Classical and Quantum Dynamics in Condensed Phase Simulations Euroconference on "Technical advances in Particie-based Computational Material Sciences". Coordinator: M. Mareschal

LERICI, Villa Marigola 7 July-18 July 1997

Editors

Bruce J. Berne Columbia Univ. USA

Giovanni Ciccotti Univ. "La Sapienza" Italy

David F. Coker Boston Univ. USA

Y f e World Scientific w b Sinqapore* NewJersey London» Singapore* New Jersey London »HongKong

CONTENTS

PREFACE v

ACKNOWLEDGEMENTS ix

ORGANIZERS xiii

LECTURERS xv

STUDENTS xxi

VISITORS xxvii

CONTENTS xxxi

PART I

A STORY OF RARE EVENTS : FROM BARRIERS TO ELECTRONS TO UNKNOWN PATHWAYS

Chapter 1

Barrier crossings: classical theory of rare but important events 3

by David Chandler

1 - Introduction 7

2 - Cyclohexane isomerization 8

3 High barriers, rare events 9

4 - Reaction coordinate, q(t) 12

5 — Rate constants and time correlation functions 13

6 - Reactive flux correlation function 14

7 — Choosing the t ransi t ion s ta te 16

8 — Lindemann-Hinshelwood and Kramers regimes 17

9 — Cyclohexane reactive flux 18

10 — Summary 21

Chapter 2

Electron transfer in water and other polar environments , how it hap-pens 25

by David Chandler

1 Marcus ' scenario 29

2 - Prom the perspective of the electron 31

3 - Prom the perspective of the solvent nuclei 33

4 — Test of t he Gaussian approximation 37

5 - Electron transfer ra te constant 40

6 - Reorganization energy and the normal and inverted regimes 42

7 - Quantum mechanical electron transfer ra te constant 43

8 - Surface hopping model and exponential decay 46

9 - Summary 48

Chapter 3

Finding t ransi t ion pathways: throwing ropes over rough mountain passes, in the dark 51

by David Chandler

1 - Locating t ransi t ion s ta tes 55

2 — Directed pa ths or chains of s tates 57

3 — Transition probabili t ies 59

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4 — I l lustrat ive e x a m p l e s 61

5 — S u m m a r y 65

P A R T II

T H E A R T O F S I M U L A T I O N

Chapter 4 \

M o n t e Carlo s imulat ions 69

by Daan Frenkel

1 — Introduct ion 73 1.1 Who should read this? 73 1.2 Monte Carlo simulations - why? 73

2 - T h e Metropo l i s m e t h o d 73

3 - Trial moves 75 3.1 'UnphysicaV moves 76 3.2 Composite moves 76

4 — Cluster moves 80 4.1 General Cluster moves 82

5 - Other ensembles 82 5.1 Isobaric-isothermal ensemble 83 5.2 Grand canonical ensemble 85 5.3 Gibbs ensemble 87

6 — Tracing coex i s t ence curves 88

7 — Virtual moves 90 7.1 Particle insertion method 90 7.2 Recursive sampling 91 7.3 Exact enumeration 93

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Chapter 5 ||

Monte Carlo: changing the rules for fun and profit 97

by John Valleau

1 — In troduct ion 101

2 — T h e r m o d y n a m i c Integrat ion 104

3 — H i s t o g r a m reweight ing m e t h o d s 105

4 Umbre l la sampl ing m e t h o d s 109

4.1 Introductory 109 4.2 Thermodynamic scaling 111

4.2.1 One-dimensional thermodynamic scaling 115 4.2.2 Two dimensional scaling: an example 119

4.3 Microscopic barriers 124

5 Conclus ion 125

Chapter 6

Molecular dynamics m e t h o d s for the enhanced sampl ing of phase s p a c e l 2 7

by Bruce J. Berne

1 Introduct ion 131

2 — M e t h o d s for deal ing w i t h t h e mul t ip le t i m e scale p r o b l e m in M o l e c ­ular D y n a m i c s and p a t h integrals 131 2.1 Background 132 2.2 Integrators generated from factorizing the classical propagator 133 2.3 Reference System propagator algorithms 135

