Production of ultracold molecules by photoassociation ...

Teoretyczne badania tworzenia zimnych molekułz wykorzystaniem spektroskopii fotoasocjacji

i techniki współchłodzenia

PRACA DOKTORSKA WYKONANA W PRACOWNI CHEMII KWANTOWEJ

WYDZIAŁU CHEMII UNIWERSYTETU WARSZAWSKIEGO POD

KIERUNKIEM PROF. DR. HAB. ROBERTA MOSZYNSKIEGO

WOJCIECH SKOMOROWSKI

WARSZAWA 2013

ii

STRESZCZENIE

Molekuły schłodzone do ultraniskich temperatur, tj. bliskich zera bezwzglednego, stanowiaobecnie przedmiot intensywnie rozwijanych, interdyscyplinarnych badan na styku fizyki i che-mii. Prace te sa motywowane mozliwosciami zastosowania zimnych czasteczek w pomiarachspektroskopowych o niespotykanej precyzji, testowaniu fundamentalnych praw fizyki, czykontroli reakcji chemicznych na poziomie stanów kwantowych reagentów. Zimne czasteczkimoga takze byc wykorzystane jako elementy architektury komputerów kwantowych orazsymulatorów kwantowych, słuzacych na przykład do modelowania przejsc fazowych.

Kamieniem milowym w badaniach nad ultrazimna materia było otrzymanie pierwszegokondensatu Bosego-Einsteina w gazie atomów rubidu w 1995 roku. Wymagało to schłodzeniaatomów do temperatur rzedu 200 nanokelwinów. Rozwój technik doswiadczalnych opar-tych na chłodzeniu laserowym i efekcie Dopplera pozwala obecnie na chłodzenie niektórychatomów (głównie metali alkalicznych) do ultraniskich temperatur. Ze wzgledu na skompliko-wana strukture stanów wewnetrznych, chłodzenie molekuł jest nieporównywalnie trudniejszeniz atomów. Rozwiniecie efektywnych metod chłodzenia molekuł do temperatur ponizej 1 mi-likelwina stanowi jeden z kluczowych celów prowadzonych obecnie badan nad zimna materia.Osiagniecie tego celu pozwoli na obserwacje nowych zjawisk o czysto kwantowym charak-terze oraz realizacje w praktyce wyzej wspomnianych zastosowan zimnych czasteczek.

Głównym celem badan teoretycznych przedstawionych w niniejszej pracy była analizamozliwosci chłodzenia wybranych molekuł oraz jonów na drodze fotoasocjacji oraz poprzezwspółchłodzenie (ang. sympathetic cooling) w gazie ultrazimnych atomów. Wybór układówmolekularnych, dla których wykonano obliczenia energii oddziaływania i dynamiki zderzenw ultraniskich temperaturach, był zwiazany z prowadzonymi pracami eksperymentalnymi.

W kontekscie zastosowania metody współchłodzenia przez termalizacje w gazie ultra-zimnych atomów w rozprawie przedstawiono wyniki obliczen kwantowo-dynamicznych dlatrzech układów: LiH + Li, OH + N oraz Ba+ + Rb. Badania mozliwosci chłodzenie wodorkulitu w zderzeniach z atomami litu obejmowały w pierwszej czesci wykonanie bardzo dokład-nych obliczen potencjału oddziaływania oraz przekrojów czynnych na zderzenia pomiedzyLi i LiH. Obliczone przekroje czynne posłuzyły nastepnie do przeprowadzenia symulacji

iii

STRESZCZENIE

procesu współchłodzenia molekuł LiH w zderzeniach z atomami Li, testujac rózne metodypułapkowania molekuł. Pokazano, ze najbardziej korzystnym podejsciem jest zastosowaniepułapki w postaci wneki rezonansowej działajacej przy czestosciach mikrofalowych. Ponad-to pokazano, ze dla układów o duzej anizotropii potencjału oddziaływania, pułapki oparteo stałe lub przemienne pole elektryczne nie gwarantuja pozadanej termalizacji ze wzgleduna szybki spadek ilosci pułapkowanych czasteczek. W ramach projektu dotyczacego analizydynamiki zderzen pomiedzy molekułami i atomami otwartopowłokowymi w zewnetrznympolu magnetycznym, wykonano obliczenia przekrojów czynnych na zderzenia pomiedzy rod-nikiem OH i atomem N. Wyniki obliczen pokazały, ze współchłodzenie rodnika OH poprzezzderzenia z atomami azotu moze byc skuteczne jedynie w stosunkowo niewielkim zakresieenergii zderzen i natezenia pola magnetycznego, co wynika z istnienia silnych mechanizmówprowadzacych do zderzen nieelastycznych w tego typu układach. Wyniki obliczen dotycza-cych mozliwosci chłodzenia jonów Ba+ w zderzeniach z ultrazimnymi atomami Rb wskazuja,ze taka metoda powinna byc skuteczna a procesy nieelastyczne zwiazane z przeniesieniemładunku pomiedzy Ba+ i Rb nie stanowia istotnej przeszkody dla pozadanej termalizacji.

W pracy przedstawiono ponadto mozliwy schemat tworzenia ultrazimnych molekuł Sr2

w mocno zwiazanych stanach rowibracyjnych, wykorzystujac do tego metode fotoasocjacji.Efektywnosc zaproponowanej sciezki opiera sie na wykorzystaniu sprzezenia spin-orbita po-miedzy silnie oddziałujacymi stanami elektronowymi dimeru strontu. To oddziaływanie umoz-liwia zarówno wydajna fotoasocjacje atomów, jak i skuteczna droge stabilizacji i tworzeniamocno zwiazanych molekuł Sr2. Proponowany schemat tworzenia czasteczek Sr2 mógł pow-stac dzieki wysoce dokładnym obliczeniom ab initio krzywych energii oddziaływania dla tejmolekuły, których wyniki istotnie rózniły sie od uprzednio opublikowanych obliczen, a jed-noczesnie były zgodne z najnowszymi danymi eksperymentalnymi.

Istotnym problemem poruszonym w rozprawie jest analiza asymptotyki oddziaływaniadalekozasiegowego pomiedzy liniowymi molekułami otwartopowłokowymi a atomami w sta-nie podstawowym o symetrii S. Waga tego zagadnienia wynika z faktu, iz tego typu układysa szeroko rozwazane w kontekscie zastosowania metody współchłodzenia, a jednoczesniewłasciwy opis oddziaływania dalekozasiegowego jest kluczowy w badaniach dynamiki kwan-towej w niskich temperaturach. W pracy wyprowadzono ogólne wyrazenia na współczynnikivan der Waalsa dla oddziaływania atomów w stanie S z molekuła liniowa w dowolnym zdege-nerowanym stanie elektronowym. Współczynniki te zostały w pełni wyrazone poprzez włas-nosci oddziałujacych układów (momenty multipolowe, polaryzowalnosci, itd.), co nadaje imjasna interpretacje fizyczna. Jednym z zastosowan tej teorii był wspomniany układ OH + N.

Rozprawa składa sie z dwóch głównych czesci. Pierwsza czesc zawiera wprowadzenie doporuszanej tematyki, nakreslenie celów i motywacji podjetych badan, oraz zwiezłe omówieniestosowanych metod i otrzymanych wyników. Druga czesc stanowia kopie siedmiu oryginal-nych artykułów naukowych, opublikowanych w czasopismach o miedzynarodowym zasiegu,zawierajacych szczegółowy opis uzyskanych wyników.

iv

Production of ultracold molecules byphotoassociation spectroscopy and sympathetic

cooling: a theoretical study

A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN CHEMISTRY

THESIS SUPERVISOR: PROF. DR. ROBERT MOSZYNSKI

WOJCIECH SKOMOROWSKI

FACULTY OF CHEMISTRY, UNIVERSITY OF WARSAW

WARSAW 2013

ACKNOWLEDGEMENTS

This thesis would have not been possible without great support from many people. Firstaf all, I would like to express genuine gratitude to my supervisor Professor Robert Moszynskifor his guidance, patience, and constant encouragement during my doctoral years.

I would like to thank Professor Jeremy Hutson (Durham University, UK) and ProfessorChristiane Koch (Universität Kassel, Germany) for giving me the opportunity to work in theirgroups and learn from them. Undoubtedly, stays in Durham and Kassel were of paramountimportance for my work.

I am very grateful to my young collaborators and comrades from “the ultracold commu-nity”, in particular Dr. Piotr Zuchowski, Dr. Maykel Leonardo González-Martínez, Dr. FilipPawłowski, Dr. Liesbeth Janssen, and Michał Tomza. Thank you for helping me with theprojects, for many fruitful discussions, and for great time I have spent with you.

I would also like to thank all the members of the Quantum Chemistry Laboratory at War-saw University for creating highly supportive and stimulating atmosphere in the Lab. Matesfrom room #502 and frequent visitors to this office are particularly acknowledged for theirexcellent company and help.

Finally, I am very grateful to my family and friends from outside the scientific communitywhose support has been invaluable.

Warsaw, April 2013Wojciech Skomorowski

2

Contents

1 Introduction to ultracold molecules 51.1 Foundations of the ultracold regime . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Getting molecules cold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Current and future applications . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 General objectives and motivation of the present research 112.1 Motivation and context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Objectives of the research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Modelling of the sympathetic cooling processes 173.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Long-range interactions between S-state atom and an open-shell linear molecule 20

3.2.1 Theoretical developments . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Collisions of an open-shell S-state atom with a 2Π molecule in a magnetic field 25

3.3.1 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Method of close-coupling equations . . . . . . . . . . . . . . . . . . 27

3.3.3 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Interaction and collisions between the LiH molecule and a Li atom . . . . . . 31

3.4.1 Computational approach . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.3 Simulations of the sympathetic cooling . . . . . . . . . . . . . . . . 36

3.5 Interaction and collisions of Ba+ and Rb . . . . . . . . . . . . . . . . . . . . 38

4 Photoassociation spectroscopy: theoretical approach 434.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Electronic structure and rovibrational dynamics of Sr2 . . . . . . . . . . . . . 46

3

CONTENTS

4.3 Photoassociative formation of deeply bound ultracold Sr2 molecules . . . . . 49

5 Conclusions 55

Bibliography 57

A Paper I 67

B Paper II 83

C Paper III 97

D Paper IV 115

E Paper V 125

F Paper VI 143

G Paper VII 157

4

CHAPTER 1

INTRODUCTION TO ULTRACOLD MOLECULES

1.1 Foundations of the ultracold regime

The fast-growing field of cold and ultracold physics was triggered by the first experimen-tal realization of the Bose-Einstein condensate in dilute atomic gases in 1995 [1, 2]. Thisphenomenon, predicted by Albert Einstein and Satyendra Nath Bose as early as in 1925, is aquantum phase transition occurring in a system consisting of identical bosons when particles,under special conditions, tend to occupy a single quantum state. Bose-Einstein condensatecan be formed only if the thermal de Broglie wavelength of the particles is comparable (oreven larger) to the average distance between them. This may happen when the particles areconfined in a trap and cooled to the nanoKelvin temperatures. Therefore, the way to the firstatomic Bose-Einstein condensate was actually paved by the important developments in thecooling and trapping techniques for atoms that started in the early 1980s [3–7].

For atoms, the routine method of reaching the microKelvin regime is the laser coolingtechnique utilizing the Doppler effect [8]. This method is based on a repeated process of ab-sorption and emission of photons in which the radiation pressure is used to decelerate atoms.The key requirement for the laser cooling to be efficient is that the absorption/emission pro-cesses are in a closed-cycle, i.e. after the emission of light atoms return to the same initial stateas before the absorption. In this way atoms can be cooled to temperatures in the microKelvinrange. This would be not enough, however, to observe the quantum degeneracy. In order tocool atoms further down, evaporative cooling is usually applied [9]. In this process the mostenergetic atoms are removed from the trap and the rest of the sample thermalizes to a lowertemperature which can already be in the nanoKelvin regime, needed for the quantum degen-eracy, provided that that the atomic density is sufficiently high. Observation of the quantumdegeneracy and other low-temperature quantum phenomena involving molecules, in particu-lar polar molecules, is one of the major objectives of the current experiments in the ultracold

5

CHAPTER 1. INTRODUCTION TO ULTRACOLD MOLECULES

physics.

The intrinsic feature of the ultralow temperature is that the kinetic energy of the particlesis very low. The energy splitting associated with the fine or hyperfine structure (of the order ofMHz or µK) can be larger than the available kinetic energy (kBT ). This means that even thehyperfine state of the species may be crucial for the dynamics and control. Ultracold atomsor molecules can be prepared and controlled in a well characterized single quantum state in-cluding the hyperfine structure. For molecular interactions taking place at the submicroKelvintemperatures, even the smallest activation energy exceeds the available thermal energy. Thisopens new possibilities for controlling also the pathways of chemical reactions [10].

Another important characteristics of the ultracold regime is the quantum statistics of themolecular states. In the full quantum regime only the lowest partial waves contribute to thecollisional dynamics. Identical bosons or distinguishable particles undergo only s−wave colli-sions, while for indistinguishable fermions only p−wave collisions are accessible. Therefore,the dynamics is much simpler to analyse than in the hot regime where many partial waves andquantum states are involved in the collision. For low-energy collisions simple analytical anduniversal laws for the cross sections and rate coefficients can be obtained since the collisionsare to a large extent governed by the well-known long-range forces [11–14].

Last but not least, in the ultracold regime the influence of the scattering resonances on thedynamics can clearly be observed [15–17]. The resonances are not covered by contributionsfrom many partial waves as it is at high temperatures. In most cases positions of these reso-nances can be tuned by applying external fields and may serve as effective knobs of controlover the collisional processes in the ultracold regime.

1.2 Getting molecules cold

Laser cooling, the most successful method for cooling atoms, is not applicable to molecules.The obstacle is a very complex structure of the internal states due to the rotational and vibra-tional degrees of freedom which would require many lasers to make a closed cycle of transi-tions. Albeit, it has been shown recently that there are a few exceptions where the laser coolingmay also work for molecules [18, 19].

In general, methods developed during the last decade to produce cold molecules can bedivided into two classes, direct and indirect. Direct methods tend to cool stable, preexistingmolecules from high to low temperatures. This approach is realized in practice through, amongothers, the buffer gas cooling [20, 21], Stark and Zeeman deceleration [22, 23], and velocityfiltering [24]. The idea of the indirect methods is to form cold molecules from atoms which arealready cold. Photoassociation [25] and magnetoassociation [26] are the two main techniquesin this direction. The basic limitation of the associative methods comes from the fact that theycan be applied only to molecules built of atoms that are easily cooled. Nowadays it implicatesthat in the process of magnetoassociation or photoassociation only the molecular dimers built

6

1.2. GETTING MOLECULES COLD

of alkali-metal and alkaline-earth-metal atoms can be formed, such as LiCs [27], KRb [28],Sr2 [29], or Cs2 [30]. It is worth noting, however, that molecules formed from the ultracoldatomic substrates reach the temperature regime of the order of 1 µK or even lower.

By contrast to the associative techniques, direct methods are quite versatile and can beapplied to a large set of molecules, including polyatomic and chemically interesting species.The process of Stark or Zeeman deceleration in which fast moving molecules are sloweddown and lose kinetic energy by rapidly switching inhomogeneous electric or magnetic fieldsis simply based on the interaction of the molecular electric or magnetic dipole moments withthe external electric or magnetic field. Thus, the deceleration techniques can be applied tomany molecules that have permanent electric or magnetic dipole moments. As a matter offact, a large variety of polar molecules have been decelerated thus far, for instance CO [22],ND3 [31], LiH [32], NH [33], OH [34], H2CO [35], and YbF [36]. Even a more versatile directmethod for preparing cold molecules is the buffer gas cooling technique. Here, molecules arecooled by collisions with cryogenically cold helium atoms. Since the cooling mechanism doesnot depend on the internal structure of the species, buffer gas cooling can be applied to nearlyany atom or small molecule [37], and has already been used to cool a variety of atoms andmolecules, such as CaH [20], CaF [38], NH [39], Cr [40], N [41]. Similarly, velocity filteringin which slow molecules are filtered out of a thermal gas, is also quite universal but at the sametime a technically challenging method. It utilizes the fact that even at room temperature thegas contains slow and therefore cold molecules, and the problem is how to select and isolatethem from the sample. For polar molecules this can be achieved by exploiting the Stark effectin a bent electric guide [42].

