” (as opposed to “chaotic”) state of a metastable molecule can be described as the dissociation of a Feshbach (compound state) resonance. These states decay exponentially, but neither the pattern of their decay rates nor their values agree with the predictions of RRKM theory. The existence of Feshbach resonances in the unimolecular decay of many van der Waals complexes is well documented (see Hase, 1986 for references). For quantal calculation of these resonant states see Waite and Miller (1980, 1981, 1982); Waite et al. (1983), Hedges and Reinhardt (1983), Bai et al. (1983), Basilevsky and Ryaboy (1983), Christoffel and Bowman (1982, 1983), Chuljian et al., (1984), Moiseyev and Bar-Adon (1 984), Skodje et al. (1 984a,b), Swamy et al. (1986) The work of Dai et al. (1985) indicates the possibility of Feshbach resonances in the unimolecular decay of formaldehyde, and it is surmised (Hase, 1986) that the decay of formyl (HCO) and hydroperoxl (HO,) radicals will involve Feshbach resonances also. Complex coordinate methods, reviewed by Reinhardt (1982), have proven very useful in these calculations.

IX. Concluding Remarks

A very promising venue of research, which nevertheless is somewhat peripheral to the theme of this review, is work on probing and manipulating the transition state. The role of initial conditions in elementary gas processes involving intermediate complexes has been reviewed by Buelow et al. (1986), and the reader is referred to that article for an exposition of the experiments of the Wittig group, who successfully manipulate the transition state by

432 T. Uzer

changing the alignment and orientation of the reagents, as well as references to Brooks' pioneering work in this area.

The emission spectroscopy of simple, unstable species with lifetimes shorter than a picosecond has a distinguished past (Polanyi, 1987). Early attempts to study transition states in absorption have been reviewed by Brooks et a!. (1982). Radiation emitted during collisions of He' with neon has been observed (Johnsen, 1983), and the spectrum of this radiative charge transfer process has been calculated (Cooper et al., 1984). The potential utility of radiation emitted during collisions as a noninvasive diagnostic probe of reaction mechanisms has yet to be realized fully. The aim in studying transition states that are formed as a result of collisions of reagents or of optical excitation is to analyze the spectra for clues about the dynamics of the reaction from the transition state to the products. The dissociation again takes place in about one vibrational period, and it is the emission during this period that is the subject of this novel spectroscopy. The process is again intertwined with two other fundamental processes with their own intricacies: the intramolecular energy flow that converts the energized molecules into transition states, and the molecular processes that convert transition states into products.

The link between the dynamics of reactive full collisions and the dynamics of half collision events is the outcome of the experimental and theoretical efforts of the last decade. The increasing state- and time-resolution of current experiments is enabling scientists to probe the molecular dissociation process in unprecedented detail, and developments in the classical and quantum theory of nonlinear systems are establishing a connection between experiment and theory in regions of excitation never treated before. The most apparent trend in the study of molecular dissociation is the ever-closer collaboration between quantum theorists, molecular reaction dynamicists, and laser spectroscopists to take on the central challenge of low-energy physics; namely, to unravel the dynamics of molecular dissociation.

ACKNOWLEDGMENTS

First and foremost, I would like to express my gratitude to Alexander Dalgarno for introducing me to the subject of this review. His understanding of atomic and molecular processes, scientific discrimination, and his foresight have affected profoundly everyone who came in contact with him. I am also grateful to M. R. Flannery and D. M. Wardlaw for their valuable comments on the manuscript. As a perusal of the sparse list of references shows, completeness and breadth have been callously abandoned for the sake of brevity, and only a few specific issues and primary references in the vast field of molecular dissociation have been

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touched upon here. Since this review omits much more than it includes, I apologize to those whose work was slighted in the process. The writing of this review was supported by the National Science Foundation through grant CHE86-19298.

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Viswanathan, R., Thompson, D. L., and Raff, L. M. (1984b). J . Chem. Phys. 80,4230. Viswanathan, R., Raff, L. M. and Thompson, D. L. (1985). J . Chem. Phys. 82, 3083. Wagner, A. F. and Wardlaw, D. M. (1988). J. Phys. Chem. In press. Waite, B. A. and Miller, W. H. (1980). J. Chem. Phys. 73, 3713. Waite, B. A. and Miller, W. H. (1981). J. Chem. Phys. 74, 3910. Waite, B. A. and Miller, W. H. (1982). J . Chem. Phys. 76, 2412. Waite, B. A., Gray, S. K., and Miller, W. H. (1983). J . Chem. Phys. 78, 259. Wardlaw, D. M. and Marcus, R. A. (1984), Chem. Phys. Lett. 110,230. Wardlaw, D. M. and Marcus, R. A. (1985). J . Chem. Phys. 83, 3462. Wardlaw, D. M. and Marcus, R. A. (1986). J. Phys. Chem. 90, 5383. Wardlaw, D. M. and Marcus, R. A. (1987). Ado. Chem. Phys. 70, 231. Wigner, E. P. (1932). Phys. Reo. 40, 749. Wittig, C., Nadler, I., Reisler, H., Noble, M., Catanzarite, J., and Radhakrishnan, G. (1985). J .

Wittig, C., Nadler, I., Reisler, H., Noble, M., Catanzarite, J., and Radhakrishnan, G. (1986). J.

Yau, A. W. and Pritchard, H. 0. (1979a). Can. J . Chem. 57, 1723. Yau, A. W. and Pritchard, H. 0. (1979b). Can. J. Chem. 57, 2458.

Chem. Phys. 83, 5581.

Chem. Phys. 85, 1710.

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 25 I1 PHOTODISSOCIATION PROCESSES IN DIATOMIC MOLECULES OF ASTROPHYSICAL INTEREST KATE P. KIRBY Harvard-Smithsoninn Center for Astrophysics Camhridqe, Massachusetts

EWINE F. V A N DISHOECK Princeton University Observatory Princeton, New Jersey

I. Introduction . . . . . . . . . , . . . ,

A. Historical Perspective . . . . . . . . . B. Photodissociation Mechanisms . . , . . ,

11. Direct Photodissociation . . . . . , . . . A. Quantum Mechanical Formulation , . . ,

B. Examples of Direct Photodissociation. . . . 111. Spontaneous Radiative Dissociation . , . . . IV. Predissociation . . . . . . . . . , . . .

A. Quantum Mechanical Description . , . . . B. Examples of Predissociation . . . . . . .

V. Coupled States Photodissociation . . . , . . A. Quantum Mechanical Formulation. . , . . B. Examples of Coupled States Photodissociation

VI. Near-Threshold Photodissociation , . . . . . A. Introduction . . . . . . , . . . . . B. Examples of Near-Threshold Photodissociation

VII. Concluding Remarks . . . . . , . . , . . Acknowledgments . . . . . . , . . , . . References. . . . . . . . . . , . . , .

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437 438 439 442 442 445 453 456 456 459 464 464 466 469 469 410 473 473 473

I. Introduction

The process of molecular photodissociation or “half-collision events” is currently a very active area for study. A great many elegant experiments and theoretical techniques have been developed to examine various aspects of this process, and it is impossible to do justice to all of them in this chapter. In limiting this discussion to photodissociation processes in diatomic molecules



438 Kate P . Kirby and Ewine F . van Dishoeck

that are of interest in astrophysics (atmospheric physics is included under this more cosmic umbrella), we have chosen to focus on an area in which theoretical ab initio calculations have provided most of the available data. Indeed, the recognition by Bates, Dalgarno, and others that molecular photodissociation processes are important to an understanding of astrophysical environments has provided a major impetus for the theoretical work in this area. The reader is referred to the excellent papers by Ashfold et al. (1979), Leone (1982), Helm (1984), and Simons (1984) for reviews of experimental work on the photodissociation of small molecules. Overviews of theoretical calculations on the photodissociation of small polyatomic molecules include those by Gelbart (1977), Freed and Band (1977), and Shapiro and Bersohn (1982).

The interstellar medium is the ideal laboratory for the quantum chemist, for it offers an environment in which the two-body processes that the theorist can attempt to quantify occur, and in which the three-body processes that often dominate in the terrestrial laboratory do not. The most abundant neutral molecule, H,, is also the simplest, and the vast majority of other. molecular species are made up of first-row atoms. Many of these molecules are difficult to produce and their spectra are difficult to measure in the laboratory; one must turn to the theorist for quantitative explanations and predictions.

A. HISTORICAL PERSPECTIVE

The 1920s and 1930s were fertile years in the development of both aspects of the present topic: first, the recognition of the importance of photodissociation and its inverse process, radiative association, in astrophysical and spheric applications; and second, the first accurate quantum mechanical treatment of a continuum process. As early as 1926, Eddington had suggested that molecules, if they existed in the interstellar medium, might be destroyed by photodissociation, and estimated a lifetime of 500 years for such species-a value that is now known to be fortuitously correct for many interstellar molecules (e.g., Solomon and Klemperer, 1972; van Dishoeck, 1987a; 1988). In cometary atmospheres, Wurm (1934, 1935) suggested that photodissociation was responsible for the production of many of the free radicals observed and noted that considerable velocities would be imparted to the fragments in the process. In 1942 Swings suggested that the process of radiative association was a source of interstellar molecules; Kramers and ter Haar (1946) developed a model for the interstellar medium in which specific molecular formation rates were equated to the destruction rates, but they assumed photodissociation took place only through transitions to repulsive

PHOTODISSOCIATION PROCESSES 439

potential curves. Bates and Spitzer (1951) were able to improve this model greatly by obtaining more reliable rate coefficients and allowing for the then known interstellar molecules CH and CH’ to photodissociate through bound electronic states as well.

There was also a great deal of interest in the same period in continuum spectra (e.g., Finkelnburg, 1938; Hogness and Franck, 1927; von Wartenberg et a/., 1931; Stueckelberg, 1932; Gibson et al., 1933). The existence of the continua showed for the doubting physicist that the repulsive molecular states that arose in quantum mechanics were “just as real as the stable molecular states” (Herzberg, 1950). With the recent publication of transition probabilities of the a 3 Z l - b3Z; system in molecular hydrogen by Kwok et al. (1 986), one of the first bound-continuum problems in molecular physics, investigated in accurate numerical detail in the years 1927-1939 (Winans and Stueckelberg, 1928; James et al., 1936; Coolidge et al., 1936; James and Coolidge, 1939), has been revisited.

B. PHOTODISSOCIATION MECHANISMS

Photodissociation of a molecule can proceed in several ways. These are illustrated in Fig. 1 a-d with schematic (Born-Oppenheimer) potential energy curves of a diatomic molecule, AB. Unless specified otherwise, we will assume absorption of radiation originating from the lowest rotational level of the u = 0 vibrational level of the ground electronic state of the molecule.

1. Direct Photodissociation

The simplest dissociation process is through direct absorption into a repulsive upper state as shown in Fig. la. This absorption also may take place onto the repulsive wall of a bound excited electronic state (not shown). As spontaneous emission back to the ground state is relatively slow compared to the time frame for movement along the nuclear coordinate, all absorptions lead to dissociation of the molecule. The photodissociation cross section is continuous as a function of photon energy, and its energy dependence is governed to first approximation by the Franck-Condon principle in that its maximum value is at the vertical excitation energy indicated by the arrow in Fig. la. This is the predominant photodissociation pathway of molecules such as CH+, OH’, and NH.

2. Indirect Photodissociation Processes

In contrast to direct photodissociation, which involves continuous absorption and therefore can occur over a range of wavelengths, the indirect photodissociation mechanisms each involve discrete transitions to bound

440

t 0 LI W 2 W

Kate P. Kirby and Ewine F . van Dishoeck

( C )

A"+ B

A + B

( b '

R

FIG. 1. Electronic potential energies E, , as functions of internuclear distance R illustrating various processes of photodissociation for a diatomic molecule AB. (a) direct photodissociation; (b) predissociation; (c) photodissociation through coupled states; and (d) spontaneous radiative dissociation (from van Dishoeck, 1987a).

vibrational levels of an excited electronic state as a first step. This has profound implications for the transfer of radiation because line absorption can be saturated much more readily than continuous absorption. Thus, molecules lying at greater column depths into an interstellar cloud see greatly reduced fluxes at the line wavelengths, and are effectively shielded from dissociation. Whereas the photodissociation cross section of a molecule and its isotopes will be very similar in the direct process, substantial differences may be observed for the isotopes in the indirect mechanisms due to sensitivities in the coupling matrix elements to small shifts in the vibrational energy levels.

a. Predissociation. In this case, the bound levels of the excited electronic state are coupled to the vibrational continuum of a third state of different

PHOTODISSOCIATION PROCESSES 44 1

symmetry. This third state usually crosses the excited electronic state within the adiabatic Born-Oppenheimer approximation, as shown in Fig. lb. Transition to the dissociating state occurs without emission of radiation, and can in most cases be described by first order perturbation theory. This mechanism is thought to be the predominant way of photodissociating interstellar CO and HCl. The spectral signature of this process appears as broadening of the discrete absorption peaks due to the interaction.

b. Coupled states photodissociation. In this mechanism, the vibrational levels of the excited electronic state are affected by the continuum of a repulsive state of the same symmetry which does not cross the bound excited state, as shown in Fig. lc. The interaction is often strong and requires a coupled states description of the process. The spectral features in this case vary depending on the strength of the coupling and the relative sizes of the transition dipole moments involved, but they may consist, for instance, of a broad continuous absorption background on which is superposed a series of resonances. This mechanism appears to play an important role in the photodissociation of CH and OH in interstellar clouds.

c. Spontaneous radiative dissociation. In this process, spontaneous emission of radiation occurs as the molecule makes a transition from the bound vibrational levels of the excited electronic state into the vibrational continuum of either the ground state (as shown in Fig. Id) or a lower lying repulsive state, or into the predissociating level of a lower bound state. The radiation emitted in the process appears as a series of peaks, broadened to some degree by rotational motion, and varying in appearance and extent depending on the number of vibrational levels of the bound excited state that have been populated. The photodissociation of H, in the interstellar medium takes place through this mechanism.

3. Astrophysical Considerations

All the above photodissociation processes are more than just mechanisms to destroy a molecule. They can be a significant source of opacity-both line and continuum-thereby modifying the radiation field and hence, the physical characteristics of an astrophysical environment. In producing atoms with significant kinetic energy, these processes heat the ambient gas. In some cases, electronically excited fragments may be formed, which then radiate producing characteristic emission spectra. In any astrophysical environment, the effectiveness of the various photodissociation mechanisms depends on the characteristics of the radiation source. Photodissociation channels with large cross sections, but at energies for which the photon flux is small, may be less

442 Kate P. Kirby and Ewine F . van Dishoeck

significant than channels with smaller cross sections at the peak of the photon flux. For example, the solar radiation field, which is responsible for the photodissociation of molecules in cometary and planetary atmospheres, has its peak intensity at visible wavelengths and decreases rapidly in strength towards shorter wavelengths. At 5000 A, its intensity is five orders of magnitude larger than that at 1500 A. In contrast, the interstellar radiation field varies more gradually, and has substantial intensity at (vacuum) ultraviolet wavelengths down to the cut-off wavelength of 912 A (13.6 eV), known as the Lyman limit. Therefore, only the lowest lying photodissociation channels are effective in the photodissociation of a cometary molecule, whereas higher lying channels often dominate the photodissociation of the same molecule in an interstellar environment. The relative importance of different channels may also change with depth in an interstellar cloud or planetary atmosphere due to the modification of the radiation field by grains and/or the ambient gas.

11. Direct Photodissociation

A. QUANTUM MECHANICAL FORMULATION

In the Born-Oppenheimer approximation, the wavefunction of a diatomic molecule is separable into a product of two functions, one of which, $JrIR), depends only on the electron coordinates r and, parametrically, on the internuclear separation, R, and the other function, tnUc(R), which describes the nuclear motion:

V(r, R ) = $eLrlR)<nuc(Rh (1)

where R = (R , d,+) is the internuclear vector. The electronic wavefunction is a solution of the fixed nuclei electronic energy eigenvalue equation, which is solved to varying degrees of approximation using standard ab initio techniques such as configuration interaction (see, for example, Hirst, 1982; Bruna and Peyerimhoff, 1987). The energy eigenvalue E,,(R), obtained at a number of internuclear distances, describes the electronic energy dependence on R and forms what is called a potential energy curve. tnuc(R) is a product of a rotational part OhJMJ(O, +), which is a spherical top eigenfunction, and a vibrational part, xuj(R)/R. For an electronic state with an appreciable potential well, the nuclear motion is bound, and the vibrational wavefunction is the solution of the radial nuclear eigenvalue equation


in which p is the reduced mass of the system, v and J are the vibrational and rotational quantum numbers, and E,, , sometimes written as Kl, functions as a potential within which the nuclei vibrate. Atomic units are used throughout, in which e = me = h = 1. This second-order differential equation can be solved exactly for the energy eigenvalues E,, by standard Numerov techniques (Cooley, 1961; Blatt, 1967; Johnson, 1977). The greatest limitation to accuracy is in the electronic potential energy, E,,(R). The bound vibrational functions are normalized to unity, and vanish at zero and infinity.

In the case that an electronic state has no potential well or that the energy regime of interest lies above the dissociation limit of an attractive potential energy curve, the radial nuclear Schrodinger equation to be solved is

where Ek = k2/2p is the relative kinetic energy of the dissociating atomic fragments. The continuum wavefunctions Xk,(R) behave asymptotically as

xkJ(R) - (g)’’’ sin(kR + 7) (4)

where 7 is an elastic scattering phase shift. In practice, the rotation of the molecule is usually ignored, and J is set equal to A, the quantum number corresponding to the projection of the total electronic orbital angular momentum along the internuclear axis.

For a discrete electronic transition from an initial vibrational level u” of the ground state i to a vibrational level u’ of an electronically excited state f, the band absorption oscillator strength is given by

fu *u ,s = 3s A E u ’ u , , I < ~ u r ( R > I Dfi (R> I xu, , (R>) I 2 ( 5 )

where g is a degeneracy factor equal to ( 2 - dO,A!+A?!)/(2 - and AEvru,, is the transition energy. The integration in Eq. ( 5 ) is over the internuclear distance R, where the vibrational wavefunctions are solutions of Eq. (2) .

The electric dipole transition moment function Dfi (R) , which is usually responsible for the absorption from the initial into the final electronic state, is

If the z-axis lies along the internuclear axis, the operator d = C j ejrj simplifies in atomic units to - c j z j for transitions between states with the same A quantum number, and to -(l/,/2)Cj ( x j + i y j ) for transitions in which A changes by f 1. Transition operators other than the electric dipole operator coupling states with different selection rules may become significant in special cases and will be mentioned in the next section.

444 Kate P . Kirby and Ewine F. van Dishoeck

Similarly, the direct photodissociation cross section for absorption into the vibrational continuum of electronic state f can be written (Allison and Dalgarno, 1969)

where Xk!(R) is a solution of Eq. ( 3 ) at energy E,. = AE,.,.. - ( E f ( R + co) - Eu,,). If both the matrix element and the transition energy are in atomic units, the factor 2.ne2/3mc has the numerical value 2.69 x lo-’’.

Because the variation of the dipole transition moment with internuclear distance is often not well known, Jarmain and Nicholls (1 964, 1967) used an approximation to the above cross section formula in which Franck-Condon densities, q k ‘ u ” = I ( X k ’ I X u ” ) 1 2 , are defined, and the transition moment at the R-centroid of the transition, Dfi(R,), is taken such that

(8)

Another simplification introduced by Winans and Stueckelberg (1 928) is the ‘‘&function approximation” or the “reflection approximation” in which the continuum wavefunction Xk!(R) is replaced by a &function at the classical turning point.

Allison and Dalgarno (1971) demonstrated a continuity relationship across the dissociation threshold in molecular absorption, relating discrete absorption oscillator strengths to photodissociation cross sections. Using an expression for the differential oscillator strength for absorption into the vibrational continuum.

b(AEkTU,,) = 2.69 x lo-’’ g AE,,,., D;i(Rc)qk,u,, cm’.

they showed by explicit calculations for the B”C: -XIXl transition of molecular hydrogen and the B3Z:;-X3C, transition of molecular oxygen that the discrete values f,.,,.(dv’/dE) lie on a smooth extrapolation of the continuous curve df,,../dE, except for a small region near threshold.

In a radiation field with mean intensity I in photons crn-’s-’ A-1, the direct photodissociation rate due to absorption from initial vibrational level v n is

where o,..(I) is given by o(AE,,,,,) in Eq. (7) and the integration is carried out over the entire range of I for which both cv,, and I are nonzero.


B. EXAMPLES OF DIRECT PHOTODISSOCIATION

Several of the earliest explicit calculations of cross sections for direct photodissociation were performed for astrophysically interesting species. Bates (1952), Buckingham et al. (1952) and Dunn (1968) computed cross sections for absorption into the repulsive 2pa state of the H: ion. Jarmain and Nicholls (1964, 1967) reported cross sections for absorption from the ground state of 0, into the Schumann-Runge and Herzberg continua, and Allison and Dalgarno (1969) studied absorption of vibrationally excited H,, HD, and D, into the continua of the Lyman and Werner systems. In subsequent papers, the direct photodissociation of systems such as NaH and LiH (Kirby and Dalgarno, 1978), MgH (Kirby et al., 1979), C, (Pouilly et al., 1983), OH + (Saxon and Liu, 1986), OH (van Dishoeck and Dalgarno, 1983), CH (van Dishoeck, 1987b), NH (Goldfield and Kirby, 1987), and CN (Lavendy et al., 1987) in the interstellar medium, and in cometary and planetary atmospheres has been studied, The following examples were chosen for discussion because they are particularly illustrative of various aspects of the photodissociation process.

1. CH'

One of the most interesting interstellar chemistry problems concerns the formation and destruction of the CH' ion. Steady state models of the interstellar chemistry based on sequences of gas-phase chemical reactions predict CH' abundances much smaller than those observed (Bates and Spitzer, 1951; Dalgarno, 1976; see also Black, this volume). Elitzur and Watson (1978) proposed that the ion is formed in shock-heated gas, where the temperature is high enough that the endothermic reaction between C +

and H, to form CH' proceeds rapidly. Photodissociation is a significant destruction pathway of the ion, and accurate cross sections are needed to test both the steady state and the shock models.

Figure 2 shows the potential curves relevant to the photodissociation of CH'. Uzer and Dalgarno (1978) calculated cross sections for photodissociation from u" = 0, 1, and 2 for J" = 0 into the vibrational continuum of the A ' I l state. The cross section for the u" = 0 level was found to be very small, 5 x cm2 at threshold, because Franck-Condon factors favor overlap with the bound vibrational levels of the A'H state. Using potential curves obtained from experiment (Helm et al., 1982), Graff and Moseley (1984) repeated the calculation of the direct photodissociation and obtained a peak value of only 7 x cm2 at the threshold of 3033 A. The difference in these two values illustrates the extraordinary sensitivity of the cross section to


12 -

10 -

8 -

h

Y 2 6- W

4 -

2 -

FIG. 2. Potential energy curves of the CH' ion.

the potential curves, in particular, the form of the A'll inner wall near threshold. Interesting threshold effects have been observed in photofragment spectroscopy experiments on CH' which will be discussed in Section VI.B.2.

The sensitivity of the continuum cross section to the shape of the potential curves is also illustrated by a comparison of the A'H-X'X+ photodissociation processes for CH ' and the isovalent SiH+ ion. The potential energy curves for these two systems are presented in Fig. 3. In SiH+, the energy minimum of the A'l l state is shallower and shifted to larger distances, so that the overlaps between the ground state vibrational wavefunction and the excited state continuum wavefunctions are greatly increased compared with the CH' case. The SiH' A'll-X'X' (u" = 0) cross section has a peak value of 2.6 x lo-'' cmz just above the threshold of 3700 8, (Kirby and Singh, 1983), almost four orders of magnitude larger than the CH + continuum cross section. Thus, for SiH+, the bulk of the oscillator strength associated with the A-X system is in the continuum, whereas for CH' it is mostly in the discrete transitions.


8 I I 1

6 - -

I I 1 I

4-

2 -

0 I

R ( a o l R (ao)

Comparison of the X'Z+ and A'I l potential energy curves of CH' and SiH' FIG. 3.

At energies above 8 eV, other 'Z' and ln photodissociation channels become accessible (see Fig. 2). Saxon et al. (1980) computed a number of these CH ' potential curves, as well as transition moments connecting them with the ground state. They found three states accessible with photon energies < 13.6 eV and unbound with respect to nuclear motion. The largest cross section for direct photodissociation, shown in Fig. 4, comes from the 3lZ' state, which peaks at 7 ~ 3 x cm2 at 12.6 eV (Kirby et al., 1980). The shape of the cross section reflects the shape of the v" = 0 vibrational wavefunction of the initial state. Dissociation through the 2 lZ+ , 3 lZ+, and 2ll-I states produces neutral carbon in excited electronic states, C('D) and C('S), and protons. The products of dissociation through the A ' n state are C'('P) and H. These different products can have an important effect on the chemistry in shocked interstellar regions. In the unshielded interstellar radiation field (Draine (1978) for h ~ 2 3 0 0 A and Witt and Johnson (1973) for 1 > 2300 A), the photodissociation rate through these higher lying channels is 3.2 x lo-'' s-' (Kirby, 1980), whereas through the A'H state it is only 1 x s - l (Graff and Moseley, 1984). Excited neutral carbon atoms and protons are therefore the dominant photodissociation products in the interstellar medium. As the depth into an interstellar cloud increases, the shorter wavelength radiation is absorbed preferentially by the grains, so that the A ' n channel becomes relatively more important.

2. HCl

Although there have been numerous experimental and theoretical studies of photodissociation of neutral diatomics, only in the case of HC1 has there


40 r I 1 I I 1 I I

hu (eV)

FIG. 4. Computed photodissociation cross section for absorption from the LJ" = 0 level of the CH' X'E' state into the 3lE' state as a function of incident photon energy (from Kirby et al., 1980).

been an opportunity to make a quantitative comparison of measured versus theoretically predicted cross sections. The neutral fragments resulting from photodissociation are difficult to identify unless they are electronically excited and therefore emit radiation. Rarely is there a situation as there is in HCl in which one photodissociation channel can clearly be isolated from other channels. As shown in Fig. 5, the low-lying A 1 l l state of HCl which separates to ground state atoms is purely repulsive, and absorption in the wavelength range 1400-2100 A is due entirely to transitions from the X'X' ground state into this dissociating state. HCl is a constitutent of the atmospheres of the planets Earth and Venus, and is known to exist in interstellar clouds. A good knowledge of its photodissociation processes is important in understanding the observed abundances.

Potential curves and electric dipole transition moments relevant to the photodissociation of HCl have been calculated by van Dishoeck et al. (1982). Cross sections for direct photodissociation from the u" = 0 level of the ground state through the A'II state are shown in Fig. 6, together with the experimental measurements of Inn (1975). Below a photon energy of

PHOTODISSOCIATION PROCESSES

r I I I I I 1 I

0

+ t

4 6

449

R(o,)

FIG. 5. Potential energy curves of the HCI molecule.

~ 8 . 5 eV, excellent agreement is obtained between theory and experiment. At higher energies, small discrepancies between theory and experiment exist. They may be removed by slightly modifying the potential curves and transition moment function (Givertz and Balint-Kurti, 1986), but such ad hoc adjustments would have to be justified by more extensive quantum chemical calculations.

3. 0,

Because of the importance of oxygen chemistry in any understanding of the terrestrial atmosphere (Nicolet, 1981), experimental and theoretical studies of the photodissociation of molecular oxygen abound and to review them all fairly might well take an entire book. Good summaries of the spectroscopic characteristics of O2 can be found in Huber and Herzberg (1979) and Krupenie ( 1 972). In this section, we will briefly discuss several continua that


I I I

6 7 8 9 10

E ( e V ) FIG. 6. HCI photodissociation cross sections as functions of incident photon energy for

absorption into the All7 state from the X'C' (u" = 0) state. The full, long-dashed and short-dashed curves refer to various calculations described by van Dishoeck et al. (1982). The open circles are the measured values of Inn (1975).

are important in producing oxygen atoms, both O(3P) and O('D). The potential curves relevant to this discussion are shown in Fig. 7.

The absorption of ultraviolet radiation by molecular oxygen in the wavelength range 1270-1750 A occurs primarily in the Schumann-Runge (S-R) continuum, adjoining the progression of Schumann-Runge bands produced in transitions from the ground X3X; state, to the excited B3X; state. The B3Cu--X3C:, transition is electric dipole allowed and it was noted early on that significant oscillator strength, f z 0.2, was associated with this continuum (Ladenburg et al., 1932). Extensive work on the discrete absorption bands of this system has been carried out by Yoshino et al. (1984, 1987)

Ii

a

IC

8

- > w 2 6

4

2

0


1 I I I I

45 1

I 2 3 4 5 R ( a , )

FIG. 7. Potential energy curves of the O2 molecule

and improved spectroscopic constants for the B3C; state have been obtained (Cheung et al., 1986a). Substantial predissociation occurs in the S-R band system, but this will be discussed in Section IV.B.2. An important aspect of dissociation through the S-R continuum is production of O( 'D), which radiates at 6300 A giving rise to the well-known oxygen red line seen in both the airglow and aurorae.

One of the first detailed theoretical studies of the 0, S-R continuum was reported by Jarmain and Nicholls (1964). Using empirical Klein-Dunham potential curves in the calculation of the relevant bound and continuum wavefunctions and Franck-Condon densities, they deduced the variation of the B-X transition dipole moment function from measured values of the absorption coefficient for 0, between 7 and 10 eV. Allison et al. (1971) used R-centroids to derive the transition moment as a function of nuclear separation for R > 2 . 2 ~ ~ and obtained good agreement with the gross experimental features of the continuous cross section. Ab initio calculations (Buenker and Peyerimhoff, 1975; Buenker et al., 1976; Yoshimine et al., 1976; Allison et al., 1986), which take account of valence-Rydberg mixing in the

452 Kate P. Kirby and Ewine F. uan Dishoeck

excited state wavefunction, give evidence of an avoided crossing with the 23C; state in the region near the Re of the ground state. This change of character in the B state wavefunction causes a rapid decrease in the B-X transition dipole moment function at small internuclear distances, a feature that is difficult to deduce from continuum measurements.

Photoabsorption measurements such as those of Ogawa and Ogawa (1975) show some structure below approximately 1450 A which cannot be explained by dissociation through the B3X; state alone. The production of O( ‘D) has been used by Lee et al. (1977) to distinguish dissociation through the B3X; state from that through nearby states, such as the 13rI,, state which dissociates to O(3P) + O(3P) (c$ Fig. 7). Lee et al. (1977) suggested that several of the features in the photoabsorption cross section between 1300 and 1400 A, which appear to be part of the S-R continuum, are due to dissociation through the 13rIu state; this was later confirmed by Allison et al. (1982). Later, Allison et al. (1986) showed that with small adjustments to the ab initio potential curve of the B state and the transition moment function, and with inclusion of the absorption into the lowest 3rIu state, good agreement can be obtained with the experimental cross sections, as is illustrated in Fig. 8.

The Herzberg continuum in molecular oxygen has a threshold of 2410 A and extends to wavelengths less than 1600 A. It consists of a superposition of three continuum transitions that can be ascribed to absorptions from the X3Xg- state to three excited electronic states, A3X:, A’ 3A,,, and clXu-. Because the transitions are spin- and/or spatially forbidden, the continuous absorption is very weak, peaking around 1970 A at a value of n w 7 x cm2 (Cheung et al., 1984 and 1986b, containing references to many measurements). Despite the small magnitude of the cross section, the Herzberg continuum plays an important role in the oxygen chemistry of the Earth’s atmosphere because the solar flux is large in the relevant wavelength region, and the absorption cross section of the major atmospheric absorber, ozone, passes through a minimum near 2020 A. The continuum absorption in the 0, Herzberg system produces oxygen atoms, O(3P), that are needed in the three-body recombination with 0, to form ozone. The calculation of the transition moment for a spin-forbidden transition between electronic states i and f requires inclusion of the spin-orbit interaction, which couples intermediate electronic states from which the intensity is “borrowed.” Saxon and Slanger (1986) have used the transition moments computed by Klotz and Peyerimhoff (1 986) to calculate the absorption cross sections as functions of wavelength. The A3X;-X3X; transition contributes the largest cross section by almost a factor of five, whereras the c’Xu--X3Xg- contribution is almost negligible. When scaled by a small factor, the cross sections of Saxon and Slanger (1986) compare well with the measured and calculated cross sections of Cheung et al. (1986b).


O2 SCHUMANN RUNGE CONTINUUM

I 6 0 1 -1

I25 I30 135 140 145 150

X (nm)

FIG. 8. 0, photodissociation cross sections as functions of photon wavelength for absorption into the Schumann-Runge continuum. Full line: experimental results of Ogawa and Ogawa (1975); Dotted line: best calculation of Allison et al. (1986); dashed line: 1311-X3Z; cross section added to the B3Z; -X3Zq- cross section.

Another continuum cross section in molecular oxygen, recently studied by Dalgarno and McElroy (1986) as well as Saxon and Slanger (1986), is the Af3Au-a1AQ system. The resulting cross section, obtained with the transition moment of Klotz and Peyerimhoff (1986), has a maximum value of 0 x 1 x

cm2 at 2600 A, several orders of magnitude too small to contribute significantly to production of O(3P) in the stratosphere.

111. Spontaneous Radiative Dissociation

The best-known example of spontaneous radiative dissociation, but with the upper state excited by electron impact rather than photon absorption, is the continuous spectrum observed in electrical discharges containing H,. The


transitions take place from discrete vibrational levels of the a3C: state to the vibrational continuum of the repulsive b3X: state, producing a spectrum extending from the extreme ultraviolet into the visible region. The inverse process, absorption from the b3C: by colliding hydrogen atoms, contributes to the opacity in stellar atmospheres. Using variationally determined wavefunctions and potential energy curves for both states (Coolidge and James, 1938), James and Coolidge (1939) calculated the electric dipole transition moment as a function of internuclear distance and integrated over the bound and continuum vibrational wavefunctions. Their results for the transition probabilities for several vibrational levels of the a3Xl state are in excellent agreement with Kwok et al. (1986), who greatly extended the range of vibrational levels studied.