2.3.1 Long and short ränge forces 135 2.3.2 Fast and slow processes 136 2.3.3 Combining force subdivision and dynamic subdivision 138

2.4 The path-integral representation of quantum Systems 138

3 - N e w M o n t e Carlo m e t h o d s 140 3.1 Hybrid Monte Carlo 141 3.2 Jump Walking 142 3.3 Smart Walking 144 3.4 Generalized ergodic measures 145

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4 - Applications of MD based Monte Carlo schemes 146 4.1 One dimensional random potential surface 146 4.2 Application to two protein Systems 150

5 Summary 154

Chapter 7

Constrained and nonequilibrium molecular dynamics 157

by Giovanni Ciccotti and Mauro Ferrario

1 — Introduction 161

2 — Molecular dynamics with constraints 161 2.1 The mechanical System with constraints 162 2.2 SHAKE optimization procedure and the numerical algorithm called SHAKE 165

2.2.1 Optimization 165 2.2.2 The numerical algorithm SHAKE 165

2.3 The Statistical ensemble 166

3 - Rare events via blue moon ensemble 169 3.1 Computing reversible work by blue moon ensemble 170 3.2 Compute time correlation functions by blue moon ensemble 173

4 — Non equilibrium molecular dynamics 174

5 Concluding remarks 176

Chapter 8

From Eyring to Kramers: computation of diffusive barrier crossing rates 179

by Maria J. Ruiz-Montero

1 - Introduction 183

2 - Transition rate 183

3 — Statistical accuracy 185

4 — Square barrier 186

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5 — General case 189

Chapter 9

Monte Carlo methods for sampling of rare event states 195

by Wolfhard Janke

1 — Introduction 199

2 — Systems with rare event states 199

3 — Multicanonical reweighting 201 3.1 Enhancing rare events 201 3.2 Avoiding rare events 203

4 — Multicanonical multigrid algorithm 204 4.1 Multigrid techniques 204 4.2 Multicanonical multigrid Updates 204 4.3 Results 205

5 — Multibondic algorithm 205 5.1 Random Cluster representation 206 5.2 Multibondic reweighting 206 5.3 Results 207

6 Conclusions 208

Chapter 10

Transition-state theory investigations of small molecule diffusion in glassy polymers 211

by Doros N. Theodorou

1 — Introduction 215

2 — Diffusion as a sequence of elementary jumps 219

3 — Molecular modeling of amorphous polymer glasses 221

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4 — T S T in p o l y m e r matr ices undergo ing isotropic "elastic" m o t i o n : t h e G u s e v - S u t e r approach 224 4.1 The elastically moving matrix assumption 224 4.2 States and dividing surfaces 225 4.3 Calculation of the diffusivity 227

5 — Mul t id imens iona l T S T approach to gas diffusion in glassy po lyn iers 227 5.1 States, macrostates and reaction paths 227 5.2 Calculation of interstate transition rate constants 236 5.3 Coarse-graining to the macrostate level 241

5.4 Generation of a network of macrostates and estimation of the diffusivity . 242

6 — Conc lus ions and out look 243

A Mul t id imens iona l T S T in general ized coordinates 246

Chapter 11

S o m e appl icat ions of the configurational-bias M o n t e Carlo technique 251

by Berend Smit

1 — Introduct ion 255

2 — P h a s e equil ibria of n-alkanes 256

3 — Adsorpt ion and diffusion of alkanes in zeol i tes 258

4 — B e y o n d chain molecu le s 262 4.1 Mixtures of colloids and polyniers 262 4.2 Parallel Monte Carlo simulations 264

5 — Conc luding remarks 264

Chapter 12

El iminat ion of fast variables v ia fictitious latt ice part ic le d y n a m i c s 267

by Sauro Succi, Gino Bella, Hudong Chen, Chris Teixeira, and Kim Molvig

1 Introduct ion 271

2 — Basics of digi tal phys ics 272

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3 Linearized collision Operator 274