An essential disadvantage of the direct cooling methods is that they can only bring mole-cules to the cold regime (from 1 K to 1 mK) and not to the ultracold regime (below 1 mK).Indeed, buffer gas cooling enables to reach final temperatures of around 0.1 - 0.5 K, whilethe beam deceleration techniques can achieve temperatures from 10 to 100 mK. Therefore,second stage cooling methods are necessary to reach the ultracold regime and observe thequantum degeneracy in a molecular sample [1, 2]. A promising candidate for this purposeis the sympathetic cooling. In the process of sympathetic cooling the sample of pre-cooledmolecules is introduced into an ultracold atomic gas and thermalize with it. Alkali atoms arethe most obvious coolants for the sympathetic cooling since they can easily be cooled to ul-tralow temperatures. In principle, sympathetic cooling is analogous to the helium buffer gascooling, however it takes place at much lower temperatures. Also the technical aspects arequite different for the two methods. We mention here two main conditions for the sympatheticcooling to work. First, sufficiently large densities have to be accumulated inside the trap toprovide high collision frequency when compared to the trap lifetime. Second, elastic colli-sions have to largely dominate over the inelastic collisions, as only the elastic events lead tothe desired thermalization, while the inelastic collisions most likely end with removing thecolliding species from the trap. In Chapter 3 we will discuss this topic in more detail.

7


At the end of this part we would like to emphasize that direct methods such as the Stark/Zee-man deceleration or the velocity filtering techniques cannot strictly be considered as coolingtechniques. In these processes, a subset of molecules with low relative or absolute velocitiesis only isolated from a larger ensemble without increasing the phase-space density ρ, definedas ρ = nλ3

T where n is density of molecules and λT is the thermal de Broglie wavelength.Although such processes lead to samples of cold molecules, they are limited, and have tobe combined with a ’true’ cooling method such as the evaporative or buffer gas/sympatheticcooling to increase the phase-space density ρ if one aims to reach the quantum degeneracy.

1.3 Current and future applications

Ultracold molecules offer a much broader spectrum of possible applications in diversefields of science than ultracold atoms. Why is it so? First of all, molecules have a very rich in-ternal structure due to the rotational and vibrational degrees of freedom, thus various energeticscales can be utilized for different purposes. Second, polar molecules with permanent dipoleand/or magnetic moments can easily be manipulated and controlled with external electromag-netic fields. Moreover, external fields and confined traps can be employed to engineer long-range anisotropic interactions between polar molecules. The main possible applications ofcold molecules include: exploring molecular dynamics and chemistry in the ultracold regime,precision measurements, high-precision spectroscopy, quantum computing and simulations ofcondensed matter phenomena. Let us briefly depict each of these fields.

Collisions taking place at ultralow temperatures are mostly driven by the quantum effects,such as resonances, tunneling, and quantum statistics. One can thus expect that chemical reac-tions in the ultracold regime would in general be suppressed. Calculations and measurementsshow that this is not the case. Actually, due to the resonance enhancement the rates of chemi-cal reactions can be very large. This was shown for the reaction F + H2→ HF + H [43,44] orexchange reactions between the alkali-metal atoms, Cs + Cs2 [45] and Na + Na2 [46]. In theultracold regime, effective control over the reaction rates can be obtained by applying exter-nal electric or magnetic fields as it was experimentally demonstrated in collisions between theKRb molecules [47] and proposed in numerous theoretical works [10, 48, 49]. Similar effectsof the control of the dynamics can be achieved by confining molecular systems in traps withreduced dimensionality, where only certain arrangements are possible [50]. Quantum statis-tics is another factor that may drastically modify the outcome of ultracold collisions. This wasobserved in the landmark experiment with the KRb molecules, where depending on nuclearspin state of the colliding species, whether they are in identical or different states, the reactionrate is highly changed [51].

To date, the most recognized applications of ultracold molecules are high-precision spec-troscopy and precision measurements aiming at testing the fundamental laws of physics. Thekey features making the ultracold molecules ideal systems for high-resolution spectroscopy

8

1.3. CURRENT AND FUTURE APPLICATIONS

are high quantum state purity, long coherence time for the measurements, and lack of strongperturbations if the molecules are confined in traps or optical lattices.At ultralow temperature molecules can be prepared almost in a single quantum state. This ispossible since the techniques of production of cold molecules are highly state-selective and thethermal distribution at low temperatures ensures a very narrow population of different internalstates. This is in striking contrast to the room temperature when only a very small fraction ofmolecules occupy the same quantum level. In the same way, low velocity of molecules enablesto extend the time of observation which increases the accuracy of the measurements. More-over, molecules confined in optical lattices with a single occupancy per site are effectivelysheltered from any perturbations. This results in further narrowing of the observed linewidthsof the spectroscopic transitions.Cold molecules are particularly suitable for a precise determination of very weakly boundlevels just below the dissociation limit. These levels are obtained by photoassociation spec-troscopy for molecules like Sr2 [52, 53] or Yb2 [54]. One can also observe purely long-rangemolecular levels, i.e. levels for which both classical turning points are located in the long-range part of the interaction potential [55, 56]. Exact positions of such very weakly boundrovibrational levels with large amplitudes of the wave function in the long range, are verysensitive to tiny changes of the pair interaction potential, or to the nonadiabatic effects, hyper-fine interactions, QED retardation effects. This was shown for Na2, where only after includingthe retardation correction it was possible to reproduce theoretically the observed molecularresonances [57]. Similarly, for Sr2 the positions of the most weakly bound levels are stronglymodified by the nonadiabatic (Coriolis-type) couplings between two close-lying electronicstates. As positions of these weakly bound molecular levels are resolved with ultra-high pre-cision (of the order of a few kHz or 10−7 cm−1), such a data is an excellent benchmark fortesting theoretical models describing the electronic structure, relativistic, or even quantumelectrodynamics effects on the molecular spectra.

Complex energy level structure of molecules opens entirely new perspectives for preci-sion measurements related to testing some fundamental physical laws. This is possible sincemolecules, in particular with unpaired electrons, have often closely lying energy levels ofdifferent parity, resulting from the couplings between the rotational, electronic, or nuclear an-gular momenta. Transitions involving such levels are very promising in the context of precisespectroscopic measurements that may shed some light on the time variation of the physicalconstants (the fine structure constant, proton to electron mass ratio), the parity violation inmolecules, or the shape of the electron. In all these directions there are currently intensivelydeveloped proposals based on high precision spectroscopy with cold molecules. For example,precise measurements of transitions in the Stark-decelerated OH [58], CH [59] or CO [60]molecules are suggested to constrain the time variation of fine-structure constant. Rovibra-tional spectra of ultracold Cs2 [61], Sr2 [62], CaH+ [63], CaH or MgH [64], ND3 [65] aresuggested as promising candidates for constraining the time variation of the proton to electron

9


mass ratio. Heavy open-shell diatomic molecules with a single valence electron, such as YbFor SrF, have the most favourable properties to study the parity violation (PV) due to the elec-troweak forces [66]. High-resolution spectroscopic measurements with cold YbF moleculeshave already been utilized to put the most accurate constraints on the value of the electrondipole moment [67].

Quantum information processing and quantum computing are the other fields where coldmolecules can be used in practice. It is generally thought that cold polar molecules can be em-ployed as efficient two-level qubits or even multi-level qudits [68]. Qubits can be encoded intotwo rotational energy levels of a single polar molecule [69], while logical operations on themare carried out by using classical microwave fields. Quantum memory register made of suchmolecules can be formed by trapping molecules in an optical lattice [70]. Polar molecules,due to their long-range interactions that help to speed up the logic operations on the remotequbits, are more attractive for a practical realization of the quantum computer than any previ-ously considered methods for this purpose [71].

Strong and tunable long-range interactions can also be very beneficial to employ cold polarmolecule in the simulations of the condensed-matter phenomena [72–74]. The general idea isto design a system composed of polar molecules, usually trapped in an optical lattice, in sucha way that its Hamiltonian will resemble the Hamiltonian of the many-body system that is thegoal of the simulation. It is assumed here that the degree of complexity of the many-body sys-tem is so large that classical computer simulations are not possible. Then, instead of computersimulations, one can engineer a model Hamiltonian realized in practice by confined ultra-cold molecules. For instance, it has been shown that lattice-trapped molecules with unpairedelectrons can be employed to engineer an arbitrary Hamiltonian for the interacting spins in alattice [72], i.e. models which are commonly encountered in the condensed-matter physics.

10

CHAPTER 2

GENERAL OBJECTIVES AND MOTIVATION OF THEPRESENT RESEARCH

This thesis is concerned with theoretical studies of the production of ultracold moleculesby sympathetic cooling and photoassociation. The reported studies involve both the theoreticaldevelopments and numerical applications carried out with the existing codes for high accu-racy electronic structure and quantum dynamics calculations. The theory developments willaddress the long-range interactions between open-shell atoms and diatomic molecules andquantum dynamics of such systems in an external magnetic field. Numerical calculations areperformed for systems being investigated experimentally or potentially prospective for futureexperiments.

2.1 Motivation and context

As it was stated in the Introduction, sympathetic cooling is considered as one of the mostpromising ways to bring molecules from the cold to the ultracold regime. This type of coolinghas already been successfully applied to cool several atomic isotopes and ions, but it has notbeen demonstrated yet for a neutral molecular system. It is conceptually very simple, basedon the thermalization of the sample of cold molecules in the bath of ultracold atoms. From theexperimental perspective, however, the realization of sympathetic cooling is very challenging.The technical challenges result from the necessity to work with two cold samples, molecularand atomic, which usually have very different physical properties concerning their lifetimes,methods of detection, trapping fields or densities in the trap. In principle, one has to designthe trapping environment for both species in such a way that they do not interfere with eachother when overlapping in space to enable collisions with a sufficiently high rate. Moreover,experimental techniques have to be developed to see the possible outcomes of the collisionsbetween the two samples.

11

CHAPTER 2. GENERAL OBJECTIVES AND MOTIVATION OF THE PRESENTRESEARCH

In this context the role of the theoretical calculations is significant in several aspects. First,theory helps to select a combination of molecules and atoms that should be most promising forthe sympathetic cooling to work. This is based on the calculations of the cross sections for theelastic and inelastic collisions. For a successful cooling, the rate of the elastic collisions mustbe high enough to allow for a quick thermalization and it should significantly dominate overthe inelastic collision rates. Most often, the second requirement is the decisive factor sincemolecules are confined most easily in the electrostatic or magnetostatic traps. Such traps canwork only if the molecule is in its low-field seeking state, i.e. a state which gains potentialenergy with the increase of the field strength and therefore is attracted to the minimum of theelectro or magnetostatic potential. Such a state is never the ground state of the molecule whichmeans that there are some open inelastic channels to which the molecule can relax after thecollision with an atomic coolant. Inelastic collisions always result in losing the molecule fromthe trap. Apart from the change in the internal state of the colliding particles, there are otherpossible inelastic events such as chemical reactions or charge exchange processes that alsolead to the trap loss.The second aspect, in which the theoretical input is helpful, is the guidance of the experimentin order to optimize the process of cooling for a given atom/molecule system. This may in-clude the design of the optimal trap and the presence of additional external fields that modifythe dynamics. All these points of the cooling process can theoretically be modeled and theresults of such simulations can save time and money for preparing the experimental setup.Calculations of the rate constants, even if their accuracy is limited due to the inherent ap-proximations of the theoretical methods, give at least a qualitative information about possibleoutcome of the collisions, thus it can help to answer questions about the detection techniquesthat can be applied. Finally, theory is essential to understand and rationalize the experimentalobservations.

In the present thesis we analyze prospects for sympathetic cooling in three systems: LiH+ Li, OH + N and Ba+ + Rb. Let us briefly explain why these systems were chosen.

LiH + Li. Cooling of the LiH molecule by collision with the Li atoms was one of thecentral goals of the large collaborative programme “Collisions of Cold Polar Molecules”(CoPoMol), established in 2007. On the experimental side it was pursued by the group ofDr. Michael Tarbutt at the Imperial College in London. In 2007 the London group demon-strated the production of a beam of the LiH molecules with a translational temperature of 1 Kby using the supersonic expansion technique [75]. In the next step, the molecular beam ofthe LiH molecules (in the low-field seeking component of the excited rotational state j = 1),was decelerated using the Stark deceleration technique [32]. The further goals of the projectwere (i) to trap the decelerated molecules using an electrostatic trap and (ii) to overlap themolecular trap with the atomic trap filled with ultracold lithium atoms in order to observecollisions between the two samples and cooling of the LiH molecules. The LiH molecule asa target for a joint experimental-theoretical study of the sympathetic cooling is attractive for

12

2.1. MOTIVATION AND CONTEXT

a few reasons. First, it has a large dipole moment (5.88 D) and relatively low mass, whichmakes it easy to manipulate with the electric field. From the theoretical point of view it hasa simple electronic structure which enables to carry out very accurate quantum-chemical andquantum-dynamical calculations, which are the goals of our study.

OH + N. The second studied system is OH + N in the presence of a magnetic field. This isa prototype system to develop the theoretical framework and analyse the possible mechanismsof the inelasticity in collisions between an S-state open-shell atom and a 2Π state molecule.Since the two colliding species have unpaired electrons and the molecule is in a spatiallydegenerate electronic state, the dynamical calculations are much more involved than in thecase of systems composed of closed-shell molecules. Open-shell molecules are most oftenused in the Stark deceleration experiments, since they exhibit the first-order linear Stark effectand therefore are easier to control with an electric field than the closed-shell Σ-state moleculeswith only quadratic Stark effect at moderate field strengths. Likewise, atoms routinely cooledto ultralow temperature are open-shell S-state atoms, namely alkali-metal atoms, N(4S), orCr(3S). Thus the theory developed here will be applicable to a broad set of experimentallyimportant systems, including interactions of molecules such as OH, NO, CH and CN withalkali-metal and other magnetically trappable atoms.

Calculations were carried out for the OH molecule colliding with the nitrogen atom. TheOH radical is very well explored experimentally as it was one of the first molecules to be Starkdecelerated [34], and many pioneering experiments were done with the decelerated OH radi-cal [76–79]. A few years ago spin-polarized nitrogen atom was suggested as a possible coolantin experiments on sympathetic cooling [41] as a alternative to alkali-metal atoms, which havesome drawbacks due to their high polarizability and readiness to form ionic states. Coolingand trapping of the spin-polarized nitrogen atoms has been demonstrated in 2008 [80]. There-fore, the OH + N system is experimentally feasible and has some features that have not beentheoretically studied before.

Ba+ + Rb. The last system addressed in the thesis in the context of the sympathetic coolingis Ba+ + Rb. Ions become more and more popular for ultracold matter studies. Ultracoldions have very prospective applications for the quantum information processing, precisionmeasurements, and chemical reactions at ultralow temperatures. Ongoing experiments arefocused on the collisions in mixed cold clouds of atoms and ions or on the dynamics of a singleion immersed in an atomic Bose-Einstein condensate [81–83]. For our work we selected theBa+ + Rb system, investigated experimentally in Freiburg, Ulm and Basel [83, 84]. Our goalwas to determine the elastic, spin changing, and charge transfer collision cross sections, whichwould be useful to assess the prospects for sympathetic cooling of the Ba+ ion by collisionswith Rb atoms.