The same process, occurring in the singlet manifold of H, and originating with a photon absorption from the X'C; ground state, is the primary photodestruction mechanism for molecular hydrogen in the interstellar medium. The process is illustrated in Fig. 9 and takes place through a series of discrete absorptions into vibrational levels of the B'C: and C'n, states, corresponding to the Lyman and Werner systems, respectively. Transitions from vibrational levels of these excited electronic states take place to the vibrational continuum of the ground state. Dalgarno and Stephens (1970) developed the proper quanta1 description of this bound-free emission process, in which the total probability for radiative decay into the continuum from a particular vibrational level, u', is given as

A,. = jOm W(E)dE s- '

where

Dalgarno and Stephens performed accurate calculations of the radiative transition probabilities, A",, into both the discrete and continuum vibrational levels of the ground state and showed that, for a uniform radiation field with a cut-off at the Lyman limit of 912 A, the fraction of absorptions that lead to dissociation is 22.9% in the Lyman system and 8 x lo-'% in the Werner system. In realistic interstellar radiation fields, the average dissociation fraction through both systems is about 11 %.

Concurrently, Herzberg observed an interesting continuous emission spectrum in discharges of H, and D, in the region 1200-1600 A, underlying the Lyman bands. Dalgarno, Herzberg, and Stephens (1970) showed that this continuous spectrum could be quantitatively accounted for by spontaneous radiative dissociation (Stephens and Dalgarno, 1972); both the observed and theoretically calculated spectra show striking agreement as can be seen for


14-

12-

10-

- 2 8 - - W

6 -

4 -

2 -

455

FIG. 9. dissociation.

Potential energy curves of H, illustrating the process of spontaneous radiative

H, in Fig. 10. In D,, the positions and relative intensities of the maxima are very different from those in H,, and again, theory and observation are in harmony, Experimental data for H, have been obtained by No11 and Schmoranzer ( 1987).

Spontaneous radiative dissociation is a source of both continuous luminosity and heat. The kinetic energies of the dissociated pair of hydrogen atoms produced by absorption in the Lyman system range from 0.04 to 1.0 eV with a mean value of 0.25 eV for H, in a uniform radiation field (Stephens and Dalgarno, 1973).

In most molecules, spontaneous radiative dissociation must be considered a possible dissociation channel, but its significance in terms of the total photodissociation process depends on several factors. First, since it is a line absorption process, the species can rapidly become self-shielded. In addition, for absorptions with large oscillator strengths, the most probable radiative


b

A

I

1400 I500 1600 I700 ii

FIG. 10. Fluorescent continuum spectrum emitted by H, in the B'C:-X'Z:: (a) Phot- ometer curves corresponding to 3 different exposure times; (b) Calculated intensity distribution for zero rotation (from Dalgarno et al., 1970).

decay route is radiation back to the bound vibrational levels of the ground state. In transitions to excited states that have small absorption oscillator strengths with the ground state, decay to a lower lying repulsive state that is coupled by a large transition moment to the excited state may be quite favorable. The process is limited in significance, however, by the small absorption oscillator strength from the ground state. These effects have been investigated quantitatively for a number of states in OH by van Dishoeck and Dalgarno (1983) and van Dishoeck et al. (1983).

IV. Predissociation

A. QUANTUM MECHANICAL DESCRIPTION

The processes of predissociation have been characterized, explored, and reviewed in a number of excellent sources (e.g. Fano, 1961; Ben-Aryeh, 1973; Child, 1974), only a few of which will be mentioned in this very simplified


summary. Lefebvre-Brion and Field (1986) distinguish two types of predissociation, “rotational” and “electronic.” Rotational predissociation occurs when a molecule, excited to quasi-bound levels of the centrifugal potential of the rotating molecule, decays by tunneling through the centrifugal rotational barrier. This process is less common than “electronic” predissociation and will be mentioned only in Section VI.B.2 with reference to CH’ threshold effects. In this section, we describe “electronic” predissociation, which is essentially a form of perturbation in which the perturbing electronic state is unbound with respect to nuclear motion. An excellent review of perturbations, predissociation, and their effects on the spectroscopy of diatomic molecules is given by Lefebvre-Brion and Field (1986).

The predissociation process begins with line absorption, usually by an electric dipole transition, into bound vibrational levels of an excited electronic state which is coupled by small terms often neglected in the Hamilto- nian to the vibrational continuum of another electronic state. Hence, there is a competition between radiation back down to the ground state (or lower excited states) and nonradiative decay through dissociation. In the predissociation process, part of the absorbed photon energy is converted into kinetic energy and in some cases, excited electronic energy of the fragment atoms.

One of the primary experimental indicators of a predissociation is spectral line broadening. The usual way of quantifying the predissociation is through measurement of the full width at half maximum of the spectral lines, Tpr. The width of the lines is related to the predissociation lifetime, rPr, by TPr = h/rPr = 5.3 x 10- 1 2 / ~ p r , if Tpr is in cm - ’ and rpr in s. The predissociation rate, kpr, is l/zPr. Radiative lifetimes for electric dipole allowed transitions are normally of the order of lo-’ s. As this lifetime corresponds to a line width of x 5 x cm-’, only predissociations with significantly shorter lifetimes can be observed with standard spectroscopic techniques. Predissociation not only causes line broadening but also level shifts which may be comparable to the line broadening and thus difficult to observe. Weaker predissociations can be detected by the breaking-off of an emission series, and they can be measured by monitoring the lifetimes over a range of vibration-rotation levels. Finally, the presence of predissociations can be inferred from theory if computed radiative lifetimes are significantly longer than measured lifetimes. The various experimental techniques have been summarized in Table 6.1 of Lefebvre-Brion and Field (1986). In order to determine the dissociating state and the coupling matrix element responsible for a predissociation, it is helpful to identify kinetic energies and angular distributions of the fragments with laboratory techniques such as laser photofragment spectroscopy (see, for example, Moseley and Durup, 1980 and Helm, 1984 for reviews).

We now relate the predissociation width and rate to the interaction of bound and continuum molecular states. Fano’s treatment (1961) of predissociation within the framework of configuration interaction is an excellent


theoretical basis from which to begin any detailed study of the process, but cannot be reviewed here due to space limitations. The complete Hamiltonian for a diatomic molecule can be written (Julienne and Krauss, 1975)

(13)

He,(r I R ) is the fixed-nuclei electronic Hamiltonian, which determines the electronic wavefunctions and potential energy curves. TR is the radial part of the kinetic energy operator of the nuclei (c$ Eq. (2) and (3) of Section 11). H,,, is the kinetic energy of the rotating molecule given explicitly by Hougen (1970) but often written as B(R) (J - L - S)’, where J is the total angular momentum vector of the molecule (excluding nuclear spin), and L and S are the total electronic and spin angular momentum vectors, respectively. Ha represents the relativistic part of the Hamiltonian, and H, is the mass polarization operator which is usually neglected. The relativistic terms in H, include the spin-orbit, spin-spin and spin-rotation interactions. The detailed microscopic forms of these operators have been given by Langhoff and Kern (1977), and Lefebvre-Brion and Field (1986). Because many of the terms in H above may be very small, they are often neglected in calculating electronic energies and wavefunctions, and included only afterwards in the framework of first-order perturbation theory. The rate for predissociation of level i, v’, J’ by a final state f, k, J is then given by the Fermi-Wentzel Golden Rule formula (Wentzel, 1927; 1928; Rice, 1933),

H = He,(rlR) + + H,,, + Ha + H,.

271 h kE!J, = - l(yiurJ’(r* R)IHintI y’J.kJ(r, R ) > 1 2 , (14)

where and YfkJ are products of an electronic wavefunction and a bound (or continuum) nuclear function (cf: Eq. (l)), and Hint stands for any of the small operators in H. The vibrational parts xVrJ. and xkJ of the nuclear wave function are solutions to the unperturbed radial Schrodinger Eqs. (2) and (3) given in Section 11. The integral in Eq. (14) over the nuclear coordinates may be extremely sensitive to the relative positions of the bound and dissociating states. A bound state crossed by a repulsive curve is not automatically predissociated. The selection rules and vibration-rotation dependence are determined by the particular coupling operator Hint responsible for the predissociation.

At this point, we distinguish between predissociation caused by states of the same and different symmetry and multiplicity. States of the same symmetry that interact strongly in the adiabatic representation exhibit an avoided crossing and are coupled by the radial component of the nuclear kinetic energy operator. These are discussed specifically in Section V on coupled states photodissociation. The dominant predissociation mechanism discussed here is the coupling of states of different spin multiplicity through


the spin-orbit interaction, the strength of which increases with the nuclear charge, Z , of the atoms.

Because of the competition between radiative and nonradiative (predissociative) decay, it is useful to define an efficiency factor for predissociation,

which lies between zero and unity. Then the photodissociation rate in a radiation field of intensity I in photons cm-2 s-' A- ' due to absorption from a lower level u", J" into an excited state vibrational level u', J' is

S - l ,

where the numerical value of ne2/mc2 is 8.85 x if A is in A.

B. EXAMPLES OF PREDISSOCIATION

Predissociation can be an exceptionally important decay mechanism, especially if direct photodissociation channels are lacking in the relevant wavelength region, or if the predissociation pathways are available at energies where the photon flux of the radiation field is largest. Two examples, 0, and OH, will be discussed in some detail, but we note in passing several other interesting systems in which predissociation is significant.

In HCI, the C'll state is crossed by a repulsive 13X+ state (cf: Fig. 5) and undergoes very efficient predissociation through spin-orbit coupling, with k P r z lOI3 s - ' (van Dishoeck et al., 1982). In an optically thin interstellar cloud, the C state photodissociation rate is more than double that of the rate through the low-lying A ' l l direct channel discussed in Section II.B.2, due to the large C'I'-X'C+ oscillator strength.

For the CH molecule, the u' = 0 and higher levels of the C2X+ state are predissociated fairly efficiently (Hesser and Lutz, 1970; Herzberg and Johns, 1969; Brzozowski et al., 1976). Potential energy curves for CH have been calculated most recently by van Dishoeck (1987b) and are presented in Fig. 11. The state responsible appears to be the B2Z-, again through the spin-orbit operator; the repulsive 1411 state crosses the C2Cf state near its dissociation limit and is therefore not effective in predissociating low-lying vibrational levels (van Dishoeck, 1987b). The C2C + channel dominates the photodissociation of CH in comets. Another interesting predissociation in CH, although not of astrophysical significance, has been observed for the A2A state (Brzozowski et al., 1976). The predissociation is extremely weak

460 Kate P. Kirby and Ewine F. van Dishoeck

12 -

I0 -

a -

- - 2 6 - w

4 -

2 -

0 2 4 6 R (a,)

I

FIG. 11 . Potential energy curves of the CH molecule (from van Dishoeck, 1987).

with kP' -= lo6 s - l , and can only be caused through interaction with the continuum of the X2H state through the rotational part of the nuclear kinetic energy operator (van Dishoeck, 1987b).

A similar situation is encountered for NH, for which potential energy curves have been calculated by Goldfield and Kirby (1 987) and are illustrated in Fig. 12. Weak predissociations have been observed for the low-lying levels v' = 0 and 1 of the A311 state by Smith et al. (1976), and have been ascribed by them to the interaction with the 1%- state. The calculations by Goldfield and Kirby (1987), however, clearly indicate that the 15X- state crosses the A311 state at vibrational levels too high to cause significant predissociation of the lower levels. The only alternative explanation is again through rotational nuclear interaction with the X3Z- state. These findings for CH and NH raise the interesting question as to whether such weak predissociations of excited levels lying just above the ground state dissociation limit are a common phenomenon.

PHOTODISSOCIATION PROCESSES 46 1

12

10

8

- Y 2 6 W

4

2

0 ,

R(a,) FIG. 12. Potential energy curves of the NH molecule (Goldfield and Kirby, 1987).

The major photodestruction mechanism for one of the most ubiquitous interstellar molecules, CO, must be predissociation. There are no repulsive states that dipole-connect to the X'C' ground state within 13.6 eV of the u" = 0 level (Cooper and Kirby, 1987). In transitions to Rydberg states, B'C', C'C', E'II, and higher-lying states, significant predissociation has been observed to occur (Lee and Guest, 1981; Letzelter et al., 1987), leading to a large CO photodissociation rate in interstellar clouds (Viala et al., 1988; van Dishoeck and Black, 1987; 1988). The states responsible for these predissociations and the coupling mechanisms have not yet been identified, however.

I . OH

The lowest energy photodissociation channel for the OH molecule, starting from the lowest rotational level of the X2H u" = 0 state, is through the u' = 2 level of the A2Z' state at 2616 A. The oscillator strength for this transition


has been calculated by Langhoff et al. (1982) to be only 5 x This predissociation channel, however, dominates the photodestruction in comets because of the large solar flux at the longer wavelengths.

Predissociations have been observed to occur for the higher rotational levels of A Z X + u’ = 0 and 1, and for all rotational levels with u’ 2 2 (see, for example, Gaydon and Wolfhard, 1951; Smith et al., 1974; Brzozowski et al., 1978). As can be seen from Fig. 13, the A2C+ state of OH is crossed by three repulsive states, 14Z-, l zX- , and 14n. The strong predissociations for u’ > 4 in both OH and OD have been shown to be caused by spin-orbit interactions with the ‘l-I state (Czarny et al., 1971). The 4111,2-A2X&z spin-orbit matrix element of 47 cm-’ estimated by Czarny et al. assuming pure precession of the relevant orbitals, is in reasonable agreement with the value of 70 cm-’ at the crossing point obtained in subsequent extensive ab initio calculations by Langhoff (1980). Using this coupling matrix element together with the Golden rule formula, satisfactory agreement is obtained between the measured and computed line widths. Similar calculations have been performed by

12 -

10 -

8 -

- 3 6- W Y

4-

-

2- 1 O l I I 1 I I

0 2 4 6 8

FIG. 13. Potential energy curves of the OH molecule.


Czarny et al. for the lower vibrational levels, which are predominantly predissociated by spin-orbit interactions with the a4C- state (Michels and Harris, 1969; Smith et al., 1974).

Sink et al. (1980) used multichannel scattering theory to calculate linewidths and energy level shifts for the v' 2 3 vibrational levels of the A2Z+ state due to the effects of the three repulsive states. They confirmed that the contributions from the three states are additive. Using semiclassical expressions, they also computed predissociation probabilities for the lower levels v' = 0-2 due to the 4C- state. The resulting line widths and level shifts agree well with measured values.

Interesting differences occur in the photodissociation of OH and O D through the A2Ef state. The low4 levels of the v' = 2 state of O D do not undergo significant predissociation (Bergeman et al., 1981), and it is absorptions into the u' = 3 and 4 levels that play a role in the photodissociation of cometary OD. The oscillator strengths for these transitions are even smaller than that for absorption into v' = 2, leading to a substantially longer lifetime of O D in comets compared with OH (van Dishoeck and Dalgarno, 1984).

2. 0,

The transmittance of solar radiation through the upper atmosphere in the 1750-2OOO A wavelength region is strongly affected by the predissociative broadening of the vibration-rotation lines in the Schumann-Runge (S-R) band system of molecular oxygen. The B3Xu--X3E:, transition, the photodissociation continuum of which was discussed in Section II.B.3, continues to be one of the best-studied molecular band systems, both theoretically and experimentally. Murrell and Taylor (1969) showed that a single repulsive curve, crossing the B3Z; state near v' = 4 could account for the observed line widths. Schaefer and Miller (1971) suggested that four repulsive curves, In,,, %,,, TI,,, and 5Cu-, might all contribute to the predissociation of the B state through spin-orbit interaction (cf: Fig. 7).

The most extensive theoretical treatment of the predissociation has been carried out by Julienne and Krauss (1975), who used ab initio calculations to obtain the relevant repulsive curves and spin-orbit matrix elements. They concluded that the dominant state causing the predissociation of most levels is the 5rIu, with the 'rI, and 3rI,, contributing only in a minor way and the 'Xu- not contributing at all. The ab initio results were subsequently used as starting points in the variation of model parameters, such as the coupling matrix elements and the crossing points and slopes of the repulsive curves, to best fit the measured level shifts of Ackerman and Biaume (1970). Satisfactory agreement was obtained for most of the level shifts up to v' = 12, but the measured values for the linewidths for v' = 5 to 11 could not be fully reproduced in this model.


Julienne (1976) investigated the role of another state crossing the B3Z; state. This 23X: state causes negligible level shifts, but affects the line widths of the u’ 2 5 levels significantly. Inclusion of this state brings the theoretical line widths for u’ = 5 to 11 into much better agreement with experiment. The best model parameters for each of the states which cross the B3Z; are listed by Julienne (1976).

In a series of six papers, Lewis et al. (1986a,b, 1987) have reported measurements of oscillator strengths and predissociation line widths with instrumental resolution of 0.05 for the S-R bands of molecular oxygen and its isotopes, l80, and 160180. They derived model parameters analogous to those of Julienne (1976) and found excellent agreement with the observed trends for the variation of the line width with u and J . Their oscillator strengths for bands (1,O) to (12,O) agree well with the uitra-high resolution (0.013 A) measurements of Yoshino et al. (1987) for 1 6 0 2 . Cheung et al. (1988) are completing similar experiments on the isotopes and have obtained more accurate predissociation line widths, stimulating further theoretical work by Dalgarno and Friedman (1987).

V. Coupled states photodissociation

A. Quantum Mechanical Formulation

The coupled states photodissociation mechanism differs from the predissociation mechanisms mentioned above in various details and therefore merits a separate discussion. In this case, the interaction between the adiabatic Born-Oppenheimer states is so strong that a first order perturbation treatment (cJ Eq. (14)) is not applicable, and a coupled states formalism must be applied. An example is provided by a bound and a repulsive state of the same symmetry for which the adiabatic potential energy curves do not cross. If the states undergo an avoided crossing-i.e., if their electronic wavefunctions exchange character-the states will couple strongly through the radial component of the nuclear kinetic energy operator. Since transitions to both states may be electric dipole allowed, additional structure in the cross sections may arise.

The formulation for the process of absorption into two states coupled by the radial component of the nuclear kinetic energy operator has been given by van Dishoeck et al. (1984). The total wavefunction, Y, of the coupled excited states can be expanded in terms of the adiabatic (Born-Oppenheimer) wavefunctions

(17) w, R ) = 1 $iad(ri R)X;~W/R j


where rotation has been neglected. The adiabatic nuclear wavefunctions xj"d satisfy in this case a set of coupled differential equations which can be expressed in matrix form as (Smith, 1969; Heil, Butler and Dalgarno, 1981)

I - 2pVad(R) + B(R) A(A + 1)

R2

d dR

+ 2A(R) - xad(R) = 0

where I is the unit matrix, K2 a diagonal matrix with elements kf = 2p[E - i5j(w)], Vad is a diagonal matrix with the adiabatic potential energies V$(R) as elements, and A and B contain the off-diagonal coupling elements

The nuclear wavefunctions satisfy the boundary conditions specified in Section 1I.A for bound and continuum states, respectively.

Although Eq. (18) can be solved by numerical integration, it is often computationally more convenient to transform first to a diabatic representation in which the matrix of dldR vanishes. A diabatic representation can be obtained from the adiabatic representation by a unitary transformation

Jrd = qPC, with

dC dR - + A C = O ,

where usually the condition C + I at infinity is imposed. In the diabatic representation, the coupled equations for the nuclear motion take the form

I - 2pVd(R) + C-' (23)

d2 A(A + 1) i , . I + K 2 - R2

where V d = C ' V a d C . For a complete basis wd, the relation B = A' + dA/dR holds, so that the last term in Eq. (23) vanishes and the diabatic states are coupled only by the off-diagonal elements of the diabatic potential matrix V d .

The photodissociation cross section for absorption from vibrational level Y" of an uncoupled initial state, i, into the coupled states becomes

C(AEk,", , ) = 2.69 X 9 AEk,",,1E (XjkT(R)IDji(R)IxiVt,(R))I2 Cm2. (24) j


The excited state wavefunction can be expanded either in adiabatic functions (cf: Eq. (17)), or in diabatic functions. The cross section is independent of the representation employed.

B. EXAMPLES OF COUPLED STATES PHOTODISSOCIATION

1. OH

One of the few species for which the process of coupled states photodissociation has been investigated in detail is the astrophysically important molecule OH (van Dishoeck et al., 1984). The adiabatic potential energy curves have already been illustrated in Fig. 13 of Section IV.B.l, where the discussion was centered on the predissociation of the A’X’ state. Although this process is the dominant photodissociation channel of OH in cometary atmospheres, higher excited states become more important in interstellar clouds. In particular, the electric dipole transition moments between the 3% and X’II states are substantial (van Dishoeck and Dalgarno, 1983). An efficient dissociation pathway for the 3’H state is provided by the 2’H state with which the 3’H state has an avoided crossing around 2.2 a,. Coupling of the two states by the radial component of the nuclear kinetic energy operator is thus expected to be large in this region.

The process of photodissociation of OH by absorption from the X’H state into the coupled 2 and 3’II states has been investigated, both in the adiabatic and in a diabatic representation. All the necessary potential energy curves V;:(R), transition dipole moments Dji(R) and nuclear coupling matrix elements Aif (R) were obtained from ab initio calculations. The resulting photodissociation cross section is presented in Fig. 14. It consists of a smooth background cross section underlying a series of resonances with Beutler- Fano line profiles. Such asymmetric line shapes were first observed in the ionization spectra of atoms (Beutler, 1935) and subsequently explained (Fano, 1935; 1961) by the interference between the excited state bound and continuum wavefunctions, which both have a nonzero transition moment with the ground state.

Although the adiabatic and diabatic representations give equivalent results in the coupled states formulation, they provide completely different views of the photodissociation process if a first order perturbation treatment (cf: Section IV) is employed. The adiabatic and diabatic excited ’H potential energy curves in the crossing region are illustrated in more detail in Fig. 15. In the adiabatic picture, strong absorptions occur into the 3% vibrational levels which are subsequently predissociated by the 2% state. The spectrum consists in this case of a series of broad discrete peaks near the positions of


-

0 9 10

t lev] FIG. 14. Photodissociation cross section for absorption into the OH coupled excited 2n

states from the Xzn (d’ = 0) state. The full arrows indicate the positions of the vibrational levels of the uncoupled diabatic bound potential. The dashed arrows indicate the positions of the first two vibrational levels of the uncoupled bound adiabatic curve (from van Dishoeck st al., 1984).

the adiabatic levels. Because of the small 2’n-X’II transition moment, the absorption continuum into the 2211 state is very weak. In the diabatic picture, however, direct photodissociation occurs by strong absorptions into the diabatic repulsive ’ll potential, whereas the absorptions into the bound diabatic vibrational levels are comparatively weak. If the positions of the resonances in the coupled states spectrum of Fig. 14 are compared with the positions of the adiabatic and diabatic vibrational levels of Fig. 15, it appears that they occur close to the energies of the diabatic levels. Moreover, the background cross section of Fig. 14 corresponds well with the direct photodissociation cross section for absorption into the uncoupled repulsive diabatic ’II state. It is therefore concluded that, for the case of OH, a diabatic

468 K a t e P . Kirby and Ewine F . van Dishoeck

1 2

11

10

- > llJ

W - 9

8 1 2 3 L

R (a,) 5

FIG. 15. Enlargement of the crossing region of the OH excited 'II potential curves. The full lines indicate the adiabatic potential curves and vibrational levels of the 3211 state. The dashed lines indicate the diabatic potentials and the vibrational levels of the diabatic bound curve (from van Dishoeck et d., 1984).

representation provides a more realistic first order perturbation description of the process than the adiabatic representation.

2. Other Examples

Avoided crossings between excited molecular states are a common phenomenon. Examples of molecules for which the interactions between the adiabatic states through the radial component of the nuclear kinetic energy operator have been studied include BeH, where rotational couplings also play a role in the interactions between the B2H, 22C+ and 32C+ states (Lefebvre-Brion and Colin, 1977) and N: with the B2C: and C2C: states (Roche and Tellinghuisen, 1979). No photodissociation cross sections for absorptions into these coupled states have yet been calculated, however. A


detailed study of the photodissociation of CH by absorption into the coupled excited 2, 3, and 4211 states (see Fig. 11) has been reported by van Dishoeck (1987b). Another example is provided by the CH photodissociation through the coupled 22C+ and C2C+ states, although in this case two continuum channels are involved instead of one bound and one continuum channel. Other astrophysically interesting molecules for which the nuclear kinetic energy operator may play a role in the photodissociation include 0, with the coupled 3Cu- and 311,, states, C , with the 3(F)'llu and 2lIIU states (Pouilly et al., 1983), and NH with the coupled 2 and 33C- states (Goldfield and Kirby, 1987). For O, , inclusion of the radial nuclear coupling of the B and 23Cu- states (cf Fig. 7) may improve comparison between theory and experiment for the Schumann-Runge continuum and for the discrete bands seen at shorter wavelengths. Because both 3Cu- states correlate with the same products, O('D) + O(3P), a proper treatment of the problem requires the solution of the coupled equations for two open channels.

Although Beutler-Fano line profiles are commonly observed in the ionization spectra of atoms and molecules, they are much less often seen in molecular absorption spectra below the first ionization threshold. In fact, the only case where asymmetric line profiles have been identified in absorption spectra is provided by the H, D'Q-X'Z: and B'''X;-X'Z: systems, in which the excited state interacts with the B' '2; state (Comes and Schumpe, 1971; Glass-Maujean et al., 1979; Rothschild et al., 1981). Further experimental searches for these asymmetric line profiles will provide valuable information on the importance of nuclear couplings between excited molecular states in the photodissociation of small molecules.

VI. Near-T hreshold P hotodissocia tion

A. INTRODUCTION

In recent years, photofragment spectroscopy experiments have revealed the presence of interesting effects in the near-threshold photodissociation of diatomic molecules to open shell atoms, if at least one of the atoms has fine structure splittings. At very low energies, the nuclear and spin-orbit interactions between the various molecular states approaching the same atomic limits become comparable to the energy differences between the states and may cause additional resonances to occur in the photodissociation cross section, as well as affecting the branching ratios to the fine-structure states (Band et a!., 1981; Durup 1981; Singer et al., 1984~ ; 1985). Although the


models of cometary atmospheres and interstellar clouds are not yet sophisticated enough to take into account the population distribution over the atomic fine structure levels resulting from photodissociation, such effects may play a role in future applications. In addition, a good understanding of the long range interactions is crucial to the study of the inverse process of photodissociation, i.e., radiative association, in the formation of molecules in interstellar clouds.

The general theory of photodissociation into open-shell atoms has been discussed in detail by Singer et al. (1984a, 1985). The interactions between the final states lead to a set of multichannel close-coupled equations similar to those of Eq. (18), but containing rotational nuclear interactions and spin- orbit couplings as well. In order to correctly describe the individual fragment atomic fine structure levels, the molecular basis used at short internuclear distances must be transformed to an atomic basis at large distances. This can be accomplished by a distance independent transformation matrix derived by Singer et al. (1984a). Total and state-specific photodissociation cross sections are obtained from formulae similar to Eq. (24).

B. EXAMPLES OF NEAR-THRESHOLD PHOTODISSOCIATION

Detailed calculations of either the interactions in the near-threshold photodissociation and/or the resulting fine structure branching ratios have been performed for the CH' A'II-X'Z' system by GraFfet al. (1983), Graff and Moseley (1984), and Williams and Freed (1986a,b); for transitions in NaH by Singer et al. (1984b) and in Na, by Struve et al. (1984); for the b4X;-a411, and f4l3,-a4III, systems of 0; by Dump (1981); for the A'n-X'Z+ transition of HCl by Givertz and Balint-Kurti (1986); and for the AZX+-X211 system of OH by Lee et al. (1986) and Lee and Freed (1987).

1. OH

The predissociation of the OH A2Z+ state by the 14X-, l'X-, and 1411 states has already been considered in Section IV.B.l. Although Sink et al. (1980) took into account the spin-orbit interactions between the A state and the dissociative states, they neglected the radial and rotational nuclear interactions among the three repulsive states. Lee et al. (1986) and Lee and Freed (1987) include the nuclear couplings in the calculations. Although these couplings do not affect the total predissociation rates from the AZX+ state to the dissociative continua, they may significantly affect the branching

PHOTODISSOCIATION PROCESSES 47 I

TABLE I

FOLLOWING ABSORPTIONS INTO THE OH AZEi BRANCHING RATIOS FOR O(3P0, U, J = 7+ LEVELS FROM THE x2n 0 = 0, J = LEVEL^

AZZ + -14n AZE+ - 1 2 ~ - A Z ~ + - 1 4 ~ -

u Multichannelb Multichannel' onlyb*d onlybVd onlyb.

1 0.058 0.137 0.805

4 0.132 0.240 0.628

8 0.004 0.379 0.617

0.190 0.195 0.615 0.158 0.176 0.666 0.00 1 0.332 0.667

0.021 0.247 0.732 0.001 0.371 0.628 0.002 0.368 0.630

0.154 0.433 0.414 0.132 0.358 0.519 0.125 0.362 0.512

0.073 0.113 0.814 0.196 0.163 0.641 0.169 0.187 0.644

~ ~~ ~ ~ ~

" From Lee et al. (1986); the first line for each u indicates the branching ratio for 3P0; the second line for 'P,; the third line for 3P2.

With asymptotic couplings included. No asymptotic couplings. Branching ratios for single channel coupled to A2Z+ if no asymptotic couplings

were included: 14n: (O.OO0, 0.333, 0.667) for (3P0, 3P,, 3P2); l ? - : (0.111, 0.333, 0.556); I4Z-: (0.222, 0.167, 0.61 1).

ratios to the various fine structure levels of the product atoms. This conclusion is illustrated in Table I where the full multichannel results are compared with those obtained without the asymptotic couplings. The branching ratios for the lowest levels, u I 4, are most strongly affected by the asymptotic couplings, but small effects persist even for higher levels, where the kinetic energies are more than two orders of magnitude larger than the fine structure splittings of the oxygen atom. The results in Table I are for a final total angular momentum J = 73; the influence of the rotational nuclear couplings increases with J . Also included in Table I are the branching ratios obtained if only one of the three states is permitted to have spin-orbit couplings to the A2C+ state, but with all asymptotic couplings retained. Consistent with the multichannel results of Sink et al. (1980), the contributions from each of the three channels are found to be additive. The 14C- state is the most important dissociation pathway for u = 1-3, whereas the 1411 state dominates for u = 5-9. The u = 4 predissociation rate has comparable contributions from each of the three levels. Because the resulting branching ratios depend strongly on the predissociation pathway, measurements of the fine structure population ratios may be used to infer the repulsive potential that is primarily responsible for the predissociation.


2. CHf

In contrast to the case of OH, many detailed experiments have been performed on the near-threshold photodissociation of the CH + ion (Carr- ington and Sarre, 1979; Cosby et al., 1980; Helm et al., 1982; Sarre et al., 1986; Carrington and Softley, 1986). As discussed in Section II.B.l, direct photodissociation of CH + can occur by the electric dipole allowed absorption into the A ‘ l l vibrational continuum. The All7 state correlates adiabatically with the upper C+(2P3,z) fine structure level which lies 63.4 cm-’ above the C+(zPl,z) level. At energies above the Cc(2P3/z) + H limit, a large number of shape resonances have been observed, arising from tunneling through centrifugal barriers for high rotational quantum numbers. Uzer and Dalgarno (1979) and Helm et al. (1982) have calculated locations, widths, and absorption oscillator strengths for these high4 quasi-bound levels. More interesting are the resonances that are seen at energies between the two dissociation limits C+(2P1,z) + H and C+(2P3,z) + H for low angular momenta. The A ’ l l state can support a number of quasi-bound rovibrational levels in this range which can couple to the continua of other electronic states correlating to the lower fine structure state. Graff et al. (1983) computed predissociation rates in a first order perturbation treatment for two possible interactions: the A’H, levels coupled by the rotational component of the nuclear kinetic energy operator with the X’C; state, and by the radial nuclear operator with the a317 continuum. The calculations indicate that these nuclear couplings cause significant predissociation of almost all levels between the two dissociation limits, and even affect substantially the lifetimes of higher lying levels. Reasonable agreement with the measured widths was found.

More extensive close-coupling calculations, which also take the long range interactions between the various CH+ continua into account, have been performed by Singer et al. (1984~) and Williams and Freed (1986a,b). These studies show that, in addition to the All7 resonances identified by Graff et al. (1983), a number of resonances occur that can be associated with other electronic states correlating to the upper dissociation limit, such as the a3n,,, and c3C+ states. The transitions to these states are in first order forbidden, but they can “borrow” intensity by the spin-orbit and nuclear couplings to other states. Evidence for the existence of resonances belonging to the a311 state has been given by Sarre et al. (1986) and Carrington and Softley (1986). The computed photodissociation spectra have an extremely complicated structure, even at low temperatures where the number of participating rotational levels is reduced. Detailed comparisons with experiments are still hampered by the highly excited and unknown population distribution of CH’ in the experiments, and by uncertainties in the CH+ potential energy curves and coupling matrix elements used in the theories.


VII. Concluding Remarks

In this review, we have attempted to illustrate the fruitful interaction between theoretical studies of photodissociation processes and their atmospheric and astrophysical applications, and emphasized the many contributions of Alex Dalgarno to this field. The authors are fortunate to have had the opportunity to collaborate with him on numerous projects. His insight into the processes themselves, as well as his overview of the wide range of potential applications continue to be most stimulating for future research in this area.