4 — Outline of the basic hydrodynamics derivations 275

5 — Reacting flows 278

6 — Discussion 280

Chapter 13

Density functional techniques for Simulation of chemical reactions 285

by Michiel Sprik

1 — Introduction 289

2 The Kohn-Sham exchange-correlation functional 290 2.1 Hohenberg-Kohn theorem 290 2.2 Kohn-Sham equations 292

3 — Separation of exchange and correlation 293

4 — The local density approximation 294

5 — Generalized gradient approximation 295 5.1 The exchange correlation hole 295 5.2 Non-empirical GGA's 297 5.3 Dynamical and non-dynamical correlation 298 5.4 GGA 's with adjustable parameters 300

6 - Optimized effective potential 301

7 — Implications of exact exchange 303 7.1 Selfinteraction and electronic stability 303 7.2 Exact exchange in atoms and solids 304

8 - Hybrid methods 305

Chapter 14

Path integral molecular dynamics: a computational approach to quan-tum Statistical mechanics 311

by Mark E. Tuckerman and Adam Hughes

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1 Introduct ion 315

2 T h e dens i ty matr ix and q u a n t u m Statistical mechanics 316

3 - P a t h integrals in the canonical e n s e m b l e 319 3.1 Derivation of the path integral 319 3.2 The path integral as a functional integral 321

4 E x p e c t a t i o n values and t h e r m o d y n a m i c propert ies derived from p a t h integrals 322

5 — Molecular dynamics m e t h o d s appl ied to p a t h integrals 326 5.1 Variable transformations 326 5.2 Diagonalizing the harmonic coupling 328 5.3 Ensuring ergodicity with thermostatted equations of motion 331 5.4 The equations of motion for path integral molecular dynamics 333 5.5 Integration methodology 335 5.6 Many-particle generalization 336 5.7 A convergence test 337

6 — P a t h integral calculat ions in t h e N P T e n s e m b l e 338 6.1 Equations of motion: primitive form 340 6.2 Equations of motion: Reduced form 343

7 - Appl icat ions 344 7.1 Charged water complexes and aqueous proton transfer 344 7.2 Hydrogen chloride trihydrate crystal 348

7.3 Fluid and solid para-Hydrogen 349

8 — Conc lus ion 353

A Der ivat ion of the p a t h integral virial t h e o r e m 354

Chapter 15

P r o t o n transfer in ice 359

by Dominik Marx

1 — P r o t o n transfer and phase trans i t ions in ice 363

2 -Ab initio pa th integrals 367 2.1 Basic ideas 368 2.2 Ergodicity, adiabaticity, and efficiency 371 2.3 Details of the ice simulations 373

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3 - Squeezing ice up to dissociation 374 3.1 Changing structure with pressure 374 3.2 Where are the protons? 376 3.3 Free energy profiles 378

4 — Concluding remarks 379

Chapter 16

Nudged elastic band method for finding minimum energy paths oftran-sitions 385

by Hannes Jönsson, Greg Mills, and Karsten W. Jacohsen

1 Introduction 389

2 - Chain-of-states methods 391

3 - The NEB method 394

4 Implementation of the NEB method 396

5 — Application to an adatom hop on a surface 397

6 What happens if the Springs are skipped? 399

7 - An object function for NEB 400

8 — Summary 402

A The two-dimensional test problems 403 A.l Model I: LEPS potential 403 A.2 Model II: LEPS + Harmonie oscillator potential 403

Chapter 17

RAW quantum transition state theory 405

by Greg Mills, Greg K. Schenter, Dmitrii E. Makarov, and Hannes Jönsson

1 — Introduction 409

2 — Feynman path integrals 410

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3 — H o w should a q u a n t u m trans i t ion s ta te b e defined? 410

4 — T h e act ion surface and the m i n i m u m act ion p a t h 411

5 - T h e conical d iv iding surface of R A W - Q T S T 416

6 - T h e prefactor for R A W - Q T S T 418

7 - Appl i ca t ion of R A W - Q T S T to a tes t p r o b l e m 419

C h a p t e r 18

D y n a m i c s of pept ide folding 423

by Ron Eiber, Debasisa Mohanty, and Carlos Simmerling

1 — Introduct ion 427

2 — Kinet i c s of prote in folding 427 2.1 The two steps of protein folding 427 2.2 An ideally efficient folder 428 2.3 Difficulties in the design of fast folders 430