A proper description of the interaction potential in the long range is particularly importantfor any theoretical study of cold collisions. If we apply the multipole expansion of the interac-tion operator and the perturbation theory, the long-range part of the interaction potential V can

13

CHAPTER 2. GENERAL OBJECTIVES AND MOTIVATION OF THE PRESENTRESEARCH

be expressed as a series in inverse power of the intermolecular distance R, V =∑nCn/R

n

where the van der Waals coefficients Cn can be related to the monomer properties such aspermanent multipole moments or polarizabilities. Calculations of the long-range constantsCn become complicated if one of the monomers is in a spatially degenerate electronic state,like Π or ∆. Some preliminary works addressed the issue of long-range interactions in sys-tems containing open-shell molecules, however there were limited only to the C6 coefficientwithout relating them explicitly to the monomer properties [85–87]. Thus, we found interest-ing and useful to present a comprehensive study on the theory of long-range interactions insystems composed of a ground state S atom and a linear molecule in a degenerate electronicstate. Timeliness and importance of this work is mainly due to the interest in such systems forexperimental and theoretical studies on the sympathetic cooling of polar molecules. One ofthe applications of this theory will be the previously mentioned OH + N system.

To date, the most successful way to produce ultracold molecules is by association of twoultracold atoms by means of the magnetic or optical fields. Magnetoassociation and photoas-sociation are well established to form ultracold bialkali dimers such as KRb, LiCS or Cs2 evenin their rovibrational ground state [88]. An open issue is how to extend the applicability ofthese techniques to other elements and molecules. Regarding the photoassociation, the bestcandidates are molecules formed from the alkaline-earth atoms or atoms with a similar elec-tronic structure like Yb. In our study we will focus on the photoassociative production of theSr2 dimers, molecules of interest for both the ultracold and conventional spectroscopy.Back in 2005 the photoassociative production of Sr2 in highly excited vibrational state wasdemonstrated using the dipole-allowed atomic transition 1S→ 1P [89], while in 2006 a simi-lar experiment was performed utilizing a dipole-forbidden transition near the 1S→ 3P1 inter-combination line [90]. The unsolved problem is how to transfer highly vibrationally excitedmolecules to the electronic and rovibrational ground state, as only then the molecules are sta-ble and may be applied in further experiments. In 2011 the group of E. Tiemann publishednew experimental data from Fourier-transform spectroscopy concerning the excited electronicstates of Sr2 [91]. These new data at some points were in a strong disagreement with the pre-viously published results of ab initio calculations [92,93]. These qualitatively wrong ab initio

data for Sr2 were used to predict and explain some properties of Sr2 molecules important inthe ultracold regime.Motivated by all new spectroscopic findings on Sr2, both from the ultracold and hot experi-ments, we started theoretical work in order to provide state-of-the-art theoretical results con-cerning the interaction and dynamics in the Sr2 molecule in the ground and excited states.The first step was to calculate the potential energy curves, together with the spin-orbit, andnonadiabatic coupling matrix elements. In the second step, we consider possible schemes foran efficient production of ultracold Sr2 molecules in the rovibrational ground state using thephotoassociation method.

14

2.2. OBJECTIVES OF THE RESEARCH

2.2 Objectives of the research

The primary objectives of the presented thesis are the following:

1. To explore the interactions and dynamics of ultracold collisions in selected atom-moleculesystems, and their implications for the prospective sympathetic cooling experiments.This task includes:

• development of the theory of the long-range interactions between an open-shelllinear molecule and atom in a ground S state, expressing the van der Waals con-stants in terms of the monomer properties, and pilot calculations for a few repre-sentative systems,

• formulation of the effective Hamiltonian for collisions between a 2Π-state moleculeand an S-state open-shell atom in an external magnetic field, numerical illustra-tion for collisions between the magnetically trapped N(4S) atoms and OH(2Π)molecules,

• study of the interaction between the LiH molecule and a Li atom with high accu-racy ab initio methods, and quantum-dynamical calculations of the cross sectionsfor collisions between the Li atom and the rotationally excited LiH molecule,

• study of the interactions, and elastic and charge transfer scattering cross sectionsin ultracold collisions between Ba+ and Rb atom.

2. To investigate interactions and rovibrational dynamics of the Sr2 molecule, and theirimplications for the photoassociative production of ultracold strontium molecules. Thetask includes:

• calculations of the potential energy curves and nonadiabatic and spin-orbit cou-pling matrix elements between various electronic states of the Sr2 dimer and theirtests by comparison with high-resolution spectroscopy,

• calculation of photoassociation rates and proposal of a scheme for an effectiveproduction of ultracold ground state Sr2 molecules.

In Chapter 3 we outline the results concerning the long-range interactions in open-shellsystems and numerical calculations for OH + N, LiH + Li, and Ba+ + Rb in the context ofthe sympathetic cooling. Chapter 4 contains a compact description of the results related to theelectronic structure and photoassociation of Sr2. Chapter 5 concludes and lists main achieve-ments of the thesis. The last part of the thesis contains reprints of seven papers published ininternational scientific journals which describe in detail all the obtained results. We will referto these papers as Paper I to VII.

15

16

CHAPTER 3

MODELLING OF THE SYMPATHETIC COOLINGPROCESSES

3.1 Introductory remarks

Sympathetic cooling as a method of production of cold species by bringing them into thethermal contact with another much colder species was originally developed for trapped ions.In the pioneering experiments of this kind simple ions, like Hg+ or Cd+ were cooled to tem-peratures below 1 K by thermal interaction with laser-cooled ions like Be+ or Mg+ [94, 95].Later on, a similar technique based on the ion-ion Coulomb interaction was applied to coolmore complex polyatomic ions such as H+

2 , H+3 , HCO+, or N2H+ [96, 97]. High efficiency

of the ion-ion sympathetic cooling is due to the very long-range electrostatic interaction. Re-cently, the possibilities for the sympathetic cooling of a single ion by thermalization in anultracold atomic bath have been demonstrated [81]. The group of M. Köhl, by combining anion trap and magneto-optical trap for the neutral atoms, showed sympathetic cooling of the174Yb+ ion immersed in the Bose-Einstein condensate of rubidium atoms 87Rb. Similar exper-iments have been performed for the Ba+ and Rb+ ions [83]. In atom-ion systems the coolingprocess is facilitated by large elastic scattering cross sections caused by the long-range R−4

polarization potential.

Neutral alkali-metal and alkaline-earth-metal atoms are routinely cooled by means of laserand evaporative cooling. However, in case of some isotopes these methods are not productivedue to either tiny scattering length and thus small rate of thermalization or due to the quantum-statistics effects since for identical fermions s-wave collisions are forbidden. In such cases thesympathetic cooling of certain atomic isotopes by thermal contact with other isotopes turnedout to be very useful. For instance, the group of F. Schreck demonstrated sympathetic coolingof the fermionic isotope 6Li below the Fermi temperature by collisions with evaporativelycooled bosonic isotope 7Li [98]. Similarly, the first Bose-Einstein condensation of the potas-

17

CHAPTER 3. MODELLING OF THE SYMPATHETIC COOLING PROCESSES

sium atoms 41K was achieved by sympathetic cooling with evaporatively cooled rubidiumatoms 87Rb [99]. Also, the first Bose-Einstein condensation of the strontium isotope 88Sr wasobtained by sympathetic cooling with the 87Sr atoms [100]. This method was necessary sincethe isotope 88Sr has an exceptionally small, negative s-wave scattering length (as = −2 bohr)which precludes efficient evaporative cooling.

A natural development of the successful experiments with sympathetically cooled ionsand atoms would be application of this method to molecular systems. Sympathetic cooling ofneutral molecules has not been demonstrated yet in practice and there are many challengesto overcome. The challenges are related to both purely technical aspects of the complicatedexperimental setup and to more fundamental issues concerning the nature of the interactionsbetween the atomic coolant and the molecules to cool. From a technical perspective, there is anecessity to produce and trap dense molecular source which can effectively be combined witha bath of ultracold atoms, preserved in excellent vacuum conditions. Possible traps for atomsand molecules, in particular polar molecules, differ significantly and may interfere with eachother which should be avoided.

The crucial issue for the efficiency of the cooling process is the character and rate ofcollisions between the atomic refrigerant and the molecules to cool. Polar molecules are mosteasily trapped using static electric fields [101], which is only possible if the molecule is inthe low-field seeking state. However, the low-field seeking state is never the ground stateand apart from the elastic collisions the exothermic inelastic events with transitions to thehigh-field seeking ground state are possible. Such transitions almost always result in ejectingmolecule from the trap, either because the molecule transfers to a non-trappable high-fieldseeking state or the released kinetic energy exceeds the trap depth. Thus, the goal is to suppresssomehow the losses due to the inelastic collisions. A possible way to do it is to carefully selectthe atom + molecule system in which the inelastic collisions are strongly reduced and inthis area theoretical guidance based on the quantum dynamics calculations is indispensable.A second solution to circumvent the problem of inelastic losses is to build traps based onthe alternating electric field (the so-called AC traps) which can confine polar molecules inthe ground state [102], thus any transitions to the lower states are automatically eliminated.AC traps have significantly lower depths than the DC traps, which is unfavourable since lessmolecules can be trapped and lower molecular densities are expected. The other drawback ofthe alternating electric trap is that it provides a dynamically stabilized confinement and elasticcollisions with ultracold atoms, even though they cool the motion, may transfer molecules intounstable orbits leading to the loss from the trap. Again, theoretical analysis and simulationsof the dynamics in the AC trap can help very much the experimentalists to assess the prospectfor sympathetic cooling for a given molecular system and geometry of the trap.

Using the hard-sphere collision model it can be shown [103] that the temperature of thetarget species T kM (hot molecule with mass M and initial temperature T (0)

M ) changes after k

18

3.1. INTRODUCTORY REMARKS

collisions with cold particles (with mass m and a constant temperature Tm) as:

T(k)M = Tm + (T (0)

M − Tm) e−k/κ where κ =(M +m)2

2Mm(3.1.1)

and about 10–100 collisions are needed to get T kM to fall within 10% of the buffer temperatureTm. The lower κ, the thermalization process is more effective if the masses of the collidingpartners are similar. Thus, the mass matching between the molecule and the atomic coolantshould also be taken into account. From the calculations of the mean time between the subse-quent collisions it can be estimated that it takes about 1 to 5 seconds for the cooling process tooccur, which puts restrictions on the trap lifetime. Assuming that after any inelastic collisionthe molecule is lost from the trap, one can estimate from the hard-sphere model that the ratioof the elastic to inelastic cross sections should be at least about 100 for the cooling process tosucceed.

Theoretical studies of the interactions and dynamics in the context of sympathetic coolingwere started by Hutson and collaborators. In their pioneering work they investigated the inter-action between the rubidium atoms and the NH(3Σ−) radicals and showed the existence of thedeeply bound ion-pair state with the dissociation limit Rb+(1S) + NH−(2Π), in addition to theelectronic states that correspond to the Rb(2S) + NH(3Σ−) dissociation threshold [104]. Theion-pair state may have important consequences for the results of sympathetic cooling since itprovides additional mechanisms for inelastic collisions and three-body recombination. Lateron, field-free scattering calculations for the Rb + NH and Cs + NH systems were carried outby employing two-dimensional spin-stretched quartet potential energy surfaces by Tacconi et

al [105]. The authors found fairly large elastic cross sections in the nanoKelvin regime sug-gesting efficient sympathetic cooling of the NH molecule by collisions with ultracold alkaliatoms, providing that there are no inelastic events. In an extensive work, Lara et al. studiedthe interaction and dynamics of the ultracold collisions in the Rb + OH system [106,107]. Theauthors did pioneering investigations of the importance of the nonadiabatic effects and hyper-fine structure on the field-free ultracold collisions. The dynamical calculations were performedwith five coupled potential energy surfaces, including the ion-pair state Rb+ + OH−. Resultsof this work showed that sympathetic cooling of the OH molecule by collisions with the Rbatoms might be challenging due to large inelastic losses driven by the significant anisotropyof the interaction potentials. This conclusion was true even when neglecting the ion-pair statein the calculations. The authors suggested that beneficial for sympathetic cooling would beemploying (i) closed-shell coolants such as alkaline-earth atoms which should generate moreisotropic potential energy surfaces than in the case of alkali-metal atoms and (ii) moleculesthat conform well to the Hund’s case (b) coupling scheme with a weak coupling of the electronspin to the internuclear axis. To a large extent these suggestions were confirmed in the studyon the the ultracold collisions between Mg and NH in the magnetic field [108]. Wallis andHutson showed that in the Mg + NH system the ratio of elastic to inelastic cross sections is

19


large enough to favour sympathetic cooling over a wide range of collision energies and mag-netic fields. This theoretical result, although significant, is hampered somehow by the factthat cooling and trapping of magnesium atoms to submilliKelvin temperatures is extremelydifficult and Mg atoms cannot yet be considered as accessible coolants in the experiments onsympathetic cooling.

Concerning molecules larger than diatomic radicals, prospects for sympathetic cooling ofammonia by collisions with rubidium atoms were thoroughly analysed [109]. A detailed studyof the interactions in the Rb–NH3 system revealed large anisotropies: the well depths for thetwo C3v geometries with the Rb atom located on the C3 axis of the NH3 molecule differ bynearly 2000 cm−1, which indicates a possibility for a strong rotation-tunnelling inelasticity,whenever inelastic collisions are allowed. Indeed, quantum-scattering calculations for the ul-tracold collisions showed that the ratio of elastic to inelastic cross sections is too small toachieve sympathetic cooling of NH3 in the low-field seeking state.

To conclude, little is known about the interactions and dynamics of ultracold molecularsystems that can be applied in the experiments on sympathetic cooling. The abovementionedarticles revealed some important features of the systems comprising alkali-metal atom and po-lar molecule such as the presence of low-lying ion-pair states or relatively strong anisotropy ofthe interaction potentials. However, these theoretical works should be considered as prelimi-nary rather than a comprehensive survey due to very few examples investigated and numerousapproximations involved, especially when calculating the interaction potentials. In most pi-lot studies the multireference configuration interaction method (MRCI) was employed whichis not size-extensive and the obtained interaction energies do not have a proper asymptoticbehaviour. Moreover, for systems containing heavy atoms, such as Rb or Cs, the pseudopo-tentials had to be used to approximately account for the relativistic effects. This results in anadditional uncertainty of the potential energy surfaces. We emphasize that the accuracy of theinteraction potential, in particular in its long range, is crucial for a quantitative description ofthe dynamics in the ultracold regime. Therefore, quantum-dynamical calculations for bench-mark systems, employing accurate interaction potentials, with special care of the long-rangetail, are still needed to understand the physics of the collisions between atoms and simplemolecules, and their implications for possible cooling.

3.2 Long-range interactions between S-state atom and anopen-shell linear molecule

We start the summary of the results by presenting theory of long-range interactions be-tween an atom in its ground S state and a linear molecule in a degenerate state with a non-zeroprojection of the electronic orbital angular momentum (Π, ∆, etc.). We show how the long-range coefficients describing the induction and dispersion interactions at large atom–diatom

20

3.2. LONG-RANGE INTERACTIONS BETWEEN S-STATE ATOM AND ANOPEN-SHELL LINEAR MOLECULE

distances can be related to the first and second-order atomic and molecular properties forsuch systems. The final expressions for the long-range coefficients are written in terms ofall the components of the static and dynamic multipole polarizability tensors, including thenondiagonal terms connecting states with the opposite projection of the electronic orbital an-gular momentum. We also show that for the interactions of molecules in excited states thatare connected to the ground state by multipolar transition moments, additional terms in thelong-range induction energy appear.

Calculations of the asymptotic constants Cn are nontrivial when one of the interactingspecies is in an open-shell electronic degenerate state. This problem has only been addressedby Spelsberg [86] for the CO + OH system, and by Nielson et al. [85] and Bussery-Honvaultet al. [87] for an atom interacting with an open-shell molecule. The latter considerations werelimited, however, to the C6 coefficients and did not relate the Van der Waals constants to themolecular properties of the monomers.