ACKNOWLEDGMENTS

E.v.D. acknowledges support from NSF grant RII 86-20342 to Princeton University.

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THE ABUNDANCES AND EXCITATION OF INTERSTELLAR MOLECULES JOHN H. BLACK Steward Observatory University of Arizona Tucson, Arizona

I. Introduction . . . . . . . . . , . . . 11. Molecular Hydrogen . . . . , , . . . ,

A. Abundance of H, . . . . , . . . . . B. Collisional Excitation of H2 . , . . . . . C. Radiative Excitation of H, . , . . . . .

111. Ion-molecule Chemistry . . . . , . . . . A. Chemistries of Carbon, Nitrogen, and Oxygen B. Chemistries of Minor Species . , . . . . C. Ionization Balance and Large Molecules . . D. Negative-Ion Chemistry . . . . . . . . E. Deuterium Fractionation . . . . , . . .

IV. Chemistry of Shock-Heated Gas . . . . . . V. The CH' Problem . . . . . . . . . . .

VI. The Excitation of Interstellar CN . . . . . . VII. Models of Interstellar Clouds . . . . . . .

VIII. Summary . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . References . . . . . . . . . . . . . .

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477 479 479 48 1 482 483 484 487 488 492 493 495 497 50 1 503 505 505 506

I. Introduction

Interstellar chemistry, the study of the composition, physical state, and evolution of interstellar clouds, has grown to be a vast subject. Despite the efforts of many investigators, this subject has developed a large reservoir of incompletely interpreted observational results and an enormous deficit of detailed information on spectra and microscopic processes. Given the number of recent reviews relating to interstellar chemistry, it would be pointless to attempt another broad survey of this field. It seems more useful to consider selected problems of current interest and to derive from them clues



478 John H . Black

to future directions. Because the work of Alexander Dalgarno has ranged throughout all aspects of the subject-excepting the purely observational ones-it is possibe to take his investigations as points of departure for discussions of many current and future developments.

The fundamental importance of interstellar chemistry lies in its relation to the formation and evolution of galaxies, stars, and planetary systems. Its particular hazards are that interstellar clouds lie so far from Earth and terrestrial experience and that they evidently exist in states very far from thermodynamic equilibrium. The remarkable chemical activity shown by some of these systems at temperatures as low as 10 K and densities as low as 100-1000 ~ r n - ~ challenges us to explain their molecular composition. The interpretation of spectroscopic observations and the effects of the chemistry on the structure and evolution of clouds require detailed understanding of excitation processes by which the atoms and molecules couple to their extended environment.

The earliest theoretical work (Eddington, 1926; Swings and Rosenfeld, 1937; Kramers and ter Haar, 1946; and Bates and Spitzer, 1951) was motivated by observations at visible wavelengths of three molecules and a handful of minor atomic species in the interstellar medium. Subsequent developments can be traced through a number of reviews and books (Dalgarno, 1975, 1976a, 1976b, 1977, 1979, 1980, 1981, 1982, 1985a, 1985b, 1986, 1987a, 1987b, 1987c, 1987d; Dalgarno and Black, 1976; Duley and Williams, 1984; Watson, 1976, 1978; Winnewisser and Herbst, 1987; Green, 198 1 ; van Dishoeck, 1988).

The study of interstellar chemistry has stimulated much experimental and theoretical work on atomic and molecular spectra and processes. Interest in the spontaneous radiative dissociative of H, in the interstellar gas (Solomon, as quoted in Field, Somerville, and Dressler, 1966; Solomon and Wickramas- inghe, 1969; Stecher and Williams, 1967) led to theoretical descriptions of the process and its spectrum (Dalgarno and Stephens, 1970; Stephens and Dalgarno, 1972) that culminated in the identification of the continuous spectrum in the laboratory (Dalgarno, Herzberg, and Stephens, 1970). In other instances, the interstellar medium has provided the laboratory in which transition frequencies or reaction rate coefficients have been measured or inferred long before such measurements were performed in a terrestrial laboratory (Dalgarno, 1979; Green, 1981). A good example is the discovery of “X-ogen” (Buhl and Snyder, 1970) and its eventual identification with HCO’ on the basis of isotope shifts in astronomical spectra (Klemperer, 1970; Snyder et al., 1976; Hollis et al., 1976), ab initio theoretical calculations (Wahlgren et al., 1973; Kraemer and Diercksen, 1976; Bruna, 1975), and spectroscopy in the laboratory (Woods et al., 1975).

ABUNDANCES AND EXCITATION OF INTERSTELLAR MOLECULES 479

11. Molecular Hydrogen

A. ABUNDANCE OF H,

It is generally accepeted that H, in molecular clouds is formed by association on the surfaces of interstellar dust grains. Although a sensible value for the formation rate coefficient can be inferred from model analyses of diffuse cloud observations (Jura, 1974b; Black and Dalgarno, 1977; van Dishoeck and Black, 1986), there is no agreement about the microscopic nature of the formation process. In particular, different studies come to quite different conclusions about the distribution of newly formed molecules among excited vibrational and rotational states (Hollenbach and Salpeter, 1970; Hunter and Watson, 1978; Leonas and Pjarnpuu, 1981; Allen and Robinson, 1976; Duley and Williams, 1986). There is also some evidence from observations that the effective formation rate may be systematically lower in the coldest, most quiescent regions where grains are larger than in the diffuse medium (Snow, 1983). The primary destruction process in regions exposed to ultraviolet starlight is spontaneous radiative dissociation where absorptions in the Lyman and Werner lines are followed by fluorescence into the vibrational continuum of the ground electronic state. The dissociation efficiencies of the various vibrational levels, u’, of the upper electronic states have been computed by Dalgarno and Stephens (1970) and Stephens and Dalgarno (1972). The fraction of absorptions leading to dissociation, weighted by the absorption probabilities and the wavelength dependence of the interstellar radiation field, is approximately 10 %.

External starlight is excluded from the interior regions of thick clouds by absorption due to dust and to molecules in the outer layers. In these cases, H, is destroyed by penetrating cosmic rays (see Section 1II.C) and to a smaller extent by chemical reactions. In theoretical models, molecular abundances are customarily calculated either in steady state or with explicit time dependence for minor species but a fixed abundance of H,. Owing to the relatively inefficient formation process, the formation time for H, must be rather long compared with other relevant chemical and dynamical time scales in some regions; more elaborate calculations of the time dependent behavior of the H, abundance in evolving clouds need to be performed (Tarafdar et al., 1985).

In molecular cloud surfaces exposed to intense ultraviolet radiation from very nearby stars, the description of the H, becomes more complicated. Significant populations of vibrationally excited H, can be maintained when the intensity of the radiation exceeds approximately 100 times its mean

480 John H . Black

background value (Shull, 1978; Black and van Dishoeck, 1987; Sternberg, 1986). Not only is vibrationally excited H, (Ht) more reactive than its ground state counterpart (Stecher and Williams, 1974; Freeman and Williams, 1982; Tielens and Hollenbach, 1985a, 1985b, Black and van Dishoeck, 1988), but it is vulnerable to destruction by photons with wavelengths A > 91 1.7 A through continuous photodissociation (when u > 3) and photoionization (when u 2 4) (Stecher and Williams, 1967, 1978). Cross sections of these processes are known (Ford, Docken, and Dalgarno, 1975; Allison and Dalgarno, 1969; Glass-Maujean, 1986).

H, has been observed directly in a number of diffuse interstellar clouds by means of ultraviolet absorption lines in the spectra of background starts (Spitzer and Jenkins, 1975). Oscillator strengths for the Lyman and Werner system bands (cf: Allison and Dalgarno, 1970) are needed to derive H, abundances and population distributions among its vibrational and rotational levels from such observations; indeed, the quality of some of the existing observational data (Morton and Dinerstein, 1976) demands line oscillator strengths that incorporate the effects of nonadiabatic coupling of the B'C: and C l n , states (Ford, 1975). Ford's calculations have been extended to a wider range of transitions (Abgrall et al., 1987), which will be valuable in the future in the analysis of possible observations of vibrationally excited H, with the Hubble Space Telescope. It is important to realize that, although H, is generally believed to be the principal constituent of molecular clouds, its abundance and excitation have never been measured directly in any quiescent, cold cloud of appreciable thickness that is opaque to ultraviolet radiation. Such clouds are translucent in the infrared and it would be barely feasible with existing spectrometers to observe lines of the fundamental vibration-rotation band of H, in absorption toward background or embedded infrared stars (Black and Willner, 1984). Owing to the lack of a permanent dipole moment, the electric quadrupole oscillator strengths of the (1,O) S(0) and S(1) lines are extremely small, 9.37 x respectively, based on the transition probabilities of Turner, Kirby-Docken, and Dalgarno (1977). This means that a column density of cold H,, N(H2, J = 0) x 4 x loz2 cm-', would be required to produce a resolved absorption line with a Doppler width of 2 km s-' and a central depth of 10 % of the continuum.

In shock-heated interstellar regions at high temperature, H, can also be destroyed by collisional processes. Interest in the dissociation of H, in such regions led to the recognition that the collisional process at low densities cannot be characterized by an equilibrium dissociation rate derived from laboratory measurements at high pressures (Dalgarno and Roberge, 1979; Roberge and Dalgarno, 1982). A large contribution to the equilibrium dissociation rate arises from excitations out of levels close to the dissociation

and 5.46 x

ABUNDANCES AND EXCITATION OF INTERSTELLAR MOLECULES 48 1

limit, whose populations tend to be highly nonthermal under interstellar conditions. Rates for collisional dissociation at low densities have also been calculated by Lepp and Shull (1983). Specific rates of H-impact dissociation of ground-state H, have been computed by Blais and Truhlar (1983) and more recently by Dove and Mandy (1986), who note that H is likely to be more effective in destroying H, than He or another H,.

B. COLLISIONAL EXCITATION OF H,

The distribution of H, molecules among their excited vibrational and rotational levels can be used to probe temperature, density, and intensity of ultraviolet light in interstellar clouds. The population distribution is governed by the distribution upon formation, relaxation by slow vibration- rotation transitions, absorption of ultraviolet radiation and subsequent fluorescence, inelastic collisions, and reactive collisions. As mentioned above, the details of the formation of H, on grain surfaces are poorly understood, while the vibration-rotation transition probabilities are well determined (Turner, Kirby-Docken, and Dalgarno, 1977). State-specific formation probabilities for the gas phase process

H- + H + H,(v, J ) + e (1)

can be calculated explicitly (Browne and Dalgarno, 1969; Dalgarno and Browne, 1967; Bieniek and Dalgarno, 1979), although this is an important source of H, only in special circumstances (Dalgarno and McCray, 1973; Black, Porter, and Dalgarno, 1981 ; Dalgarno and Lepp, 1987). Rotational excitation of H, by collisions with H, H,, He, and electrons has been studied extensively and most theoretical descriptions of such processes derive from the theory of Arthurs and Dalgarno (1960). Cross sections and thermal rate coefficients for low temperature H + H, collisions used in the astrophysical literature include those of Allison and Dalgarno (1967), Wolken, Miller, and Karplus (1972), and the smaller ones of Green and Truhlar (1979). There remain unresolved questions about the size of the cross sections for H + H, collisions. Allison and Dalgarno (1 967) also presented excitation rates for H, + H, collisions. Schaefer (1985, and in preparation) has recently computed cross sections and rates based upon a very accurate ab initio potential surface for the H, + H, system, and similar results have been reported by Danby, Flower, and Monteiro (1 987). At high temperatures, vibrationally inelastic collisions become important and reactive collisions with significant energy barriers, like

(2) H + H,(J) + H + H,(J f l),

482 John H . Black

can proceed at appreciable rates. Cross section and rate data relevant for the temperatures of interstellar molecular shocks ( T x 300 - 3000 K) have been summarized by Draine, Roberge, and Dalgarno (1983).

An interesting feature of the ultraviolet absorption line studies of H, in diffuse clouds is that the populations of the lowest levels of the ortho ( J = 1) and para ( J = 0) modifications appear to be nearly thermalized at the low kinetic temperature, T x 30 - 100 K, of the gas. The two modifications differ in nuclear spin orientations and thus couple extremely weakly to each other by normal inelastic collisions or radiative processes. Without interchange, the thermal populations are also inconsistent with grain-surface formation mechanisms in which the H, binding energy goes into internal excitation of the new molecules, thus mimicking high-temperature formation, or in which the rotation of newly formed H, is tied to the very low dust grain temperature, x 10 - 15 K. The answer seems to reside in proton transfer reactions

H + + H,(J) P H,(J k 1) + H +

H: + H,(J) e H,(J f 1 ) + H l

(3)

(4)

that are sufficiently rapid, even in neutral clouds of low fractional ionization, to compete with the slow rate of formation and a reduced, shielded rate of photodissociation (Dalgarno, Black, and Weisheit, 1973a). Time-dependent effects may be important (Flower and Watt, 1984). The rate coefficients originally estimated for Reaction (3) (Dalgarno, Black, and Weisheit, 1973a; Black and Dalgarno, 1977) appear-with small exceptions-to be in harmony with subsequent detailed studies (Gerlich and Bohli, 1981; D. Gerlich, in preparation) and with data on the deuterated analogue

( 5 ) D+ + H, F?HD + H f

(Villinger, Henchman, and Lindinger, 1982).

c . RADIATIVE EXCITATION OF H,

In diffuse clouds and in the outer parts of thick, opaque clouds, the excitation and abundance of H, are closely linked through the absorption of ultraviolet starlight and subsequent fluorescence to the vibrational continuum or to the various bound levels of the ground state (Black and Dalgarno, 1976). The ultraviolet fluorescent excitation is thought to be responsible for the nonthermal rotational population distributions ( J = 4 - 7) observed in diffuse clouds (Black and Dalgarno, 1973b, 1977; Spitzer and Zweibel, 1974; Jura, 1975) and is predicted to maintain measurable populations of vibrationally excited H, in some clouds (van Dishoeck and Black, 1986; Black,

ABUNDANCES A N D EXCITATION OF INTERSTELLAR MOLECULES 483

1987). Ultraviolet fluorescence to excited vibrational levels will produce infrared line emission in the ensuing cascade of quadrupole vibration- rotation transitions. Although this emission was anticipated (Gould and Harwit, 1963) and first sought some time ago (Werner and Harwit, 1968; Gull and Harwit, 1971), the first unambiguous observations of the radiatively excited infrared lines were not reported until the mid 1980s for a region in the Orion Nebula (Hayashi et al., 1985), several reflection nebulae (Sellgren, 1986; Gatley and Kaifu, 1987; Gatley et al., 1987; Hasegawa et al., 1987), and a star-forming complex in the galaxy M 33 (E. F. van Dishoeck, private communication). The fluorescent line emission is characterized by high vibrational excitation temperatures that exceed the rotational excitation temperatures, resulting in relatively high fluxes in the lines at wavelengths I I < 2 pm, which help to distinguish it from thermally excited emission in shock-heated molecular gas. The fluorescent line intensities are of considerable diagnostic value; their interpretation requires accurate transition probabilities for both the ultraviolet and infrared transitions and a means of modeling the depth-dependent abundances and excitation rates for realistic cloud properties (Black and Dalgarno, 1976; Sternberg, 1986; Black and van Dishoeck, 1987).

111. Ion-Molecule Chemistry

Despite the assertion above that the most abundant interstellar molecule, H,, forms on grain surfaces, and despite the extensive literature on interstellar grain chemistry (see Tielens and Allamandola, 1987), gas phase formation of other species is demonstrably important and can be studied in some detail. Dust grains play several roles in the chemistry. They help exclude ultraviolet light from the interiors of clouds. In quiescent regions, they probably act as a sink for virtually all gaseous species except H, and He. They provide local sources of long wavelength radiation to which polar molecules couple by absorption and stimulated emission. The smallest grains or largest molecules (e.g. polycyclic aromatic hydrocarbons) may alter the ion chemistry as discussed in Section 1II.C.

That gas phase ion-molecule reactions are important in the interstellar medium follows from (1) direct observations of molecular ions, e.g., CH', HCO', N,H', HCS', and (2) enhanced deuterium abundances in molecule pairs like DCO'/HCO' and DCN/HCN that can be explained by temperature-sensitive exchange reactions. The abundances of molecules other than H, in diffuse clouds, where photodissociation lifetimes are short, also seem to require rapid gas phase chemistry, as does the ortho/para thermalization in H2.

484 John H . Black

The essential difficulty of interstellar gas phase chemistry is that most of the ions are produced by light from stars with effective temperatures T, 2 30000 K and by inherently nonthermal cosmic rays with energies greater than 1 MeV/nucleon, while the gas itself has a kinetic temperature T w 10 - 100 K. At the low densities of interest, abundances are governed by the detailed balance among a multitude of microscopic processes. There exist numerous reviews of ion-molecule chemistry (Section I) and various tabula- tions of reaction rate coefficients (e.g. Anicich and Huntress, 1986; Herbst and Leung, 1986). The following subsections summarize only a few aspects of the chemistry.

A. CHEMISTRIES OF CARBON, NITROGEN, AND OXYGEN

Carbon, nitrogen, and oxygen are the most abundant elements aside from hydrogen and helium : for reference, their abundances relative to hydrogen by number in the Sun are 4.7 x lop4, 1.0 x and 8.3 x respectively. It is thought that simple carbon-bearing molecules can be built up in a network of reactions initiated by a radiative association process

C + + H , - + C H ; + h v (6)

(Black and Dalgarno, 1973a). Although this hypothesis “bears the seeds of its own destruction” (in the words of A. Dalgarno) by being vulnerable to experimental test, it has not yet been disproven. The rate coefficient of Reaction (6), inferred from the abundance of CH in diffuse clouds, k , z 7 x

cm3 s-’(van Dishoeck and Black, 1986), while comfortingly consistent with theory (Herbst, 1982a), lies perilously close to the best experimental upper limit, k , < 1.5 x cm3 s-’ (Luine and Dunn, 1985). The basic operation of the carbon chemistry could be tested by observation of other species, CH,, CH,, CH:, or CH;, formed in the same cycle of reactions. Unfortunately, the oscillator strengths of electronic transitions in CH, and CH, are not known, and the electronic spectra of the corresponding ions have not been identified.

In dense, dark clouds, the carbon chemistry can also be started by

H : + C - - + C H + + H , , (7)

although the importance of this process in relation to Reaction (6) depends upon the H l abundance and the balance among C, C+, and CO, which are expected to be the main forms of carbon in most circumstances. The extent to which the chemistry of small carbon-bearing molecules proceeds to very complex species depends in large part on the radiative association process

(8) CH; + H, + CH; + hv.


This is one of the few radiative association processes for which there is a direct measurement of the rate at low temperature in the laboratory (Barlow, Dunn, and Schauer, 1984). This process therefore poses a major test for the theory of radiative association and close theoretical examination of it has revealed considerable subtlety (Bates, 1987; Herbst and Bates, 1988). The formation of complex molecules has been reviewed by Dalgarno (1987~).

There is significant uncertainty about the mechanisms that drive the nitrogen chemistry. Proton transfer,

(9) H: + N + N H + + H,,

H l + N - + N H l + H,

is endoergic, and the alternative reaction,

(10)

is likely to have substantial energy barriers at low temperature, based on theoretical calculations (Herbst, DeFrees, and McLean, 1987). The low- temperature behavior of the reaction

(1 1) N C + H, + N H f + H

is an interesting case study in laboratory astrophysics (Marquette et al., 1985; Luine and Dunn, 1985; Adams and Smith, 1985; Bates, 1986). The contribution of Reaction (1 1) is further complicated by the recognition that N + ions produced in the reaction

H e + + N , + N + + N + H e (12)

will have excess translational energy as large as 0.14 eV ( & k T ) (Adams, Smith, and Millar, 1984). Because energetic N + ions are thermalized by elastic collisions not much faster than they react via Reaction (1 l), they will have a non-Maxwellian velocity distribution in molecular clouds that must be evaluated explicitly in order to estimate the effective rate of Reaction (11) (Yee, Lepp, and Dalgarno, 1987). While these considerations alleviate problems in the production of interstellar NH,, they leave unanswered the question of the origin of the N, involved in Reaction (12). In some model calculations, N, arises primarily from NO, which is often predicted to be more abundant than CN (Millar, Leung, and Herbst, 1987). Extensive observations of NO and N2H+, which are most closely linked chemically to the undetectable N,, would be instructive. At the present time, NO has been weakly detected in only two interstellar clouds, neither of which is a cold, dark cloud where NH, is abdundant.

Other nitrogen-bearing species, such as CN, HCN, HC,N, and so forth, can apparently be formed by reactions of hydrocarbon molecules and molecular ions with atomic nitrogen. The interesting suggestion that C,N+

486 John H . Black

will be the most abundant nitrogen-containing ion (Hartquist and Dal- garno, 1980) has not yet been subjected to an observational test. The nonsymmetrical structure CCN+ is predicted to have a large dipole moment ( Ipel x 2.4 D) and to have a J = 1 --t 0 rotational transition at v l 0 = 24.2 f 0.1 GHz (Kraemer et al., 1984). If, however, the reaction

C + + H C N + C 2 N + + H (13)

is the principal source of the ion, then only the symmetric, nonpolar isomer CNC+ is formed (Daniel et al., 1986), and C2N+ may be unobservable by radio techniques.

The basic oxygen chemistry in diffuse clouds is perhaps the best understood, because many of the important reaction rates have been measured and some crucial processes like the near resonant charge transfer,

H + + 0 $0' + H

OH + hv -0 + H

(van Dishoeck and Dalgarno, 1984), have been studied in accurate ab initio theoretical calculations.

In thick, dense interstellar clouds, the oxygen chemistry and the identity of the predominant oxygen-bearing species are less well understood. A large abundance of atomic oxygen would be expected to inhibit the growth of observed, complex molecules (Millar et al., 1987), although the census of possible oxygen-containing hydrocarbons is far from complete. For conventional assumptions about elemental abundances, CO can account for, at most, half of the total oxygen. The abundances of 0, and H,O cannot be determined directly by ground-based observations owing to the large opacities of their spectral lines in the terrestrial atmosphere. Searches for 160180 lines (Black and Smith, 1984) suggest indirect upper limits on the abundance ratio OJCO x 0.1 - 1 (Goldsmith et al., 1985; Liszt and Vanden Bout, 1985). Although the interpretations of observations of HDO (Moore et al., 1986; Plambeck and Wright, 1987) and of H 3 0 + (Hollis et al., 1986; Wootten et al., 1986) are ambiguous, it appears unlikely that most of the oxygen resides in H,O.

The chemistries of carbon and oxygen converge on the deeply stable and very abundant CO molecule through such processes as

(14)

(15)

(Chambaud et al., 1980), and photodissociation of OH,

C + + OH +CO+ + H

C + + OH +CO + H +

CO' + H, + HCO' + H

( 164

( 16b)

(17) and

HCO' + e + H + CO. (18)


Although the basic processes that form CO (Oppenheimer and Dalgarno, 1975; Langer, 1976,1977) have been known for some time, the photodissociation of CO is only now beginning to be understood in detail (see Kirby and van Dishoeck, 1988, this volume). Prasad and Tarafdar (1983) noted an important effect on dense cloud chemistry due to penetrating cosmic rays; secondary excitations of H, produce a significant internal source of ultraviolet photons that can enhance the photochemistry in heavily obscured regions. Sternberg, Dalgarno, and Lepp (1987) calculated in detail the cosmic-ray induced spectrum in the Lyman and Werner systems of H,, integrated the resulting rates of photodissociation into a chemical model, and pointed out that these molecular destruction rates may be large enough to stunt the growth of complex molecules in molecular clouds. Gredel, Lepp, and Dalgarno (1987) considered the contributions of other states of H, to the local ultraviolet spectrum and discussed in detail the destruction of CO inside molecular clouds and the steady state balance among C, C', and CO. The description of the CO destruction is complicated by the fact that CO photodissociation occurs through line absorptions (van Dishoeck and Black, 1988) and that the cosmic-ray induced ultraviolet spectrum is concentrated predominatly in lines: only a relatively small number of H, and CO line pairs overlap. That overlap may be sensitive to such effects as the velocity dispersion of emitting molecules, which could be governed by momentum transfer in the excitation event. The enhanced CO destruction can help account for the surprisingly large amounts of atomic C observed in molecular clouds (Phillips and Huggins, 1981; Keene et al., 1985; Jaffe et al., 1985); moreover, enhanced abundances of C and C' drive the gas phase chemistry more rapidly toward the buildup of complex molecules, thus compensating for their destruction by cosmic-ray induced photons. CO is widely used as a surrogate tracer of H, or of the total molecular content of interstellar clouds; therefore, a full theoretical understanding of its abundance is important for the interpretation of data on the structures and distributions of molecular clouds (van Dishoeck and Black, 1987).

Another very widespread molecule, H,CO, also arises from both the oxygen and carbon chemistries (Dalgarno, Oppenheimer, and Black, 1973d). There remain some questions, however, about explaining its relatively high abundance in clouds of relatively low total density. Again, observational searches for its chemical precursors-notably CH, - would be valuable.

B. CHEMISTRIES OF MINOR SPECIES

The element sulfur is present in a number of interstellar molecules: SO,, CS, SiS, OCS, H,S, HNCS, H,CS, H,CSH, HCS', SO', NS, SO, CIS, and C,S. The gas phase sulfur chemistry originally proposed by Oppenheimer

488 John H. Black

and Dalgarno (1974a) has been augmented by Prasad and Huntress (1982), Millar et al. (1985,1986), Pineau des For&, Roueff, and Flower (1986), Watt and Charnley (1989, and discussed with reference to grain surface processes by Miller (1982a, 1982b). Silicon chemistry, represented only by SiO and SiS in interstellar clouds so far, has been discussed by Turner and Dalgarno (1977) and Millar (1980), but lacked observational tests until the late 1980s (Ziurys, 1988).

Chlorine chemistry has drawn an inordinate amount of attention, probably because Cl' is one of the few atomic ions that can react directly with H, at low temperature (Jura, 1974a; Dalgarno, de Jong, Oppenheimer, and Black, 1974; Blake, Anicich, and Huntress, 1986). The absence of HCl in diffuse clouds has been enigmatic, although the worst discrepancies between observation and prediction have been removed by laboratory measurements (Smith and Adams, 1981; Cates, Bowers and Huntress, 1981) and ab initio calculations (van Dishoeck, van Hemert, and Dalgarno, 1982). A submilli- meter emission line of HCl has been identified in the Orion Molecular Cloud (Blake, Keene, and Phillips, 1985).

No interstellar molecules containing metals like Mg, Ca, Na, Fe, or Ti have been found, except for a possible detection of MgO (Turner and Steimle, 1985). In part, this must be due to their being rather unreactive and having relatively small abundances in the gas phase. Calcium, iron, and titanium, in particular, are evidently severely depleted from the gas by being in solid form in dust grains. Even so, a major role in the overall ionization balance has been attributed to minor metal atoms (Oppenheimer and Dalgarno, 1974b; see Section III.C), and it is of interest to understand their chemistry better.

c . IONIZATION BALANCE A N D LARGE MOLECULES

Although interstellar molecular clouds are predominantly neutral with ionization fractions x, = n(e)/n, 6 the small concentrations of charged particles play important roles in the chemistry and in the dynamical coupling of the gas to magnetic fields. In diffuse clouds which are translucent to ultraviolet starlight, the ionization is maintained by starlight photoionization of species with ionization potentials less than that of atomic hydrogen. Rates of photoionization in the mean starlight background have been computed by Roberge, Dalgarno, and Flannery (1981) as functions of depth through interstellar clouds, based on a rigorous treatment of the effects of scattering and absorption by dust particles. Although the extinction curves (the sum of scattering and absorption cross sections per hydrogen nucleus as a function of wavenumber) are measured for many locations in the galaxy to wavelengths as short as 1100 A, the optical properties that govern the penetration


of radiation into a cloud-e.g., the scattering phase function and albedo-are poorly determined, especially at the shortest wavelengths. Moreover, data are limited on the intensity of starlight in the crucial wavelength interval, A = 912 - 1100 A, that controls directly the ionization of C, C1, and Ca' and the photodissociation of H, and CO.

In the cores of thick clouds, where starlight is excluded, the level of ionization is controlled by penetrating cosmic rays, by chemiionization processes, and ultimately, by minute traces of radioactive elements. The description of the heating and ionization of interstellar gas by cosmic rays is complicated by the myriad, discrete energy-loss processes that occur (Cra- vens, Victor, and Dalgarno, 1975; Cravens and Dalgarno, 1978). Low-energy cosmic rays (kinetic energy 5 10 MeV per nucleon) can also be excluded from the interiors of clouds to some extent by magnetic fields (Cesarsky and Volk, 1978), although the Alfven waves generated in this manner may add to the heating rate (Hartquist, 1977). It has been recognized recently that excitation by secondary electrons-primarily of H,-can provide a significant local source of ultraviolet radiation deep inside molecular clouds (Prasad and Tarafdar, 1983; see Section IIIA).

Some interstellar molecules like O H and HD are thought to be formed by processes that require H + , O', D', and H: ions produced by cosmic ray ionizations; therefore, their observed abundances can be used in combination with chemical models to infer the interstellar ionizing frequency of cosmic rays, lo (Black and Dalgarno, 1973b; ODonnell and Watson, 1974; Hartqu- ist, Black, and Dalgarno, 1978a). Now that important intermediate processes like H + + 0 charge transfer (Chambaud et al., 1980) and O H photodissociation (van Dishoeck and Dalgarno, 1984) are better understood, observations and models of diffuse clouds should be usable with more confidence in constraining lo (van Dishoeck and Black, 1986). Hartquist, Doyle, and Dalgarno (1978b) also have shown how the cosmic ray ionizing rate inside clouds should be related to its value in the intercloud medium. In thick, dense clouds, limits can be placed on lo from observed abundances of OH (Lepp and Dalgarno, 1987a; Lepp, Dalgarno, and Sternberg, 1987).

The H: ion is formed by cosmic ray ionizations of H, followed by an H atom abstraction reaction and is removed primarily by reactions with abundant forms of carbon and oxygen. It has been shown that ground-state H: undergoes dissociative recombination with electrons at a very low rate (Michels and Hobbs, 1984; Smith and Adams, 1984; A d a m and Smith, 1987), although there remains some question about the magnitude of the upper limit on this rate and the level at which radiative recombination should dominate (Dalgarno, 1987d). The H l ion is thought to play a pivotal part in almost all interstellar ion-molecule chemistry and it is predicted to have column densities as large as 1014-1015 cmP2 both in diffuse (van Dishoeck

490 John H . Black

and Black, 1986) and dark (Lepp and Dalgarno, 1987a; Lepp, Dalgarno, and Sternberg, 1987) molecular clouds. Oka (1981) originally suggested searching for the infrared vibration-rotation lines of interstellar H i in absorption towards background or embedded infrared sources. Although the initial searches for H i have been unsuccessful so far, the predicted abundances should produce absorption lines detectable with existing infrared spectrometers; further searches will provide a direct and significant test of a foundation of interstellar chemical theory. Unlike H:, which has only a forbidden pure rotational spectrum (Pan and Oka, 1986), HzD+ has an allowed radiofrequency spectrum (Dalgarno, Herbst, Novick, and Klem- perer, 1973b; Bogey et al., 1984; Saito et al., 1982; Warner et al., 1984). Phillips et al. (1985) have reported a possible detection of one line of H2D+ in the NGC 2264 molecular cloud. The enhancement of deuterium in HzD+ relative to H: by the temperature-sensitive exchange reaction

H: + HD F? H2D+ + H, (19)

is thought to drive much of the observed deuterium fractionation in interstellar clouds.

Metal atoms and atomic ions may be very important in the interstellar ionization balance. As pointed out by Oppenheimer and Dalgarno (1974b), atomic ions recombine more slowly with electrons and are generally less reactive than molecular ions; therefore, the atomic ions of metals of low ionization potential might account for much of the ionization in molecular clouds if the corresponding neutral atoms undergo charge transfer with molecular ions. The original analysis of Oppenheimer and Dalgarno was complicated by two developments: first, the recognition that H i recombines unusually slowly; second, the possibility that large molecules such as the polycyclic aromatic hydrocarbons (PAHs) could be even more effective than metal atoms in neutralizing ions and monopolizing the excess charges in neutral clouds if they are indeed present in the amounts suggested by some observations (Omont, 1986; Lepp and Dalgarno, 1987b; Lepp, Dalgarno, van Dishoeck, and Black, 1988). Both of these factors can modify the overall ionization balance and the role of metals within it. Although abundances of metal atoms are well measured in many diffuse clouds, virtually nothing is known about their gas phase abundances, degree of ionization, and presence in molecular form in thick, dense clouds. The chemistries of the simplest metal hydrides, LiH, NaH, MgH, and AlH, seem to be controlled by rapid photodestruction and by inefficient or nonexistent paths of formation (Kirby and Dalgarno, 1978; Smith et al., 1983; Millar, 1982c; Leung, Herbst, and Huebner, 1984; Cooper et al., 1983). Observational searches for hydrides and oxides such as MgH, NaH, LiH, CaO, FeO, and T i 0 have so far been unsuccessful. Low limits on atomic sodium in molecular clouds have interest-


ing implications for the sodium chemistry (Turner, 1987). The possible identification of MgO and the detection of several alkali halides in the circumstellar envelope of the carbon star IRC + 10216 (Cernicharo and Guelin, 1987) should stimulate more work on metal species in interstellar clouds.