3 - Folding of pept ides 431 3.1 Searching for peptides with a significant tendency to form a structure:

methodology 431 3.2 A summary of the features of LES (Locally Enhanced Sampling) 432

3.2.1 A qualitative discussion 432 3.2.2 A discussion with formulas 434

3.3 Determination of peptide structure 436 3.3.1 CA4C 436 3.3.2 CHDLFC 437

3.4 SYPFDV and variants 438

4 — F inal remarks 443

Chapter 19

Theoret ica l s tudies of act ivated processes in biological ion Channels 445

by Benoit Roux and Serge Crouzy

1 - In troduct ion 449

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2 — Theoret ica l formulat ion 449 2.1 Potential of mean force technique 451 2.2 Activated trajectory technique 452 2.3 High friction limit 453

3 — Appl icat ions to biological Sys tems 454 3.1 Ion mouvement in the GA Channel 455 3.2 Gating transition in a dioxolane-linked GA Channel 457 3.3 Extension to quantum Systems: centroid rate theory 460

4 — Conclus ions 460

Chapter 20

Ion Channels: a chal lenge for Computer s imulat ions 463

by Micheal L. Klein, Qingfeng Zhong, and Thomas Husslein

1 — Introduct ion 467

2 — Bas ic propert ies of ion Channels 467 2.1 Examples of ion Channels 468 2.2 Structures of ion Channels 468 2.3 Function of ion Channels 469 2.4 Peculiarities of proton Channels 470 2.5 Atomistic modeis of ion Channels 470

3 — S imulat ion s tudies of ion Channels 471 3.1 Moleeular dynamics 471 3.2 Building blocks of a model ion Channel 472

3.2.1 Water 472 3.2.2 Lipids 472 3.2.3 Channel proteins 474 3.2.4 Protons 478 3.2.5 Membrane-bound ion Channels 480

4 Conclus ions 483

P A R T III

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Q U A N T U M D Y N A M I C S

Chapter 21

M i x e d quantum-class ica l dynamics : mean-f ie ld and surface-hopping 489

by John C. Tully

1 - In troduct ion 493

2 Ehrenfest (mean-field) approach 494

3 E x a m p l e : energy transfer at m e t a l surfaces 500

4 — Surface-hopping 503

5 E x a m p l e : proton transfer in Solution 509

C h a p t e r 22

Non-ad iabat i c q u a n t u m d y n a m i c s Simulat ion us ing classical baths 515

by Peter J. Rossky

1 - In troduct ion 519

2 — M i x e d quantum-class ica l dynamica l evo lu t ion 520

3 — Q u a n t u m decoherence and non-adiabat ic d y n a m i c s 523 3.1 Determination of the decoherence time 527 3.2 Implementation in non-adiabatic Simulation and application to the hy-

drated electron 532

4 — Conc lus ions 535

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Chapte r 23

Thermal average t ime correlation functions from non-adiabatic M D : application to ra te constants for Condensed phase non-adiabatic reac-tions 539

by David F. Coker, Hsiao S. Mei, and Jean-Paul Ryckaert

1 Introduct ion 543

2 - Theory 544 2.1 Semiclassical evolution of wavefunctions 544 2.2 Thermally averaged time correlation functions for mixed quantum-classical

Systems 550

3 — General Implementat ion 558 3.1 Initialization 558 3.2 Surface hopping methods to generate trajectories distributed according to

\Tml(R (s))|2 '. 559

4 — Electronically nonadiabatic barr ier crossing rates 564 4.1 The nonadiabatic reactive flux correlation function 564 4.2 Initial amplitude sampling in the transition State / nonadiabatic coupling

region 566

5 - Model Application 569 5.1 Model system - bath problem and transition State theory results 570 5.2 General behavior of nonadiabatic reactive flux correlation functions, com-

parison with direct calculations 573 5.3 Effects of solvent friction on nonadiabatic transmission coefficient, com-

parison with adiabatic results and approximate theory 577

6 — Conclusions 580

Chapter 24

Chemical ra te laws and ra te constants 583

by Raymond Kapral, Styliani Consta, and Liam McWhirter

1 — Introduct ion 587

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2 — Derivation of chemical rate laws 587 2.1 Phenomenological description 587 2.2 Nonequilibrium initial ensemble 589 2.3 Rate law derivation 590 2.4 Structure of the rate kernel 591