3.2.1 Theoretical developments

We consider the interaction of an atom A in the ground S state |S〉A and a linear moleculeB in a state |Λ〉B, where Λ is the projection of the electronic orbital angular momentum ofthe molecule on the molecular axis. The resulting spin multiplicity of the complex does notplay any role in our developments, and will be omitted. The electronic Hamiltonian H of thecomplex AB can be written as:

H = H0 + V, (3.2.1)

where H0 is the sum of the Hamiltonians describing isolated monomers A and B, H0 = HA +

HB, and V is the intermolecular interaction operator collecting all Coulombic interactionsbetween electrons and nuclei of the monomer A with the electrons and nuclei of the monomerB. Assuming that the electron clouds of the monomers do not overlap, V can be representedby the following multipole expansion [110, 111]:

V =∞∑

lA,lB=0

ClA,lBR−lA−lB−1

lA+lB∑m=−lA−lB

(−1)mY −mlA+lB(R)[QlA ⊗ QlB

]lA+lB

m(3.2.2)

where the constant ClA,lB is given by

ClAlB = (−1)lB[ 4π2lA + 2lB + 1

]1/2(

2lA + 2lB2lA

)1/2

, (3.2.3)

Y ml (R) is the normalized spherical harmonic depending on the spherical angles R of the

vectorR connecting the centers of mass of the monomers A and B in a space-fixed coordinatesystem, Qm

l denotes the multipole moment operator in the space-fixed frame. We made also

21


use of the tensor product of two spherical tensors:

[QlA ⊗ QlB

]LM

=∑

mA,mB

〈lA,mA; lB,mB|L,M〉 QmAlA

QmBlB. (3.2.4)

where 〈lA,mA; lB,mB|L,M〉 is the Clebsch-Gordan coefficient.

A state |Λ〉B of a linear molecule with Λ 6= 0 is doubly degenerate, and so is the stateof the complex AB at R = ∞, which is just a product |S〉A|Λ〉B. For the interaction of aground state S atom with a molecule, the first-order electrostatic energy vanishes identicallyin the multipole approximation, so the degeneracy is lifted in the second-order, leading to thesplitting of the two states |S〉A| ± |Λ|〉B, into the A′ and A′′ states of the complex AB at finiteR. Thus, to obtain the long-range behavior of the A′ and A′′ states we have to diagonalize thesecond-order interaction matrix:

V(2) =

V(2)

Λ,Λ V(2)

Λ,−Λ

V(2)−Λ,Λ V

(2)−Λ,−Λ.

(3.2.5)

The elements V (2)Λ,Λ′ of V (2) are given by the standard expressions of the polarization theory

[112] and can be decomposed into the induction and dispersion parts:

V(2)

Λ,Λ′ = −∞∑k

′ 〈S|A〈Λ|B V |k〉A|Λ〉B〈k|A〈Λ|B V |S〉A|Λ′〉BωS,k

−∞∑k

′∞∑n

′ 〈S|A〈Λ|B V |k〉A|n〉B〈k|A〈n|B V |S〉A|Λ′〉BωS,k + ωΛ,n

, (3.2.6)

where ωS,k = Ek−ES is the excitation energy of the atom from the ground state |S〉A to theexcited state |k〉A characterized by the set of quantum numbers k, and ωΛ,n = En−EΛ

is the excitation energy from the state |Λ〉B to the excited state |n〉B of the molecule withthe set of quantum numbers denoted by n.

It can easily be shown [85] that in the case of interaction with an atom in an S statethe matrix elements V (2)

Λ,Λ and V (2)−Λ,−Λ are equal, and the same holds for off-diagonal elements

V(2)

Λ,−Λ and V (2)−Λ,Λ. Eigenvalues of the matrix (3.2.5) can simply be constructed on the symmetry

basis only as:E

(2)(±) = V

(2)Λ,Λ + (−1)Λ+pA+fV

(2)Λ,−Λ (3.2.7)

where pA defines the spatial parity of the atomic wave function in the S state and f = 0 for A′

and f = 1 for A′′ state.

The multipole expansion of VΛ,Λ′ is readily obtained by inserting the multipole expansion(3.2.2) of the interaction operator V into Eq. (3.2.6) and transformation of the multipole op-erators from the space-fixed to the body-fixed frame of each monomer. Details are given inSec. II of Paper I. Final formulas for the long-range coefficients expressed in terms of the

22

3.2. LONG-RANGE INTERACTIONS BETWEEN S-STATE ATOM AND ANOPEN-SHELL LINEAR MOLECULE

irreducible components of the polarizabilities are the following:

V(2)

Λ,Λ′ = −∑

lA,lB ,l′B

∑L,K

ξL,KlAlB l′BR−(2+2lA+lB+l′B)

[Λ,Λ′CL,K

lAlB l′B

(ind) + Λ,Λ′CL,KlAlB l

′B

(disp)]P|K|L (cos θ),

(3.2.8)where the constant ξL,KlAlB l

′B

is given by:

ξL,KlAlB l′B

=[

(2lA + 2lB + 1)!(2lA + 2lB′ + 1)!(2lA)!(2lB)!(2lA)!(2lB′)!

]1/2lA + lB lA + l′B L

0 0 0

×( 2L+ 1

2lA + 1

)1/2√√√√(L−K)!

(L+K)!

lB lA + lB lA

lA + l′B l′B L

, (3.2.9)

and the expressions in the round and curly brackets are the 3j and 6j coefficients, respectively.PKL (cos θ) denotes the associated Legendre polynomial and θ is the angle between the vectorR and the axis of the molecule B.Combining all terms with the same power n = 2lA + lB + l′B + 2 in the above expansion, weget standard long-range coefficients CL,K

n :

CL,Kn =

2lA+lB+l′B+2=n∑lA,lB ,l

′B

ξL,KlAlB l′B

[Λ,Λ′CL,K

lAlB l′B

(ind) + Λ,Λ′CL,KlAlB l

′B

(disp)], (3.2.10)

by means of which the asymptotic expansion of Eq. (3.2.7) simply reads:

E(2)(±) = −

∞∑n=6

∑L

[CL,0n

RnP 0L(cos θ) + (−1)Λ+pA+f C

L,2Λn

RnP|2Λ|L (cos θ)

]. (3.2.11)

The dispersion part Λ,Λ′CL,KlAlB l

′B

(disp) is proportional to the Casimir-Polder integral over theatomic and molecular polarizabilities calculated at imaginary frequencies:

Λ,Λ′CL,KlAlB l

′B

(disp) =1

2π

∫ ∞0

α(lAlA)00 (iω) Λ,Λ′α

(lB l′B)LK (iω)dω, (3.2.12)

where for an open-shell linear molecule we have introduced extra superscripts Λ and Λ′ inthe definition of the polarizability tensor to distinguish between the diagonal and off-diagonalcomponents:

Λ,Λ′αll′

mm′(ω) = 2∑n

′ ωΛ,n〈Λ|Qm

l |n〉〈n|Qm′l′ |Λ′〉

ω2Λ,n − ω2

, (3.2.13)

and the irreducible polarizabilities are obtained by Clebsch-Gordan coupling:

Λ,Λ′α(ll′)LK (ω) =

∑m,m′〈l,m; l′,m′|L,K〉 Λ,Λ′α

ll′

mm′(ω). (3.2.14)

The induction term Λ,Λ′CL,KlAlB l

′B

(ind) is the product of the static polarizability of the atom and

23


permanent multipole moments of the open-shell molecule:

Λ,Λ′CL,KlAlB l

′B

(ind) = α(lAlA)00 (0)

[〈Λ|QlB |Λ〉 ⊗ 〈Λ|Ql′B

|Λ′〉]LK. (3.2.15)

General considerations which components Λ,Λ′α(ll′)LK do not vanish and how they can be

transformed from the spherical to the Cartesian representation are given in Sec. II of Paper I.Here we only add that the expression for the diagonal term V

(2)Λ,Λ is the same as for the interac-

tion between atom and closed-shell linear molecule in a Σ state. The additional term emerg-ing when Λ 6= 0 is the off-diagonal V (2)

Λ,−Λ term. It has both the induction and the dispersionparts. The source of the induction energy in V (2)

Λ,−Λ comes from the fact that open-shell linearmolecules have an additional independent component of the permanent multipole moments,〈Λ|Q2Λ

l |−Λ〉, in contrast to the Σ-state molecules that only have one independent compo-nent, namely 〈Λ|Q0

l |Λ〉. Likewise, the dispersion terms in V (2)Λ,−Λ result from the presence of

additional components of the polarizability for the open-shell linear molecules.

Equation (3.2.12) is strictly valid only when the molecule is in its ground electronic state.If the molecule is in an excited state that is connected to any other state lower in energy bymultipolar transition moments then the Casimir-Polder integral is no longer valid and an extraterm has to be added to the energy. This additional term depends on the dynamic polarizabilityof the atom calculated at frequency ω equal to the energy of all possible deexcitations in themolecule, and we denote it byCL,K

lAlB l′B

(corr, deexc). Its form is slightly similar to the inductionpart:

Λ,Λ′CL,KlAlB l

′B

(corr, deexc) =∑n−

α(lAlA)00 (ωn−,Λ)

[〈Λ|QlB |n−〉 ⊗ 〈n−|Ql′B

|Λ′〉]LK.

(3.2.16)The summation in the above equation runs only over states of the molecule ψB(n−) withenergy lower than the reference one, i.e. ωn−,Λ = EΛ − En− is positive, and hence ωn−,Λcorresponds to a possible electronic deexcitation of the molecule. This term does not have asimple physical interpretation, but it leads to a different QED retardation of the long-rangepotential than given by the classical Casimir-Polder formula [113].

3.2.2 Numerical illustration

Theoretical developments have been illustrated with the numerical results for systemsof interest for the sympathetic cooling experiments: interactions of the ground state Rb(2S)atom with CO(3Π), OH(2Π), NH(1∆), and CH(2Π) and of the ground state Li(2S) atom withCH(2Π). For these systems we have calculated the long-range coefficients CL,K

n up to andincluding n = 10. At present no generally available ab initio code allows for the calculationsof all components of the dynamic polarizability tensor for open-shell linear molecules. There-fore, in our calculations we computed the polarizabilities appearing in Eq. (3.2.12) from the

24

3.3. COLLISIONS OF AN OPEN-SHELL S-STATE ATOM WITH A 2Π MOLECULE INA MAGNETIC FIELD

sum-over-states expansion, Eq. (3.2.13). The appropriate transitions moments to the excitedstates and excitation energies were calculated using linear response formalism with referencewave function obtained from the multiconfigurational self-consistent field method (MCSCF)with large active spaces.

The quality of the computed long-range coefficients has been analyzed in three differ-ent ways, by comparison with (i) the interaction energy obtained from the supermolecularmethod, (ii) the induction and dispersion energies obtained from the symmetry-adapted per-turbation theory (SAPT) calculations, and (iii) the values of CL,K

n derived from the fits of thepotential energy surfaces published previously. In all cases we noted a satisfactory agreementbetween the long-range interaction energies based on our CL,K

n coefficients and those derivedfrom the reference data. Any stated discrepancies have been discussed and possible reasonsfor them have been given. A detailed analysis of the obtained results together with tables con-taining numerical values of the CL,K

n coefficients and illustrative plots are presented in Sec.III of Paper I.

3.3 Collisions of an open-shell S-state atom with a 2Πmoleculein a magnetic field

In this part we present rigorous quantum-dynamical studies of collisions between an open-shell S-state atom and a 2Π-state molecule in the presence of a magnetic field. We analyse thecollisional Hamiltonian and indicate possible mechanisms for the inelastic collisions in suchsystems. This is an extension of the theory presented in Refs. [114,115] covering only systemswith closed-shell atoms. The theory is applied to the collisions of the nitrogen atom with theOH molecule in their fully spin-stretched incident state, i.e. assuming that the interaction takesplace exclusively on the quintet (high-spin) potential energy surfaces.

3.3.1 Effective Hamiltonian

We consider the case of an atom A(2s+1S) interacting with a diatomic molecule BC(2Π),in the presence of an external magnetic field B. The field direction is used to define thelaboratory (space-fixed) Z-axis, B = (0, 0, B). The system A–BC is described in the Jacobicoordinates, with the vector r connecting the heavier and the lighter of the atoms B and C,and R connecting the center of mass of BC and the atom A. By convention, lower-case andcapital letters are used to represent the quantum states of the atom A and of the moleculeBC, respectively. The subscripts 1 and 2 refer to the monomers A and BC, respectively. Forsimplicity, the diatom will be treated as a rigid rotor in the vibrational state v, although ageneralization to include the vibrations is straightforward.

The Hamiltonian describing the nuclear motions of the complex A-BC in the presence of

25


magnetic field can be written as:

H = − ~2

2µR−1 d2

dR2R +

L2

2µR2+ Hmon + H12, (3.3.1)

where L is the space-fixed angular momentum operator describing the end-over-end rotationand µ is the reduced mass of the complex. The Hamiltonian Hmon contains all terms describingthe isolated monomers, i.e. Hmon = H1 + H2. Finally, H12 describes the interaction betweenthe monomers:

H12 = Hs1s2 + V (R, θ). (3.3.2)

Here, Hs1s2 accounts for the direct dipolar interaction between the magnetic moments dueto the unpaired electrons of the monomers, and V is the intermolecular interaction potential.Equation (3.3.2) neglects the small dipolar interaction between the spin of A and the orbitalmagnetic moment of BC.

If s1 6= 0 and hyperfine terms are neglected, the Hamiltonian for an isolated atom in thestate 2s1+1S is fully determined by the electron Zeeman interaction with the external magneticfield:

H1 = gSµB s1 · B, (3.3.3)

where gS is the electron g-factor, µB the electron Bohr magneton, and s1 is the spin operator.

The analogous Hamiltonian for a 2Π molecule can be written as [116]:

H2 = Hrso + HZ,2 +HΛ, (3.3.4)

where the rotational and spin-orbit contributions were collapsed into the first term:

Hrso ≡ Bv n · n+ Av l · s2, (3.3.5)

Bv and Av are the rotational and spin-orbit constants, respectively, n is the operator of themechanical rotation of BC which can be expressed as j − l− s2, where j, l and s2 are the op-erators of the total, electronic orbital, and spin angular momenta of the molecule, respectively.Finally, Hrso can be rewritten as:

Hrso = (Av + 2Bv) lz sz,2 +Bv

[j · j − 2j · s2 − 2jz lz

]. (3.3.6)

The terms l · l, s2 · s2 and lz lz have been omitted in the equation above as they only shift allthe levels by a constant value. The term HΛ responsible for the Λ-doubling of the rotationallevels of BC is represented by the effective Hamiltonian:

HΛ =∑q=±1

e−2iqφr[(pv + 2qv)T2

2q(j, s2)− qvT22q(j, j)

], (3.3.7)

26


where φr is the azimuthal angle associated with the electron orbital angular momentum aboutmolecular axis defined by r, while pv and qv are empirical parameters. In Eq. (3.3.7) thestandard textbook definition of the second rank tensor T2

q coupling two vectors k1 and k2 isused:

T2q(k1,k2) =

∑q1,q2

〈1, q1; 1, q2|2, q〉T1q1

(k1) T1q2

(k2), (3.3.8)

where 〈1, q1; 1, q2|2, q〉 is the Clebsch-Gordan coefficient, and the first-rank tensor compo-nents are T1

0(k) = kz and T1±1(k) = ∓(kx ± i ky)/

√2. Finally, if only the electron spin and

orbital contributions are taken into account, the Zeeman term reads:

HZ,2 = gSµB s2 · B + g′LµB l · B, (3.3.9)

where g′L is the orbital g-factor. For diatomic molecules of multiplicity higher than 2 (forexample 3Π) an additional term describing the intramolecular spin-spin interaction has to beincluded in the monomer Hamiltonian of Eq. (3.3.4). The dipolar spin-spin interaction canconveniently be written as [116]:

Hs1s2 = −g2Sµ

2B(µ0/4π)

√6∑q

(−1)q T2q(s1, s2) T2

−q(C), (3.3.10)

with T2q(C) = C2

q (θ, φ)R−3, where C2q is the spherical harmonic function in the Racah nor-

malization, (R, θ, φ) is the set of relative spherical coordinates of the ‘composite’ atomic anddiatomic electronic spins in the space-fixed frame, and µ0 is the magnetic permeability of thevacuum.