An additional small source of ionization in interstellar clouds is provided by chemiionization processes like

CH + 0 + HCO' + e (20)

(Dalgarno, Oppenheimer, and Berry, 1973c) and

Ti + 0 + TiO' + e (21)

(Oppenheimer and Dalgarno, 1977), which are exothermic at low interstellar temperatures. In a note added in proof in their original discussion, Oppen- heimer and Dalgarno called attention to revised thermochemical data suggesting that Reaction (21) might be endoergic or only barely thermoneutral (Hildebrand, 1976). The existing thermochemical data on titanium- bearing species are based on measurements at high temperature ( T x 2000 K), but with appropriate corrections to zero temperature, the data from Chase et al., (1982) and Pedley and Marshall (1983) suggest that chemiionization to form TiO' is exoergic by 0.3-0.6 eV, even with allowance for the unresolved discrepancy in the ionization potential of T i 0 (Hildenbrand, 1976; Rauh and Ackermann, 1974). The chemistry of titanium is particularly interesting, despite its low abundance, 1.2 x by number of atoms, where dTi 5 is the factor by which the reference solar abundance is depleted in dust grains in the interstellar medium. The TiO' formed in Reaction (21) can be converted into T i 0 by charge transfer with neutral metal atoms and with large molecules (PAHs), but cannot be removed by dissociative electron capture. If a rate coefficient of kzl x lo-'' cm3 s-' is adopted and if the rates of photodestruction of T i 0 and TiO' are assumed to be no larger than the rate of photoionization of Ti, then abundance ratios Ti : T i 0 : T i 0 ' = 1 : 0.007 : 0.2 and 1 : 10 : 0.1 could be expected in diffuse and dark clouds, respectively, similar to the results suggested originally by Oppenheimer and Dalgarno. Gaseous T i 0 has been sought in interstellar clouds by means of its X3A, J = 3 + 2 pure rotational transition (Church- well et al., 1980) at a detection level of 5 1.3 x 10" cm-" in column density. Now that it is possible to probe some dark clouds by means of optical absorption line techniques, T i 0 could be observed with 10-30 times higher sensitivity through its potential interstellar absorption lines at 5169, 6161, and 7131 A wavelength, based on a limiting equivalent width of W, = 1 mA. At the same detection limit, column densities of the atomic species Ti and Ti' as low as 3-4 x 10" cmW2 could be measured. These detection limits become

492 John H. Black

quite interesting for any diffuse or translucent cloud with N,, 2 2 x loz1 cm-' even when the depletion is severe, aTi x It might thus be possible to use titanium species as sensitive probes of both the ionization balance and metal chemistry in translucent clouds. The spectrum of TiOf is apparently unknown; thus, further laboratory investigations would be valuable.

D. NEGATIVE-ION CHEMISTRY

In the earliest discussion of interstellar negative-ion chemistry, McDowell (1961) noted that H, could form as a result of radiative attachment

e + H + H - + hv

followed by associative detachment of the negative ion (Reaction 2). Al- though this sequence of reactions can be an important source of H, when formation on grain surfaces is absent or inhibited, it is only a very minor source of H, in normal interstellar clouds. The initiating reaction is slow, k , , = 1.04 x cm3 s - ' at T = 100 K, and H - is extremely vulnerable to photodetachment

(22)

H - + hv + H + e

owing to the long threshold wavelength, do = 1.6pm, and the large flux of detaching photons even in obscured regions. In the standard background starlight, the unattenuated rate of Reaction (23) is k23 = 1.1 x lO-'s-'. Deep inside molecular clouds, where Reaction (23) is a minor sink of H-, the free electrons needed in Reaction (22) are scarce, and production of H - by cosmic ray ionizations

(24)

(23)

H, + C R + H + + H- + CR

is relatively inefficient, k24 = 5.4 x (Cravens and Dalgarno, 1978). Dalgarno and McCray (1973) extended the discussion to a much greater range of negative ions and processes that might be important in the interstellar medium. Herbst (198 1) pointed out that relatively large molecules with high electron affinities might have radiative attachment rates that approach the electron-molecule collision rate. Even so, he concluded that the negative ions themselves might not be very abundant owing to the high rates of associative detachment

X- + H + X H + e (25)

x - + y + + x + y . (26)

and mutual neutralization


Sarre (1 980) also considered the observability of the particularly strongly bound ions C;, C,H-, and CN- in interstellar clouds.

As pointed out by Dalgarno and McCray (1973) and re-emphasized by Herbst (1981) and Sarre (1980), many negative ions could be important intermediates in the gas phase chemistry. Proving this is very difficult in general. First, there exist alternative-if sometimes speculative-mechanisms for forming most observed molecules. Second, the negative ions themselves are likely to have small abundances, even if they participate actively in the chemistry. Third, very few negative ions have known spectra, particularly at radio frequencies. Some observational tests may be possible. It appears that

(27)

could be the only important source of interstellar HO,; if so, and if the 0, abundance could be determined, then observational searches for HO, would provide a specific probe of negative ion chemistry (Black and van Dishoeck, 1988). Although nonpolar 12C’2C- lacks a radiofrequency spectrum and the trivially polar 12C’3C- ion will be much rarer (Sarre, 1980), C ; does have a well-studied electronic spectrum. Searches for the C ; B-X(0, 0) lines near 5400 A have been made in a few carbon stars (Wallerstein, 1982; Black and van Dishoeck, unpublished) without success. Such searches could be carried out as well in diffuse or translucent interstellar clouds, although probably with insignificantly higher prospects of detection.

H - + 0, +HO, + e

E. DEUTERIUM FRACTIONATION

Deuterium has been observed in the forms of D and H D in diffuse clouds and in more than 12 molecules in thick, dense clouds. The best estimates of the overall deuterium abundance in the interstellar medium are [D]/[H] rc 1.5 x 10- by number of nuclei. Nuclear processes in stars tend towards a net destruction of deuterium; therefore, the observed abundance is thought to provide a lower limit on the primordial abundance. In the context of “standard Big Bang cosmology,” this primordial deuterium fraction is sensitive to the density of the universe at the time when the equilibrium abundances of light nuclei were fixed (Yang et a!., 1984; Boesgaard and Steigman, 1985) and-with some extrapolations-to the current mean density. Understanding the deuterium chemistry and fractionation is important not only for the cosmological interest in the deuterium abundance but also for the diagnostic information on electron fraction and ionization rate provided by some deuterated molecules.

Only one deuterated molecule, HD, has been observed in diffuse clouds. Its abundance relative to that of H, is affected by opposing processes in a

494 John H. Black

complicated manner. While interstellar H, forms primarily on grain surfaces, the sequence of charge transfer,

(28) H + + D P D + + H,

and ion exchange,

D + + H , e H D + H + ,

(Eq, ( 5 ) ) reactions is a more significant source of HD than direct formation on grain surfaces (Dalgarno, Black, and Weisheit, 1973; Black and Dalgarno, 1973b). High abundances of H i , which result from its slow dissociative recombination, lead to another important source of HD through

H: + D+H,D+ + H

H: + HD P H2D+ + H,.

(29)

(30)

Although Reaction (30) is one of the principal routes towards fractionation of deuterium in molecular clouds (Watson, 1976), the temperature-sensitive ratio of reverse and forward rates, k$,/k&, 2 0.1 at T 2 60 K, will be fairly large in warm, diffuse clouds (Adams and Smith, 1981; Herbst, 1982b; Smith, Adams, and Alge, 1982). Thus, where photodissociation overwhelms Reac- tion (30) in destroying HD, the reverse of Reaction (30) is actually a net source of HD. Once H, is present, HD forms more rapidly than does H, itself. This would lead to an enhancement of deuterium in molecular form, except that the destruction rate of HD by spontaneous radiative dissociation remains larger than the corresponding dissociation rate of H, throughout most diffuse clouds. The isotope shifts in the wavelengths of the Lyman and Werner system lines are large enough that H, absorption does not effectively shield HD. The effect of self-shielding in HD is much smaller than in H,, owing to the much smaller abundance of deuterium. In most diffuse clouds where HD is directly observed, a ratio of column densities 4 x 5 N(HD)/N(H,) 5 3 x lo-' is found.

In thick, dense clouds, widely observed species like DCO+ and DCN show abundances relative to those of HCO+ and HCN that are apparently enhanced by factors of 10, or more in relation to [D]/[H] = 1.5 x One of the most extreme cases of fractionation is in the ethynyl radical, for which the observed abundance ratio CCD/CCH =- 0.01 in two sources (Herbst, Adams, Smith, and DeFrees, 1987, and references therein). At low temperatures, the forward rate of Reaction (30) greatly exceeds the reverse rate; consequently, deuterium can be enhanced in H2D+ relative to H:. This enhancement can be transferred to other species by reactions such as

(31)

followed by the reverse of

H2D+ + CO + DCO' + H,.


Dalgarno and Lepp (1984) have called attention to the important role of atomic deuterium through reactions like

HCO' + D + DCO+ + H. (32)

Croswell and Dalgarno (1985) showed that OD/OH will be particularly sensitive to the atomic deuterium through the neutral-neutral exchange reaction

D + OH + O D + H

and predicted abundance ratios as large as OD/OH w 0.01. Further observational searches for the ground-state A-doubling transitions of O D in regions where the corresponding OH lines are strong will be of interest.

The details of deuterium fractionation driven by Reactions (5 ) , (29), and (30) can be very complicated owing to the sensitivity of the ratio of reverse and forward rate coefficients, kr/kf, to the rotational populations of the reactants (Herbst, 1982b). The energy differences between the ground and first excited rotational levels can be comparable to the forward exoergicities. Under some conditions, these populations can be highly nonthermal; indeed, the exchange reactions themselves help govern the population distributions. It is feasible to observe HzD+ by its submillimetre emission and H: by infrared absorption in the same regions. Moreover, measurements of several transitions of H: could provide information on the rotational populations. In this way, a basic fractionation mechanism could be studied directly and in detail.

(33)

IV. Chemistry of Shock-Heated Gas

Gas phase chemistry in quiescent interstellar clouds is restricted to reactions that occur at low temperatures ( T 5 100 K) and very low densities. Shock-heated regions of the interstellar medium provide a venue for otherwise endothermic processes to affect the interstellar chemistry. Whether the high-temperature modifications will be observable depends upon the amount (i.e., the column density) of gas that can be maintained at an elevated temperature before cooling to its pre-shock temperature. The structure of the cooling zone of compressed gas behind a shock is governed, in part, by the abundances of coolants and their energy loss rates and by the abundances of ions when magnetic fields are involved. Thus, the structure of an interstellar shock and its effects on molecular abundances are closely linked.

Interstellar shocks are widespread: they often accompany the formation of stars, they are associated with the expansion of supernova remnants and of

496 John H . Black

stellar winds, and they can be driven by density waves and galaxy-galaxy collisions on very large scales. Molecular shocks can be revealed by collisionally excited emission lines, e.g., the quadrupole vibration-rotation lines of H, in the 2pm wavelength region (see, e.g., Gatley and Kaifu, 1987). Some molecular abundances are also expected to serve as diagnostics of shocks (Hartquist, Oppenheimer, and Dalgarno, 1980). In particular, shocks in diffuse clouds have been invoked to explain the remarkably high abundance of CH', as discussed in detail in Section V. One of the very best studied molecular clouds, OMC-1 in Orion, evidently has a substantial high- temperature component whose chemistry differs from that of the quiescent gas. Magnetohydrodynamic (MHD) shock models have been devised to account for a range of high-temperature phenomena there (Draine, Roberge, and Dalgarno, 1983; Chernoff, Hollenbach, and McKee, 1982).

Earlier work on shock chemistry has been reviewed by Dalgarno (1981, 1985b). The deluge of new work on this subject in the last two years requires mention. The formation and excitation of simple molecules, notably CH' and OH, in diffuse cloud MHD shocks have been discussed independently in a series of papers by Draine and Katz (1986a,b), Draine (1986a) and by Flower, Pineau des Forkts, and Hartquist (1985, 1986), Pineau des Forkts, Flower, Hartquist, and Dalgarno (1986), Pineau des Forkts, Flower, Hartqu- ist, and Millar (1987). The two sets of investigators do not always agree in their treatments of MHD shock structure; some differences in the two approaches have been discussed by Draine (1986b) and Flower and Pineau des For& (1986). The shock chemistry of sulphur has received particular attention (Mitchell, 1984; Millar et al., 1986; Pineau des Forcts, Roueff, and Flower, 1986; Leen and Graff, 1988) and the SH' abundance has been proposed as a significant (if observationally difficult) test of theory. High abundances and distinctive Doppler velocity signatures of molecules like C,H and C,H, might supply more easily measurable probes of MHD shocks with ambipolar diffusion (Pineau des For&ts and Flower, 1987). Nonequili- brium rates and effects of excited fine structure have been investigated for crucial processes in the oxygen chemistry (Wagner and Graff, 1987), and their consequences in interstellar shocks have been explored (Graff and Dalgarno, 1987). The one supernova remnant, IC 443, that displays clear evidence of shock chemistry in associated molecular gas (DeNoyer and Frerking, 1981 ; Burton, 1987) has recently been shown to have more shocked, neutral cloudlets than previously suspected (Huang, Dickman, and Snell, 1986). Bipolar outflow sources in molecular clouds are of great interest for their dynamical interaction with their surroundings and for their relations to young stellar objects. Excepting the unusual outflow in Orion (Blake et al., 1987), almost nothing has been written about their effects on cloud chemistry; yet some of the bipolar outflows involve 10-100 solar masses of


molecular gas (Lada, 1985). The enhanced abundance of CS in the bipolar outflow in NGC 2071 (Takano, 1986) certainly suggests shock chemistry.

More work is needed on nonequilibrium rate coefficients for important molecular processes. Shock chemistry in the bipolar outflow sources needs to be investigated both observationally and theoretically. Despite the attention given to diffuse cloud shocks, there has not been a self-consistent combined model of a shock and its associated quiescent cloud.

V. The CH' Problem

One of the oldest and most serious challenges for interstellar chemistry is the explanation of the high abundance of CH+ in diffuse clouds. Not only is CH + extremely widespread, but its abundance is comparable to that of CH in many clouds. Numerous attempts have been made to account for CH'. The earlier work is reviewed by Black, Dalgarno, and Oppenheimer (1975) and Dalgarno (1976a). The interstellar problem has stimulated much effort on molecular processes. Reactions of CH+ with various atoms and molecules have been investigated. The processes of radiative association,

C+ + H +CH+ + hv (34)

C H + + h v + C + H + (354

CH' + hv+C+ + H (35b)

(Uzer and Dalgarno, 1978, 1979; Kirby et al., 1980) have been studied in detail.

The problem is summarized as follows. In diffuse clouds, CH is destroyed most rapidly by photodissociation at a rate of approximately s- ' in the unattenuated background starlight (van Dishoeck, 1987). If CH and CH+ were formed in steady state at comparable rates, then the ion should be rather less abundant than the neutral at the densities and temperatures inferred for most diffuse clouds, because CH + reacts rapidly with the most abundant interstellar species,

(Bates, 1951; Graff et al., 1983), and photodissociation

CH' + H + C+ + H,, (36)

at a rate k,, z 1.3 x lo-' exp(-209/T) cm3 s- ' based on the rates of Gerlich et al. (1987) for the reverse process, and

CH' + H, +CH: + H (37)

498 John H. Black

at a rate k 3 , = 1.4 x cm3 s - l at T = 80 K (Smith and Adams, 1981). Dissociative recombination will be relatively less important in predominantly neutral regions and there are indications that this process does not operate at all in ground-state CH' (van Dishoeck, 1987). All attempts to explain the CH+ abundance in steady state in diffuse clouds have thus failed. The magnitude of the difficulty can be illustrated with results from a steady state model of the i Ophiuchi cloud (Model G of van Dishoeck and Black, 1986) which reproduces well the observed column density of CH. The observed column density of CH' is N(CH') = 2.9 x ern-'. In the model, N(CH') = 2.8 x 10" cm-', a factor of lo3 too small. The model also can be used to evaluate the contributions of additional possible sources of CH'. It has been suggested that reactions of C' with vibrationally excited H, ( u 2 l),

Cf + H,(v 2 1) -+ CH+ + H, (38)

may be an important source (Stecher and Williams, 1974; Freeman and Williams, 1982). The concentration of H, (u 2 1) produced by ultraviolet fluorescent excitation is computed as a function of depth through the model, and the resulting CH' production rate can be evaluated for k3* = 2 x lop9 cm3 s- '. This additional source can approximately double the previous CH+ production rate in the ( Oph cloud model (Black and van Dishoeck, 1988). If large molecules like PAHs are present in this cloud, then some fraction of their reactions with C' might lead to formation of CH'. Assuming equal probabilities of CH + formation and charge transfer in reactions of C+ with neutral PAHs and taking the maximum PAH abundance suggested by the analysis of the [ Oph cloud by Lepp et al. (1988), we find a factor of four possible increase in the CH' production rate. Unlike the analogous discussion of nitrogen chemistry, hot carbon ions produced in the reaction

He' + CO -+ He + C' + 0

provide a negligible source of CH' in direct reactions with H,, even if the most generous assumptions are made about the rates of thermalization and reaction. If the above additional sources are included in the 5 Oph cloud model, a revised column density N(CH+) = 5.8 x 10" cm-' is predicted. This still fails, by a factor of 50, to reproduce the observations. There may exist some regions, however, in which a quiescent component of CH+ would be detectable.

The remaining viable explanation of the CH+ abundance at the present time involves formation at high temperature (T > 1000 K) in shock-heated gas by the reaction

(39)

C + + H, -+ CH' + H - (0.369 eV) (40)


as proposed by Elitzur and Watson (1978, 1980). The amount of CH' that can be produced in a diffuse cloud shock is then limited principally by the cooling length of the shock, i.e., the distance over which the temperature is high enough that Reaction (40) proceeds at an appreciable rate. The shock also increases the density and the rates of destruction of CH' by Reactions (36) and (37) so that the CH+ abundance is also sensitive to the pre-shock molecular fraction and total density. The effective rate of Reaction (40) in an interstellar shock is affected by nonequilibrium level populations in H, and by non-Maxwellian velocity distributions (Herbst and Knudson, 1981). Gerlich, Disch, and Scherbarth (1987) have measured state-selective reaction rates for Reaction (40) as functions of translational energy and have analyzed the results thoroughly in a way that is directly applicable to interstellar shocks.

A significant factor in the production of CH+ is the effect of a magnetic field on the structure of the shocked gas and on the relative velocities of ions and neutrals (Hartquist and Dalgarno, 1982). In general, nonionizing MHD shocks tend to have lower peak temperatures but thicker cooling zones than nonmagnetic shocks of the same speed. The investigations by Draine (1986a), Pineau des For&, Flower, Hartquist, and Dalgarno (1986), and Mitchell and Watt (1985) all suggest that nonmagnetic shocks can produce N(CH+) x lo', crn-', but fail to explain the higher column densities, N(CH+) x 10'3-10'4 cm-', that are observed in some regions. The MHD shock models, however, yield CH + column densities in good agreement with measured values in the best studied diffuse clouds for reasonable assumptions about shock speed and pre-shock density. Such shocks have other observable consequences such as their substantial contributions to the column densities of rotationally excited H, and to the abundances of various other molecules. Indeed, it is a prediction of the MHD models (Draine and Katz, 1986a, 1986b; Draine, 1986a; Pineau des Forets, Flower, Hartquist, and Dalgarno, 1986) that much of the H, in levels J = 3 - 5 observed in diffuse clouds with CHf must arise in shocked regions. In steady state models of diffuse clouds (Black and Dalgarno, 1977; van Dishoeck and Black, 1986), the excited H, is maintained by the ultraviolet fluorescence that must accompany the photodissociation of H,, which in turn balances its rate of formation on grain surfaces. The steady state models that reproduce observed amounts of rotationally excited H, in the u = 0 vibrational state also predict detectable concentrations of vibrationally excited H, with a vibrational excitation temperature that increases with u. The MHD shocks cannot maintain such large amounts of vibrationally excited H,: therefore, sensitive searches for H, in u 2 1 with the Hubble Space Telescope will provide a clear test of the effects of shocks and of steady state excitation of H, in diffuse clouds.

Kinematic information from line shifts and line profiles provides another kind of observational test of the shocked-gas origin of CH'. Absorption lines

500 John H. Black

of species formed in shocks are expected to be typically broader than those formed in the quiescent gas and to be shifted relative to them. The amount of the shift will depend on the speed and structure of the shock and on its inclination to the line of sight. In MHD shocks, the neutral and ionized species within the disturbed region are predicted to have mean Doppler radial velocities that differ by as much as 3 km s - l (Draine, 1986a). Direct identification of distinct shocked and quiescent components in species like CH, and unambiguous observation of ion-neutral velocity differences between CH and CH' in the shocked component would provide confirmation of the theory.

Lambert and Danks' (1986) observations of CH' give further support to the idea that it is formed in shocks. Lambert and Danks demonstrate an empirical correlation between CH' column desnity and the column density of rotationally excited ( J = 3 - 5) H,; moreover, they show that

where T,, is the excitation temperature that characterizes the relative populations of H, in J = 3 - 5. The kinematic information is more ambiguous. Although there is an apparent tendency for the CH' lines to be broader than those of species ascribed to the quiescent gas, velocity shifts between CH' and CH or Ca are small. In the past, such investigations have been plagued not only by inadequate resolution, but also by inaccurate rest wavelengths. The measurement by Carrington and Ramsay (1982) of the position of the CH' A-X (0,O) R(0) line, v,,, = 23619.780 cm-' or ;lair = 4232.5478 A, differs by +0.62 km s - l in Doppler velocity from the wavelength adopted in much of the earlier astronomical work (;lair = 4232.539 A). It now appears that the mean position of the unresolved CH A-X (0,O) Rf(l/2) A-doublet is v,,, = 23247.5803 cm-' or ,lair = 4300.3132 A (Black and van Dishoeck, 1988) rather than the conventional value of ,lair =

4300.321 A, a difference of -0.54 km s - l in Doppler velocity, based on the measurements of other CH line positions of Brazier and Brown (1984) and the ground-state term energies of Bernath (1987). New, more accurate line positions for the B-X bands of CH will be available soon (P. F. Bernath, private communication). Very recent observations of CH and CN (Palazzi, Mandolesi, and Crane, 1988) and of CO (Langer et al., 1987) towards 5 Ophiuchi appear to rule out the presence of distinguishable shocked and quiescent components of these species in this particular cloud.

In summary, formation of CH' in shock-heated gas is the only surviving, viable explanation of its high abundance at the present time. More sensitive observational tests of the theory are urgently needed.


VI. The Excitation of Interstellar CN

Interstellar CN molecules in diffuse clouds can be used as remote radiometers for sensing the brightness temperature (intensity) of the cosmic background radiation at the wavelengths of its lowest rotational transitions, N = 1 -+ 0 and N = 2 + 1, at 2.65 and at 1.32 mm, respectively (Shklovsky, 1966; Field and Hitchcock, 1966; Thaddeus and Clauser, 1966; Thaddeus, 1972). The dipole moment of CN, p = 1.450, and its rotational transition probabilities, e.g., A,, = 1.2 x lop5 s-’ for N = 1 + 0, are large enough that its excitation might approach the low density limit in diffuse clouds. In this limit, the rotational populations are governed only by the rates of absorption and of spontaneous and stimulated emission in the ambient millimeter wavelength radiation. In this limit, the measured excitation temperature

should equal the brightness temperature Tan of the

(42)

radiation field at the . _ _ frequency vN”” of the transition between levels of rotational quantum number N’ and N”, where N N , is the measured column density of molecules in state N‘. How closely T N , , N , + Tad depends upon the local, collisional contribution to the excitation of CN. Meyer and Jura’s (1985) and Crane et al. (1986) observations of CN in diffuse clouds determine To, to f 1 %; therefore, small corrections for local excitation may be important in deriving the true value of Tad.

This is one of the rare instances in which the collisional excitation of an interstellar molecule is probably controlled by electron impact. Although the cross sections for rotationally inelastic scattering of H and H, by CN are not known, reasonable estimates suggest that neutral impact contributes no more than 10- 15 % of the electron impact excitation in the centers of diffuse clouds where the density ratio n(e)/n(H,) % n(e)/n(H) x 3 x Thermal rate coefficients for N = 0 -+ 1 and N = 0 + 2 excitation by electron impact have been published by Allison and Dalgarno (1971) and these supersede the earlier results of Crawford, Allison, and Dalgarno (1969). Jamieson, Kalaghan, and Dalgarno (1975) also demonstrated that proton impact excitation is unlikely to be important in this context.

The electron density and kinetic temperature are not measured directly in any diffuse cloud, but must be inferred from careful model analyses of a variety of observational data. In order to illustrate the local effects, results are presented for the calculated excitation of CN at the centers of three different

502 John H . Black

TABLE I

EXCITATION OF CN IN THE [ OPH DIFFUSE CLOUD

Model VRA Model 6 vDB Model B vDB Model G Observation

n(e) [cm -’I 0.25 0.10 0.044 T CK1 40 20 30 T a d CK1 2.70 2.773 2.785 2.74 & 0.02“ To, CKI 2.800 2.800 2.800 2.800 & 0.027 TI2 CKI 2.730 2.780 2.789 2.757:;; T’( 113.5 GHz) [K] 0.0141 0.0042 0.0025 TA( 113.2 GHz) [K] 0.0074 0.0024 0.0015 T”(226.9 GHz) [K] 0.0009 0.0003 0.0002

a Mean brightness temperature of the microwave background in the wavelength range 0.1-50 cm. The most precise single measurement gives Tad = 2.783 0.025 K.

models of the 5 Ophiuchi cloud (Table I). The densest model is VRA 6 of Viala, Roueff, and Abgrall(l988). Two less dense models, vDB B and vDB G of van Dishoeck and Black (1986) are shown for comparison. The excitation has been computed for a 5 level CN molecule including all radiative and electron impact transitions at the indicated densities and temperatures. Radiative transfer in the rotational lines is treated in a fully self-consistent manner by an escape probability method. Line center opacities in the N = 1 -+ 0 transition are z = 0.1 - 0.2. Hyperfine structure has been ignored in the excitation and line formation calculation. In all cases, the column density, N(CN) = 2.9 x 10l2 cm-2 and line width A V = 1.46 km s-l, inferred for a single cloud component from the measurements of Crane et al. (1986), have been adopted. Results are presented for the radiation temperatures that give the observed value To, = 2.800 K in each model. Note that a weighted mean value T a d = 2.74 k 0.02 K refers to direct measurements at wavelengths 1 = 0.1 - 50 cm and is in harmony with all three models (Smoot et al., 1987). If the very accurate radiometric observation of Johnson and Wilkinson (1987) at 1 = 1.2 cm is adopted, Tad = 2.783 k 0.025 K, then the model of lowest density is slightly favored. The presence of local excitation means, of course, that the rotational lines of CN will appear with intensities slightly in excess of the background intensity. Thus, the table also lists the computed Rayleigh-Jeans antenna temperatures, T’(v), expected for selected spin components of the CN rotational transitions in each model. These values of TA(v) refer to the signal that would be measured by a perfect antenna viewing an extended source and they sum over the hyperfine components. A search for these lines would provide an observational test among the competing models and would help calibrate the local excitation correction to Tad directly.


It can be noted in passing that electron impact excitation of CO (Crawford and Dalgarno, 1971) is much less important than H or H, impact in most interstellar regions. The electron impact cross sections for CO are much smaller than those for CN owing to the large difference in diplole moments.

VII. Models of Interstellar Clouds

The identification of major processes of molecular formation and excitation is only one step in the interpretation of astronomical observations. Models of interstellar clouds must be constructed in order to test theory in comparison with observation and in order simply to extract physical information (i.e., temperature, density, intensity of radiation, and abundances) from the observations. The modeling of interstellar clouds is a very complex problem. It is complicated by the need to describe the physical state of a nonequilibrium system in terms of a vast number of microscopic processes. Comparison with observation is further hampered by the fact that most real clouds are irregular in shape with a hierarchy of structures over a large range of linear scales. The literature on models of molecular clouds has been reviewed for example, by Prasad et al. (1987) and van Dishoeck (1988). Only some aspects of the subject will be addressed here.

One general issue that has not been resolved fully in practice is the division of responsibility between theorists and observers for presenting “observable” properties that relate to molecular abundances. In the case of radio emission lines from thick, dense clouds, fractional abundances do not come directly from the observations. Relatively weak (i.e., unsaturated) lines have measurable intensities that are proportional to the column densities of emitters: a molecular column density can be derived provided that basic data on line strengths are available and that corrections can be made for excitation (i.e., for molecules in unobserved energy levels). The abundance of H, is generally not observable, however; therefore, the column density of a minor species can be related to its fractional abundance only through indirect estimates of the total molecular column density. In many cases, the uncertainties involved are small enough to provide quite useful tests of theoretical models. The comparison of column densities of various minor species with each other would seem to avoid the necessity of knowing the total hydrogen abundance. This is true only to the extent that the observations sample the same volumes of space and that the abundances are insensitive to fluctuations or gradients in temperature, density, and radiation field. In some cases, emission lines of a molecule will be weak not because its abundance is low but because it couples so strongly to the local radiation by absorption and stimulated emission that

504 John H. Black

collisions are unable to make its own emission perceptible above the intensity of the background radiation. This effect can lead to errors of a factor of ten or more in derived abundance and will be particularly severe for molecules like the metal hydrides MgH and NaH, which have large dipole moments and fundamental rotational transitions at high frequencies (cf. Bernath et al., 1 98 5).

The interpretation of saturated (i.e., optically thick) emission lines of abundant molecules is a complicated problem. The measurable intensity is not proportional to the total column density of emitters and the line formation is governed in a complex way by the cloud structure. In an idealized case, where the macroscopic Doppler velocity gradient is large compared with the microscopic velocity dispersion on the scale of a photon’s mean free path, molecular abundances can be extracted from measured line intensities. This is conventionally called the “large velocity gradient” approximation (de Jong, Chu, and Dalgarno, 1976); in its pure form, it makes rather special assumptions about the structure and kinematics of a cloud. Other kinds of assumptions can be used to relate line intensities to abundances: For example, the kinematic structure can be treated as microturbulence or some details of radiative transfer can be avoided by treating the line formation with mean escape probabilities. An alternative approach to the interpretation of abundance data is for the theorist to compute the line intensities and profile shapes that are actually implied by a model and to compare these with observations rather than to rely on the observer to extract “abundances” from the data. This requires, of course, that the model represents cloud structure and provides abundances as functions of depth. In principle, the emergent spectrum can be calculated explicitly for a model with specified velocity field and gradients of density and temperature. In a few cases, this approach to modeling dark clouds has been carried out (de Jong, Dalgarno, and Boland, 1980; Boland and de Jong, 1984). This method is of special value in treating issues of isotope fractionation; not only are fractionation effects often likely to be sensitive to temperature, density, and depth-dependent photochemistry, but the observational data often require comparison of both saturated and unsaturated lines.

In the case of diffuse clouds that can be studied by means of optical absorption line techniques, the extraction of column densities from measured line intensities is more straightforward, although not lacking in subtleties. The problem of modeling these regions remains complex (Black and Dal- garno, 1977; Viala, 1986; van Dishoeck and Black, 1986; Viala, Roueff, and Abgrall, 1988). In part, the complexity is a response to the demands of accurate observations: for example, some molecular column densities can be determined to an accuracy of 10% or so, limited by the uncertainties in oscillator strengths rather than by observational error or uncertainties about


cloud structure. In part, the complexity is also inherent. The backbone of diffuse cloud chemistry is the variation with depth of the abundances of the principal constituents, H and H,. We have already seen that the abundance of H, in dilute interstellar gas is inextricably linked to its excitation by a combination of collisional and radiative processes. Indeed, a full description of the hydrogen in a diffuse cloud can involve treating a statistical equilibrium for hundreds of energy levels and thousands of radiative transitions. Models of diffuse clouds have had mixed success in reproducing all details of the best observations. Their predictions, however, are indisputably useful in identifying critical tests of theory and thus, in extending our understanding of processes in all interstellar clouds.

In the future, detailed models will be important for describing the evolution of interstellar clouds. Although much work has been done on isolated treatments of time-dependent chemistry, the combined treatment of dynamical evolution and chemistry is in its infancy (cf. Gerola and Glassgold, 1978; Tarafdar et al., 1985).

VIII. Summary

Some problems in interstellar chemistry have been investigated by now in great detail if not solved satisfactorily. The subject benefits from the fascina- tion of giant molecular clouds, exotic molecules, and a great variety of interesting molecular processes. At the same time it suffers from the inherent complexity of the molecular cloud systems; unlike stars they are neither spherical nor in thermodynamic equilibrium to first order. Some issues have remained vital from the very beginning of interstellar chemistry as a subject: the explanation of the CH' abundance is a good example. In other cases, such as the excitation of CN, it is a 1 % effect that is demanding interpretation. Certainly, there is no shortage of unsolved problems that require contributions from a variety of disciplines.

The reader who has reached this point will have noticed that virtually every topic mentioned has seen some contribution by Alex Dalgarno. This is not accidental, indeed it is unavoidable as his influence on this field has been pervasive.

ACKNOWLEDGMENTS

Dr. E. F. van Dishoeck is thanked for many helpful comments and discussions. The support of the National Aeronautics and Space Administration through astrophysical theory grant NAGW-763 to the University of Arizona is gratefully acknowledged.

506 John H . Black

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493-51 1.