3 — Classical Systems 593 3.1 Diffusive barrier crosssing: projected versus ordinary dynamics 596 3.2 Ion solvation dynamics in Clusters 599

4 — Mixed quantum-classical Systems: adiabatic dynamics 602 4.1 Proton transfer in molecular Clusters 605

5 — Mixed quantum-classical Systems: nonadiabatic dynamics 609 5.1 Nonadiabatic proton transfer 612

6 — Conclusions 614

Chapter 25

The semiclassical initial value representation for including quantum effects in molecular dynamics simulations 617

by William H. Miller

1 — Introduction 621

2 — The semiclassical initial value representation 622

3 — Electronically nonadiabatic processes 624

4 - Concluding remarks 626

Chapter 26

Tunneling in the Condensed phase: barrier crossing arid dynaniical con-trol 629

by Nancy Makri

1 — Introduction 633

2 - Methodology 634

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3 - Tunne l ing effects in barrier crossing 636

4 — Control of d iss ipat ive tunne l ing 640

5 — Conc lus ions and out look 644

Chapter 27

F e y n m a n p a t h centroid m e t h o d s for Condensed phase q u a n t u m d y n a m ­ics 647

by Gregory A. Voth

1 - In troduct ion 651

2 — T h e centroid molecular dynamics m e t h o d 652 2.1 Theory 652 2.2 Algorithms for CMD 653

2.2.1 MD-based methods 654 2.2.2 "Adiabatic" CMD 656 2.2.3 Pa th integral Monte Carlo approaches 657 2.2.4 Hyper-parallel implementation 657

3 - Represen ta t ive appl icat ions of C M D 660 3.1 Self-diffusion in liquid hydrogen 660 3.2 Quantum water simulations 660

4 — A n e w perspec t ive on centroid d y n a m i c s 660 4.1 Centroid dynamics justified 660 4.2 Exact formalism for centroid dynamics 664 4.3 Approximate Urne evolution and the centroid molecular dynamics method 665

5 — Conc lud ing remarks 665

Chapter 28

Q u a n t u m molecular dynamics us ing W i g n e r representa t ion 667

by Vladimir S. Filinov, Sara Bonella, Yuri E. Lozovik, Alex V. Filinov, and Igor Zacharov

1 Introduct ion 671

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2 — T i m e dependent q u a n t u m averages 672 2.1 Wigner's formalism 672 2.2 The evolution equation 673 2.3 The mean value as an average value 675

2.3.1 Dynamics transfer 676 2.3.2 The stochastic process 677 2.3.3 The algorithm 681

3 Q u a n t u m stat is t ics: t i m e d e p e n d e n t correlat ion funct ions 682 3.1 The writing of a correlation function 682 3.2 W 686

3.3 The average value 688

4 - P r o b l e m s 689

5 — Conclus ions 690

A Evo lut ion equat ion for W 691

Chapter 29

Nonad iabat i c molecular d y n a m i c s m e t h o d s for diffusion 697

by Daniel Laria, Giovanni Ciccotti, David F. Coker, Raymond Kapral, and Mauro Ferrario

1 — Introduct ion 701

2 - Seniiclassical d y n a m i c s for m i x e d quantum-c lass ica l Sys tems 701 2.1 The Pechukas force 702 2.2 Quantum molecular dynamics algorithm 703

3 — Q u a n t u m express ion for the diffusion coemc ien t 706

4 — Molecular dynamics i m p l e m e n t a t i o n 708 4.1 Diagonalization of the S ehr ö ding er Equation 708 4.2 Sampling procedure 710