3.3.2 Method of close-coupling equations

Let us briefly sketch how we proceed to calculate elastic and inelastic cross sections,quantities that determine the outcome of the collisions. We start from the definition of thebasis functions that are used to define the state of the two colliding species, the atom A and themolecule BC, and the state of the collisional complex A–BC. The state of the BC moleculecan conveniently be described by using the Hund’s (a) basis functions |Λ〉|SΣ〉|JΩMJ〉, aproduct of the electronic orbital, electronic spin, and rotational parts, where S is the electronicspin with the projection Σ on the molecular axis (body-fixed z axis), Λ is the projection ofthe electronic orbital angular momentum on the molecular axis, and J is the total angularmomentum of the BC with projections Ω on molecular axis and MJ on the space-fixed axis.The rotational part |JΩMJ〉 is proportional to the Wigner D-function. For the body-fixedprojections we have Ω = Λ + Σ. The state of an atom is characterized exclusively by theelectronic spin function |sms〉. The primitive basis set for the A–BC collisional system isconstructed as |sms〉 |Λ;SΣ; JΩMJ〉 |LML〉, where |LML〉 describes the relative motion ofA and BC in the space-fixed reference frame.

27


In the presence of a magnetic field, the projection M of the total angular momentum,M = ms + MJ + ML, and the parity P are the only conserved quantities. In order to takeadvantage of these symmetries, a parity-adapted basis set φα(R, r) is defined:

φα(R, r) = |sms〉∣∣∣S; JΩMJ ε

⟩|LML〉 (3.3.11)

where the functions of the molecule are:

∣∣∣S; JΩMJ ε⟩≡ 1√

2

[ ∣∣∣Λ = 1;S Σ = Ω− 1; JΩMJ

⟩+ε(−1)J−S

∣∣∣Λ = −1;S Σ = 1− Ω; J − ΩMJ

⟩], (3.3.12)

and Ω = |Ω|, ε = ±1. In this basis set the parity of BC is given by p2 = ε(−1)J−S , while thatof the triatomic complex is simply P = p2(−1)L. Moreover, we use the index α to representthe set of all quantum numbers s,ms, S, J, Ω,MJ , ε, L,ML. Functions φα(R, r) are thechannel functions in the absence of external fields. The total wave function Ψ describingnuclear motion of the complex A–BC can be expanded as (the bond length of BC, r, is fixed):

Ψ(R, r) =1R

∑α

Fα(R)φα(R, r) (3.3.13)

By substituting Eq. (3.3.13) into the Schrödinger equation H Ψ = E Ψ with the Hamilto-nian of Eq. (3.3.1), multiplying from the left by φ?α′(R, r), and integrating over the angularcoordinates (R, r) we end up with the following set of equations for the unknown functionsFα(R): [

− ~2

2µd2

dR2− E

]Fα(R) = −

∑α′Wαα′(R)Fα′(R) (3.3.14)

where the elements of the coupling matrixW (R) are given by:

Wαα′(R) = 〈φα|L2

2µR2+ Hmon + Hs1s2 + V (R, θ)|φα′〉. (3.3.15)

Exact expressions for the elements Wαα′(R) are presented in Sec. 3.2 of Paper II 1. The wavefunction Ψ must be regular at the origin which determines the short-range boundary condition:

Fα(R) = 0 as R→ 0. (3.3.16)

Assuming that we have N channel functions φα, for each energy E > 0, the close-couplingequations (3.3.14) have N independent solutions which fulfill the short-range boundary con-dition (3.3.16). These solutions form a N ×N matrix F (R). From the asymptotic form of theHamiltonian for the nuclear motion, it is known that at long range the solution matrix F (R)

1Note that in Paper II lower-case letters were used to express quantum state of both monomers.

28


must satisfy the following scattering boundary conditions [117]:

F (R) = J(R) +N (R) ·K as R→∞ (3.3.17)

where J(R) andN (R) are the (known)N×N diagonal matrices with their elements given bythe spherical Bessel or modified spherical Bessel functions, depending on whether the channelα is open or closed for a given energy E. From the Nop × Nop block of the K matrix corre-sponding toNop open channels we calculate the so-called S matrix which contains amplitudesfor transitions from the channel α to the channel α′:

S = (1+ iKop−op)−1(1− iKop−op). (3.3.18)

Finally, having the S matrix we can calculate the cross sections for transitions between twochannels:

σα→α′ =π

k2α

|δα,α′ − Sα,α′|2 (3.3.19)

where k2α = 2µ(E − Eα)/~2 and Eα is the threshold energy of the channel α.

In principle, the K matrix can be obtained by propagating outwards the wave functionmatrix F (R) from the pointRmin located deeply enough in the classically forbidden region tothe final point Rmax where the interaction potential V (R, θ) is effectively zero, and matchingthe solution F (Rmax) to the form given by Eq. (3.3.17). However, it is more practical topropagate the log-derivative matrix Y (R):

Y (R) =dF (R)dR

F (R)−1. (3.3.20)

The log-derivative method was originally proposed by Johnson [117] and later improved byAlexander and Manolopoulos [118, 119] to enable a variable step propagation, which is par-ticularly useful for cold collisions as it allows very large propagation steps in the long range.

In the presence of external magnetic (or electric) field the coupling matrix W (R) is notdiagonal at R → ∞. This is caused by the fact that the channel functions φα are not eigen-functions of the monomers Hamiltonian in the presence of the field. The boundary conditionsof Eq. (3.3.17) are appropriate only in a basis set in which the asymptotic coupling matrix isdiagonal. Hence, at the ending point of the propagation the Y (Rmax) matrix has to be trans-formed from the original basis φα to a new basis φα which diagonalizes the asymptoticHamiltonian.

To simplify calculations we always take advantage of the existing symmetries to reducethe number of coupled channels. In the presence of a magnetic field, the conserved quantitiesare the projectionM of the total angular momentum onto the field axis and the total parityP . This means that the close-coupling equations and the S matrix separate into uncoupledblocks, each with defined values of the conserved quantum numbers M,P. Then the cross

29


section for a given transition α → α′ is obtained by adding contributions from each separateSM,P matrix:

σα→α′ =π

k2α

∑M,P

∣∣∣δα,α′ − SM,Pα,α′

∣∣∣2 . (3.3.21)

3.3.3 Numerical illustration

As a numerical illustration of the theory exposed above we analyze the results for cold col-lisions of the N(4S) atom with the OH(2Π) radical. We assume that both collisional partnersare initially in their fully spin-stretched states which are low-field seekers and thus magneti-cally trappable.

The interaction between N(4S) and OH(2Π) takes place on four adiabatic surfaces: 3A′,3A′′, 5A′, and 5A′′. The triplet 3A′′ surface is reactive: the reaction N + OH → NO + H isbarrierless and highly exothermic. If the collisional partners, N(4S) and OH(2Π), are initiallyin their fully spin-stretched states, it is well-justified to assume that the interaction occursonly on the high-spin (quintet) potential energy surfaces and this was done in this work. Thisassumption is equivalent to approximating the triplet surfaces by the quintet ones. Such anapproximation is reasonable for cold collisions as the high-spin and low-spin surfaces differonly in the short range due to the quantum exchange effects, having the same long-range tailand the outcome of the collisions in the cold regime is mainly determined by the long-rangeforces.

The two quintet potential energy surfaces were calculated with the unrestricted versionof the coupled-cluster method restricted to single, double, and noniterative triple excitations[UCCSD(T)]. The unrestricted model of CCSD(T) was chosen to circumvent the problem ofthe lack of size-consistency of the (partly) spin-restricted approach when the interaction be-tween two open-shell systems is considered [120]. To improve the description of the potentialin the long range, analytic representation of the large R asymptotics has been introduced. Thenecessary long-range coefficients Cn were calculated employing the theory presented in theprevious chapter of the thesis and Paper I.

Dynamical calculations were done with the MOLSCAT package [121], locally extendedwith a plug-in to implement the coupling matrix elements Wα,α′(R) discussed above for col-lisions between an S-state open-shell atom and a 2Π state molecule. The cross sections werecalculated for four strengths of the magnetic field B, 10, 100, 500 and 1000 Gauss, and colli-sional energies from 10−5 to 1 Kelvin, see Fig. 3 of Paper II . The results show a substantialinelasticity. Only for the lowest examined field of B = 10 Gauss the inelastic processes arestrongly suppressed due to the centrifugal barriers in the outgoing channels. With the increaseof the magnetic field this suppression disappears as the energy released in the inelastic colli-sions starts to exceed the height of the barrier.

We have indicated two main mechanisms leading to the inelasticity in the collisions ofan open-shell atom with a 2Π state molecule. The first one is the direct coupling through the

30

3.4. INTERACTION AND COLLISIONS BETWEEN THE LIH MOLECULE AND A LIATOM

anisotropy of the interaction potential that drives transitions to the lower Zeeman levels ofOH without affecting the Zeeman state of the atom. The second mechanism is due to the cou-pling via the intermolecular spin-spin interaction Hs1s2 . Here, the final Zeeman levels of thetwo colliding partners are changed. The mechanism involving the spin-spin term dominatesfor small strengths of the magnetic field and ultralow collisional energies, while the othermechanism with the anisotropy of the interaction potential prevails when the field strength islarger (above 100 Gauss) or the collisional energy higher (more than 1 mK). Thus, the anal-ysed system, an open-shell atom interacting with a 2Π molecule, features and combines twodirect ways of coupling between different Zeeman levels. This conclusion suggests that theinelasticity that may occur here is significantly larger than it was previously shown for N(4S)+ NH(3Σ−) [122] or He(1S) + OH(2Π) [115], which was confirmed in the obtained numericalresults.

The ratio of the elastic to inelastic cross sections is larger than 100 and thus favourable forsympathetic cooling only at collision energies below roughly 1 mK and magnetic fields of 10G or less. If either the collision energy or the magnetic field is significantly above, inelasticprocesses due to the potential anisotropy dominate and the ratio of the elastic to inelastic crosssections falls down. This suggests that sympathetic cooling of OH by collisions with the Natoms is unlikely to be successful except at collision energies below 1 mK.

3.4 Interaction and collisions between the LiH molecule anda Li atom

In this part we summarize results of high-accuracy electronic structure calculations forthe LiH molecule interacting with a Li atom. This system with only seven electrons offersunique opportunities to apply a hierarchy of the state-of-the-art ab initio methods of the elec-tronic structure theory, which was not possible for the previously studied molecular systemsin the context of sympathetic cooling experiments. High accuracy of the adopted computa-tional approach, a combination of the explicitly correlated coupled cluster F12 method withthe valence-valence full configuration interaction method, enabled us to estimate the errorbars on the potential. Our analysis of the interaction in the Li–LiH system includes the groundand first excited state potential energy surfaces, the nonadiabatic coupling matrix elements be-tween the ground and excited surfaces, transformation to the diabatic basis, possible reactivechannels, and their implications for the quantum dynamics.

3.4.1 Computational approach

We calculated the ground state interaction potential between the LiH molecule and a Liatom employing a combination of the explicitly-correlated coupled cluster method with single,double, and approximate noniterative triple excitations [CCSD(T)–F12] for the core-core and

31


core-valence correlation with the full configuration interaction method (FCI) for the valence-valence correlation. Two dimensional potential energy surface V (R, θ), with the LiH bondlength kept fixed at its equilibrium value, was constructed according to the following expres-sion:

V (R, θ) = V CCSD(T)−F12(R, θ) + δV FCIv−v (R, θ) (3.4.1)

where V CCSD(T)−F12 is contribution obtained from all-electron CCSD(T)-F12 calculationsand δV FCI

v−v is the correction for the valence-valence correlation beyond the CCSD(T)-F12level. Both terms, V CCSD(T)−F12 and δV FCI

v−v , were obtained from the standard expressions forthe supermolecule interaction energy, corrected for basis set superposition error.

The CCSD(T)–F12 is a variant of explicitly correlated methods [123], i.e. methods inwhich the interelectronic distance is explicitly introduced into the wave function. In the CCSD(T)–F12 method the cluster operator S from the standard parametrization of the coupled clusterwave function Ψ(CC) = eS|HF〉 has the following form:

S = T1 + T2 + T2′ (3.4.2)

where apart from the usual single T1 and double T2 excitation cluster operators, responsiblefor excitations into set of unoccupied orbitals, there is an additional operator T2′. The opera-tor T2′ excites into the correlated pair functions (geminals) and thus introduces r12-dependentterms into the wave function. The exact equations of the CCSD-F12 method are much morecomplex than those of the standard CCSD method due to additional matrix elements contain-ing the explicitly correlated geminal factor f(r12), and extra terms entering the CC equationsfor the cluster amplitudes. The complexity of the exact CCSD-F12 formulation triggered de-velopment of numerous approximate CCSD-F12 models. In the present work we have em-ployed a variant with approximate noniterative triple excitations dubbed as CCSD(T)-F12band introduced by Adler et al. [124, 125] In this method the geminal factor f(r12) that en-ters expressions for the amplitudes of the operator T2′ is approximated by a linear combina-tion of Gaussian functions f(r12) =

∑β e−βr212 . The CCSD(T)-F12b variant is characterized

by (i) using fixed amplitudes of the T2′ operator so that the wave functions has short-rangeasymptotic behavior of the correlation cusp and (ii) omitting some terms in the intermediatesthat appear in the CCSD-F12 equations and do not appear in the simpler MP2-F12 model.These approximations make the CCSD(T)-F12b method an excellent compromise betweenthe computational cost and accuracy towards the complete basis set limit and its efficient im-plementation in the MOLPRO package has been used in our calculations of the V CCSD(T)−F12

term.

Results of explicitly correlated calculations with the CCSD(T)-F12 method have beenconfronted with those from the conventional CCSD(T) approach employing the families ofthe Dunning’s correlation-consistent polarized valence basis sets, cc-pVXZ, with the cardinalnumber X = D, T, Q, 5. In addition, results based on the conventional orbital basis sets

32


have been used to apply a simple two-point extrapolation schemes for the correlation energy,according to the following formula [126]:

Ecorr(X−1)X = Ecorr

X +EcorrX − Ecorr

(X−1)

[1− (X)−1]−α − 1, (3.4.3)

where Ecorr(X−1)X is the extrapolated correlation energy, while Ecorr

(X−1) and EcorrX denote the cor-

relation energies obtained for two consecutive cardinal numbers, (X−1) and X , respectively.The final extrapolated interaction energy at a single grid-point was obtained by subtracting theLi and LiH extrapolated total energies from the Li–LiH extrapolated total energy and addingthe Hartree-Fock interaction energy calculated with the cardinal number X . We used α = 2

and α = 3 in the extrapolations, since these values were indicated by Jeziorska et al. in theirhelium dimer study [127, 128] as the most universal ones for extrapolating all components ofthe interaction energy.

In the course of studying the interaction between Li and LiH, it turned out that at linearLiH–Li geometries the ground state potential shows a close avoided crossing with the excitedion-pair potential. In order to investigate how far the excited state may affect the scatteringdynamics, we have computed the full potential energy surface for the excited state in questionby means of equation-of-motion coupled-cluster method with single and double excitations(EOM-CCSD) [129–131] implemented in the QCHEM code [132] for open-shell referencesstates.

Since the two electronic states are strongly coupled in the vicinity of the avoided crossing,it was advisable to calculate the nonadiabatic coupling matrix elements τ12 between the twostates that could then be used to make the transformation from the adiabatic to the diabaticrepresentation of the potential energy surfaces for subsequent quantum-dynamical studies.Diabatic representation enables to eliminate the nonadiabatic coupling matrix elements fromthe equations for the nuclear motion at the expense of introducing an extra diabatic couplingpotential. Such a procedure can be indispensable for numerical integrations of the equationsfor the nuclear motion since the nonadiabatic couplings, which enter the equations for nuclearmotion in adiabatic representation, are strongly varying functions close to the avoided crossingwhile diabatic coupling potential is a smooth function of nuclear coordinates.