INDEX

A action integral, 314 action-angle variables, 423 actual eigenvalues. 402 adiabatic

(Born-Oppenheimer) wavefunctions,

basis, 335 behavior, 65 Born-Oppenheimer approximation, 441 Born-Oppenheimer states, 464 core-polarization

model, 171 potential, 171

equations, 336 formulation, of equations, 336 functions, 466 levels, 467 nuclear wavefunctions, 465 potential energies, 465 potential energy curves, 464, 466 potentials, 260 representation, 142, 458, 465, 466, 468 states, 468

aeronomy, 267 airglow, 23, 24, 451 Airy functions, 334 AlfvCn waves, 489 algebraic equations, 386 algorithms, 324, 326, 327, 331, 333, 334,

335, 331, 338 numerical, 338 stable efficient. 339

464-465

alkali halide beam, deflection of, 55 alkali halides, 491 alkali-metal dimer, 189 ambient gas, 441, 442 ambient radiation field, 3 I ambipolar diffusion, 32-33, 496 amplitudes, 314, 351, 356

length form, 384

of transitions, 375, 379 velocity form, 383

angle of deflection, 368 angular coupling factor, 380 angular distribution, 457

experimental, 364 for electron capture, 368 measurements of, 364

angular factors, 380 angular momentum, 140, 383, 384, 385, 403,

421. 423 algebra, 272 coefficients, 385 components, 381 of the continuum electron state, 408 quantum numbers, 369 techniques, 38

angular part, of Dirac equation, 396 anharmonic systems, 426 anharmonicities, 422 approximate wave functions, 210 approximation methods, 422 approximation. 257, 320, 327, 335, 339, 401,

439 finite different, 271 first order, 349 in transition state, 420 without, 313

associative detachment, 492 reactions, 69

associative-ionization process, 15 astronomical observations, 503 astronomical spectra, 478 astronomy, see infrared astronomy astrophysical

applications, 438

considerations, 441 environments, 441, 438 literature, 481 plasmas, 251

of photodissociation processes, 473

513

514 INDEX

astrophysics, 62, 67, 252. 267, 414, 438

asymmetric line profiles, 469 asymmetric line shapes, 466 asymmetric model, 353 asymmetry parameter, 234 asymptotic

importance of hydrogen in, 283

behavior, 246, 324 of resonance wavefunction, 242 oscillatory, 328 boundary conditions, 142 condition, 3 16 couplings, 47 1 final state, 308 form, 306, 307, 360 for arbitrary capture cross section, 367 of scattered radial function, 269 plane, 311-312 region, 326, 329, 330, 332, 335 series, 169 solution, 323 for long range potentials, 330

ATI, 154, see also above threshold ionization Atmosphere Explorer Satellite Project, 25, see

Atmosphere Explorer, 25 atmosphere

also Atmosphere Explorer

cometary, 442 of Earth, 448 of Venus, 448 planetary, 442 structure and composition of, 24

chemistry, 425 of photodissociation processes, 473

atmospheric parameters. 25 electron energy distribution function, 25 electron temperature, 25 ion composition, 25 neutral composition, 25 ultraviolet solar radiation, 25

atmospheric physics, 62, 438 atom abstraction reaction, 489 atom, excited state of, 381 atom-rare-gas interaction, 189 atomic basis set, 357 atomic beam

atmospheric applications, 438

measurement, of polarizabilities, 4 1 methods, 39 techniques, 41

atomic beams, 46, 50 atomic channel state, 142 atomic collisions, 8, 106

charge-transfer processes, 8 fine-structure excitation, 8 low-energy, 8 spin-changing, 8

atomic core, 167 atomic deuterium, important role of, 494-495 atomic electronic eigenfunctions, 14 1 atomic excitation

problems, 128 rate of, 117 spectrum, 389

atomic excited state, 381 atomic fine structure levels, 470 atomic form factor, 1 18 atomic fragments, 428 atomic ground state, 383

atomic hydrogen, 488 atomic ions, 488, 490 atomic limits, 469 atomic line broadening, 140 atomic line shape theory, 146 atomic line shapes, statistical mechanics

atomic lines, overlapping, theory of, I43

atomic operators, 156 atomic oscillator strengths, 196 atomic parameters, methods of calculating,

Hartree-Fock description of, 375

of, 143

wings of, 140

nonperturbation, 37 perturbation, 37

atomic perturbation theory, 38 atomic perturbers, 140 atomic physics, 223, 405

calculations, 4 1 1 data, 267 nonrelativistic, 393

atomic polarizabilities, 38. 41, 57 atomic processes, 25 1 atomic projection operator, 387 atomic radial wavefunctions, 367 atomic Rydberg states, 368 atomic shells, 376 atomic species, minor, 478 atomic spectral lines, 133, see also spectral

atomic spectroscopy, I54 lines, atomic

INDEX 515

atomic state, of system, 358 atomic structure, 39 atomic structure code SUPERSTRUCTURE,

atomic systems, 164. 289 182

determination of properties, 183 of core electrons and valence electrons,

one-electron, 173 atomic transition moment, I38 atomic transitions, 146 atomic zone, 308, see also Zone I atomospheric effects, 84 atoms

166-167

highly ionized, 292-293 minor metal, 488 neutral, 344

multielectron, 353 attractive potential energy curve, 443 attractive well, 89 attractive well depth, 74 aurora, 23, 24, 451 Autler-Townes

absorption, 148 effect, 147 splitting, 147

auto-double-ionization, 290 autocorrelation function, I 1 I , 112 autoionization, 9-10, 292

time-independent variational approach to, 9-10

autoionizing levels, 175, 176 averaging procedures, 423 Avogado’s number, 40 avoided crossings, 16, 464, 466, 468

B B state wavefunction, 452 B-splines, 388, 389, 404-405

calculation, 407 eigenvalues, 405 method, 405 pseudospectrum, 388

6-X bands, 30, 500 B-X transition dipole moment function, 451 BIB amplitudes, 351, 361-362 BIB approximation, 351, 353, 354, 356, 362,

363, 364, see also boundary corrected first Born

BIB cross sections, 353, 364-365

scaling properties of, 367 BIB, relativistic version of, 370 B2Bo approximation, 364, 366, 369 B2Bo cross secions, 364-365, 366 B2B amplitude, numerical calculation of,

365 B2B cross sections, 365 background

cross section, 467 intensity, 502 noise, 92 radiation, 504 starlight, 492, 497 state, 426

balance method, 44 balanced basis set, 401 band absorption oscillator strength, 443 band oscillator strengths, 189 basis expansion approach, 386 basis functions, 394 basis idea, 405 basis set, 238, 394, 395, 397, 398, 399, 401,

402, 404, 413, 414 expansion, 388 parameters, variation of, 41 1

basis-spline method, 404-405, see also

Bates approximation, 363, 367

Bates distorted wave approximation, 357, 363, Bates distorted wave model, 356-357 BBGKY hierarchical formalism, I03 beam deflection profile, monomeric, 54 beam deflection, slope method, 48 beam foil technique, 179-180 beam gas spectroscopy, 179 beam intensity, 45-46, 54 beam measurements, methods, 41, 42, 44

B-spline

cross sections, 365

atomic, 42, 57 molecular, 57 deflection profile analysis, 41-42

beam polarizability, 52 beam profile,

deflected, 54 of deflected dimers, 54

beam techniques, 57 beam width, 43, 48 beam-foil spectroscopy, 177 beams,

deflected, 53

516 INDEX

beams, (Continued) dimer, 53 monomer, 53

bending vibrational-molecular rotational

Bessel functions, 333, 385

Bethe’s generalized oscillator strength, I18 Bethe-Bloch result, 110 Beutler-Fano line profiles, 466, 469 Big Bang cosmology, 493 bimolecular reactions, 418 binary collision

approximation, 145 matrix, 122 theory, 145

motion, highly coupled, 421

spherical, 383

binary thermoneutral reaction, 67-68 bipolar outflow, 497

bispinors, 400, 412 Bohr magnetons, 46 Bohr radius, 97 Boltzmann factors, 114 Boltzmann’s constant, 273 bond breaking, 0-0, as prototype, 427 bond fission process, 430 bond polarizability, 56 bond-selective photochemistry, 429 Born approximation, 17-18, 123, 124, 224,

225, 282, 324, 344, 345 Born calculations. 124 Born cross sections, 124 Born model, 303 Born series,

sources, 496

behavior of, 366 convergence of, 366

Born term, 225 Born terms, first and second, 236 Born treatment, 15 Born-Oppenheimer

approximation, 442 calculations, 142

Born-type series, 361 bosons, 96, 97 bound channel, one, 469 bound diabatic vibrational levels, 467 bound levels, 440 bound nuclear function, 458 bound state, 222, 239, 240, 242, 247, 282,

458, 464, 465

eigenvalue, 403 electronic, 439 energy, 394 equation, 242 excited electronic, 428, 429 excited, 441 molecular, 457 solutions, 396 targets, 276 theory, 230 wave functions, 137, 272, 275

bound vibrational functions, 443 levels, 439-440, 441, 445, 456, 457 wavefunctions, 454

bound wavefunctions, 45 1 bound-bound spectroscopy. 137 bound-bound-free resonance-fluorescence

bound-continuum problems, 439 bound-free

spectra, 141

emission process, 454 integrals, 232 line shape function, notation for, 136-

transitions, 142, 179

energy, 223, 241-242 integrals, 23 I properties, 173 spectra, 170

137

bound-state

boundaries, in density-temperature plane, 104 boundary,

conditions, 325, 326, 327, 328, 329, 330, 336, 348, 352, 353, 358, 401 cavity, 388 MIT bag model, 388 relativistic, 412 semi-infinite, 326

approach, 364 first Born, 351, see also BIB model, 370 second Born approximation, 365, see also B2B

function method, 310 outer, 31 1 value, 326

corrected

bounded wavefunction, 241 boundness, necessary condition of, 398

INDEX 517

bounds, 247, 395, 400, 414 for multichannel scattering, 226 for phase shift, table of, 223 for scattering parameters, 22 1-222

branching ratios, 470-471 Breit interaction, 377

corrections, 413 Breit-Pauli operators, 183 Breit-Rabi formula, 44 Breit-Wigner, 242

method, 246 widths, 244

bremsstrahlung, 127 brightness temperature, 501 broad band continuous emission, 30 broadened emission lines, 282 broadening, 144

mechanisms, 134 of allowed atomic lines, 138 of Cs resonance line, 142 of discrete absorption peaks, 441

Bubnovaalkerkin eigenvalue equation, 240

C c-functional, 236 C-functional formalism, 247 calculation of interaction energy, accurate, 263 capture amplitude, 35 I , 352 carbon, 484

chemistry, 484, 487 ions, hot, 498 stars, 493

coordinates, 423 momentum components, 423

negative-energy projection operators, 402 positive-energy projection operators, 402

Cartesian

Casimir

Cauchy propagator, 337, 338 causal coordinate, 3 12 causality, 31 I caustics, 314, 320 cavity radius, 388 cavity spectrum, 388 CD W

amplitude, 359, 360, 363 approach, relativistic, 371 approximation, 360, 363, 367 calculations, 364 model, 359-360

perturbation series, 365-366 wave functions, 360

cellular model, 110 central potential, arbitrary, 405 centrifugal barriers, 472

centrifugal potential, 457 centrifugal terms, 283, 330, 332 channels, 245, 283, 323, 442

rotational, 457

coupling, 234 final, 363 functions, 174, 382 inelastic, 91

charge distributions, final state, 356 initial state, 356

charge exchange, 343, 344 charge imbalances, 114 charge transfer, 16, 92, 353, 357, 361, 366,

489, 490, 491, 494 cell, 84 collisions, 353 cross section, 95, 367 for thermal collisions, 17 measurements of, 94 neutrals, 92 radiative, 17 reactions, 357 theory, 364

charge-charge structure, 128 charge-density disturbance, 109 charge-density fluctuation, 112 charge-particle scattering, 230 charge-transfer process, 62 charge-transfer reactions, 63, 66, 69 charge-transfer, constraints on, 63-64 chemical

activation, 425 bonding, 66 model, 487 physics, 420 reaction dynamics, 417 reactivity, 417

chemiionization processes, 489, 491 chlorine chemistry, 488 circular polarization, 156 circumstellar envelope, of carbon star IRC +

10216, 491 classical behavior, 431 classical distribution functions, 423

518 INDEX

classical double scattering mechanism, 368 classical dynamics, 135

classical kinematics, 368 classical mechanics, 313, 423 classical orbits, 3 12 classical path

of coupled oscillator systems, 431

approximation, 144 formalism, 142 methods, 139, 140, 142

derivations of, 139 of proton, 255 theory, 141

classical phase-space arguments, 42 I classical region, 335 classical systems, 423 classical theory, 419 classical trajectory calculations, 156 classical unimolecular dynamics, 423 close collision, 257 close-coupled

calculations, 258, 260 equations, 324, 327 quantal calculations, 263 result, 258, 260 semiclassical formulations, 258 theories, 255 of one-perturber line shape, 141 quantum mechanical, 142

close-coupling, 257, 261, 265 approach, 339 approximation, 290

equations, 268 results, 269

calculation, 124, 283, 295, 297, 472 equations, 238, 272, 283, 323 expansion, 268, 270, 272, 275, 276, 280,

methods, 189 results, 275

treatment, 18 closed atomic subshells, 380 closed channels, 238, 328, 338

281. 282, 291

for transitions, 288

configurations, 238 solutions, 330

closed form expression, 360 closed-channel components, 237 closed-shell atom, 379-380, 410 closed-shell many-electron system, 377

cloud chemistry, 496 cloud, structure and kinematics of, 504 cloud, translucent, 492 clouds,

cores of thick, 489 dense, 484 interior of, 479, 483 thick, 490 see also diffuse clouds; opaque clouds

CM scattering angles, 96-97 coefficients, 196, 204, 283, 333, 405

angular momentum, 382 matrix of, 329

asymmetrical, 352 calculations, 255, 283 cross sections, 101 dynamics, 16, 124-125, 141 energies, 106, 127, 140 first with the projectile, 368 formulation, of Heil er al . , 263 induced spectra, 141 kernel, 146 lifetime matrix, 105 models, 124 (molecular), excitation transfer in, 15 perturbed by plasma environment, 122 plane, 258 process, 257 products, investigation of, 91 products, ionic, determination of, 91-92 proton-ion, 34 quantal and semiclassical treatments of, 262 radiation emitted during, 432 rate, 65

constant, 65, 66 strength, 102, see also effective collision

strength strong, 123 theory, 39, 126, 282

collision

applied to radiative process, 17 of atomic, with rotating molecules, 18

times, 16

wave function, 271 two-body, 134

collision-induced absorption Coefficients, 17 collision-induced transitions, 135, 138 collisional

contribution, 501 damping, 156

INDEX 519

dissociation, 48 I duration, 140 excitation, 252

interaction, 106 processes, 480, 505 relaxation rates, 158 stabilization, 73

of interstellar molecule, 501

collisionally excited emission lines, 496 collisionless multiphoton excitation (MPE),

146 collisionless multiphoton dissociation, 152, see

also MPD collisions, 67, 98, 106, 255, 367, 369, 481,

504, see also ion-neutral collisions atomic, 115 between ions and atoms, 343 close, 141, 260 distinct, 116 electron, 262 electron-electron. 24, 25 electron-ion, 34, 124, 268 fast, 343, 344 multiple, 142 neutral-neutral , 75 of electrons with atmospheric gases, cross

section for, 25 of molecular ions with neutrals, 71 of vibrationally excited ions, 72 one-perturber, 134 optical, 146 projectiles from, 96-97 proton, 252 single, 351 strong, 143, 145, 257 thermal, 17

three-body, 143 velocity-changing, 134, 1346 weak, 141, 143

collisions (molecular) charge transfer of, 15-16 cross sections for, 14 elastic scattering in low-energy, 15 fine-structure excitation in low-energy,

ion-ion, 15 spin change in, 14-15

collisions of electrons, with atoms, 268 collisions of reagents, 432

charge transfer for, 17

15

collisional applications, of screening concepts,

collisional ionization, dense plasma’s influence

column densities, 489-490, 498, 499, 500,

combination vibrations, 428 cometary atmospheres, 438, 442, 445, 466

models of, 470 cometary OD, 463 comets, 34 complete spectrum, 398 complete wave function, 269 completely bound potential surface, 429 completeness, 400, 414

necessary condition of, 398 complex molecules, formation of, 485 complex R matrix method, 246 complex rotation method, 246 complex-coordinate rotation methods, 23 I complex-vibrational predissociation model, 73 computational methods, 268 computer codes, 324, 335 computer programs, 270, 271-272, 423 configuration interaction, 442, 457

110

on, 128

503, 504

wavefunctions, 233 expansion, 174 terms, 270

conservation, energy and momentum,

constant transition moment, 135 continua, 446, 449-450

continuous

124

of Lyman and Werner systems, 445

absorption, 439, 440, 452 cross section, 45 1 emission spectrum, 454 heat, 455 luminosity, 455 molecular emission spectrum, of hydrogen,

radiation, 17 spectra, 134 spectrum, 453, 478

continuum, 398 channel, one, 469 channels, two, 469 components, 237 contribution from, 275 contributions, 276

141

520 INDEX

continuum, (Conrinued) cross section, 446

dissolution, 395, 41 1 distorted wave approximation, 359, see also

electron, 233 energy state, 394 importance of, 276 lowering, 116 measurements, 452 molecular state, 457 multiple scattering method, 235 nuclear function, 458 of repulsive state, 441 opacity, 441 phenomena, 23 1 process, quantum mechanical treatment of,

solutions, 408 spectra, 439

states, 39, 383, 388, 465

in molecular oxygen, 453

CDW

438

spectroscopy, I37

properties, 173 modified density of, 128 of the target, 272 threshold, 156

vibrational wavefunctions, 454 wavefunctions, 358, 443, 444, 451, 466

conventional multipole expansion, 167 convergence, 196, 198, 224, 226, 232, 238,

247, 268, 282, 310, 311, 334, 394 criterion, 413 of close-coupling approximation, 272 of close-coupling expansion, 275, 276,

of close-coupling results, 273 of the rate coefficient, 279 of the single-center expansions, 233 properties, 357 rapid, 349 rates of expansion, 204 sufficient conditions for, 398 table, 232

277

convergent solutions, 3 10 coordinate methods, 43 1 coordinate states, 423 copper vapor laser, 298 core

polarizability of, 172

spherically symmetric, 164-165, 169 unperturbed, 166

core electrons, 163, 183, 184 core excitation energies, 166, 172 core Hamiltonian, 165 core orbitals, 169, 233 core polarization, 186

potentials, 49 term, 184

core potential, 168, 183 core projection operator, 165 core states, 164

occupied, 169 core-core interactions, 187 core-electron, 189 core-polarization correction, to dipole matrix

element, 182 core-polarization effects, 14, 185, 189 core-polarization terms, 182, 185, 186 core-valence orthogonality requirements, 164 cores, few-electron spherically symmetric,

correct phase, restoration of, 344 correction terms, short-range, 174 corrections,

arising from dipole term, 179 first-order to HF potential, 378 for scattering event, 96 made for excitation, 503 relativistic, 369 successive, 325 to atomic properties, 395

188

correlate wave function calculations, 276 correlation, of ingoing and outgoing waves,

correlation corrections, 376

correlation effects, 345 correlation energies, 9 correlation functions, 269

densitydensity, 112 correlation time, 116 correspondence-principle limit, 423 cosmic background radiation intensity, 501 cosmic ray ionization, 3 I , 492 cosmic rays, 489

313

first-order, 375

low-energy, 489 penetrating, 479

cosmic-ray induced photons, 487 cosmic-ray induced spectrum, 487

INDEX 521

Coulomb aspect, of radial motion, 261 barrier, 15 boundary conditions, 346, 352, 359. 361,

case, 398-400 coupling constants, 103 distorted free electron Green's function,

364, 365 equation, 24 1 excitation, 256 forces, 103 functions, 242, 308 gauge, 383, 384 Green's function. 233 Hamiltonian, 41 1-412 interactions, 97, 113-1 14, 308, 309, 345,

362, 369. 371

346, 347, 352, 362, 377 long-range, 366 part, 315 phase functions, 35 I phases, 346, 347. 348, 352, 353 asymptotic. 348

operator, 206 long-range, 233

potential, 156, 233, 358, 362, 386, 406

repulsion, 114, 253, 255, 261 tail, 361 trajectories, repulsive and attractive, 106 wave, 233, 242 zone, 308, see ulso Zone I1

Coulomb-Born approximation, 18, 256, 263,

CoulombDirac equation, 408 Coulomb-Dirac Hamiltonian, 404 Coulombic repulsion, 173 coupled channel

approach, 345 equations, 357 formulations, 348, 360 model, 343

265, 354

coupled differential equations, 465 coupled equations, 465

coupled integrodifferential equations, 269, 27 1 coupled oscillator systems, 420, 431 coupled radial equations, 382 coupled spinor, 380 coupled states, 441, 466, 468

formalism, 464

of quantum molecular scattering, 324

formulation, 466 photodissociation mechanism, 464 photodissociation, 441 spectrum, 467

coupled, time-dependent Hartree-Fock Theory, 8, 9

coupled-state calculations, 29 1 couplings, 430

constants, 49, 103 effects, 235 matrix elements, 440, 457, 463, 466,

mechanisms, 461 of degrees of freedom, 417 operator, 458 states, 443 strength, 441 strong, 419 term, 339

CRAY vector machine, 272 CRESU, 76 critical dipole moment, 17-18 cross section, 87-88, 89, 97, 98, 106, 125,

472

252, 253, 256, 258, 260-261, 262, 263, 264, 273, 275, 305, 344, 345, 353, 441, 442, 445, 446, 448, 466, 481, 501, see ulso differential cross sections; MPD cross section; TPD cross section

absolute, 294 calculation of, 445 effective, 118, 126 experimental, 95, 365, 371 data, 84 excitation-transfer, 15 for absorption, 445 for charge transfer, 16 for de-excitation, 16 for electron excitation, 274, 275

of atomic helium, 277, 278 for excitation, 16 for high energies, 257-258 for non-resonant charge transfer, 15 for resonant charge transfer, 15 for scattering, 83 momentum-transfer, 18 of ionic transition, 116 inelastic, 14, 15 measured versus theoretically predicted,

partial, 263 447-448

522 INDEX

cross section, (Continued) semiclassical, 256 total, 119

crossed-beam results, 297 crossing points, 336, 462, 463 crossing region, 468 CTC, 84-85, 91, see also charge transfer cell curve crossing, 84

problems, 336 cusp, 317 cut-off functions, 168, 169, 172, 188, 189 cut-off parameter, 168, 188 cut-off radius, 173-174, 185, 189 cut-off wavelength, 442

D Dalgarno-Lewis sum rule, 8 damped oscillations, 160 dark clouds, 491 day glow

excitation of, 25 nature of, 24 quantitative theory of, 24

DB cross section, 123, see also differential

de Broglie wavelength, 103, 106 de-excitation rate, 285 Debye expression, 123 Debye lengths, 109, 117 Debye model, 127 Debye potential, 124 Debye screening, 105 Debye sphere, 103 Debye wavevector, 108 Debye-screened Coulomb expression, 124 decay, 292, 421, 456

by tunneling, 457 mechanism, 459 of excited nuclei, 420 of undisturbed isolated molecule, 42 I process, 420 properties, 43 1 rates, 431

data, 42, 43 method, slope method, 43-44 of molecular beam, 51 profile

Debye-Born cross section

deflection

dimeric, 55 monomeric, 56

classical, 90 of the projectile, 368

degeneracy effects, 216 degenerate eigenvalue, 202 degenerate perturbation theory, 198 dense plasma, 102, 128

collisional phenomena in, 127 effects, 126 environment, atomic collisions in, I15 environmental influence of, 102

densities, of states, 420 density, 104, 503, 504

dependence, 143 expansion, 144 of ultraviolet light, in interstellar clouds,

waves, 496 48 I

density-dependent line shape, 134, 135, 136, 138, 142, 143, 144, 145

formulas, 146 shift, 140 statistical mechanics of, 143 width, 140

depth-dependent abundances, 483 depth-dependent photochemistry, 504 destruction process, primary, 479 destruction rate, of HD, 494 destruction rates, 499

molecular, 438 detailed balance, principle of, 136 detection efficiency, 86, 95 detection limit, 491 detector, 51-52, see also ionizer, electron

bombardment; resonance induced fluorescence

path, 54 detuning, 135, 140, 143, 149 deuterated molecules, 493 deuterium, 493

abundance, 483, 493 chemistry, 493 fractionation, 495 in molecular form, 494

DeVogelaere technique, 270 DHF

case, 411, 412 equations, 410

radial, 381 potentials, 414 procedure, analytical, 41 1

INDEX 523

pseudospectrum, 388, 389 routines, 410 analytic, 41 1

diabatic basis, 336 functions, 466 levels, 467 picture, 467 potential matrix, 465 representation, 465, 466, 467-468 repulsive potential, 467 states, 465

diagnostic probe, noninvasive, 432 diagonal matrix, 465

diagonal potential, 142 diagonalization, 394

of the Hamiltonian, 397 sequence of, 394

element, 270

diatomic bound-free continua, 140 diatomic continuous spectra, 135, 136, 141 diatomic molecular electronic states, 135 diatomic molecules, 46, 238, 323, 428,

diatomic spectroscopy, 134 diatomic systems, 187, 247 diatomics, 72, 164 dielectric constant, 40, see also refractive

dielectric recombination, 127-128 dielectric response function, 107 dielectric term, 173, 185, I86 differential (plane-wave) Born cross section,

differential cross section, 86, 88, 96, 119,

437-438, 439, 458, 469

index

118

238, 360, 364, 365 calculation of, 344 classical, 91 measurements, 84 peak in, 368 quantum mechanically expressed, 88 theoretical, 89-90

differential Debye-Born cross section, I23 (see

differential equations, 325, 358, 386, 405 also DB cross section)

coupled radial, 376 coupled, 258

second-order, 261 first order, 337, 339

linear coupled, 378

ordinary, 331 radial, 379

differential operators, 27 I differential oscillator strength, 444 differential plane-wave, 119 differential probability, 317 differential scattering, 84

cross sections, absolute, 98 in ion-atom and atom-atom collisions, 83

effects, 84 peak, 91

diffraction, 314-315

diffuse clouds, 482, 484, 488, 491, 492, 494, 497, 498, 499, 501, 504

chemistry, 505 HD observed in, 493 interstellar, 493 MHD shocks, 496 models of, 489, SO5 observations, 479 shock, 499

diffusion equation, 315 dimensionality, 310, 419 dimer nuclei, 48 dimer polarizabilities, 56 dimers

alkali, 48 halide, 52 metal, 46, 49

cross section of, 182 nonpolar, 54

absorption probability, 386 amplitudes, 375 coupling strength, 149 field, external electric, 388 hyperpolarizabilities, 21 8 length transition element, 214 matrix elements, 179 moment, 49, 52, 55, 153, 486, 501, 503,

dipoles, 52

504 deflections, 52 permanent, 51, 53, 56

correction, 181 unmodified, 182

oscillator strength, 118 strength sum rule, 406

polarizabilities, 185, 186, 216, 217, 406 for hydrogenic ions, 407

operator, 172, 179, 184, 390

524 INDEX

dipoles, (Conrinued) shielding factor, 2 17 terms, 189, 283 transitions, 117, 171

amplitudes, electric, 383 elements, 217 matrix elements, 185 moment, 444

dipole-connect, 461 dipole-moment potential, 236 Dirac delta function, 124 Dirac electron, 380 Dirac energies, 402 Dirac equation, 388, 403, 404, 405

asymptotic form, 403 Dirac Hamiltonian, 396, 397, 401, 402 Dirac matrices, 370, 376 Dirac negative-energy sea, 386 Dirac quantum number, 396 Dirac spectrum, 396, 400, 403

Dirac spinor functions, 370 Dirac-Breit many-electron Hamiltonian, 376 Dirac-Coulomb Green’s function. 406 Dirac-Coulomb Hamiltonian, 41 1 Dirac-Fock orbitals. 183 Dirac-Fock-Slater potential, 398 Dirac-Green’s operator, 396, 398 Dirac-Hartree-Fock calculations, 40 1 Dirac-Hartree-Fock equations, 10, 381, 410

(see also DHF) Dirac-like equation, 403 direct absorption, 439 direct excitation, approximation for, 361 direct overtone excitation, 425 direct photodissociation, 445, 447, 448, 467

complete, 405

channels, 459 cross section, 444 rate, 444

direct reactions, 498 direction cosines, 423 discrete absorption oscillator strengths, 11 discrete basis functions, 23 1, 236 discrete quadrature, imposed on integrals, 271 discretization, in basis sets, 3 10 dispersion, 12 dissociating states, 457, 458 dissociating systems, 424 dissociation channels, 422

possible, 455

dissociation coordinate, 429 dissociation cross sections, 12 dissociation, (MPD), 146 dissociation, 418, 422, 432, 440

of hydrogen peroxide, 418, 428 of ions, 418 of rotationless molecule, 427

dissociation efficiencies, 479 dissociation energy, 427 dissociation fraction, 454 dissociation limit, 443, 459, 472, 480-481 dissociation pathway, 466. 471 dissociation process, simplest, 439 dissociation processes, 424, 430 dissociation rate, 494

thermally assisted, 428 dissociation threshold, 444 dissociation continua, 470 dissociative continuum, 430 dissociative electron capture, 491 dissociative electronic state, 430 dissociative recombination, 24, 498

with electrons, 489 dissociative states, 470 dissociative surface, 430 distance independent transformation matrix,

distorted amplitudes, first order, 349 distorted waves, 346, 357

first order, 349

470

approximations, 282, 324

basis, 358 calculation, 124, 294-295, 297

of direct ionization, 297 formalism, conventional, 360 function, 349, 358, 359 method, 349 model, first order, 354 prediction, by Younger, 291 Schrodinger equation, 362 series, 303 theory, 17

first order, 361, 367 transition amplitude, 347

distorted-wave techniques, 125 distorting potentials, 350, 351, 358 dividing surface, 419 Dodd and Greider formulation, 360 Doppler broadening, of atomic spectral lines,

133

INDEX 525

Doppler line shape, 142 Doppler radial velocities. 500 Doppler velocity, 496, 500

gradient, 504 Doppler width, 480 double perturbation theory, 8 double scattering mechanism, 369 double scattering, classical model of,

doubly-excited states, 10 downhill simplex method, 414 dressed charges, 112 dressed-atom energy, 149 duration, 106 dust grains, 483, 488, 491

367

interstellar, 479 temperature, 482

dynamic behavior, of plasma, 124 dynamic plasma response, 120, 121 dynamic polarizabilities, 13

dynamic structure factor, I 1 1 dynamical corrections, 171 dynamical coupling, 488 dynamical interaction, 496 dynamics, 134

calculation of, 14

E E-H Gradient Balance device, 44 E-H gradient balance measurements, 48 E-H Gradient Balance method, 45, 46 E-H gradient-balance spectra, 47 effective collision strengths, 273, 279, 282,

284, 285, 288, 289 for electron excitation of atomic helium,

278, 280, 281 for transitions, 281

effective core radii, 181 effective Hamiltonian, for valence electrons,

EHF, 207 (see ulso extended Hartree-Fock

eigenfunction expansions, formal apparatus of,

eigenfunctions, 141, 165, 189, 228, 310,

164

approximation)

200

378-379 expanded, 203

eigensolutions, 149 eigenstates, 31 1 , 393, 394, 400, 41 1

eigenvalue equations, 243, 378, 383 eigenvalue problems, variational treatment of,

eigenvalues, 165, 174, 221, 245, 327, 239

378-379, 381, 395, 396, 400, 402, 404, 408

discrete, 405 expanded, 203 of a matrix, 283

eigenvectors, 402 eikonal distorted wave functions, 359 eikonal function, 358, 359 Einstein coefficient, 136 ejected electron, 316 ejection, of electrons, from bound states, 389 elastic approximation, 263 elastic collisions, 298, 485 elastic Coulomb interactions, 105 elastic cross section, 298 elastic d-H scattering, 238 elastic scattering, 84, 92, 93, 223, 224, 238,

276, 283-284, 286, 356, 363 cross sections, 92, 95 differential cross section for, 92 low-energy, 17 1 of electrons, 267 partial cross sections for, 298 phase shift, 443 single-channel, 91

elastically scattered (electrons), 17 elastically scattered ions, 92 electric deflection data, 46 electric deflection measurements, 46-48, 50 electric deflection method, 44 electric dipole, 450

allowed, 464

contribution, 409, 410 excitations, 385 moment, 51 operator, 171-172, 443 transition, 209, 457

matrix elements, 214 moment function, 443 moments, 448, 454, 466

inhomogeneous, 43, 44, 51, 53 static, 38 strengths, 44, 46 occurrence of, 114

absorption, 472

electric field, 44

526 INDEX

electric moment, 44 electric multipoles, 383, 408 electric potential, 109 electric quadrupole excitation, 256 electric quadrupole oscillator strengths,

electrical discharges, 453 electromagnetic disturbance, 106

response of plasma to, 113 electromagnetic multipole potential, 379 electromagnetic transition amplitudes,

electromagnetism, 3 I 1 electron bombardment, 84 electron capture, 344, 356

by protons, 353, 354, 355 electron cloud, 257, 260 electron collision theory, 267 electron cross sections, 252 electron densities, 109-1 10, 501

in planetary nebulae, 252 electron energies, 273

low, 284 electron exchange, 65-66 electron excitation, 25, 267 see also

480

379

excitation, electron calculations, 282 cross section, 284

of hydrogen, theoretical, 285 for transitions, calculated, 288-289

of atomic hydrogen, 283 of helium, 276 rates, 124, 283

low energy, 289 resonances in, 282

electron fluctuation term, neglected, 128 electron fraction, 493 electron hydrogen scattering, 283 electron impact, 267, 289, 453, 501

cross sections, 503 excitation cross sections, 298 excitation, 503

rates, 268 ionization, 290, 292, 293, 294, 295, 296,

297 calculation of, 303 of ions, 297

transitions, 502 electron momentum, 118 electron OCP values, 121

electron scattering, 9-10, 268 cross section, 49

low energy behaviour of, 300 electron states, excited, 428 electron systems, one- or two-valence, with

electron temperature, 25, 298 electron transfer interaction, 65 electron transport, in copper vapor laser

spherically symmetric core, 164

discharges, 298 incoming, 289 inner shell, 289, 290 ion's, 255 scattered, 298

electron-open-shell-core interaction, 190 electron-atom/molecule scattering, 222 electron-cloud penetration, 265

electron-cloud polarization, at long range,

electron-core interactions, 187 electron-impact excitations, 124 electron-impact ionization, studies, 101 electron-ion collision events, 125 electron-ion collisions, 10, 105, 124, 262 electron-ion coupling, 122 electron-ion excitation collision, I 18 electron-ion excitation rates, 126 electron-ion interactions, 122-125, 126, 127 electron-ion problems, 27 1 electron-molecule collision rate, 492 electron-molecule scattering, 232 electron-molecule/ion scattering, 23 1 electron-positron pairs, 386 electronic angular momentum, 137 electronic angular momentum vector, 458 electronic configurations, 413 electronic energies, 370, 458 electronic energy dependence, 442 electronic energy resonance, 16 electronic excitation, 429, 430 electronic excited state, 39 electronic Hamiltonian fixed-nuclei, 458 electronic motion, 141 electronic potential energies, 440, 443 electronic predissociation, 457 electronic spectra, 484, 493 electronic state, 442, 443, 472

electronic structure, 290

at short range, 264

264

repulsive, 428

INDEX 521

electronic transitions, 484

electronic wavefunctions, 344, 458, 464 electronic-energy resonance, 15 electrons

as cause of excitation, 252 binding with target, 369 dynamic structure factor of, 118 ejection of loosely bound, 289 interaction with atoms, 267 interaction with ions, 267 unperturbed core, 165

electrostatic forces, 65 electrostatic interaction, 255 elemental abundances, 486 elementary particle physics, 404 emission, 141, 143, 430, 432

broad band continuous, 30 emission lines, 251, 290, 488. 503

interpretation of, 504 emission line shape, 138 emission of radiation, 393, 441 emission series, 457 emission spectra, 441 emission spectroscopy, 432 end points, 337 endothermic reaction, 32, 33 energy

discrete, 443

electronic, 429 rapid randomization of, 419 vibrational, 429

energy barriers, 481, 485 energy coefficients, 199 energy content, 425

functions of, 424 energy conversion, intramolecular, 429 energy dependence, 439 energy disposal, 425 energy eigenvalue, 393, 442, 443 energy expansion, traditional, 197-198 energy functional, 401 energy interval, probability in, 307 energy level shifts, 463 energy levels, 101, 164, 175, 184, 393,