5 A n e lectron in a m o l t e n salt 711

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Chapter 30

M o d e l l i n g pro ton transfer in Solution us ing non-addi t ive va lence -bond force fields 721

by Rodolphe Vuilleumier and Daniel Borgis

1 — Introduct ion 725

2 — Simple p ic tures for p r o t o n transfer in Solution 726 2.1 Relevant variables 726 2.2 Proton transfer along weak'H bonds 726 2.3 Proton transfer along strong H bonds 727

3 E V B descr ipt ion of p r o t o n transfer 729 3.1 Two-state model 729 3.2 Extended EVB model for proton transfer in water 731

4 — Molecular d y n a m i c s at finite t e m p e r a t u r e 732 4.1 Molecular dynamics algorithm 732 4.2 Simulated annealing for the H+ (H2 0)n Clusters 734 4.3 Proton transfer in H*, O^ and H13 O j 734 4.4 Vibrational spectroscopy of the H+ (H2 0)n Clusters 736 4.5 Simulation of an excess proton in liquid water 737

5 — Q u a n t u m dynamics 738

Chapter 31

T h e q u a n t u m d y n a m i c s of interfacial hydrogen in meta l s : an introduc­t ion 743

by Jim D. Doli, Dongsup Kim, Maria Eleftheriou, Bin Chen, Chinsung Bae, and David L. Freeman

1 — In troduct ion 747

2 - Background 747

3 — Formal d e v e l o p m e n t s 750

4 - Appl i ca t ions 752

5 — Current and future research direct ions 756

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Chapter 32

M o d e l i n g the solvent efFect in e lectronic t rans i t ions 759

by Piero Procacci and Marc Souaille

1 - In troduct ion 763

2 — Semi-class ical calculat ion of molecu lar absorpt ion and fluorescence spec tra 765 2.1 Absorption 765 2.2 Fluorescence 768

3 — Parameter i za t ion of the B O surfaces 770 3.1 Electro-negativity equalization principle 770 3.2 Fitting the BO surfaces in polyatomic molecules 773

4 — A n appl icat ion: the absorpt ion and fluorescence s p e c t r u m of formalde-hyde 775

5 Conc lus ions and perspec t ives 778

Chapter 33

M i g r a t i o n of hydrogen on a sol id surface: t h e phys ics of t h e process and t h e m e t h o d o l o g y 781

by Horia Metiu

1 — A physical descr ipt ion of t h e processes to b e s tud ied 785

2 - T h e correlat ion funct ion theory 786 2.1 The equations 786 2.2 The physical interpretation of the correlation function 787

3 — T h e phys ics of a t o m migrat ion o n the surface 789 3.1 The atom can be its own heat bath 789 3.2 Mean field theory, fluctuations, and dynamics 793 3.3 Why do we need simpler modeis? 794 3.4 How to define rate constants without including lattice dynamics? 794 3.5 How large are the errors of the simplified modeis? 795 3.6 Multiple jumps 796 3.7 Quantum Systems 799

1

C h a p t e r 34

T h e rate of p h o t o n absorpt ion 805

by Horia Metiu

1 — I n t r o d u c t i o n 809

2 T h e s y s t e m and t h e p r o b l e m 809 2.1 An electron in sodalite 809 2.2 The time-dependent method for calculating the absorption spectrum. . . . 809 2.3 The QC version of the absorption cross section theory 810 2.4 Why should the QC method work? 811 2.5 Why the QC method shouldn't work 812 2.6 How bad is the QC method? 814 2.7 Why is the spectrum so hard to compute by a classical method? 814 2.8 A more detailed look at this interference 815 2.9 These comments are relevant to curve crossing also 817

3 — H o w t o c o m p u t e t h e s p e c t r u m 817 3.1 Introduction 817 3.2 The absorption cross-section revisited 818 3.3 We only need C(t) for a short Urne 818 3.4 The implications of this finding 819 3.5 The gaussian wave packet (GWP) method 820 3.6 The adiabatic time-dependent Hartree method 821

4 — S o m e resu l t s for an e l ec tron in zeol i tes 822 4.1 The problem 822 4.2 Photon-induced migration 823 4.3 Spectra of complicated Compounds are simple 823

A U T H O R I N D E X 829