The (vectorial) nonadiabatic coupling matrix elements τ12 are defined as τ12 = 〈Ψ1|∇Ψ2〉[133, 134], where ∇ is the gradient operator of the position vector R and Ψ1 and Ψ2 arethe wave functions of the two states. On the two-dimensional surface, we may specify radialτ12,R = 〈Ψ1|∂Ψ2/∂R〉 and angular τ12,θ = 〈Ψ1|∂Ψ2/∂θ〉 components of the vector τ12.

The transformation from the adiabatic representation to the diabatic representation maybe expressed in terms of a mixing angle γ,

H1 = V2 sin2 γ + V1 cos2 γ, H2 = V1 sin2 γ + V2 cos2 γ, H12 = (V2 − V1) sin γ cos γ,

(3.4.4)

33


where V1 and V2 are the ground-state and excited-state adiabatic potentials, H1 and H2 are thediabatic potentials, and H12 is the diabatic coupling potential. In principle, the mixing angleγ can be obtained by performing line integration of the nonadiabatic coupling τ12:

γ(R) = γ(R0) +∫ R

R0τ12 · dl, (3.4.5)

where R0 is the starting point of the integration. For polyatomic molecules, however, themixing angle γ obtained by integrating this equation is non-unique. The source of the problemis the contribution from higher excited states, which make the integral path-dependent. Tocircumvent the problem of the path dependence, one may assume that we deal with an idealtwo-state model. It can be shown [133] that in this case Eq. (3.4.5) has a unique result, whichcan be written in the form:

γ(R, θ) = γ(R0, θ0) +∫ R

R0τR(R, θ0) dR +

∫ θ

θ0τθ(R, θ) dθ. (3.4.6)

Equations (3.4.4) to (3.4.6) are valid only for a two-state model and we checked that such amodel is applicable for the Li–LiH system because the nonadiabatic coupling matrix elementswith any other states are spatially well separated from τ12.

We evaluated the nonadiabtic matrix elements τ12 for all (R, θ) geometries by means ofthe multireference configuration interaction method limited to single and double excitations(MRCISD) [135, 136], using the MOLPRO code [137]. Then the mixing angle was obtainedaccording to Eq. (3.4.6) with the starting point of the integration at R = 20 bohr and θ = 0.Finally, the diabatic potentials were generated according to Eq. (3.4.4) by inserting in placeof V1 and V2 our most accurate interaction potentials for the ground and excited states.

3.4.2 Numerical results

Figures illustrating the comparison between the interaction energies in the Li–LiH systemobtained by employing the conventional orbital basis sets, extrapolation schemes, and explic-itly correlated approach are presented in Sec. III of Paper III. Likewise, contour plots of theground and first excited electronic states, and of the diabatic potentials can be found in Sec.IV of Paper III. Here we sum up our main findings concerning the interaction between Li andLiH.

The Li–LiH system is strongly bound. The potential energy surface has a global minimumof 8743 cm−1 at a distance R = 4.40 bohr from the lithium atom to the center of mass ofLiH, at a Jacobi angle of θ = 46.5. The potential for the ground state of Li–LiH is also verystrongly anisotropic, which is clearly seen from the expansion of the potential in terms ofthe Legendre polynomials. Around the distance of the global minimum, the first anisotropiccontribution, V1(R), is almost two times deeper than isotropic part V0(R). This will havedecisive consequences for the calculations of the cross sections.

34


At the linear LiH–Li geometry, the ground state potential shows a close avoided crossingwith the first excited state potential. The energy difference between the ground and excitedstates at the avoided crossing is only 94 cm−1. However, it is only near the linear LiH–Ligeometry that the two states come very close together. If we distort the system from the lineargeometry, the excited state goes up in energy very rapidly, and around the global minimumenergy, θ ≈ 45, it is almost 6000 cm−1 above the ground state. Such an interaction betweenthe ground state and the first excited state potential is also confirmed by the shape of theradial nonadiabatic coupling matrix element τ12,R. The coupling τ12,R has the largest valueand approaches the Dirac delta form near the avoided crossing point, while it becomes abroad function with an approximately Lorenzian shape at bent geometries. This behaviour ofthe coupling τ12,R implies that the mixing angle γ, Eq. (3.4.5), shows an accumulation pointnear the avoided crossing between the ground and excited states. As expected, the diabaticpotentials H1, H2, and the coupling potential H12, Eq. (3.4.4), are smooth functions also inthe proximity of the avoided crossing. From the shape of the adiabatic and diabatic potentialswe can conclude that the dynamics will be strongly nonadiabatic in the region of the avoidedcrossing, and to take this rigorously into account would require a full two-state treatment ofthe dynamics. However, at low collision energy there are no open channels that involve thesecond surface, and any collisions that cross onto it must eventually return to the originalsurface. Its effect in collision calculations will therefore be at most to cause a phase change inthe outgoing wave function.

If we allow to vary the bond length of the LiH molecule we come across a seam of conicalintersection at the C2v geometries, with the two LiH bond lengths equal. This behaviour issimilar to the potential energy surfaces for homonuclear triatomic systems composed of thehydrogen [138] or lithium atoms [139]. At the linear LiH–Li geometry, the conical intersectionoccurs for an Li–H distance which is only slightly larger than the equilibrium distance of theLiH monomer. This explains why we encountered an avoided crossing and not a true crossingfor a fixed bond distance in LiH.

The Li–LiH system has several possible reaction channels: an exchange reaction to formproducts identical to the reactants, and two insertion reactions that produce Li2(a3Σ+

u ) andLi2(X1Σ+

g ) plus a ground-state hydrogen atom. The insertion reactions are highly endother-mic, with the energy difference between the entrance and exit channels of the order of 12000cm−1 and 22500 cm−1 for Li2(X1Σ+

g ) + H and Li2(a3Σ+u ) + H, respectively. Large endother-

micity implies that these reactive channels are completely irrelevant at low collision energies.

At the end of this part we comment on the performance of the conventional CCSD(T) ap-proach against the explicitly correlated CCSD(T)-F12 method. To this end, we have analyzedfour characteristic points of the Li–LiH potential: the global minimum, the saddle point, thelocal minimum, and one point very close to the avoided crossing. For all the analyzed pointsthe interaction energies from the CCSD(T)/cc-pVXZ (X = T, Q, 5) calculations smoothlyconverged towards the CCSD(T)-F12 result. Moreover, the extrapolation, Eq. (3.4.3), acceler-

35


ated the convergence which was particularly efficient when taking the coefficient α = 3. Forthe global minimum the extrapolation with α = 3 based on two largest orbital basis sets (X =Q, 5) led to an error as small as 0.01% when compared with the CCSD(T)-F12 result.

Smooth convergence of the extrapolation based on the conventional orbital basis sets to-wards the result from the CCSD(T)–F12 allowed us to think that the error related to the basisset in the CCSD(T)–F12 calculations is negligible and we may try to estimate the error barson the potential constructed according to Eq. (3.4.1). We assumed that the missing part of thepotential from Eq. (3.4.1) is due to the effect of the core-core and core-valence correlation inthe FCI calculations:

δV FCI = δV FCIall−all − δV FCI

v−v , (3.4.7)

where the subscript “all” refers to all electrons correlated. The δV FCI term was in turn es-timated to be no larger than an analogue contribution due to the triple excitations in theCCSD(T) calculations. Based on these assumptions we estimated that the potential from Eq.(3.4.1) has an uncertainty of 5% of the FCI correction δV FCI

v−v . Auxiliary calculations at the FCIlevel with all electron correlated in a small basis set (cc-pVDZ) for the characteristic pointsof the LiH–Li potential confirmed our uncertainty estimation.

3.4.3 Simulations of the sympathetic cooling

The best ground state adiabatic potential energy surface for Li– LiH has been used in asubsequent dynamical calculations. In this study we neglected the excited state surface, asour analysis showed that the interaction between the two surfaces is confined to a very smallregion of the configurational space. Moreover, the inclusion of the excited state surface wouldnot introduce any new open inelastic channels for low-energy collisions.

We have carried out field–free scattering calculations on 7Li + 7LiH collisions for col-lision energies up to 1 K, which is the relevant energy range for sympathetic cooling. Thecalculations were done by solving the close-coupling equations, as summarized in the previ-ous Chapter, however in a coupled rotational basis set with a defined total angular momentumJ :

φjlJ(R, r) =∑mj ,ml

〈jmj; lml|JM〉|jmj〉|lml〉 (3.4.8)

where |jmj〉 and |lml〉 are describe the rotation of the molecule and the rotation of the wholecollisional complex, respectively. The Li atom was treated as structureless, since in the colli-sions with a closed-shell 1Σ molecule only terms which could cause a change of the atomicspin state are the hyperfine couplings that are negligible [109]. Close-coupling calculationswere carried out for fixed values of the total angular momentum J , which in the absence of ex-ternal fields is a conserved quantity, and the resulting partial wave contributions were summedto form the state-to-state cross sections. It is worth noting that due to the strong anisotropy ofthe Li–LiH potential, we had to include rotational states of LiH up to and including j = 36 to

36


reach the convergence. Our main focus was on the calculations for the LiH molecules initiallyin the first excited rotational state j = 1. In the presence of the electric field this state has alow-field seeking component (j,mj) = (1, 0), and has already been utilized to decelerate theLiH molecules, so LiH in the j = 1 state can be trapped electrostatically.

Scattering calculations showed that above 10 mK the ratio of the elastic to inelastic crosssections stabilizes at a factor of 5 to 10, see Fig. 1 of Paper IV. Below roughly 1 mK bothelastic and inelastic cross sections follow the Wigner threshold laws. The elastic cross sectionbecomes constant and the inelastic cross sections are approximately proportional to the inverseof the collision energy, E−1. Figures showing the computed cross sections together with adetailed analysis of different contributions to the cross section can be found in Sec. 2 of PaperIV.

The computed cross sections were then used to simulate the sympathetic cooling of LiHmolecules co-trapped with ultracold Li atoms. Three types of traps were considered. Thefirst was a static electric trap for molecules in the weak-field seeking state (j,m) = (1, 0).Here, elastic collisions with the Li atoms cool the molecules, whereas any inelastic collisiontransfers a molecule to a lower-lying high-field seeking state causing the trap loss. The twoother traps under consideration were an AC trap and a microwave trap, both of which can trapground-state molecules so that inelastic losses are avoided. In our simulations a hard-spherecollision model has been used. Trajectories were calculated by solving classical equations ofmotion, with the force acting on the molecules, F = −∇W , where W is the Stark shift.From the calculated cross sections the probabilities of elastic and inelastic collisions weredetermined.

In all traps, collisions would take place in the presence of an electric field, while in thesimulations we used collision cross sections calculated in zero field. For the electrostatic trapthis approximation can be justified by the fact that in both cases, with zero and nonzero field,the main sources of inelasticity are transitions to the rotational ground state (j,mj) = (0, 0)

if the initial state of the molecule is (j,mj) = (1, 0). For the AC and microwave traps onlythe elastic cross section matters and this cross section was shown to be insensitive to thefield [122].

We should also add that full close-coupling calculations in the presence of the electric fieldturned out to be unfeasible due to a huge number of the coupled channels. In contrast to thefield-free case, the channels corresponding to different total angular momenta J are coupledin the electric field. The very strong anisotropy of the interaction potential makes it necessaryto include many channels to get a converged result.

In the first considered case of the static electric trap, the LiH molecules are confined in theweak-field seeking (j,m) = (1, 0) state and may undergo inelastic transitions to the ground(j,m) = (0, 0) state. The simulations showed that although after about 20 collisions themolecules with an initial temperature of 100 mK, have reached a temperature of 1 mK, only0.7% of them remain in the trap because of the inelastic collisions. The ratio of the elastic to

37


inelastic cross sections was too small for the sympathetic cooling in a static trap to be feasible.

A possible solution to eliminate inelastic collisions is to trap molecules in their groundstate by means of an AC electric field [102]. In the AC trap the molecules move on a saddle-shaped potential and the stability of motion relies on a specific correlation between the posi-tion and speed of the molecules. In such a confinement even the elastic collisions may ejectmolecules from the trap by pushing them into unstable trajectories. The simulations showedthat indeed the resulting trap losses were too large for sympathetic cooling to be feasible. Thereason for these large trap losses are nonlinear forces which are unavoidable in realistic ACtraps.

The last considered confinement was a microwave trap [140]. The ground state moleculesare attracted to the electric field maximum of the standing wave microwave field inside a res-onant cavity. In this trap each time a molecule collides with an ultracold atom its energy isreduced. Nevertheless, it is possible for a collision to transfer energy between axial and ra-dial motions, so that it has enough energy in one direction to leave the trap. The simulationsshowed that the coupling between the axial and radial motions in the microwave trap is rel-atively weak and does not provide additional trap losses, so this environment appears to besuitable for a successful sympathetic cooling. Moreover, results of the simulations showedthat a suitable choice of the atomic cloud size may improve the effectiveness of the coolingprocess. If the trap lifetime is long enough, it is best to use a large atom cloud to maximizethe number of cold molecules obtained. If the lifetime is short, it is better to use a small atomcloud to maximize the cooling rate and then to remove the molecules that remain hot forexample by lowering the trap depth.

Although in this work we have focussed oni the LiH molecules sympathetically cooledwith Li atoms, we expect our general conclusions to apply to a wide range of other systems,including other alkali-metal atom and polar molecule systems.

3.5 Interaction and collisions of the Ba+ ion and Rb atom

In this part we summarize the results of the electronic structure calculations together withpilot dynamical studies on the interactions and ultracold collisions between the Ba+ ion andthe Rb atom. Dynamical studies aimed to estimate prospects for the sympathetic cooling ofthe Ba+ ions by collisions with ultracold rubidium atoms. The Ba+ + Rb system is of pri-mary interest for experiments combining dual hybrid traps for the simultaneous cooling andtrapping of ions and atoms, and for subsequent studies of ion-neutral interactions at ultralowenergies [83, 84]. Theoretical calculations can indicate what kind of physical and chemicalphenomena (chemical reactions, charge transfer processes, cooling or heating) are likely tooccur in such experiments.

In the first step we performed extensive electronic structure calculations for the molecular(BaRb)+ ion. Specifically, we calculated:

38

3.5. INTERACTION AND COLLISIONS OF BA+ AND RB

1. the electronic ground X1Σ state potential dissociating into Rb+ ion and Ba(1S) atom,

2. the manifold of the excited electronic state potentials with the dissociation limits cor-responding to the Ba+(2S) ion and Rb(2S) atom (1Σ, 3Σ), into Ba(3D) and Rb+(1S)(3Σ,3 Π,3 ∆), and into Ba(3D) and Rb+(1S) (1Σ,1 Π,1 ∆),

3. nonadiabatic coupling matrix elements between the low-lying electronic excited states.

4. electric transition dipole moments between the ground X1Σ state and the three excitedelectric dipole-allowed states, two 1Σ and one 1Π,

5. spin-orbit coupling matrix elements for the lowest dimer states that couple to the rela-tivistic Ω = 0+ and Ω = 1 electronic states of (BaRb)+, with the spin-orbit operatorHSO defined within the Breit-Pauli approximation. These couplings were then used togenerate relativistic Ω = 0+ and Ω = 1 states through diagonalization of appropriatenonrelativistic + spin-orbit Hamiltonian,

6. leading long-range coefficients of the nonrelativistic and relativistic potentials up to andincluding the C6 term. Depending on the dissociation limit, the lowest Cn coefficientin the ion-atom system is due to the induction [C4, for states dissociating into Ba(1S)+ Rb+(1S) and Ba(2S) + Rb+(2S)] or due to the electrostatic interaction [C3, for statesdissociating into Ba(1D) + Rb+(1S) and Ba(3D) + Rb+(1S)]. Appropriate equations canbe found in Sec. II A of Paper V.