505 table for lithium atom, 175

energy loss rates, 495 energy maximation method, 10 energy regions, three, 265

high, 257

intermediate, 257 low. 257

energy resolution, width of, 290 energy resonance, 63 energy spectrum, 400 energy transfer, 424 energy transfer process, 78-79 energy-dependent solutions, 228 energy-independent basis functions, 27 1 energy-level spectra, predictions, 173 energy-sharing, inside molecules, 419 equation, linear and homogeneous, 33 1 escape probabilities, 504 escape probability method, 502 ethynyl radical, 494 Euler equation, 200, 221 equilibrium dissociation rate, 480 exact amplitude, 352 exact fixed-dipole calculation, 18 exact initial state, 308 exact wavefunctions, 11 exchange potential operator, 206 exchange potential terms, 269 exchange potential, short range, 236 excimer transitons, theory of, 134 excitation

atomic-fine structure, I8 by electron fluctuations, 122 by secondary electrons, 489 coherent, 426 electron, 25 electronic, 429 inner shell, 290, 293 molecular rotational, 18

of helium, 267 rotational, 429

of Hz. 30

of molecules by atom impact, 18 of molecules by ion impact, 18

vibrational, 429 excitation amplitude, I17 excitation and abundance of H2 in clouds, 482 excitation calculation, 502 excitation channels, 381, 388 excitation correction, local, 502 excitation cross sections, 273, 276, 285

excitation dissociation, 428 excitation energies, 185, 252, 376

for hydrogen, 284

vertical, 439

528 INDEX

excitation event, 487 excitation frequency, 379 excitation of CN, 502 excitation probability curves, 119-120 excitation probability, 118, 121, 122 excitation processes, 290, 478 excitation rate

calculations, 121 characteristics, 128 coefficient, 118, 285

excitation rates, 265, 285, 481, 483 excitation scheme, 424 excitation spectrum, 378 excitation temperature, 500, 501 excitation threshold, 283, 284, 285-286, 289,

excitation transfer, 15 excitation-autoionization, 294, 295

290

contributions to, 295, 297 effects, 293 energy region of contribution, 291 inner shell, 292 process, 289, 292

excitation-transfer cross section, 15 excited bound states, 383 excited electronic energy, 457 excited electronic state, 440-441, 447, 457 excited fine structure, 496 excited ground singlet state, 429 excited molecular states, 468 excited rotational levels, 481 excited rotational states, 479 excited species, highly vibrationally, 425 excited state, 375, 384, 456

initially, 426 in a dense plasma, 116

excited state bound wavefunctions, 466 excited state continuum wavefunction, 446 excited state energies, of the beryllium

sequence, 2 I 1, table, 2 1 1 excited state potential, 430 excited state wavefunction, 452, 466 excited vibrational levels, 48 1 excited vibrational states, 479 exothermic associative detachment, 69-70 exothermic process, 70 exothermic reactions, 425 exothermicity, 68 expansion, 351, 396. 398, 408, 412

for the phase shift, 223

free jet, 426 in basis sets, 310 of amplitude, 348 separable, in the Lippmann-Schwinger

equation, 229 expansion coefficients, 203, 212, 213, 386,

387, expansion method for calculating atomic

properties, 9 expansion parameter, 197 expansion procedures, 269 expectation value, 208, 262, 393

experimental cross sections, 452 experimental diatomic continuous

experimental differential cross sections,

experimental distributions, 429 experimental technique, for studying

unimolecular reactions, 424 experiments, for measuring cross section,

exponential growth, of solutions, 334 exponential parameters, 413 exponential potential, 225, 230 exponential propagator, 337 exponential solutions, 337 exponentials, decaying, 328 extended Hartree-Fock approximation, 207 see

external electric fields, 203 external fields, effects of, 216 external potential, 377 extinction curves, 488

for neutral helium, table of, 213

spectroscopy, 139

352

86-87

also EHF

F FA, 61, 71 (see also flowing afterglow)

studies, 63 system, 69 technique, 69

FA-SIR instrument, construction of, 71 far zone, 308 (see also Zone 111) FC factors, 66 (see also Franck-Condon

factors) FDT, 71, 72 Fermi energy, 103 Fermi-Wentzel Golden Rule formula, 4598 fermions, 97 Feshbach method, 246

INDEX 529

Feshbach resonance, 170, 298, 431 dissociation of, 43 I

field strengths. 114, 125 field, inhomogeneous, 54 field-free Hamiltonian operator, 203 field-free limit, 126 field-free nonrelativistic unstricted Hartree-

Fock, 206 see also UHF fifteen-state calculation, 273-275 fifteen-state close-coupling calculation, 283 fine structure branching ratios, 470 fine structure population ratios, 471 fine-structure changing collisions, 253 fine-structure constant, 118, 183 fine-structure excitation, 15, 252, 256 fine-structure levels, I89

fine-structure splittings, 183, 252, 469, 471 fine-structure states, 469 fine-structure transitions, I89

proton-induced, 25 I finite basis sets, 408

expansions, 407 final test of, 408 method, 41 1

finite impact velocities, 351 finite interval, 405 finite nuclear size, 414 finite threshold. 284 finite-basis-set approach, 10 finite-rank approximation. 227, 23 1 first Born approximation, 366 first derivative terms, 337, 338, 339 first-order approximations, 255, 258, 349 first-order Born formula, I16 first-order coefficient, 217 first-order corrections, 205 first-order correlation function, 156 first-order cross section, for low energies, 257 first-order differential condition, 441 first-order energies, 199 first-order equations, 31 1 firs-order expansion coefficient, 2 12 first-order formulae, 265 first-order matrix element, 202 first-order perturbation

of hydrogen-like ions, 267

description, 468 operator, I98 theory, 117, 126, 345, 441, 458 treatment, 446, 472

first-order results, 265 first-order screening approximation, 2 13,

first-order semiclassical theory, 256 first-order solutions, 207 first-order theory, 255. 257 first-order transition probability, 257 fixed nuclei electronic energy eigenvalue

flexible transition theory, 421 Floquet Hamiltonian, 149

method, 154 Floquet-Liouville super-matrix, 158 (see also

FLSM) flow reactors, 62 flow techniques, 63 flow tubes, 63

219

equation, 442

aeronomical applications of, 63 astrophysical applications, 63 studies, 62 technique, 61 technology, 62

flowing afterglow, 61 (see also FA) FLSM approach, 158 FLSM, 158 (see also Floquet-Liouville

super-matrix) fluctuation phenomena, in a plasma, 11 1 fluctuation spectra, in plasmas, I 1 1 fluid mechanic, application in, 310 fluorescence, 428-429, 479, 481, 492

power spectrum, 158, 159 fluorescent continuum spectrum, 456 fluorescent light, nature of, 158 fluorescent line emission, 483 fluorescent line intensities, 483 flux, 313

conservation of, 298, 314 of detaching photons, 492 photon, 441, 442

fly-bys, 26 Fock operator, 206, 207 forbidden gap, 396 forbidden transitions, 414 forbidden-line emission, 252 force constant, 422 foreign gas broadening, 134 formation

of CO, 487 paths of, 490 process, 479

530 INDEX

formation, (Continued) rate, effective, 479 rates, molecular, 438

forward diffraction peak, 91 forward exoergicities, 495 four-state close-coupling calculations, 298 Fourier components, 107, 109, 112

of fluctuations, 11 I Fourier transform, 113. 139, 144, 156 fractional abundance, 503 fractional ionization, 482 fractionation. 493, 494

of deuterium in molecular clouds, 494 mechanism, 495

fragments, 457, (see also atomic fragments; molecular fragments)

atomic, 428 electronically excited, 441 neutral, 448

Franck-Condon barriers, 64 Franck-Condon densities, 444, 451 Franck-Condon factors, 63, 66, 445 Franck-Condon principle, 140, 430, 439 Frauenhofer diffraction, theory of, 305 free electron Green’s function, 364 free jet expansion, 426 free radical addition, 425 free radicals, 438 free state wave function, 137 free-bound line shape, 137 free-bound process, I39 free-bound-free resonance-fluorescence

free-free absorption coefficients, I7 free-free line shape function, 137 free-free matrix, 138 free-free process, 17, 139 free-state wave functions, 137 freedom, degrees of, 419, 420 frequency eigenvalues, 378, 379 frequency solutions

negative, 388 positive, 388

systems, 9

spectra, 141

frequency-dependent properties of atomic

frequency-dependent refractive index, 13 frequency-dependent shift parameter, 145 frequency-dependent width parameter, 145 frozen-core approximation, 233 frozen-core photoionization cross sections, 233

full beam deflection profile, two-dimensional

full multichannel results, 471 full scattering wavefunction, 236 fully coupled, multichannel description, 375 fully quantal solution, 3

analyses of, 43

G galaxies, formation and evolution of, 478 galaxy-galaxy collisions, 496 Gamow states, 228 gas, interstellar, 29-30 gas phase

abundances, 490 chemistry, 483-484, 487, 493, 495 formation, 483 process, 481 sulfur chemistry, 487-488

gas phase-ion-molecule reactions, 483 gas processes, elementary, 431 gaseous nebulae, 34 gauge independent, 375 Gaussian functions, 23 1 Gaussian line shape, 133

Gaussian wave basis set, 237 GE

of Doppler broadening, 134

amplitude, 363 approximation, 361, 362, 363

geomagnetic disturbance, 84 Glauber eikonal approximation, 361 (see also

glory effects, 84 glory scattering, 91 Golden rule formula, 462 grain surfaces, 481, 492, 494

grains, 442 Gram-Schmidt process, 329 Green’s function. 166, 225, 227, 228, 231,

GE)

processes, 488

236, 237, 240, 241, 245, 246, 270, 307, 308, 309, 316, 348, 351, 358

matrix, 243 method, 152 operator, 223, 240 projectile, 365 target, 365 unbounded, 242 unbounded smooth, 242

Green’s lemma, 309

INDEX 53 1

ground electronic state, 429, 430, 479

ground state, 284, 298, 375, 384, 393, 395, 400,439, 441, 443,454, 456, 457, 461

of molecule, 439

charge transfer, 366 configuration, 297 cross section, 186 energy, 200

of the helium sequence, table of, 210 excitation of helium from, 288 expectation values, helium sequence, table

vibrational wavefunction, 446 transitions to, 390

of, 212

ground-state energy, of neutral helium, 195

H H-impact dissociation, 48 1 H-KV anomalies, 226 H-KV expressions, 225 H-KV methods, 221, 222, 224, 226, 228,

230, 231, 247, (see also variational methods, HulthCn-Kohn)

half collision events, dynamics of, 432 half-collision events, 418, 437 half-vibrational period, 430 Hamilton’s equations, 422, 423 Hamiltonian formulations, 226 Hamiltonian matrix elements, 222, 225 Hamiltonian matrix, 394 Hamiltonian methods, 240 Hamiltonian operator, 270 Hamiltonian, 92, 165, 184, 198, 203, 225,

307, 320, 376-377, 387, 393, 402, 457, 458

relativistic part of, 458 Hamiltonian-based resonance methods, 242 Hamiltonian-Jacobi theory of mechanics, 221 Hankel function, 309 Hank effect, 179 harmonic systems, 426 Harris Michel variational method, 226 Hartree screening functions, 382 Hartree screening potentials, 380 Hartree-Fock

approximation, 196 calculations, 177 core, 233 core potential, 174 energies, 9

isolated core wave function, 168 potential, 236

static, 188 scheme, 394 theory (see coupled, time-dependent

Theory, Coupled, Time-Dependent, 9 value, 206 wavefunction, 9

HCL. constituent of atmospheres, of Earth and Venus, 448

HD, source of, 494 He-like ions, 398 hearing rate, 489 heavy atom problem, 424 heavy-particle scattering, 422 helium, 484 Heller’s wave packet, application to molecular

Hellmann-Feymann theorem, 21 2 Herzberg continuum, 26, 452 Herzberg system, 452 HF (see also Hartree-Fock)

Hartree-Fock Theory)

dissociation, 430

approximation, 207-208, 209, 212, 213,

calculations, 2 I3 components, 387 eigenfunctions, 387 equations, 208, 379-380, 387 Hamiltonian, radial, 380 orbital approximation, 209 orbitals, 208, 382 projection operators, 387 pseudospectrum, 388 states, 388 wave-function, 207 2-expansion, calculations, 207

Hibridon code of Alexander, 338 high energy

214, 217

behavior, 366 region, 265

high impact energies, 253 high overtones, 426 high-density plasma environments, 106 ’

high-density plasmas, 103 high-energy formulae, 260 high-energy scattering, 230 high-order 2-expansion perturbation procedure,

high-temperature formation, 482 10

532 INDEX

high-temperature phenomena, 496 higher bond states, 275 higher-order SV expressions, 236 Hilbert space, 231 homogeneous equilibrium plasmas, 105, 107 homogeneous operator, 209 Honl-London factor, 136, 137 hot, 101 Hubble Space Telescope, 480, 499 Hulthen results, 224 HulthCn-Kohn method, 222 hydrocarbon molecules, reactions of, 485 hydrodynamic shocks, 32, 33 hydrogen, 484

atoms, colliding, 454 molecular, 454

hydrogen peroxide, 426, 427 hydrogenic 2p-orbital, 260 hydrogenic atom, 344 hydrogenic eigenfunctions, 200, 346 hydrogenic ions, 398

dipole polarizabilities for, 407 hydrogenic orbitals, 198-199, 207 hydrogenic spectrum, I98 hydrogenic systems, 394 hydrogenic value, 209, 214 hydrogenic wavefunction, 412

Hylleraas functional method, 200 Hylleraas-Undheim theorem, 394, 397, 400 hyperbolic Coulomb trajectory, 255 hyperfine structure. 502 hypergeometric functions, 402 hyperpolarizability, 167, 204 hyperspherical coordinates, 305, 307

bound state, 367

I ice mantles, 33 ICF, 101 (see also inertial confinement)

conditions, 103 laser experiment, 128

IDE, 152, 153 (see also inhomogeneous differential equation)

ideal gas distribution, 105 imaginary number of magnitude, 144 impact energy, 252, 256, 303

impact parameters, 90, 97, 105-106, 255,

impact theory, 143

range, 353

256, 251, 262, 265, 344, 347

impact velocity, 369 IMPACT, 270, 271, 272 (see also computer

programs) impact, 324

atom, 17 electron, 17

implicit formulae, 338 IMSL, 331 inactive electrons, 13 incident ion, 353 incoming wave, 383 independent perturber, 145

approximation, 144, 146 formalism, 145

independent-particle approximation, 377 induced dipole moment, 17 industrial chlorofluorocarbons, 26 inelastic calculations, 238, 239 inelastic collisions, 481, 482 inelastic component, 156 inelastic ion-ion collisions, 126 inelastic scattering, 84, 102, 237

of electrons, 267 problems, 298

infinite summations, 9 infrared

absorption, 495 astronomy, 30, 480 line emission, 483 multiphoton excitation, 425 photons, 425 sources, 490 spectrometers, 490 transitions, 483 vibration-rotation lines, 490

ingoing wave, 3 I3 inhomogeneous differential equation, 152, 395

(see also IDE) inhomogeneous equation, 199 inhomogeneous term. 242 initial conditions, 43 1 initial excitation, 429 initial states, 420 initial vibrational level, 443 initial-state correlation, 233 inner shell excitation

autoionization contributions, 268 threshold, 297

inner shell vacancy production, 357 inner wall, 446

INDEX 533

lnnsbruck measurements, 64 instabilities, 338-339 integrable caustic, 3 17 integral equations, 270, 309, 325, 335, 348,

358, 359 approach, 332 coupled two-dimensional, 227 one-dimensional, 227

integral expressions, 308 integral operators, 271 integral, first, kernel of, 332 integral, second, a constant, 332 integrals, 232 integrated probability, 319 integration, 443, 444, 328, 330, 333

angular, 384 numerical, 386

integrodifferential equations, 270 coupled sets of, 270

intensity, measurable, 504 intensity, of ultraviolet light, in interstellar

clouds, 48 1 intensity-dependent AT1 electron energy

spectrum. figure, 155 intensity-dependent fluorescence power

spectral patterns, 158 intensity-ratios, 25 1 interaction potentials, 272 interaction well depth, 65 interactions, 468, 470

acurate, 263 long range, 260, 345 overlapping strong, 122 short range, 261, 265 weak, 116-122

interatomic potential, 137 determination of, 134

interchange theorems, 8, 201, 202, 204, 214 intercloud medium, 489 interelectronic repulsion energy, 21 2 interference, 466 interference effects, 146 intermediate coupling, 260

on transitions, 261 intermediate electronic states, 429, 452 intermediate energies, 258, 276, 343, 345, 363 intermediate energy region, 265, 353-354 intermediate processes, 489 intermediate states, 365, 394, 395, 405 internal conversion, 425

internal excitation, 482 internuclear distance, 444, 452, 454, 470 internuclear potential, 344, 345, 352 interstellar, components in, 33 interstellar chemical theory, 490 interstellar chemistry, 445, 477, 478, 495,

497, 505 model, 31-32 problems, 445

interstellar clouds, 30, 34, 67, 440, 441, 442, 447, 448, 459, 461, 466, 470, 477, 478, 485, 488, 491, 492, 493, 505

chemistry of, 31 diffuse, 480 emission, 30 initial absorption, 30 models of, 503 molecular content of, 487 quiescent, 495 thick, 486

interstellar CN molecules, 501 interstellar conditions, 481 interstellar environment, 442 interstellar gas, 29-30, 478, 505

cooling of, 33 heating of, 33 heating of by cosmic rays, 489 ionization of by cosmic rays, 489

interstellar grain chemistry, 483 interstellar ion-molecule chemistry, 489 interstellar ionization balance, 490 interstellar ionizing frequency, of cosmic rays,

interstellar medium, 31, 438, 441, 445, 447, 489

454,478, 483, 491, 492, 493

cosmic ray flux, 31 elements, relative abundances of, 31 ionization, level of, 31 radiation field, 3 1 rotational emission spectra, 31

parameters of

interstellar molecular clouds, 488 interstellar molecular shocks, temperatures of,

interstellar molecules, 438, 439, 461, 488

interstellar molecules, CO, 461 interstellar molecules, H2, 483 interstellar negative-ion chemistry, 492 interstellar neutral matter. 30

482

formation of, 489

534 INDEX

interstellar NH3, production of, 485 interstellar radiation field, 442, 454, 479 interstellar regions, 503 interstellar shocks, 31, 32, 495, 496, 497,

interstellar species, 497 interstellar temperatures, 491 interval, 405 intramolecular bottlenecks, 420-42 1 intramolecular conversion, 429 intramolecular dynamics, 418, 426, 431 intramolecular energy

499

conversion, electronic to vibrational, 429 flow, 432 redistribution, 418, 431 (see also IVR) sharing, 417, 419 sharing process. 428

intramolecular relaxation rates, 420 intramultiplet transition, 25 1 inverse chemi-ionization, 16 ion

beam, 297 composition, 61 core-excited, 289 cyclotron resonance spectronomy, 63 exchange, 494 flow tube techniques, 62 formation, 445 fully stripped, 344 mass analysis, 71-72 microfields, 125, 126 most abundant nitrogen-containing, 485-

production, 484 scattering signal, 92 temperature, 25 traps, 179 vibrational quenching, 73, 75 vibrational relaxation, 72

ion-atom interchange, 24 reactions, 63, 69

ion-ion collision, 128 studies, 126

ion-ion interactions, 123, 126-127, 128 ion-microfield effects, I26 ion-molecule

486

chemistry, 484 collisions, 75 flow tube studies, 78-79 interactions, 62

reactions, 62, 65, 66, 67, 73 exothermic, 63, 76-77

reaction rate constants, 61, 62 reaction studies, gas discharge physics

approach to, 61 ion-neutral attraction, 65 ion-neutral collisions, 91 ion-neutral flows, 32 ion-neutral velocity differences, 500 ion-sphere

calculations, 127 model, 109, 126-127 potential, 116 radius, 127

ionic core, 186 ionic polarizabilities, 56 ionic products, 25 ionization, 253, 317, 357, 488, 489

(MPI), 146 by electron impact, 289 degree of, 490 direct, 289, 293, 294, 297

cross section, 289 Younger theory of, 291

292 direct and indirect, interference between,

direct collisional, 293 direct scaled, Younger’s, 297 in interstellar clouds, 491 indirect, 289, 293 indirect inner shell excitation process,

293 net, 297 of helium, 267 of the neutral, 65 of unpolarized alkali atoms, 180 pathways to, 290 scaled direct, 291

ionization amplitudes, 305, 308 ionization balance, 488, 492 ionization calculations, of Younger, 290 ionization cross sections, 290, 297

behavior of, 304 for Na isoelectronic sequence, 294

ionization data, 294 ionization equilibrium, in plasma, 252-253 ionization level, 489 ionization mechanism, indirect, 289 ionization potential, 65, 186, 490, 491

for valence electrons, 165

INDEX 535

ionization probability, 307 ionization process, energy dependence of,

ionization rate, 493 ionization results, direct, of Lotz, 295 ionization spectra, 466

ionization threshold, 469 ionization values, direct, 290 ionized systems, 215 ionizers, electron bombardment, 52 ionosphere, 23, 68

290

of atoms and molecules, 469

electron temperature in, 24 StNCtUR Of, 24

ionospheric chemistry, 25 ionospheric physics, 62 ions

autoionizing metastable, 294 excited, 72, 293 fully stripped, 353 highly charged, 289, 297 hydrogen-like, 252 negative, 492, 493 positive, 114 production of, 489 strongly bound, 493

IRAS observations, 252 isoelectronic ions, 195 isoelectronic sequence, 185, 186, 195, 198,

413 lithium, 290 Na, 293

isoenergetic charge-transfer reaction, 66 isoenergetic crossing, 425 isomerization, of methyl cyanide, 425 isotope exchange, 71 isotope shifts, 478 isotopes, 440, 464 isotopic factor, 77 isotopic parameters, 77 isotopic results, 26-27 iteration potential, 97 iteration procedure, 223, 232 iteration process, 413, 414 iterations, 414

successive, 230 iterative method, 334 iterative process, 326, 334 iterative Schwinger method, 233, 235, 236 IVR, 418, 430

J JB approximation, 89 JS approximation, 345, 351-352, 353, 356 JS cross section, 353 Jupiter, 26 JWKB approximation, 89

K Kato variational principle, 334 kinematic information, 499 kinematic information, 500 kinetic energy, 309, 441, 457, 471, 489

couplings, 142 of electron, 376 of rotating molecule, 458 operator, of nuclei, 458

kinetic temperature, 482, 484, 501 Klein paradox, 388 Klein-Dunham potential curves, 45 1 Kohn correction, 231 Kohn method, 226 Kohn phase shift, 226 Kohn principle, 238 Kohn variational method, 227, 310 Kohn variational principle, 269 Kohn’s principle, amplitude-independent form

Kramers-Kronig relations, 144 Kronecker delta function, 327

of, 225

L laboratory astrophysics, 485 laboratory plasmas, 25 1, 252 Lagrange multiplier formalism, 233 Lagrange multipliers, 270, 378 LAM, 271, 272 (see also linear algebraic

large angle scattering, 344 large velocity gradient, 504 large-amplitude motions, 422 laser-Stark-spectroscopy, 4 I laser, 63, 428

continuous, 426 pulsed, 426 transform-limited pulse, 426

method)

laser beam, of finite cross section, 3 15 laser excitation, 179, 428 laser fields

intense, 146 strong, 153

536 INDEX

laser photofragment spectroscopy, 457 laser pulse, intense, 425 laser techniques, 421

high-power, 154 laser-induced fluorescence, 427 laser-induced phenomena, 418 laser-induced processes, 420 least-squares method, 226 Legendre polynomial, 89. 203 length form, 386 length gauge, 385 LHS, 310 lifetimes, 49, 432

from trajectory calculations, 427 of OD in comets, 463 of triatomic collision complexes, 424 statistical, 427

light scattering spectral pattern, 157 Light, 334 line absorption, 440, 457, 487

line broadening, 189, 457 line core, intensities in, 143 line formation, 504

calculation, 502 line intensities, 504 line opacity, 441 line oscillator strengths, 480 line positions, 500 line profiles, 143, 499

unified theories to describe, 143 line shapes, 134

analysis, 149 density-dependent, 142 formulas, 143 functions, 134, 135, 139, 143-144 normalized, 135 theory, 139

process, 455

line shift parameter, 143 line shifts, 144, 499 line strengths, 503 line wavelengths, 440 line widths, 457, 462-464 line wing spectra, 141 line wings, intensities in, 143 line-ratio, 252 linear algebraic equations, 270, 271 linear algebraic method, 235, 27 1 linear behaviour, 288 linear coefficients, 394

linear dependence, 329 linear differential equations, 325

second order, 328 linear equations, 325-326, 387 linear HF equation, 387 linear momentum magnitude, 423 linear multistep methods, 333 linear operator, 310 linear problem, 325-326 linear reference potential method, 337 linear response regime, 107 linear Stark term, 125 linear threshold behaviour, 288 linearized Boltzmann transport equation, 11 1 lines, isolated, theory of, 143 Lippmann-Schwinger equation, 224, 226-228,

229, 231, 232, 233, 237, 240, 358 Lippmann-Schwinger scattering equation, 242 log-derivative, 333, 334

matrix, 338 method, 335, 336, 337, 338 propagator, 338

logarithmic derivative, 271 long-range Coulomb interaction, 110 long-range dipole interaction, 16 long-range interactions (molecular), 12-1 3 long-range adiabatic interactions, 13 Lorentzian core, of pressure-broadened line,

Lorentzian line shape, of natural broadening,

Lorentzian peaks, 156-157 low density limit, 501 low energies, 276 low order calculations, 207 low order FT, 196 low order truncations, 198 low-density values, 122 low-energy behaviour, 300 low-energy formulae, 260 low-energy nucleon-nucleon scattering,

low-energy physics, 417 low-energy region, 265 low-energy, first-order approximation, 260 low-frequency modes, 428 low-temperature behavior, 485 lower repulsive wall, 89 LS multiplets, 260 LS-coupling, 260, 261, 265

138

134

229-230

INDEX 537

Lyman bands, 454, 30 Lyman limit, 442, 454, 479 Lyman system bands, 480 Lyman system, 454, 455, 487

M macroscopic system, 110, 1 I I magnetic fields, 32, 44, 50, 128, 203, 488,

489, 499 inhomogeneous, 46, 56

magnetic moments, 44, 46 magnetic multipoles, 383, 408 magnetohydrodynamic shock models, 32, 33,

496 (see also MHD shock) Malik-Rudge variational method, 226 many-body Hamiltonian, 377, 386

many-body phenomena, statistical analysis of,

many-electron systems, 394, 417 many-state calculations, 276-277, 279,

Mars, 26

nonrelativistic, 389

111

280-282, 286

history of volatiles on, 27 nitrogen escape from, 26, 34

mass spectrometry, 63 Master Equation formalism, 421 matrices. 330, 337

diagonal, 327 size of, 326

of coefficients, 329 of solutions, 329 residual, 338

matrix, 326

matrix diagonalization, 246, 271 matrix eigenvalue problem, 387-388 matrix elements, 202, 229, 258, 270, 345,

385, 390, 422, 444 of core-polarization correction, 179 of dipole operator, 385 table, 232

matrix equations, 271 matrix form, 465 matrix inverse, 331-332 matrix T, 327 maximum principle, 402 Maxwellian distribution, 105, 11 1, 273, 284,

Maxwellian velocity distribution, 54 MCDHF calculations, 41 1

288

MCDHF routines, 410 MCDW amplitude, 350 MCDW treatment, 350, 356, 357 (see also

mean density, of plasma particles, 113 mean transition-probability, 265 measured oscillator strengths, 13 mechanistic knowledge, 62 mechanistic study, 76-78 metal atoms, 490 metal chemistry, in translucent clouds, 492 metal hydrides, 490, 504 metastable ions, 290 metastable molecule

multi-channel distorted wave treatment)

chaotic state of, 431 regular state of, 431

metastable states, 297 methyl radicals, 421 MHD models, prediction of, 499 MHD shocks, 496, 500

microcanonical density of states, 419 microcanonical trajectories, 423 microfield gradients, 128 microfields

high-frequency, 114 low-frequency, 114-1 15

microscopic processes, 477 microwave transition frequencies, 171 minimax approach, 401-402 minimax strategies, 414 minimum-norm method, 226 mode specificity, 425 mode, of reactions, 424 mode-specific laser chemistry, 430 model calculations, 485 model parameters, 463, 464 model potential, 163, 164, 184, 185, 183,

nonionizing, 499

188 forms of, 170 methods, 8, 14, 17

model pseudo-potentials, 13-1 4 model triatomics, dissociating, 43 1 model,

of classical trajectory, 139 of cloud structure, 504 of dense plasma, 102 of interstellar shock, 102

approach, 1 75 model-potential

538 INDEX

model-potential, (Continued) calculations, 172, 179

for molecular properties, 189 energies, 174 energy-level predictions, 175/177 methods, 174, 177, 183

applications of, 173 in photoionization calculations, 180-182

predictions, 179 phase shifts, 170 results, 175 scheme, 14 theory, 164 treatment, 18 1 wavelengths, I77

excited thermally, 423 excited, 423 local, 426

modes

modified excitation rate formula, 122 modified-effective-range theory, 17 molar polarization, 40 molar refraction, 40 molecular absorption, 444 molecular abundances, 479, 495, 496, 504 molecular astrophysics, 62 molecular beam

investigations, of reactive scattering, 418 laser-Stark-spectroscopy , 41 measurement, of polarizabilities, 41

molecular beam, alkali halide, components of,

molecular beams, 50, 63 molecular cloud surfaces, 479 molecular cloud systems, 505 molecular clouds, 480, 485, 487, 489-490,

54

492, 505 Hz in, 479 models of, 503 probes of, 34

molecular collisions, 330 molecular destruction rates, 487 molecular dissociation, 417, 418, 422

dynamics of, 432 process, 432

molecular dynamics, 420 molecular excitation, 503 molecular formation, 503 molecular fragments, 429 molecular Hamiltonian, 426

molecular integrals, calculation of, 188 molecular ion vibrational relaxation, 74-75 molecular ions, 490 molecular line spectra, discrete, 136 molecular model potential, 190 molecular model-potential methods, 187 molecular perturbers, 140 molecular photodissociation, 437 molecular polarizabilities, 38, 48, 57

temperature dependence of, 40 molecular polarization terms, 57 molecular potential curves, 16 molecular potential energy curves, 173 molecular potentials, 420 molecular processes, 432, 505 molecular properties, 10-1 1 molecular pseudo-potential methods,

molecular rotations, 18, 423, 424 molecular scattering, 328, 334, 338, 339

molecular shocks, revelation of, 496 molecular spectroscopy, 418 molecular studies, I 1 molecular-ion vibrational

molecules, 8

187

equations, 326

excitation-quenching, 75

according to Slater’s theory, 419 asymmetric top, 323-324 cometary, 442 electronic state of, 336 energized, 432 exotic, 505 formation and excitation of, 496 formation of, 470 geometry of, 423 highly excited, 418 highly vibrational states, 431 interstellar, 438 (see also interstellar

molecules) model of, 422 optical properties of, 11

dynamic polarizability (DP), 11 oscillator strengths, (0s). 11 photoabsorption (PA), 11 photodissociation (PD), 11 photoionization (PI), 1 1 radiative lifetimes (RL), 11 Raman depolarization factor, 11 Rayleigh depolarization factor, 1 I

INDEX 539

Rayleigh scattering cross section, 1 1 Verdet constant, 1 I

rotational excitation of, in collisions with electrons, 8

rotationless, 427 spherical top, 323-324 symmetric top, 323-324

Mollow symmetric triplet spectrum, 157 momentum states, 423 momentum transfer, 118, 370, 487 momentum vector, 423, 458 momentum-transfer cross section, 1 10,

monoenergetic site-selective deposition

monomers, 52

298

scheme, 424

alkali halide, 56

monopole term, 255 Monte Carlo methods, I14 motion

polar, 54

of system, 419 relative of target and projectile, 344 rotational, 423 torsional, 423 translational, of electron, 346

MPA, 147 (see also multiphoton absorption) spectra, 152 study, theoretical techniques for, 147-149

MPD calculations, 152 MPD cross section, 153 MPD, 146, 152 (see also multiphoton

MPE, 146 (see also multiphoton excitation) MPI, 146 (see also multiphoton ionization) multi-channel distorted wave treatment, 350

multi-exponential parameters, 4 12 multicentered systems, 414 multichannel

case, 326, 331-332, 334 denominator, 243 extension, 243 resonance problems, 243 scattering theory, 463 scattering, 334 theory, 230 variational principle, 247

dissociation)

(see also MCDW ireatment)

multiconfiguration Dirac Hartree-Fock code, 184

multidimensional potential, 430

multielectron systems, 353 multiphoton

surface, 422

absorption, 147 (see also MPA) induced resonance fluorescence spectra, 158 processes, 156 transitions, 146

multiple transitions, 120 multiple-collision theory, 143 (see also

collisions , mu1 tiple) multiple-wave mixings, 146 multiplet, 251, 252, 255, 257, 258, 261 multiplicity, 458 multipole dynamic polarizabilities, 13 multipole expansion, 168, 255 multipole field, 383

electric, 383-384 multipole moment operator, 384 multipole moments, 384 multipole operator, 172 multipole polarizability, 204 multipole potential, 383 multipole shielding factor, 204 multipole transition amplitude, 379 multistate theory, 8 mutual neutralization, 492

N NAG, 331 natural broadening, of atomic spectral lines,

natural variables, 206 (see also unscaled

near resonant charge transfer, 486 near-threshold photodissociation, 469, 470,

negative charge density, 110 negative eigenvalues, 400 negative ion charge-transfer, 63 negative ion chemistry, 493 negative energy,

133

variables)

472

continua, 387, 405 eigenstates, 396 eigenvalues, 398 reaction matrix, 242 spectra, 395-396, 398, 404

negative-energy state contributions, to RRPA,

Nesbet’s anamoly-free method, 226 390

540 INDEX

net ionic charges, 127 net polarizability, 56 neutral atom, 345 neutral cloudlets, 496 neutral clouds, 482, 490 neutral molecule vibrational relaxation, 74-75 neutral-neutral exchange reaction, 495 neutralization, 16 neutralizing ions, 490 neutron-proton scattering, 223, 224 NIEM, 270, 271, 272 (see also noniterative

integral equation method) nitrogen, 484

nitrogen-bearing species, 485 no-pair approximation, to QED, 386, 390 nodal structure, 169 non adiabatic correction, 167 non-adiabatic correction, to static interaction,

non-Maxwellian velocity, 485

non-relativistic Harniltonian, 165 non-resonant charge transfer, 15 non-RRKM behavior, 424 nonadiabiatic coupling, 480 nonautodetaching curve, repulsive, 71 nonclassical region, 328, 330, 333, 334, 338 nonequilibrium rates, 496

nonequilibrium system, 503 noninteracting particles, 1 14 noniterative integral equation method, 270 (see

nonlinear

chemistry, 485

13

distributions, 499

coefficients, 497

also NIEM)

coupled TDHF equations, 377 mechanics, 420 optimization method, 413 oscillators, 424, 427 parameters, 394, 395, 398, 401,413, 414 systems, 432 variational parameter, 397

nonlocal potentials, 164 nonmagnetic shocks, 499 nonoptimized wavefunctions, 41 3 nonradiative capture process, 369 nonradiative decay, 457, 459 nonrelativistic

calculations, 402 formula, 370

Hamiltonian, 268 limit, 401, 412 SCF method, 410 Sturmian functions, 397 theory, 344, 369 three-particle systems, 227

nonresonant charge transfer cross section, 188 nonstationary theory, 230 nonthermal cosmic rays, 484 nonthermal rotational population distributions,

normalizable eigenfunctions, 394-395 normalization, 55-56

normalized line shapes, notation for, 135 normalized resonance channel wavefunctions,

nuclear

482, 483

constant, 245

244, 245

charge, 398 coordinates, 458 couplings, 470-471, 472 geometry, 430 kinetic energy operator, 469, 472 motion, 430, 442, 447, 457, 465 physics, 404 processes, in stars, 493 spin, 46, 97, 458

orientations, 482 three-body problem, theory for, 230 wavefunctions, 465 multiconfiguration DHF calculations, 414 multidimensional surface, 419 numerical integration, 465 numerical solution, of equations of

numerical solutions, to RRPA equations,

numerical techniques, 410 Numerov algorithm, 331 Numerov formula, 333 Numerov method, renormalized, of Johnson,

Numerov techniques, 270, 443

molecular scattering, 339

390

332

0 OBK

amplitude, 35 1 approximation, 345, 351-352, 369, 370 cross sections, 345, 365, 366, 367 models, 370

INDEX 54 1

OBK2 approximation, 35 1, 364, 367 cross sections. 365

OCP, 107 (see also one-component plasma) microfield distribution, 114 radial distribution functions, 113-1 14 results, 108, 114

OCS molecule, experiment on, 147 off-diagonal coupling elements, 465 OH molecule, 461 OMC- I , in Orion, 496 one-component plasma, 107 (see also OCP) one-electron

Coulomb case, 396 Dirac equation, 184, 386 Dirac Hamiltonian, 186 eigenfunctions, 174 energies, 4 1 3 function, 269 H+LC2CL system, 10-1 1 Hamiltonian, 376 ionization energies, 185 model, 182 model-potential terms, 173 operators, 172, 189, 201, 204, 205, 206,

209, 214 orbitals, 377 positive-energy Hartree-Fock projection

potential, 177 quantum numbers, 184 reduced matrix elements, 384 spectra, 174 system, 168-169

operators, 387

one-perturber absorption line shape, 142 one-perturber line shape function, 146 one-perturber line shapes, 135, 141, 142,

one-perturber spectra, 134, 135, 138, 141 one-perturber theory, 140, 143 one-photon induced resonant light scattering,

one-valence electron atom, 188 opacity, source of, 441 opacity clouds, 482 open channel, 238, 243, 328

144

158

case, 332 solutions, 330

open shell atomic systems, 41 1 open shell atoms, 469, 470

open shell DHF calculations, 414 open-shell-core molecular model-potential

operators, 314 optical absorption line techniques, 491, 504 optical excitation, 432 optical potential, 163 optical properties, 488-489 optical-potential methods, 237 optics, elementary, 305 optimization process, 394, 400-401 optimization technique, 414 optimized Kohn method, 226 orbital angular moments, 269 orbital angular momentum quantum number,

323, 327 orbital angular momentum, non-zero, 190 orbital energy, 207 orbital screening constant, 209 orbital values, 213 orbital(s), 198, 208, 291, 381, 462

method, 190

angular components of, 381 excited HF occupied, 377, 378 perturbed, 378, 381

radial components of, 381 radial RRPA, 382, 387, 388 set of, 381 single radial DHF, 388 two-component radial, 380, 381

adjacent classical, 3 17 classical, 3 16

order-of-magnitude, 117 variation, 429

ordered exponential, 313 organic positive ion chemistry, 70 original variational estimate, 236 Orion Molecular Cloud, 488 ortho/para thermalization, 483-484 orthogonal matrix, 329 orthogonality conditions, 189 orthogonality requirement, 169 orthonormal set of functions, 394 orthonormality

radial, 386

orbits, 3 1 1-3 12, 320

of HF orb, 208 relations, 379

orthonormalization, 397 oscillation strengths, 182

542 INDEX

oscillations, 94, 95, 96-97, 141 in scattering, 84 quantum, 141

oscillator strengths, 13, 14, 24, 46, 101, 171, 177-179, 181, 183, 184, 185, 209, 215, 390, 446, 450, 455, 461-462, 463, 464, 480, 484, 504

Bethe’s generalized, 118 calculated, 408 densities, 398, 408, 409, 410 discrete calculation of, 14 formula, 39-40 generalized, 119-121

anharmonic, 422 harmonic, 419 local mode, 427 nonlinear, 422

behavior, 95 structure, 95

oscillators

oscillatory

outgoing wave, 307, 313, 315 overlap integrals, 208 overlap, of line pairs, 487 overlapping lines, 146 overtone excitation, 425 overtone vibration excitation, 425 overtone-induced dissociation, of hydrogen

overtone-induced reaction, in a supersonic

overtone-induced reactions, 426 oxygen, 484

peroside, 426-428

beam, 424

chemistry, 449, 452, 486, 487, 496 molecular, continuum cross section in, 453

oxygen-bearing species, 486 oxygen-containing hydrocarbons, 486 ozone, 452

P Pad6 approximant, 205, 209, 210, 234, 236

correction method, 235-236 corrections, 236, 247

Padk-like approximants, 215 PAHs, 498 pair-wise additivity, of perturber-active atom

parabolic approximations, 3 17 parallelism, 339 parameter theory, 217

interactions, 144

parameterized Gaussian potential, 230 parameters, 38, 169, 186, 378, 403, 422

dipole moment, 73 exponential, 413 in effective Hamiltonian, 164 multi-exponential, 412 neutral polarizability, 73 probided by measurements of scattering, 83 theoretical, 420

parametric-potential method, realtivistic

parity, 384 partial collision strengths, 262 partial wave method, 93-94, 98 partial waves, 262, 265 partial wave calculations, 90 path integrals, 309, 313 Pauli equation formalism, 421 Pauli exclusion principle, 169 PBK approximation, 369

version of, 186

peaks broad discrete, 466-467 finite, 317 in fluorescence spectrum, 157 intensity, 442 of photon flux, 442 strength of central, 158 suppression, 154, 156 switching, 154

peaking approximation, 370 penetrating cosmic rays, 487, 489 penetration of radiation, into a cloud, 488-489 Penning ionization, 15 Penning process, 15 performing integrals, 395 permanent dipole moment, 480 permittivity, 108, 113 calculations, 338 perturbation, 1 11, 116, 457

atomic, 38 corrections, 335 methods, 8, I 1

of small disturbances, 9 in the target, 370 molecular, 38 parameter, 203 potential, 335 series, 360, 364 theory, 106, 116, 183, 204, 345

application techniques, 8

INDEX 543

first-order, 379 many-body, 375

time-dependent, 255 two-electron, 200

approaches, 153, 344, 347 calculation of El-MI decay rates, 398 expansion, 351, 352 models, 343, 352 techniques, 152

perturbed potential, 378 perturber density, 143 perturber motion, uncorrelated, 144 perturber-perturber interactions, 142-143, 145 perturbers, multiple, 145 perturbing electronic state, 457 phase factor, 360 phase point, motion of, 422 phase shifts, 144, 224, 225, 226, 238, 242,

perturbative

325, 333 constant, 315 results, 239

phase space, 419, 431 phase space theory, 78. 421, 429 (see also

Phillips-Kleinman pseudo-potential, 233 photo-electtic effect, 369 photoabsorption, 12

cross section, 452 measurements, 452

photochemistry, 487 photodestruction, 490, 491

mechanism, 454, 461 in comets, 462

PST)

photodetachment, 492 photodissociating interstellar CO and HCL,

441 photodissociation, 8, 12, 26, 30, 418, 422,

425, 428, 438, 439, 445, 448, 463, 466, 470,472,489, 487,494, 497,499

channels, 441, 447 dominant, in cometary atmospheres, 466 in HCL, 448 lowest energy, 461 lowest lying, 442

continuous, 480 continuum, 463 cross sections, 11, 12, 439, 440, 444, 465,

466, 467, 468, 469 computed, 449

direct, 439 in cometary and planetary atmospheres, 442 in comets, 459 indirect, 439 mechanisms, 441 pathway, predominant, 439 of CH and OH, in interstellar clouds, 441 of CH, 469 of CO, 487 of diatomic molecules, 4 I8

of HCL, 448 of ICN, 423 of molecular oxygen, 449 of neutral diatomics, 447-448 of OH by absorption, 466 of small molecules, 438, 469 processes, 34, 437, 438, 441, 445, 448,

products, in interstellar medium, 447 rate, 447, 459, 461 shielded rate of, 482 spectra, computed, 472 studies, 11 1-12

of Hz, 30, 441

455, 473

photoelectron peaks, disappearance of, 156 photoelectron problem, 25 photoelectrons, 24 photoemission, 16 photoexcitation, 11 photofragment spectroscopy experiments, 446,

photofragmentation, 429 469

dynamics, 430 spectroscopy experiments, 430

photoionization, 8, 11, 14, 179, 222, 234, 235, 236, 267, 376, 480, 491

by hard radiation, 30 cross sections, 25, 180-181, 182, 183, 186,

233 calculation of, 14, 408 in hydrogenic ions, 398

data, 180-181 of methane, 11 rates, 488 studies, 186, 235

photoionitationle- -ion scattering, 233 photolysis, 26 photon, 426

absorption, 453, 454 and electron impact experiments, 64

544 INDEX

photon, (Continued) antibunching, 160 destruction by, 480 energy, 234, 235, 429, 439, 447, 448,

flux, 441,459 interference between, 146 polarization vector, 408

457

photophysics, 425 piecewise linear, 334 piecewise polynomials, 405 Planck's constant, 136 plane wave, 345, 408 plane wave basis set, 237 plane-wave Born cross sections, 124 plane-wave Born formulae, 125 planetary atmosphere, 62, 442, 445

planetary exosphere, 26 planetary nebulae, 252 planetary systems, formation and evolution of,

478 plasma, 252

atomic and molecular processes in, 134

charge-neutral, 114 conductibility, influence of screening on,

density, 104 dispersion function, 107-108 effects, on collisions, 101-102 electron-ion, 114 environment, 116-1 17

collision perturbed by, 122 on ionic states, I16

fluctuations, 117, 128 excitations via, 119 formula, 124

110

high temperature, high density, 101 homogeneous, 102-103 internal electric potential, 112 kinetic theory, 102 of degenerate stellar cores, 110 particles, 102 permittivity, 107, 110, 1 I 1 strongly coupled, 108, 110 temperature, 104, 253 two-component, electron-ion, 113 weakly coupled, 107, 108, 1 1 1-112, 117,

weakly coupled, electron-ion, 109 1 24

Poisson's equations, 112

polar molecules, 483 deflection of, 54

polarizabilities, 13, 39, 44, 45-46, 48, 49, 50-51, 52, 57, 167, 169, 207

calculated of the elements, 58 for alkali metal dimers, 49

anisotropic, 41 as a parameter, 43 diagram of, 45 of dimer, 55 theory, 40-42

polarization, 12, 240 effect, 238 interactions, 173 potential, 171, 173

long range, 276 terms, 263, 265

long-range, 174 long-range, 17 1-172

dynamics of, 421 MPE/MPD understanding of, 152 photodissociation of, 438 photodissociation processes, 41 8

polarizabilit y

polyatomic molecules, 238, 422

polyatomic systems, 247 plycyclic aromatic hydrocarbons. 490 (see

polynomial basis functions, 224 polynomials, 229, 332, 403

finite basis set of, 402 of Sturmian type, 404 set of finite, 403

also PAHS)

pondermotive potential, 154, 156 population distribution, 48 1 position sensitive detectors, 98 positive eigenvalues, 400 positive energy

continua, 405 eigenstates, 396 eigenvalues, 397 eigenvectors, 174 HF orbitals, 387 projection operator, 386, 41 1 spectra, 395, 398 spectrum, of solutions, 402

potential curve topologies, 71 potential curves, 69, 71, 189, 190, 445, 449,

450, 452, 468 repulsive, 69

INDEX 545

potential energy curves. 141. 164, 188, 442, 446, 447, 448, 451, 454, 455, 458, 460, 461, 462, 466, 468, 472

potential energy matrix, 141 potential energy surface, 138, 139, 420

of polyatomic molecules, 429 potential energy, effective, 420 potential following, 327

algorithm, 335 methods, 330

potential interstellar absorption lines, 491 potential matrix, 323, 328, 330, 334-335 potential of interaction, 223 potential surface, 422, 424, 481 potential well, 442, 443 potential(s), 361, 402

central, 396 long-range, 247 perturbing, 379 separable, 228, 229 short-range, 170, 247, 346 square well, 224 static, 224 Yukawa, 224

potentional energy surfaces, 422 power series expansions, 196, 201 power-broadening, 147 pre-shock density, 499 pre-shock molecular fraction, 499 pre-shock temperature, 495 predictions, 426 predissociating level, of lower bound state,

predissociation, 418, 440-441, 451, 456-457, 44 1

459-460, 461, 462, 463, 466, 470, 471, 472

channel, 462 lifetime, 457 line widths, 464 mechanism, 458, 464 pathways, 459, 471 probabilities, 463 process, 457

accidental, 420 quantification of, 457 rate, 457, 458, 470, 471, 472 signficant, 463 to repulsive electronic state, 429 weak, 460 width, 457

predissociative broadening, 463 pressure broadening, 134

of atomic spectral lines, 133 quasistatic formula of, 140 theory of, 134

pressure dependence, 76 primary ion beam, 91-92 principal quantum number, 280 probability density functions, 423 probability distributions, 423 probability of emission, 430 probe temperature, 48 1 products, 419, 432

atoms, 471 detection scheme, 425 energy distribution, 430 internal states of, 430

profile shapes, 504 programmes for computing, 353 projectile, 95, 350, 351, 357 projectile nuclei, 344 projection operators, 350, 357, 401, 41 1 propagators, 337-338

concept of, 335 definition of, 337 theory of, 338

properties of helium, lithium, other sequences, 9

proton as cause of excitation, 252 as perturber, 253 cross sections, 252 excitation, 252 fast, 344 impact, 354

incident on a neutral atom, 345 transfer, 485

excitation, 501

reactions, 482 proton-induced transitions, 261 proton-mixing, 25 1, 252 PSD, 95, 96 PSD calibration, 88 pseudopotential approach, 169 pseudopotential calculations, 188 pseudopotential theory, 13, 14, 164 pseudopotential, 163, 164 pseudo-resonances, 276 pseudo-state calculations, 276 pseudo-state expansions, 276

546 INDEX

pseudo-state methods, 9 pseudo-states, 357 pseudostates, 200 PSS method, 15

Q QED, 386

QES, 149 (see also quasi-energy state) quadratic variational methods, 226 quadrature formula, 332 quadruple

excitation, 120-121 interaction, 258

polarizabilities, 187, 216 term, 255, 263

transition, 127 vibration-rotation lines, 496 vibration-rotation transitions, 483

calculations, 395

long-range, 261, 263, 264

long-range, 262

quantal behavior, 43 1 quantal calculations, 16-17, 253, 261, 431

quantal cross sections, 256 quantal description, 344, 454

quantal formalism, 323 quantal formula, 347 quantal formulation, 265 quantal interference patterns, 84 quantal results, 256, 260

quantal second Born description, 368 quantal transition probabilities. 262 quantal unimolecular dynamics, 423 quantitative rate constant measurements, 61 quantization axis, 258

space-fixed, 258 quantized state structure, in molecules, 417 quantum chemical calculations, 449 quantum defects, 171, 240-241, 242 quantum electrodynamics, 377 quantum mechanical

calculations, 141 close-coupled theory, I , 142 distribution, 423 line shapes, 139 methods, 139, 140 process, 422

for transition, close-coupled, 262

full, 14

of Faucher, 262

properties, 163 virial theorem, 2 12 wavefunctions, initial state, 423

Quantum mechanics, 135 quantum mechanics, 305, 306, 439 quantum numbers, 269, 380, 443

angular momentum, 369, 383 effective, 242 magnetic, 382, 385 principal, 242, 285, 289, 367 rotational, 443, 472, 501 vibrational, 443

quantum states, 422, 424-425 quantum theory, 4 19 quantum-defect curve, 242 quantum-defect theory, 182 quasi-bound excited state, 290 quasi-bound levels, 457 quasi-bound rovibrational levels, 472 quasi-bound states, 282 quasi-energy, 149, 152

quasi-state microfield, 126 quasi-static ionic microfield, 116 quasistatic formula, 140 quasistatic theory, 141, 143 quenching, 72 quenching rate constants, 72 quiescent cloud, 497 quiescent gas, 500 quiver kinetic energy, 154

state, 149 (see also QES)

R R-dependent factor, 260 R matrix, 328, 332

elements, 328 method, 242 results, 244

R-matrix, 336 calculation, 276 (R-matrix) methods, 337, 338 propagator, 338

Rabi condition, 43 Rabi frequency, 156, 160 Rabi position, 44 radial component, of nuclear kinetic energy

radial coordinate, 405 radial coupling term, 336 radial DHF equation, 388

operator, 458, 464, 466, 468

INDEX 547

radial Dirac equation, 396 radial Dirac Hamiltonian, 396, 403 radial distribution function, 113 radial eigenfunctions, 396, 404 radial eigenvectors, 402 radial equations, 261 radial factors, 380 radial form, 383 radial functions, 269, 271, 382, 402

radial grid, 405

radial Hamiltonian, 382 radial HF, 381

radial integrals, 272. 408 radial nuclear

hydrogenic , 1 2 1

overlapping segments of, 405

equation, 387

coupling, 469 eigenvalue equation, 443 operator, 472 Schrodinger equation, 442-443

radial part, of Dirac equation, 396 radial RRPA orbitals, 382 radial spinor, 396 radial unclear interactions, 470 radial variational eigenfunctions, 4 12 radial-factor, common, 260 radiation, 441, 448, 456, 457

collisional redistribution of, 146 emitted during collision, 432 extreme ultraviolet solar, 25 field, 31-32, 441, 442, 444, 459, 501,

intensity, 479 from a finite source, 307 interaction with molecules, 417 sources, 441

process, 484, 485 theory of, 485

rates, 492

process, 432

uniform, 454

radiative association, 17, 438, 470, 497

radiative attachment, 492

radiative charge transfer, 17

radiative charge-transfer, 16 radiative damping, 156 radiative deactivation, 16 radiative decay, 158, 252, 293, 454,

rate, most probable, 455-456 459

radiative lifetimes, 179, 189, 457

radiative mechanism, 17 radiative processes, 482, 505 radiative recombination, 489

radiative stabilization, 292 radiative transfer, 502 radiative transition probabilities, 454 radiative transitions, 502, 505 radiativity, of excited infrared lines, 483 radio emission lines, 503 radio frequencies, 493 radio frequency spectrum, 493 radio probing, 23 radio techniques, 486 radioactive elements, 489 radiometers, 501 radiometric observation, 502 rainbows, 314

angle, 95 effects, 84 scattering, 91

table, 180

coefficients, calculation of, 14

Ramsauer minimum, 238 random-phase approximation, 103, 375 (see

also RPA) relativistic, 375

randomization, assumption of, 419 rare-gas model potential, 189 rate coefficient, 439, 484, 491 rate constant, 77, 420

503 for three-body association, 73 measurement, 428

rate of formation, on grain surfaces, 499 rate of product formation, 424 rates of decay, 42 1 radio frequency resonance transition

Rayleigh scattering cross section, 11 Rayleigh-Ritz method, 222, 240 Rayleigh-Ritz variational principle, 221 Rayleigh-Jeans antenna temperatures,

computed, 502 reactance matrices, 269 reactant molecule, 424 reactant states, uncontrolled, 62 reaction coordinate, 419, 430

of hydrogen peroxide, 427 reaction dynamics, 71 reaction mechanisms. 432

wavelengths, 171

548 INDEX

Reaction Path Hamiltonian, 421 reaction products, 429 reaction progress variable, 419 reaction rate, 419, 425

calculation, 419 coefficients, 478 constants, 78-79

reaction specificity, 429 reactions

charge-transfer, 63, 64 ion-atom interchange, 63

reactive collisions, 48 I reactive full collisions, dynamics of, 432 reactive scattering, 418, 422

molecular-beam investigations of, 4 18 READI, 290, 292 (see also resonance

reagents, 419, 432 realistic potential energy surfaces, 427 rearrangement collisions, 360, 366 recoil particles, detection of, 95 recursion formula, stable, 338 REDA, 295, 297 REDA, 290 (see also resonance-excitation-

double autoionization process) REDA, 294 (see also resonance-excitation

double ionization) REDA mechanism, 295 reference potential, 335 reflection nebulae, 483 reflection structure, 141 refractive index, 11, 40

region

excitation auto-double-ionization process)

of a gas, 40

classically allowed, 314 of intense irradiation, 30 star forming, 30

relative collision velocity, 343 relative velocities, 499 relativistic

atomic structure, theory of, 10 calculation, of atomic properties, 396 continuum states, 409 contraction, 185 core contraction, 183 Dirac equations, 183 effects, 183, 376 energies, 344 energy eigenstate calculations, 398 extension, 410

field equations, 404 Hamiltonian, 395 HF equation, radial reduction of, 379 modifications, 370 pseudopotential theory, I87 random phase approximation (RRPA), 8, 10 random-phase approximation, 376 (see also

RRPA) method, 410

Sturmian basis set, 397, 404, 405, 406 sum rules, 398 variational calculations, uses of, 395 variational formulation, 397 variational representation, of atomic

systems, 396-414 virial theorem, 41 1, 413

Relativistic Quanta1 treatments, 10 relaxation, 325

times, 126 repulsive curve, 458, 463 repulsive molecular states, 439 repulsive potential, 471 repulsive potential curves, 439 repulsive states, 459, 461. 463, 464, 470

repulsive term, short-range, 169 repulsive upper state, 439 repulsive wall, 89, 439 residual electron-nucleus term, 35 1 resonance, 244, 246, 284, 285, 288, 289, 298

lower lying, 456

contribution, 285 effects, 289 energies, 242 excitation auto-double-ionization process,

fluorescence, 146, 160 290 (see also READI)

experiment, 137 processes, 156 spectrum, 156, 157

induced fluorescence, 52 line, 142 low-lying, 288 scattering, 230 series, 283

states, 175, 222, 239, 240, 242-243, 245, that converges to a threshold, 283

247 in helium, 282

structure, 298 wavefunctions, 245, 246

INDEX 549

normalized, 246, 247 normalizing, 245

widths, 242 resonance-excitation double ionization, 294

(see also REDA) resonance-excitation-double autoionization

process, 290 (see also REDA) resonance-fluorescence spectra, 14 1 resonances, 12, 466, 467, 469, 472

effect on electron excitation rates, 268 effects of, 295, 297 in electron collision experiments, 282 in hydrogen, positions, 283 in hydrogen, widths, 283 of atomic hydrogen, 283 of helium, 286-287 positions of, 298 series of, 441 that can decay via double autoionizing,

widths of, 298 with inner-shell vacancy, 295

295-297

resonant dissociative photoionization, 12 resonant excitation transfer, 15 resonant light scattering processes, 156 resonant photoabsorption, 153 resonant states, 431 response frequencies, I12 rest energy, of electron, 376 rest frame, 370 rest mass energy, 370 rest wavelengths, 500 restrictions

equivalence, 206 spin, 206

RHS, 309 Rice-Ramsperger-Kassel-Marcus theories, 4 19

ring center, 87 ring widths, 87 RMATRX, 270, 271, 272 rotation, 465

of the diatomic, 137 of molecule, 443

(see also RRKM)

rotational angular momentum, 137 rotational congestion, 427 rotational distributions, non-Maxweilian, 75 rotational energy, of molecule, 428 rotational excitation, 17, 18, 323, 499

cross sections for, 25

distorted-wave treatment of, 18 molecular, 18

of nonpolar molecules, 17 temperatures, 483

rotational fraction, 202 rotational levels, 439, 480, 495 rotational lines, 502 rotational motion, 141, 441 rotational nuclear couplings, 471 rotational nuclear interaction, 460, 470 rotational part, 442

rotational populations, 495 rotational predissociation, 457 rotational spectrum, 490 rotational states, 426 rotational transitions, 501, 504

rotational-vibrational ladder, of the ground

rotationally inelastic scattering, 501 rotator, vibrating, 324 rovibrational states, 429 rovibronic states, densities of, 419 RPA, 375

of Hz, 481

of nuclear kinetic energy operator, 460

CN, 502

state, 30

amplitudes, 375 applications of, 375-376 equations, nonrelativistic, 389 nonrelativistic treatment of, 388 permittivities, 112, 121 quantity, for one-component plasma, table,

108 RRKM theory, 419, 420, 421

assumptions of, 424 predictions of, 43 1

RRPA, 376, 386 (see also relativistic random phase approximation)

eigenvalues, 389 equations, 376, 377, 378, 379, 382, 387,

388, 389 expansion of, 388 homogeneous, 378, 379, 381, 382, 387 inhomogeneous, 379 radial, 386, 390 solutions to, 383, 390

excitation energies, 389 functions, radial, 385, 388-389 spectra, 389 transition amplitude, 384

550 INDEX

RRPA, (Continued) treatment of correlation, 390 values, 390

RSPT, 195, 196 (see also Raleigh-Schrodinger perturbation theory)

coefficients, 21 1 equations, 197, 199, 200 solutions, 196 sums, 211 treatment, 197 wavefunctions, 212

Runge-Kutta routine, 337 Rydberg electron energy, 169 Rydberg levels, 17 1 , 174 Rydberg series, 240, 242 Rydberg states, 240

Rydberg units, 195 of alkali atoms, photoionization of, 182

S S matrix, 323, 330, 334

S-matrix, 327, 328 theory, of Newton, 239

S-R bands, 451 of molecular oxygen, 464

S-R continuum, 451, 452 s-wave scattering, 226 SA, 61, 62 (see also stationary afterglow) saddle, classical, 317 scalar, 46 scalar potential, 383 scale energy, 320 scaled variables, 206 scaling behavior, 367 scaling parameter, 173-174 scattered collision products, distribution of, 83 scattered elastically, 368 scattered flux, 88 scattered particles, detection of, 95 scattered wave packet, I05 scattering, 324, 328

amplitudes, 93-94, 95, 97, 307, 316, 317, 323 apparatus need to calculate, 305

and absorption, by dust particles, 488 angles, 90, 106

large, 96 boundary conditions, 304 calculations, 127 continuum, 395

cross sections, 164, 393 elastic, 156 equations, 335 event

“textbook”, 115 recording of, 95-96

matrix, 327 methods, 231 models, direct, 303 of beams, measurements of, 83 (of electron), 17 parameters, 230 phase function, 489 phase shifts, 170

phenomena, 230 problems, 336 process, 247, 327

theories, 102 theory, 237

variational principles in, 221 variational methods, 225-226, 247 variational principle, 239 wave function, 272 wavefunction, 246

SCE, 350 (see also single-centre expansion) SCE approximation, 357 schematic (Born-Oppenheimer) potential

energy curves, 439 Schriidinger equation, 165-166, 172, 174,

196, 198, 203, 239, 268, 303, 304, 311, 313, 326, 331, 334, 357, 359, 339, 352, 345, 346

projected, 237 radial form of, 324-325 radial, 136 time-independent, 261 two-dimensional, 315

table for Helium atom, 170

multichannel description of, 327

Schriidinger Hamiltonian, 205, 206, 207 Schrodinger’s wave mechanics, 195 Schumann-Runge band system, 463 Schumann-Runge bands, 450 (see also S-R

Schumann-Runge continuum, 450, 469 (see bands)

also S-R continuum)

Schwinger denominators, 228 Schwinger expression, 223, 225, 229, 232 Schwinger formalism, 224

absorption in, 26

INDEX 55 1

Schwinger method, 222, 226, 228, 242, 246,

Schwinger phase shift, 226 Schwinger principle, 243 Schwinger T matrix, 228, 239

Schwinger variational expression, 226 Schwinger variational method, 222, 227 Schwinger variational principle, 226-227,

screened Coulomb potentials, 398 screened-Coulomb, 414 screening, 413 screening approximation, 205, 21 1, 212, 213,

214 screening constant, 205 screening functions, 382 SE

247

residues, 245 -246

229, 231

amplitude, 363 approximation, 363, 367 perturbation series, 365-366

Seaton’s approximation, modification of, 257 second Born contribution, 369 second Born cross sections, 367 second Born results, 224 second Born term, 366 second-order differential equation, 3 1 1, 443 second-order energies, 199 second-order interaction energy, 13 second-order matrix element, 202 second-order nonlinear equations, 398 Selected Ion Flow Tube technique, 62 (see

also SIFT) selection rules, 458 self-broadening, of hydrogen in absorption,

self-shielded species, 455 semi-empirical representations, of dynamic

semi-local potentials, 164 semiclassical approximation, 313, 315 semiclassical calculations, 255, 261 semiclassical close-coupled results, 263 semiclassical coupled-equations, 261 semiclassical expressions, 3 14, 463 semiclassical Floquet theory, 148 semiclassical formula, symmetrized, 265 semiclassical formulation, 265 semiclassical methods, 261, 309, 422 semiclassical models, 253

141

polarizabilities, 13

semiclassical results. 256

semiclassical solution, 3 15 semiclassical theory, 255, 256 semiclassical transition probabilities, 262 semiclassical treatment, 320 semiempirical formula, by Lotz, 291 Separable Unimolecular Rate Theory, 421 (see

separate orbital contributions, 208 Separate Statistical Ensembles, 421 (see also

shape resonances, 473 shift parameter, 144 shock chemistry, 32, 496, 497

of sulphur, 496 shock models, 445 shock speed, 499 shock tube, 41 shock

of Landman. 262

also SURT)