For the ground state potential we used the coupled cluster method restricted to single, double,and noniterative triple excitations, CCSD(T). For the first triplet (high-spin) states of the Σ, Π,and ∆ symmetries we employed the restricted open-shell coupled cluster method with single,double, and noniterative triple excitations, RCCSD(T). Calculations on all other excited statesemployed the linear response theory within the coupled-cluster singles, doubles, and lineartriple excitations (LRCC3) framework [141,142]. The electric transition dipole moments werecomputed as the first residue of the linear response function with two electric dipole opera-tors [141]. Finally, nonadiabatic and spin-orbit coupling matrix elements were obtained withthe MRCI method. In all electronic structure calculations pseudopotentials from the Stuttgartlibrary were employed to mimic the scalar relativistic effects for both species and the (BaRb)+

ion was treated effectively as a 18-electron system.

In the second part of the study the electronic structure input has been employed in the sin-gle channel scattering calculations of the cross sections for collisions between the Ba+ ion andRb atom. In the collisions of the Ba+(2S) ion with Rb(2S) atom we have basically three typesof processes: elastic scattering in the singlet and triplet potentials, spin-flip (spin-exchange)process, and the inelastic radiative charge transfer from the singlet and triplet manifolds ofBa+ + Rb to the ground state of Ba + Rb+. Calculations of the elastic and superelastic (spin-

39


flip) cross sections in the single channel model are straightforward and reduce to determina-tion of phase shifts δJ(E) from the large-R behaviour of the wave function ΨEJ(R):

ΨEJ(R) ∼( 2µπ~2k

)1/2

sin(kR− Jπ

2+ δJ(E)

), (3.5.1)

which is the solution of the radial Schrödinger equation for the relative motion of the Ba+ ionand Rb atom at a collision energy E in a given single potential V (R):

(d2

dR2− 2µ

~2V (R)− J(J + 1)

R2+

2µE~2

)ΨEJ(R) = 0. (3.5.2)

Having the phase shifts δJ for each partial wave J as a function of the energy E, we can com-pute the cross sections for the elastic and spin-flip collisions from the standard expressions:

σsel(E) =

4πk2

∞∑J=0

(2J + 1) sin2 δsJ(E), σt

el(E) =4πk2

∞∑J=0

(2J + 1) sin2 δtJ(E), (3.5.3)

σsf(E) =4πk2

∞∑J=0

(2J + 1) sin2(δsJ(E)− δt

J(E)), (3.5.4)

where the superscripts “s” and “t” on σel and δJ pertain to the singlet and triplet potentials,respectively. An exact description of the spin-flip process would require at least two coupledchannels, so the expression (3.5.4) is only approximate. However, it was shown to work rela-tively well, even at low energies [143].

More elaborate is the theoretical description of the charge transfer process between theatom and the ion. To the first-order of perturbation theory the radiative charge transfer can bedescribed by the following Fermi golden type expression [144]:

σct(E) =4π2~k2

A(E), (3.5.5)

where A(E) is the Einstein coefficient and k is the wave vector associated with collision en-ergy E. Following the approach proposed in Ref. [144] we computed the Einstein coefficientsfrom the approximate expression as an average over the initial scattering wave function ΨEJ :

A(E) =∞∑J=0

(2J + 1)〈ΨEJ |A(R)|ΨEJ〉, (3.5.6)

whereA(R) =

α3

3~e6[δV (R)]3µ2(R), (3.5.7)

and δV (R) is the difference between the excited and ground state potentials while µ(R) is thetransition dipole moment between the two states.

Figures showing the calculated interaction potentials, various couplings between the elec-

40

3.5. INTERACTION AND COLLISIONS OF BA+ AND RB

tronic states, and scattering cross sections can be found in Sec. III of Paper V. Here we onlynote that our results for the spectroscopic characteristics of the electronic states of (BaRb)+

agreed well with the previously published fully relativistic Dirac-Coulomb calculations [145].The agreement is particularly good for states with the dissociations Ba(1S0) + Rb+(1S0) andBa+(2S1/2) + Rb(2S1/2), where the differences in the well depths are of the order of a fewpercent at most. Similarly, the positions of the minima in the two calculations agree within0.1 bohr or better, see Table II of Paper V. This comparison confirms good quality of the basissets and pseudopotentials used in our calculations. It also confirms the applicability of thepseudopotential approach to take into account the relativistic effects (mainly the spin-orbitterm) in the electronic structure calculations for heavy atoms.

Results of dynamical calculations showed that the inelastic cross section corresponding tothe charge transfer from the Rb atom to the Ba+ ion is much smaller than the elastic one over awide range of energies up to 1 mK. In particular, at microKelvin and nanoKelvin temperaturesthe inelastic charge transfer cross section is four to five orders of magnitude smaller than theelastic cross section. This suggests that sympathetic cooling of the Ba+ ion by collisionswith ultracold Rb atoms should be possible. However, our single-channel calculations did notinclude the hyperfine spin states of the atom and ion, which might be crucial in the ultracoldregime. Thus, our conclusions concerning the prospects for sympathetic cooling is an initialfinding which should be confirmed from the multichannel calculations with the hyperfinestructure effects included.

41

42

CHAPTER 4

PHOTOASSOCIATION SPECTROSCOPY:THEORETICAL APPROACH

4.1 Introductory remarks

To date, the most successful method for the production of ultracold molecules is by a di-rect formation from ultracold atoms. Photoassociation [25], in which the bonding of atoms isinduced by photon absorption, is the oldest technique of this kind. First experimental observa-tion of molecules created from ultracold atomic gases was demonstrated already in the early1990s, for the Na and Rb dimers [146, 147].

In the photoassociation process two colliding atoms are transferred by resonant light intoa bound rovibrational level of the excited electronic state of the molecule:

A + B + hν → AB?. (4.1.1)

so it is a free-to-bound transition with a resonant character, see Fig. 4.1 for a scheme of theprocess. Molecules created in this way are usually in highly excited vibrational states. Actu-ally, ultracold photoassociation is most efficient to the weakly bound levels lying just belowthe atomic threshold. This is because the pair density of atoms decreases with the decreasinginteratomic distance R and transitions are more likely to occur at large R, preferring diffuseweakly bound levels. In contrast to high vibrational excitation, the photoassociated moleculesare typically in the lowest possible rotational state. The reason for this is that ultracold atomsundergo mostly s−wave collisions (with zero angular momentum of the relative motion), andselection rules for the dipole transitions allow for changes of the total angular momentum onlyby ±1.

Vibrationally excited molecules AB? have very short lifetimes (of the order of micro ornanoseconds) and decay either to free atoms or to a bound level of the ground electronic state.

43

CHAPTER 4. PHOTOASSOCIATION SPECTROSCOPY: THEORETICAL APPROACH

P

ote

ntia

l e

ne

rgy

A + B*v’=−5

v’’=−2 A + B

(1) PA(2) (2)′

Internuclear distance R

Figure 4.1: Schematic plot of the photoassociation process (PA) of two atoms A and B. After(1) the photoassociation transition to the weakly bound level (here v′ = −5), the moleculeAB? can either (2) deexcite to a bound-state molecule (here v′′ = −2) of the ground electronicpotential or (2)′ redissociate to two free atoms.

Molecules can basically decay to all possible ground electronic state levels. Distribution of thepopulated levels after the spontaneous decay can be predicted from the Einstein coefficientsAv′,v′′ which are the measure of the probability for spontaneous emission from the upper statev′ to the lower state v′′. Thus, by calculating the ratio Av′,v′′/

∑v′′ Av′,v′′ we can predict the

relative population of the bound levels v′′ in the ground electronic state after the spontaneousdecay of the photoassociated molecules.

In practice, the Einstein coefficients Av′,v′′ can well be approximated by the square ofthe Franck-Condon factor 〈v′|v′′〉 describing the overlap between the vibrational wave func-tions of the v′ and v′′ levels. As we already noted, the v′ level is most often a weakly boundlong-range state, which implies that after the spontaneous emission only the weakly boundlevels v′′ are most likely to be populated. Simply because then only the value of the overlapintegral 〈v′|v′′〉 can be substantially larger than zero. However, if we think of exploiting thephotoassociation to create ultracold molecules we would like to produce the molecules in therovibrational ground state v′′ = 0 as such molecules are stable against inelastic collisions,trapped for longer times, and useful for further purposes.

The main difficulty with the photoassociation method is how to effectively transfer mole-cules from the initially created weakly bound long-range state into the desired (short-range)rovibrational ground state. In general, spontaneous emission is not an ideal solution, althoughin some fortunate cases it may lead to a substantial population of the rovibrational groundstate, as it was shown, for instance, for LiCs [27]. A few optical techniques have already beendeveloped to transfer the photoassociated molecules into the rovibrational ground state. The

44

4.1. INTRODUCTORY REMARKS

most popular are the stimulated emission pumping [148], the stimulated Raman adiabatic pas-sage (STIRAP) [149], and optical pumping with broadband laser pulse [150]. Their efficientapplication is not an easy task, since it usually requires to identify some intermediate levelshaving good overlap with both the initial and the target levels, which are most often mutuallyexclusive conditions.

Alkali-metal homonuclear dimers (Rb2, Na2, K2) were among the first molecules formedfrom ultracold gases by photoassociation spectroscopy [146, 147, 151]. In these early experi-ments, pairs of colliding atoms were excited to molecular levels just below the atomic thresh-old 2S + 2P. Molecular electronic states dissociating into the 2S + 2P atoms have an R−3

asymptotic behaviour due to the resonant interaction. The resonant C3 long-range coefficientis proportional to the square of the dipole transition moment between the 2S and 2P states,C3 ∼ |〈2S|d|2P〉|2. Such a form of the long-range potential, R−3, guarantees the existence ofvery weakly bound molecular levels, which are most favorable for efficient ultracold photoas-sociation. On the other hand, these levels are rather unfavorable for the transfer to the groundelectronic state, since the ground electronic state potentials for these dimers behave like R−6.This suggests small values of the Frank-Condon factors for subsequent bound-to-bound tran-sitions.

The second group of molecules that were produced from ultracold atomic gases by pho-toassociation spectroscopy were alkali-metal heteronuclear dimers, such as LiCs, KRb, RbCs,or NaCs [88]. For the heteronuclear dimers there is no resonant interaction, the excited elec-tronic potentials have an asymptotic behaviour R−6, the same as for the ground electronicstate. Such a similarity of the long-range parts of the ground and excited state potentials sug-gests that is should be much easier to transfer the photoassociated molecules to the deeplybound levels of the ground electronic state. Indeed, even without any additional optical tech-niques it was possible to form ultracold LiCs molecules in the lowest rovibrational level afterthe spontaneous decay.

Another group of atoms which have been used in the photoassociation experiments arethe alkaline-earth atoms, Ca, Sr, and also Yb2 [90, 152, 153] which has a similar electronicstructure. For strontium, which is the main object of our study, two different strategies havebeen attempted to produce molecules. In first approach, the photoassociation laser was red-detuned from the dipole allowed 1S→1P atomic transition. After photoassociation near thedipole-allowed atomic transition, the majority of the excited state molecules redissociates,and only the last two bound levels of the electronic ground state can be populated [154]. Thisis due to the long-rangeR−3 nature of the electronically excited state dissociating into the 1S +1P atoms that does not provide any mechanisms for an efficient stabilization to bound groundstate levels [155]. In the second approach, the photoassociation laser was red-detuned fromthe 1S→ 3P1 atomic intercombination line [53, 90]. The atomic intercombination 1S→ 3P1

transition is weakly allowed due to the spin-orbit mixing between the 3P1 and 1P atomic states.It has a natural width of only 7.4 kHz which enables to resolve the positions of the least-bound

45


levels near the 1S + 3P1 threshold with high accuracy since the observed molecular transitionsare not covered up by a broad atomic line.

For the photoassociation near the intercombination line the excited electronic potential inthe asymptotic region behaves predominantly as R−6 with a small δC res

3 R−3 contribution:

V (R)0+u /1u ∼ C6

R6+δC res

3

R3, (4.1.2)

where δC res3 results from the spin-orbit coupling between the 3P1 and 1P atomic states:

δC res3 ∼ |〈1S|d|1P〉|2 |〈

1P|HSO|3P〉|2

E1P − E3P(4.1.3)

and E1P and E3P are the atomic energies calculated without the spin-orbit perturbation. ForSr2 the δC res

3 is 0.007537 a.u. [90]. Such form of potential in the long range, Eq. (4.1.2),represents a unique interplay between dispersion, C6/R

6, and dipole-dipole forces, δC res3 .

Large bound-to-bound transition matrix elements with the electronic ground state that behavesasymptotically as R−6 are then expected [90]. However, due to the resonant δC res

3 term, com-paratively large free-to-bound transition matrix elements will appear if the addressed excitedbound level has a long-range character. Thus, the photoassociation near the intercombinationline could be employed for an effective production of the Sr2 dimers in the ground state.

Theoretical modelling of the photoassociation process and production of molecules re-quires the knowledge of the excited state potentials and possible couplings between them.For the strontium dimer there were already a few papers reporting results of ab initio elec-tronic structure calculations [92, 93, 156]. However, the quality of these ab initio calculationswas largely questioned with the arrival of the new data from the conventional high-resolutionspectroscopy [91]. Thus, investigation of the photoassociation process and prospects for pro-duction of ultracold molecules had to be proceeded by reexamining the electronic structure ofSr2.

4.2 Electronic structure and rovibrational dynamics of Sr2

In this part we summarize our results concerning the interactions and rovibrational dynam-ics of the strontium dimer in the A1Σ+

u , c3Πu, and a3Σ+u manifold of electronic states. The A

state dissociates into the 1S + 1D atoms, while the c and a states dissociate into the 1S + 3Patoms. These electronic excited states are relevant for the modelling of the photoassociationspectra near the 1S− 3P1 intercombination line. The available experimental data are mostlylimited only to the bottom of the A state [91].

Regarding the electronic structure, our study included computations of:

1. potential energy curves for the electronic states in the A1Σ+u , c3Πu, and a3Σ+

u manifoldof Sr2 by using the LRCC3 method,

46

4.2. ELECTRONIC STRUCTURE AND ROVIBRATIONAL DYNAMICS OF SR2

2. electric transition dipole moments from the ground X1Σ+g to the singlet excited states

dissociating into 1S + 1P and 1S + 1D atoms with the LRCCSD method,

3. the spin-orbit and nonadiabatic coupling matrix elements between the A1Σ+u , c3Πu,

a3Σ+u and B1Σ+

u , B’1Σ+u states employing the MRCISD method.

4. the spin-orbit coupling matrix elements were then used to generate relativistic Ω = 0+u

and Ω = 1u states through diagonalization of appropriate nonrelativistic + spin-orbitHamiltonian matrices.

In all these calculations the core electrons were described by the ECP28MDF pseudopotential[157] from the Stuttgart library in order to mimic the relativistic effects, so the Sr2 moleculewas effectively treated as a system of 20 electrons.

Potential energy curves were fitted to physically sound analytic expressions with fixedlong-range behaviour through the C6, C8 and C10 dispersion coefficients. Analytical repre-sentations were also obtained for the spin-orbit matrix elements. Parameters of these fits andplots showing the calculated curves can be found in Secs. II and III of Paper VI. Comparisonof the spectroscopic characteristics of the computed potential energy curves for the nonrel-ativistic A1Σ+

u , c3Πu, and a3Σ+u states and of the relativistic states of 0+

u and 1u symmetryreveals significant deviations from most of the previous ab initio results [92, 93]. In the caseof the binding energy of the spectroscopically active A1Σ+

u state, the difference between ourresult and the recently published ab initio data is as large as 4000 cm−1. This means also thatthose previous ab initio calculations did not predict any interaction between the A1Σ+

u andc3Πu states. We highlight that our result agrees with the newest experimental finding [91].