SSE)

cooling length of, 499 in diffuse clouds, 33, 496 speed and structure of, 500 structure, 32

shock-heated gas, 445, 498 formation of CH+ in, 500

shock-heated interstellar regions, 480 shock-heated molecular gas, 483 shock-heated regions, of interstellar medium,

shocked interstellar regions, 447 shocked-gas, 499 short-range repulsive potential, 164 short-wavelength lasers, 101 (see also EUV

sidebands, 156, 158 Siegert method, 246 Siegert results, 244 SIW, 62, (see also Selected Ion Flow Tube

technique) SIFTDT techniques, 76 silicon chemistry, 488 Simpson’s Rule, 332 single channel case, 334 single particle distribution functions, 105 single-centre expansion, 350 (see also SCE) single-electron matrix element, 209 single-electron operators, 208, 213 single-particle excited states, 375 singlet manifold, 454

495

and X-ray lasers)

552 INDEX

singlet-state oscillator strengths, 390 single-triplet splitting, 390 singularity, 348 site specificity, 425 site-specificity, 426 site-specificity, of reactions, 424 Slater determinant, 208, 377, 378 Slater's dynamical theory, 419 SLEIGN code, 331 slope

of excitation cross section, 288 of repulsive curves, 463

small angle scattering, 84, 85 apparatus for, 85-86

sodium chemistry, 490-491 software packages, high quality, 33 1 solar abundance, 491 solar corona, 252, 260 solar emission lines, 252 solar flux, 452, 462 solar radiation, 463

solution following algorithm, 335 solution following methods, 330 space coordinates, 268-269 space-dependent response, 1 13 spatially forbidden transitions, 452 species

astrophysically interesting, 445 highly ionized, 265

processes, 477 of highly ionized heavy atoms, 186 simulated, 146

spectral congestion, 427 spectral features,

nonlinear, 149 Autler-Townes splitting, 149 dynamical Stark shift, 149 hole burning, 149 power broadening, 149 S-hump behaviors, 149

broadening, 16, 117, 457 shape analysis, figure, 151 shapes, 128, 146

MPA, 149 strong-field MPA, 149

field, 442

spectra, 136

spectral line

spectral lines, 147, 457, 486 atomic, 133, 134

quasi-static Stark broadening of, 114

spectral patterns, 147, 152-153

spectral resolution, of beam foil technique, 179-180

spectral signature, 441 spectral width, 158 spectrometers, 480 spectroscopic constants, 49, 451

spectroscopic observations, 478 spectroscopic phenomenon, 4 I8 spectroscopic polarizabilities, neutral and

spectroscopic studies, high-resolution, 152,

spectroscopic techniques, 457 spectroscopy, 432, (see also molecular

light scattering, 157

neutral and ionic, 49

ionic, 49

430

spectroscopy; traditional spectroscopy) of diatomic molecules, 457 high-resolution photogragment, 429

spectrum, 156, 401,454 emergent, 504 extreme ultraviolet, 24 for scattered light, 156 of quantized light field, 158 of radiative charge transfer process, 432

speed of light, 118 spencal hormonics, 167 spherical basis, 380, 381 spherical Bessel function, 408 spherical top eigenfunction, 442 spin angular momentum vector, 458 spin conservation, 67 spin conversion, 66, 67 spin coordinates, 268-269 spin exchange, 67 spin moments, 269 spin orbital, 206, 207 spin selection rules, 66 spin-dependent interaction, 389 spin-forbidden transitions, 452 spin-orbit, 183, 472

coupling, 459, 471 effects, 14 interaction, 183, 452, 458, 459, 462, 463,

matrix, 462 470

elements, 463

INDEX 553

mixing, 10 operator, 185, 459 splitting, 376 terms, 186

spin-other-orbit interaction, 183 spin-polarized photoelectrons, 180 spin-rotation interaction, 458 spin-spin interaction, 183, 458 spinor function, 370 spinor orthogonality, 397 splitting, 147 spontaneous emission, 156, 439, 501

spontaneous radiative dissociation, 441, 453,

spurious energy eigenvalue, 398 spurious roots, 396, 398, 400, 401, 41 I square wells, 244 square-well potential, 229, 230 square-well problem, 243 stabilisation technique, 329, 330 stability properties, 338 stability studies, 337 stabilization technique, 23 1 stable molecular states, 439 standard methods, 270 standing wave solution, 328 star-forming complex, 483 Stark broadening, 134

Stark effect, 116 Stark energy, 42

nonvanishing, 53 Stark mixed states, 125 Stark mixing, 125-126 Stark representation (parabolic coordinates),

125 starlight

of radiation, 441

454-455, 478, 479, 494

of atomic spectral lines, 133

external, 479 intensity of, 489 photoionization, 488

background, 480 formation and evolution of, 478, 495 young hot, 30

state crossing, 464 state structure, of molecule, 431 state-by-state unimolecular dissociations, 428 state-populations, 25 1 state-resolution, 432

stars, 505

state-selective preparation, true, 424-425 state-selective reaction rates, 499 state-specific formation, 48 1 state-specific photodissociation cross sections,

470 state-specificity, 426 state-to-state

coupling, 422 experiments, 420 rate constants, 420, 421 reaction rates, 430 transition, 422 unimolecular reactions, 430

static pole polarizabilities, 187 static potential, 223, 230, 363 static screening model, 123 static-exchange, 233

approximation, 226, 232, 237, 240 (???) level, 238 studies, single channel, 236

static-model-exchange-polarization studies, 238 static-screened interaction potentials, 126 static-screened potentials, 124 stationary afterglow, 61 (see also SA) statistical adiabatic channel calculation, 428 Statistical Adiabatic Channel Method, 421 (see

also SACM) statistical behavior, 431 statistical calculation, 428, 430 statistical description, 420 statistical equilibrium, 252, 505 statistical mechanics, 134, 135 statistical rate theory, 420 statistical reaction rate theories, 427 statistical space theory, 78 statistical theories, 419, 421, 429

applicability of, 428 assumptions of, 430

statistical Thomas-Femi potential, 182 statistical unimolecular reaction rate theories,

steady-state models, 445, 498 of diffuse clouds, 499

steady-state response, 122 steady-state situation, 31 stellar atmospheres, 454

stellar cores, plasma of, 110 stellar interiors, 101 stellar winds, 496

419

atomic and molecular processes in, 134

554 INDEX

step function, 348 Stern-Gerlach force, 44 Stieltjes imaging, 408-409

method, 12, 231 techniques, 398

Stieltjes-Tchebycheff moment theory, 235 stimulated emission, 136, 483, 501, 503 stimulated emission pumping, 425 stochastic electrodynamic perturbation, 102 stomic scattering theory, 141 stratospheric chlorine, 26 stratospheric ozone, 26

stratospheric research, 26 stretching vibration, 428 strong coupling, I 1 1-1 13

effects, 297 plasma regime, 128

effects of industrial activity on, 26

strong interactions, 122, 128 Strong Potential Born approximation, 353 structure factor, charge-charge, 113 structure levels, 471 Stueckelberg-Landau-Zener studies, 16 Sturmian basis sets, 394, 402, 407, 409

methods, 414 results, 409

problem, 402 Sturmian eigenvalues, 403, 404

Sturmian functions, 394 Sturmian-type basis sets, 394, 413 sulfur, 487 sum-rule technique, 8 summation calculations, 395 summations, 405 supercomputers, 381, 339 supernova

explosions, 33 remnant, IC 443, 496 remnants, 495

superposition principle, 107 supersonic beam, in polarizability

measurement, 51 supersonic beams, 57 superthermal motions, 32 sv

applications, 225 expression, 222-223, 228, 243 methods, 22 1, 222, 224, 226, 230, 240,

247, (see also variational methods, Schwinger)

principle, 223, 226, 228, 229, 230, 231, 237, 239, 240, 242-243, 246 application of, 223

symmetric reactance matrix, 328 symmetric resonance charge transfer,

symmetric systems, 84 symmetrical eikonal approximation, 362, 363

(see also SE) symmetrical eikonal model, relativistjc

version, 371 symmetry, 458 symmetry relations, 419

188

T T matrix, 227, 229, 245

channel, 245 denominator, 242, 245 method, 231 poles of, 240 results, 231

target, 344, 351 target electron, 350, 357, 367 target system, 272

target wave functions, 270, 290, 292, 298 TDHF, 376 (see also Coupled, Time-

Dependent Hartree-Fock Theory)

wave function of, 272

equations, 377-378 method, 377 potential, 377 wave function, many-electron, 379

Ternkin’s critique, 320 temperature, 503, 504

electron, 25 ion, 25

490 temperature-sensitive exchange reactions, 483,

temperature-sensitive ratio, 494 temperatures, 265

in planetary nebulae, 252 terrestrial atmosphere, 449, 486 terrestrial experience, 478 test charge screening, I 17 test particle, 109, 1 1 I

theoretical calculations, 486 theory of long-range forces, 8 thermal chemistry, 425 thermal distribution, 49, 54

dressed, 11 1

INDEX 555

thermal energy ion-atom interchange reactions,

thermal energy ion-molecule reactions, 63 thermal rate coefficients, 481, 501 thermochemical data, 491 thermonuclear fusion, controlled, 101 thermonuclear reaction rates, 110 thick clouds, 494, 503 third-order energies, 199 third-order static contribution, 167 Thomas angle, 368 Thomas peak, 368, 369 Thomas-Fermi statistical potential, 173-174 Thomas-Fermi-Dirac theory, Z-expansions

applied to, 196 Thomas-Reiche-Kuhn sum rule, 405 three-body equations, 227 three-body processes, 438 three-electron system, 394 three-particle scattering, 226 three-peak structure, asymmetric, 158 threshold, 304, 445, 446

behavior, 316, 317 cross section, 288 effects, 457 regime, 3 11 wavelength, 492

time-dependent amplitude, 106 atomic wave function, 377 behavior, 479 chemistry, 505 DHF calculations, 414 effects, 482 Hartree-Fock equations, linearized, 376 (see

also TDHF) interaction, 102 interaction, 357, 377 models, of cooling interstellar gas, 33 orbitals, 377, 378 periodic Hamiltonian, 149 response, 113 Schrodinger equation, 139, 303, 344, 346,

350 situation, 31 theory, 10 variational principle, 377

Floquet Hamiltonian, 149, 150 HF equation, 378

64

time-independent

HF potential, 378 orbitals, 377, 378 rate constant, 420 theory, 10

time-ordered products, 348 time-ordered, in classical path method, 142 time-resolution, 432 time-resolved studies, 430 timescales, 104

correlation, 104-1 05 hydrodynamic, 104-105 relaxation, 104-105

titanium chemistry, 491 Tokamak plasmas, 252 torsion, 428 total angular momentum, 47 I total cross section, 292, 304, 352. 363

experimental, 363 measurements of, 364

total density, 499 total elastic cross section, 298 total energy, 411, 413, 414

distribution, 424 spectrum, 395

total excitation cross section, 291 total photodissociation cross sections, 470 TPD, 152 (see also two-photon dissociation)

cross sections, 152, 153 processes, 153

tracers of molecular gas, 31 traditional spectroscopy, 41 8 trajectories,

average potential, on the, 139, 140 classical, 423

initial state potential, on the, 139 ionizing, 316 propagating, 423 quasiclassical, 422-423 set of, 315 straight line, 140 straight-line, 138-139 surface-hopping, 139

calculations, 422, 428 effects, 125 classical, 138, 315

straight line, 344 time-dependent, 255

applications to specific molecules, 424

trajectory,

linear, 106

556 INDEX

trajectory, (Continued)

classical, 422 quasiclassical, 422

transfer of radiation, 440 transformations

initial, 335 stabilising, 332

transformed Hamiltonian, 197 transient species, 418 transition, 126, 450

methods

allowed, 252 dipole-forbidden, 121 forbidden, 126 from lower state to upper state, 273 long-range form of, 257 optically forbidden, 285 to dissociating state, 441

in RRPA, 379 transition amplitudes, 347, 376, 383, 385, 388

transition dipole moments, 441, 466 transition energy, 276, 444 transition frequencies. 118, 478 transition matrix elements, 171-172,

transition moment function, 452 transition moments, 49, 138, 452, 467 transition operators, 443 transition probabilities, 9, 164, 179, 344, 345,

426, 439, 454, 480 transition rates, 393 transition state formulation, 420 transition state, 125, 421, 430, 432

209

in absorption, 432 manipulation of, 43 1-432 probing and manipulating, 431

transition types, 136 transition wavelengths, 177 transition state, 429 transition state theory, 419-420, 421 transitions, 138, 251, 252, 253, 261, 273,

279, 426, 439-440 allowed and forbidden, 121 between magnetic sublevels, 258 between quantum states, 138 bound-bound, 136 bound-free, 136 caused by ions, 126 free-free, I36 optically allowed, 288

sequence of, 350 spin forbidden, 288 to higher states of helium, 289 to repulsive potential curves, 438-439 to Rydberg states, 461

translational energy, 485, 499 translational equilibrium, 136, 137 translucent interstellar clouds, 493 trial functions, 189, 224, 225, 226, 230, 243,

247, 393, 395, 396 tnal wave functions. 269 triatomic collision complexes, lifetimes of,

424 triatomics, 421 triplet transitions. 390 tunneling, 472 tunneling motion, 423 two-body processes, 438 two-component plasma, 109 two-dimensional reduction, 3 17 two-electron atomic and ionic systems,

two-electron equation, 173 two-electron model potential, 177 two-electron operators, 2 13 two-electron Schr6dinger equation, 173 two-electron system, 168-169, 184, 394 two-particle coupled channels, 226 two-particle scattering, 226 two-photon decay, 10

two-photon dissociation, 152 (see also TPD) two-potential formalism, 233, 236, 247 two-potential technique, 236 two-state approximation, 135, 140, 142 two-state close-coupling results, 297 two-state coupled channel approach, 354 two-state coupled-channel calculation, 234 two-valence electron molecule, 188 two-valence-electron systems, 173, I85

173

rates, 398

U UHF, 206 (see also unrestricted Hartree-Fock) ultraviolet. 282

absorption lines, 480 studies, 482

fluorescence, 499 flurorescent excitation, 482, 498 photons, internal source of, 487 radiation, 26, 450, 479, 480

INDEX 557

absorption of, 481 inside molecular clouds, 489

solar radiation, 26 spectrum, 487 starlight, 479, 482, 463 transitions, 483

uncertainty principle, 423 uncoupled repulsive diabatic state, 467 undistorted plane wave states, 35 1 unified line shape, theories of, 145-146 uniform electric multipole field. 203 uniform radiation field, 455 unimolecular bond fission processes, 427 unimolecular decay, 431 unimolecular decomposition, 76, 425 unimolecular dissociation, 421, 428, 429

unimolecular dynamics, 426, 43 1 rates, 429

advantages of small molecules in studies, 426-427

unimolecular lifetimes, 427 unimolecular process, 420 unimolecular reaction calculations, 423 unimolecular reaction rate, 420, 424

constant measurements, 426 theory, 419, 431

unimolecular reaction studies, 422 unimolecular reactions, 417, 418, 422,

423-424, 425 mode-selective, 424 overtone-induced, 426

unitary transformation, 41 1, 465 united-ion limit, 127 universal beam detector, 57 universal detection scheme, 52 unperturbed final state functions, 370 unperturbed Hamiltonian, 236, 237 unperturbed initial state functions, 370 unperturbed kinetic energy, 309 unperturbed radial Schriidinger equations, 458 unperturbed wave functions, relativistic, 370 unsaturated lines, 503 unshielded interstellar radiation field, 447 unstable species, 432 upper atmosphere, 23, 463 upper bounds, 394, 395, 402, 411, 413 upper electronic state, 12, 479 upper triangular matrix, 329 UV emissions, 25 1 valence-core interaction, 164, 168

V valence electron(s), 1 valence-electron Schrodinger equation, 165 valence-electron wave functions, 171-172 valence electrons, 163, 164-166, 187 valence energy levels, 183 valence-orbital energy, 183 valence orbitals

contractions, 185 expansions of, 185

valence-Rydberg mixing. 45 1 valence structures, determination of, 183 valence transition energies, 166 values, calculated, 332 van der Waals

attraction, 19 coefficient, 13 complexes, 43 1 contribution, 13 interactions, 15 molecules, 73, 418 term, 13-14

indistinguishable, 328 of dipole transition moment, 444

variation

variation-perturbation results, 170 variational

basis set, 400 bound energies, 243 bounds, 222, 402 calculations, 390 collapse, 395, 396, 401, 402, 411 DHF results, 413 eigenstates, 41 I eigenvalues, 395, 397, 399, 400, 402 eigenvectors, 394 energies, 21 I expressions, 224, 225, 226 functional, 200 method, anamaly-free, 230 methods, 8, 9, 10-1 1, 221 (see nlso SV

methods) application techniques, 8 HulthBn-Kohn, 221 (see also H-KV methods) (???)

minimum, 411, 413 principle, for scattering theory, 237 principles, 230 procedures, 394, 402 projection, 398

558 INDEX

variational, (Continued) Rayleigh-Schrodinger perturbation theory,

relativistic self-consistent field calculations,

representation, 398

resonance energies, 243 results, 395 solutions, 412 spectrum, 396,401 techniques, finite basis set, 393 theory, 221, 247 treatment, 11, 13

of small disturbances, 9 virtual energies, 243 wave functions, 209, 390

vector potential, 383 vector spherical harmonics, 383 velocities, 438

velocity, 368

195 (see also RSFT)

420

of Dirac spectrum, 400

asymptotic, 366

dispersion, of emitting molecules, 487 distribution, 51 field, 504 form, 385 shifts, 500 Bohr, 343 relative collision, 343

Venus, 26 vibration, stretching, 427 vibration-rotation

band, 480 dependence, 458 excitation, 429 lines, 463 transitions, 481

vibrational continuum, 440, 441, 444, 454, 457,472, 479,482

of electronic state, 444 vibrational energy, 422, 430

distribution, 427 levels, 440 relaxation, between normal modes, 419

temperatures, 483 vibrational excitation, 429-430, 499

vibrational levels, 429, 441, 454, 459, 463, 467,479

discrete, 454 of electronically excited state, 443

vibrational overtone excitation, 426

vibrational parts, 442, 458 vibrational predissociation, 73

rate constants, table of, 74 vibrational quenching, 74, 75

collisions, free-free, 73 vibrational states, 76, 429

of diabatic curves, 12 vibrational wavefunctions, 442, 443, 447 vibrational-rotational degrees, of freedom,

vibrational-rotational state, 137 vibrationally excited, HZ. 479, 480 vibrationally inelastic collisions, 48 1 vibrations, 426, 429 virial theorem, 4 I3

method, 414 virial-DHF calculation, 414 Voyager, 26 VPS approximation, 361, 363, 367 VPS cross sections. 361

experiments, 424

430

W Wannier’s analytic predictions, 3 17 Wannier’s predictions, 304 Wannier’s theory, 305, 317, 320, 321 wave function(s), 9, 10, 175, 208, 223, 224,

226. 232, 237, 261, 268, 269, 271, 304, 357, 377, 422, 442, 458

wave function, asymptotic form of, 306 exact, 348 electronic, 442 hydrogenic , 4 12 of atomic target states, 283 of metastable states, 43 1 nonoptimized, 413 positive continuum, 408 quantum mechanical initial state,

final state, nonorthogonality of, 345 initial state, nonorthogonality of, 345 unperturbed, 345 variationally determined, 454

423

wavelength dependence, of interstellar radiation field, 479

wavelength interval, 489 wavelength radiation, 447 wavelength region, 452, 459, 463, 496

INDEX 559

wavelengths, 439, 442, 452, 478, 483, 488, 491, 500, 501

longer, 462 of Lyman and Werner system lines, 494 ultraviolet, 442

wavenumber, I24 wave-packet methods, 422 wave packet, 305 wave packets, 422

limitations of, 303 wave vector, 323, 327 wave-vectors, 142 weak-interaction constraint, 122 weakly coupled plasma, 11 1-1 12 Weinberg series, 228 Weisskopf radius, 106, 122, 146 well depth, 75 (see also attractive well depth;

well-behaved behavior, 326 Werner band, 30 Werner lines, 479 Werner system bands, 480 Werner system, 454, 487 width

repulsive well depth)

partial, 244, 245, 246, 247 total, 244, 245

Wigner Distribution Function, 423 Wigner-Seitz model (see ion-sphere model) Wronskian, 332

X x-ray emissions, 251 x-ray laser experiment, 128

Y Yukawa potential, 226, 240

Z Z charge-neutralizing electrons, 1 10 Z-dependent screening, 217 Z-expansion, 213

coefficient, 214 methods, 219 procedures, 210 theory, 205

Z-expansions, 196, 204 application of, 196 nonrelativistic, 195

Z-scaling 127 zero angular momentum, 298 zero eigenvalues, 310 zero orbital angular momentum, 190 zero-order energies, 199 zero-order Hamiltonian, 186 zero-order mixing, 2 I5 zero-order solutions, 198 Zone I, 308, 309, 311, 315 Zone 11, 308, 309, 310, 315 Zone 111, 308, 309, 310, 315


Contents of Previous Volumes

Volume 1

Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. T . Amos

Electron Affinities of Atoms and Mole- cules, B. L. Moiseiwitsch

Atomic Rearrangement Collisions, B. H. Bransden

The Production of Rotational and Vibra- tional Transitions in Encounters between Molecules, K . Takayanagi

The Study of Intermolecular Potentials with Molecular Beams at Thermal En- ergies. H . Pauly and J . P. Toennies

High-Intensity and High-Energy Molec- ular Beams, J . B. Anderson, R. P. Andres, and J . B. Fenn

Volume 2

The Calculation of van der Waals Inter- actions, A. Dalgarno and W . D. Dauison

Thermal Diffusion in Gases, E. A. Mason, R. 1 . Mum, and Francis J . Smith

Spectroscopy in the Vacuum Ultraviolet, W . R. S. Carton

The Measurement of the Photoioniza- tion Cross Sections of the Atomic Gases, James A. R. Samson

The Theory of Electron-Atom Col- lisions, R. Peterkop and V . Veldre

Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . deHeer

Mass Spectrometry of Free Radicals, S. N . Foner

Volume 3

The Quanta1 Calculation of Photoioni- zation Cross Sections, A. L. Stewart

Radiofrequency Spectroscopy of Stored Ions I : Storage, H. G. Dehmelt

Optical Pumping Methods in Atomic Spectroscopy, B. Budick

Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H.

Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney

Quantum Mechanics in Gas Crystal- Surface van der Waals Scattering, F. Chanoch Beder

Reactive Collisions between Gas and Surface Atoms, Henry Wise and Ber- nard J . Wood

c. wolf

Volume 4

H. S. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop

Electronic Eigenenergies of the Hydro- gen Molecular Ion, D. R. Bates and R. H. G. Reid

Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buck- ingham and E. Gal

Positrons and Positronium in Gases, P. A. Frazer

Classical Theory of Atomic Scattering, A. Burgess and I . C. Percival

Born Expansions, A. R. Holt and B. L. Moiseiwitsch

Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke

Relativistic Inner Shell Ionization, C. B. 0. Mohr

CONTENTS OF PREVIOUS VOLUMES

Recent Measurements on Charge Transfer, J . B. Hasted

Measurements of Electron Excitation Functions, D. W . 0. Heddle and R. G. W . Keesing

Some New Experimental Methods in Collision Physics, R. F. Stebbings

Atomic Collision Processes in Gaseous Nebulae, M . J . Seaton

Collisions in the Ionosphere, A. Dalgarno The Direct Study of Ionization in Space,

R. L. F . Boyd

Volume 5

Flow Afterglow Measurements of Ion- Neutral Reactions, E. E. Ferguson, F. C. Fehsenfeld, and A. L. Schmeltekopf

Experiments with Merging Beams, Roy H. Neynaber

Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt

The Spectra of Molecular Solids, 0. Schnepp

The Meaning of Collision Broadening of Spectral Lines: The Classical Oscilla- tion Analog, A. Ben-Reuven

The Calculation of Atomic Transition Probabilities, R. J . S. Crossley

Tables of One- and Two-Particle Coeffi- cients of Fractional Parentage for Configurations s's'"'pq, C. D. H. Chis- holm, A. Dalgarno, and F . R. Innes

Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle

Volume 6

Dissociative Recombination, J . N . Bardsley and M . A. Biondi

Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spec- tral Emission Lines, A. S. Kaufman

The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazyu Itikawa

The Diffusion of Atoms and Molecules, E. A. Mason and R. T. Marrero

Theory and Application of Sturmain Functions, Manuel Rotenberg

Use of Classical Mechanics in the Treat- ment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston

Volume 7

Physics of the Hydrogen Master, C. Audion, J . P. Schermann, and P. Grivet

Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C. Erowne

Localized Molecular Orbitals, Hare1 Weinstein, Ruben Paunez, and Maurice Cohen

General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt

Diabatic States of Molecules-Quasi- Stationary Electronic States, Thomas F . O'Malley

Selection Rules within Atomic Shells, B. R. Judd

Green's Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris

A Review of Pseudo-Potentials with Em- phasis on Their Application to Liquid Metals, A. J. Greenjield

Volume 8

Interstellar Molecules: Their Formation and Destruction, D. McNally

Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck

Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y . Chen and Augustine C. Chen


Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. 1. Schoen

The Auger Effect, E. H. S. Burhop and W . N . Asaad

Volume 9

Correlation in Excited States of Atoms, A. W . Weiss

The Calculation of Electron-Atom Exci- tation Cross Sections, M. R. H . Rudge

Collision-Induced Transitions between Rotational Levels, Takeshi Oka

The Differential Cross Section of Low- Energy Electron-Atom Collisions, D. Andrick

Molecular Beam Electronic Resonance Spectroscopy, Jens C. Zorn and Thomas C. English

Atomic and Molecular Processes in the Martian Atmosphere, Michael B. Mc Elro v

Volume 10

Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr. and Serge Feneuille

The First Born Approximation, K. L. Bell and A. E. Kingston

Photoelectron Spectroscopy, W . C. Price

Dye Lasers in Atomic Spectroscopy, W . Lunge, J . Luther, and A. Steudel

Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett

A Review of Jovian Ionospheric Chem- istry, Wesley T. Huntress, Jr.

Volume 11

The Theory of Collisions between Charged Particles and Highly Excited Atoms, I . C. Percival and D. Richards

Electron Impact Excitation of Positive Ions, M . J . Seaton

The R-Matrix Theory of Atomic Process, P. G. Burke and W . D. Robb

Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Levine

Inner Shell Ionization by Incident Nu- clei, Johannes M. Hansteen

Stark Broadening, Hans R. Greim

Chemiluminescence in Gases, M . F. Folde and B. A. Thrush

Volume 12

Nonadiabatic Transitions between Ionic and Covalent States, R. K . Janev

Recent Progress in the Theory of Atomic Isotope Shift, J . Bauche and R.-J. Champeau

Topics on Multiphoton Processes in Atoms, P. Lambropoulos

Optical Pumping of Molecules, M. Broyer, G. Gouedard, J . C. Lehman, and J . Vigue

Highly Ionized Ions, Ivan A. Sellin

Time-of-Flight Scattering Spectroscopy,

Ion Chemistry in the D Region, George Wilhelm Raith

C. Reid

Volume 13

Atomic and Molecular Polarizabili- ties-A Review of Recent Advances, Thomas M . Miller and Benjamin Bederson

Study of Collisions by Laser Spectros- copy, Paul R. Berman

Collision Experiments with Laser-Ex- cited Atoms in Crossed Beams, I . I/. Hertel and W . Stall

Scattering Studies of Rotational and Vi- brational Excitation of Molecules, Manfred Faubel and J . Peter Toennies


Low-Energy Electron Scattering by Complex Atoms: Theory and Calcula- tions, R. K . Nesbet

Microwave Transitions of Interstellar Atoms and Molecules, W. B. Somrner- ville

Volume 14

Resonances in Electron, Atom, and Molecule Scattering, D. E. Golden

The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J . Jamieson, and Ronald F. Stewart

(e, 2e) Collisions, Erich Weigold and Ian E. McCarthy

Forbidden Transition in One- and Two- Electron Atoms, Richard Marrus and Peter J . Mohr

Semiclassical Effects in Heavy-Particle Collisions, M . S. Child

Atomic Physics Tests of the Basic Con- cepts in Quantum Mechanics, Francis M . Pipkin

Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V . Bobashev

Rydberg Atoms, S. A. Edelstein and T. F. Gallagher

UV and X-Ray Spectroscopy in Astro- physics, A. K . Dupree

Volume 15

Sir Harrie Massey, D. R. Bates Negative Ions, H. S . W . Massey Atomic Physics from Atmospheric and

Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R. F.

Stebbings Theoretical Aspects of Positron Col-

lisions in Gases, J . W . Humberston Experimental Aspects of Positron Col-

lisions in Gases, T. C. Grijith Reactive Scattering: Recent Advances in

Theory and Experiment, Richard B. Bernstein

Ion-Atom Charge Transfer Collisions at

Aspects of Recombination, D. R. Bates The Theory of Fast Heavy-Particle Col-

lisions, B. H. Bransden

Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody

Low Energies, J . B. Hasted

Inner-Shell Ionization, E. H . S. Burhop Excitation of Atoms by Electron Impact,

Coherence and Correlation in Atomic

Theory of Low-Energy Electron-Mole-

D. W. 0. Heddle

Collisions, H. Kleinpoppen

cule Collisions, P. G. Burke

Volume 16

Atomic Hartree-Fock Theory, M . Cohen and R. P . McEachran

Experiments and Model Calculations to Determine Interatomic Potentials, R. Diiren

Sources of Polarized Electrons, R. J . Celotta and D. T . Pierce

Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain

Spectroscopy of Laser-Produced Plas- mas, M . H. Key and R. J . Hutcheon

Relativistic Effects in Atomic Collisions Theory, B. t. Moiseiwitsch

Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N . Fortson and L. Wilets

Volume 17

Collective Effects in Photoionization of

Nonadiabatic Charge Transfer, D. S. F.

Atomic Rydberg States, Serge Feneuille

Atoms, M . Ya . Amusia

Crothers

and Pierre Jacquinot


Superfluorescence, M. F. H . Schuurmans, Q. H. F. Vrehen, D. Polder, and H. M . Gibbs

Applications of Resonance Ionization Spectroscopy in Atomic and Molecu- lar Physics, M. G. Payne, C. H . Chen, G. S. Hurst, and G. W. Foltz

Inner-Shell Vacancy Production in Ion- Atom Collisions, C. D. Lin and Patrick Richard

Atomic Processes in the Sun, P. L. Dufton and A. E . Kingston

Volume 18

Theory of Electron-Atomic Scattering in a Radiation Field, Leonard Rosenberg

Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E . Kaup- pila

Nonresonant Multiphoton Ionization of Atoms, J . Morellec, D. Normand, and G. Petite

Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S . Dickinson and D . Richards

Recent Computational Developments in the Use of Complex Scaling in Reso- nance Phenomena, B. R. Junker

Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Sys- tems, N. Andersen and S. E. Nielsen

Model Potentials in Atomic Structure, A. Hibbert

Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W . Norcross and L. A. Collins

Quantum Electrodynamic Effects in Few-Electron Atomic Systems, G. W . F. Drake

Volume 19

Electron Capture in Collisions of Hydro- gen Atoms with Fully Stripped Ions, B. H . Brunsden and R. K . Janev

Interactions of Simple Ion-Atom Sys- tems, J. T. Park

High-Resolution Spectroscopy of Stored Ions, D. J . Wineland, Wayne M . Itano, and R. S. Van Dyck, Jr.

Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K . Blum and H . Kleinpoppen

The Reduced Potential Curve Method for Diatomic Molecules and Its Appli- cations, F. Jenajc

The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson

Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel

Spin Polarization of Atomic and Molec- ular Photoelectrons, N. A. Cherepkov

Volume 20

Ion-Ion Recombination in a n Ambient Gas, D. R. Bares

Atomic Charges within Molecules, G. G. Hall

Experimental Studies on Cluster Ions, T. D. Mark and A. W . Castleman, Jr.

Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W . E. Meyer- hofund J.-F. Chemin

Numerical Calculations on Eelectron- Impact Ionization, Christopher Bottcher

Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong

On the Problem of Extreme UV and X- Ray Lasers, I . I. Sobelman and A. V . Vinogradou

Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J . M . Raimond

Rydberg Atoms: High-Resolution Spec- troscopy and Radiation Interaction- Rydberg Molecules, J . A. C . Gallas, G. Leuchs, H. Walter, and H. Figger


Volume 21

Subnatural Linewidths in Atomic Spec- troscopy, Dennis P. O’Brien, Pierre Meystre, and Herbert Walther

Molecular Applications of Quantum De- fect Theory, Chris H. Greene and Ch. Jungen

Theory of Dielectronic Recombination, Yukap Hahn

Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-Z Chu

Scattering in Strong Magnetic Fields, M . R. C . McDowell and M . Zarcone

Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M. More

Volume 22

Positronium-Its Formation and Inter- action with Simple Systems, J . W . Humberston

Experimental Aspects of Positron and Positronium Physics, T. C. Grifith

Doubly Excited States, Including New Classification Schemes, C . D. Lin

Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H. B. Gilbody

Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K . Dolder and B. Peart

Electron Capture by Simple Ions, Ed- ward Pollack and Yukap Hahn

Relativistic Heavy-Ion-Atom Collisions, R. Anholt and Harvey Gould

Continued-Fraction Methods in Atomic Physics, S. Swain

Volume 23

Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal

Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney

Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Yan

Transition Arrays in the Spectra of Ion- ized Atoms, J . Bauche, C. Bauche- Arnoult, and M . Klapisch

Photoionization and Collisional Ioniza- tion of Laser-Excited Atoms Using Synchrotron Radiation, D. L. Ederer, F . J . Wuilleumier, and J . L Picque

Volume 24

The Selected Ion Flow Tube (SIFT): Studies of Ion-Neutral Reactions, D. Smith and N . G. Adams

Near-Threshold Electron-Module Scat- tering, Michael A. Morrison

Angular Correlation in Multiphoton Ionization of Atoms, S. J . Smith and G. Leuchs

Optical Pumping and Spin Exchange in Gas Cells, R. J . Knize, 2. Wu, and W . Happer

Correlations in Electron-Atom Scatter- ing, A. Crowe

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