The ab initio results were then used in the converged coupled channel calculations of therovibrational spectra in the A+c+a manifold of electronic states. In particular, we calculated:

1. rovibrational energy levels for odd values of the rotational quantum number number Jand spectroscopic parity e, up to and including J = 219, using the variable step-sizeFourier grid representation [158, 159],

2. line strengths for selected transitions from the rovibrational levels of the ground elec-tronic state to the rovibrational levels of the A1Σ+

u , c3Πu, and a3Σ+u manifold,

3. positions and lifetimes of the Feshbach resonances lying above the 1S0 + 3P1 dissoci-ation limit. This was accomplished by diagonalizing the Hamiltonian for the nuclearmotion with an imaginary absorbing potential VCAP [160] added to the diagonal termand identifying quasi-bound levels with complex eigenvalues stable to variation of theVCAP parameters.

After including the spin-orbit interaction, the A+c+a electronic states couple to the relativisticstates of 0+

u and 1u symmetry. Neglecting the small Coriolis coupling between the 0+u and 1u

47


states partitions the Hamiltonian matrix into two separate blocks, and each rovibrational levelcan approximately be assigned to the 0+

u or 1u symmetry. The spin-orbit interaction betweenthe 0+

u components of the A1Σ+u and c3Πu electronic states makes the spectrum strongly per-

turbed for some rovibrational levels located above the crossing between the A and c states.Such levels show a heavily mixed singlet/triplet character and cannot be assigned either tothe A1Σ+

u or c3Πu adiabatic states. The presence of heavily mixed levels is also demonstratedin (i) irregular behaviour of the rotational constant Bv as a function of binding energy, (ii)

nonlinear shape of the rotational spacing as a function of number J(J + 1) and (iii) break-down of the smooth rotational progression for transitions from the rovibrational levels of theground electronic potential X1Σ+

g to the levels in the surrounding of the crossing between theA and c states. Suitable plots illustrating all these irregularities can be found in Sec. III B ofPaper VI. Levels with mixed singlet/triplet character may have very favourable properties forphotoassociative production of ground state Sr2 molecules and this point will be addressed inthe next section of this review.

Comparison of the computed rovibrational energy levels for the 0+u state with the newest

data derived from the high-resolution Fourier transform spectroscopy [91] showed a root-mean-square-deviation (RMSD) between our theory and experiment of 10.5 cm−1. It is a goodaccuracy for such a heavy system like Sr2, in particular taking into account the large well depthof the A1Σ+

u state, namely 8044 cm−1. However, in order to enhance the predictive powerof our results for the ongoing and future spectroscopy experiments on ultracold strontiummolecules, we decided to slightly adjust the ab initio results to the existing experimentaldata to lower the RMSD. By changing two fit parameters in the analytical expression forthe A1Σ+

u potential we could reduce the RMSD for the J ′ = 1 rovibrational levels to 0.64cm−1. With the new values of the two adjusted parameters for the A1Σ+

u potential, the root-mean-square-deviation of our results for J ′ ¬ 50, as compared to the raw data of Tiemannand collaborators, was 4.5 cm−1. We have also adjusted the spin-orbit coupling between thec3Πu and B1Σ+

u states in order to reproduce the most weakly bound levels below the 1S + 3P1

threshold, resolved from the photoassociation spectra [90]. Again, this was done to improvethe quality of our electronic structure input to model production of ultracold molecules bymeans of the photoassociation.

The last point of dynamical calculations was related to analysis of the Feshbach reso-nances lying above the 1S + 3P1 dissociation limit. We found a few such quasi-bound levels,which roughly correspond to bound levels of the A1Σ+

u state located above the dissociationlimit of the c3Πu electronic state, i.e. 1S + 3P. Positions, widths and lifetimes of these statescan be found in Table VII of Paper VI. Spin-orbit coupling between the c and A states isresponsible for possible predissociation of these resonances. The resonances with the largestlifetimes (of the order of a few nanosecond) are those which have predominantly singlet char-acter and thus they decay more slowly to the triplet continuum. Relatively long lifetime ofthese resonance should make possible to observe them in high-resolution Fourier transform

48

4.3. PHOTOASSOCIATIVE FORMATION OF DEEPLY BOUND ULTRACOLD SR2

MOLECULES

spectroscopic experiments.

4.3 Photoassociative formation of deeply bound ultracoldSr2 molecules

Here we outline our predictions concerning the photoassociative formation of ultracoldSr2 molecules in arbitrary vibrational levels of the electronic ground state, based on the state-of-the-art ab initio calculations reviewed in the previous section.

Let us first mention that optically trapped ultracold strontium molecules will have someunique properties that facilitate engineering and interpretation of high precision spectroscopicmeasurements. In particular, the most abundant isotope 88Sr does not have nuclear spin. Inthe ground state, two closed-shell Sr atoms interact only on one potential energy curve. Thus,the spectra of Sr2 are relatively easy to measure and interpret with high resolution. What ismore, techniques to optically cool Sr atoms down to temperature ≈ 1µK and to trap themin an optical lattice have been well developed in the last 10 years. They are based on thenarrow intercombination line 1S → 3P1. It turns out that the width of this transition is themost favourable for an efficient cooling and trapping among all alkali-earth atoms. One ofthe considered proposals for the precision measurement based on high precision rovibrationalspectroscopy of Sr2 is the laboratory detection of possible time variation of the proton-to-electron mass ratio. The idea is to prepare tightly confined Sr2 molecules in their electronicground state by photoassociation in an optical lattice and carry out high-precision Ramanspectroscopy on the ground state vibrational level spacings which are most sensitive to thevariation of the proton-to-electron mass ratio [161, 162].

In our work we considered the photoassociation process of two ultracold strontium atomsin the manifold of the coupled c3Πu(1S + 3P) + A1Σ+

u (1S + 1D) + B1Σ+u (1S + 1P) states,

using a continuous-wave laser that is red-detuned with respect to the 1S → 3P1 intercom-bination line of Sr. The B1Σ+

u (1S + 1P) and the ground X1Σ+g electronic states needed in

the calculations were taken from the fits to high-resolution experimental data [91,163]. In thenonrelativistic approach the A1Σ+

u and c3Πu electronic states cross and this crossing is locatedapproximately 1000 cm−1 below the 1S +3 P1 atomic threshold. After including the spin-orbitterm into the Hamiltonian these two electronic states have components of the same symmetry,0+u , which strongly interact with each other, and the crossing of the potential energy curves

becomes an avoided crossing. Such a shape of the electronic potentials suggests the existenceof the rovibrational levels that are heavily perturbed by the spin-orbit coupling. Indeed, weidentified a series of such levels for J ′ = 1, the two most weakly bound have binding energies−12.9 cm−1 (v′ = −15) and −75.8 cm−1 (v′ = −26)1,2. These mixed singlet/triplet levels

1Binding energies are given with respect to the 1S + 3P1 atomic threshold.2Negative value of v′ means that the vibrational levels are counted starting from the dissociation limit, thus

v′ = −1 denotes the least-bound level.

49


have optimal properties for production of deeply bound Sr2 molecules in the ground electronicstate. Let us explain the reasons for this.

The wave function for the mixed singlet/triplet levels has a strongly nonadiabatic characterand it is a mixture of contributions from all electronic states: c3Πu(1S + 3P) + A1Σ+

u (1S + 1D)+B1Σ+

u (1S + 1P). This means that for such levels the rovibrational wave function will havemultiple maxima (corresponding to classical turning points). Some of these maxima can belocated in the long range (needed for large photoassociation rate) and some in the short range(needed for large overlap with deeply bound ground-state levels). Both singlet states, A1Σ+

u

and B1Σ+u , are connected by a dipole-allowed transition to the ground electronic state, X1Σ+

g .Thus, an effective transition dipole moment is created which for moderate and large inter-atomic separations is well approximated by:

dSO =〈X1Σ+

g |dz|B1Σ+u 〉〈B1Σ+

u |HSO|c3Πu〉Ec3Πu − EB1Σ+u

+〈X1Σ+

g |dz|A1Σ+u 〉〈A1Σ+

u |HSO|c3Πu〉Ec3Πu − EA1Σ+u

,(4.3.1)

where HSO is the spin-orbit Hamiltonian in the Breit-Pauli approximation [164]. The long-range part of dSO, dominated by the first term in the above expression, is due to the cou-pling with the B1Σ+

u state, ideally suited for photoassociation. The short-range part is dueto the coupling with the A1Σ+

u state, paving the way toward an efficient stabilization of thephotoassociated molecules to the electronic ground state. The proposed scheme for optimalphotoassociative production of ground state Sr2 molecules which utilizes the heavily mixedsinglet/triplet level of the 0+

u excited states is depicted in Fig. 4.2.

For the first step, photoassociation of two ultracold ground state Sr atoms into the proposedlevel v′ = −15, the key is the spin-orbit coupling between the c3Πu(1S + 3P) and B1Σ+

u (1S + 1P)

electronic states. It has a long-range nature and can well be approximated by its asymptoticatomic value corresponding to the coupling between the nonrelativistic 1P and 3P states. Thiscoupling is responsible for the enhancement of the free-to-bound transitions at large inter-atomic distances where photoassociation at ultracold temperatures is most likely to occur.

Quantitatively, the rate of the photoassociation step is characterized by the absorptioncoefficient, K(ω1, T ). The absorption coefficient K(ω1, T ) from the continuum state into thebound level |v′, J〉 at a laser frequency ω1 and temperature T is given by [165, 166]:

K(ω1, T ) =2πρ2

~QT

∑J ′′gJ ′′(2J ′′ + 1)

∫ ∞0

e−E/kBT |Sv′J ′(E, J ′′, ω1)|2dE, (4.3.2)

where ρ denotes the gas number density, J ′′ the rotational quantum number of the initialscattering state, gJ ′′ the spin statistical weight depending on the nuclear spin, equal to one for88Sr, and QT = (µkBT/2π~2)3/2. Sv′J ′(E, J ′′, ω1) is the S-matrix element for the transitionfrom a continuum state with the scattering energy E and the rotational quantum number J ′′

50


MOLECULES

0

5000

10000

15000

20000

25000

6 8 10 12 14 16 18 20 22 24

Ene

rgy (

cm

−1)

R (bohr)

1S+

1S

3P+

1S

1D+

1S

1P+

1S

x100

ω1ω2

∆ω1

X1Σg

+

c3Πu

A1Σu

+

B1Σu

+

0

5000

10000

15000

20000

25000

6 8 10 12 14 16 18 20 22 24

Ene

rgy (

cm

−1)

R (bohr)

1S+

1S

3P+

1S

1D+

1S

1P+

1S

x100

ω1ω2

∆ω1

X1Σg

+

c3Πu

A1Σu

+

B1Σu

+

Figure 4.2: Proposed scheme for the production of ultracold Sr2 molecules by photoassoci-ation near the intercombination line. The green wavefunction represents a scattering state oftwo Sr atoms and the red, blue and brown wavefunctions the diabatic components of the ex-cited state vibrational level with binding energyEv′=−15 = −12.9 cm−1. Spin-orbit interactionfacilitates a transition from this level to X1Σ+

g v′′ = 6 (with the corresponding wavefunction

depicted in purple) via spontaneous or stimulated emission.

into the bound level |v′, J ′〉. The square of the S matrix element for the transition from acontinuum state |E, J ′′〉 into the bound level |v′, J ′〉 in Eq. (4.3.2) can be approximated by theresonant scattering expression for an isolated resonance [165]:

|Sv′J ′(E, J ′′, ω1)|2 =γsv′J ′(E, J

′′)γdv′J ′(E −∆v′J ′(ω1))2 + 1

4 [γsv′(E, J ′′) + γdv′J ′ ]2, (4.3.3)

where γsv′J ′(E, J′′) is the stimulated emission rate, γdv′(E, J

′′) is the rate of the spontaneousdecay, both in units of ~, ∆v′J ′(ω1) is the detuning relative to the position of the bound rovi-brational level |v′, J ′〉, i.e., ∆v′J ′ = Ev′J ′ − ~ω1, where Ev′J ′ is the binding energy of thelevel |v′, J ′〉. The spontaneous emission rates γdv′J ′ are obtained from the Einstein coefficientsAv′J ′,v′′J ′′:

γdv′J ′ =∑v′′J ′′

Av′J ′,v′′J ′′ , (4.3.4)

where the Einstein coefficient Av′J ′,v′′J ′′ is given by:

Av′J ′,v′′J ′′ =4α3

3e4~2HJ ′(Ev′J ′ − Ev′′J ′′)3

∣∣∣∣∑n′〈χX

v′′J ′′ |d(n′ ← X)|χn′v′J ′〉∣∣∣∣2. (4.3.5)

Here, HJ ′ is the so-called Hönl-London factor equal to (J ′ + 1)/(2J ′ + 1) for J ′ = J ′′ − 1

and J ′/(2J ′ + 1) for J ′ = J ′′ + 1, d(n′ ← X) denotes electric transition dipole moment

51


from the ground state X to the excited electronic state n, χXv′′J ′′(R) is vibrational function

of the |v′′, J ′′〉 level in the ground electronic potential X1Σ+g , and χn′v′J ′(R) is the analogous

vibrational component of the |v′, J ′〉 level in the manifold of c3Πu + A1Σ+u + B1Σ+

u states, thusn′ = c,A,B. Finally, the stimulated emission rate γsv′J ′(E, J

′′) at low laser intensity I isgiven by the Fermi’s golden rule expression:

γsv′J ′(E, J′′) = 4π2 I

c(2J ′ + 1)HJ ′

∣∣∣∣∣∣∑n′〈χX

EJ ′′ |d(n′ ← X)|χn′v′J ′〉

∣∣∣∣∣∣2

, (4.3.6)

where χXEJ ′′(R) are the energy normalized continuum wave functions of the ground electronic

state with scattering energy E.

The calculated photoassociation rates K(ω1, T ) for two temperatures, T = 2 and T =

20 µK, are plotted in Fig. 5 of Paper VII. The levels with strong spin-orbit mixing, such asv′ = −15, have K(ω1, T ) coefficients that are very similar to that of very weakly boundv′ = −6 (already observed in the experiment), which shows that photoassociation into theselevels should also be experimentally feasible.

For the second step, the transfer of the excited Sr2 molecules into the deeply bound lev-els of the ground electronic state, the spin-orbit coupling between the c3Πu(1S + 3P) andA1Σ+

u (1S + 1D) states is essential. It has a short-range nature, and it is responsible for gov-erning the spontaneous (or stimulated) emission to the ground electronic state. Quantitatively,the distribution of the populated levels after the spontaneous decay is obtained from the thebranching ratio:

P (v′′ ← v′J ′) =∑J ′′ Av′J ′,v′′J ′′∑v′′J ′′ Av′J ′,v′′J ′′

, (4.3.7)

which describes the probability for the spontaneous decay from the level |v′, J ′〉 of the elec-tronically excited state to rovibrational levels |v′′, J ′′ = J ′± 1〉 of the ground electronic state.The calculated branching ratios for the the spontaneous decay from the v′ = −15 and v′ = −6

levels are shown in Fig. 8 of Paper VII. It turns out that the large short-range amplitude of theA1Σ+

u component in the rovibrational wave function for the v′ = −15 level makes possible topopulate, after spontaneous or stimulated emission, deeply bound levels such as v′′ = 6. Thisis in a sharp contrast with weakly bound non-perturbed levels (for instance v′ = −6), whichhave a predominant c3Πu(1S + 3P) character, where the spontaneous decay leads to popula-tions of the least bound levels of the electronic ground state potential. Similar enhancementof bound-to-bound transition probabilities to form deeply bound molecules in their electronicground was already shown for a few other molecules [167–169].

In conclusion, excited state levels with strong singlet/triplet mixing below the intercom-bination line 1S → 3P1 should exhibit large bound-to-bound transition moments to deeplybound ground state levels and simultaneously a decent photoassociation probability. Thesefeatures make them optimal for the photoassociative production of ultracold deeply bound Sr2

molecules. The goal is now to identify in the experiment the strongly perturbed levels of the

52


MOLECULES

c3Πu, A1Σ+u , B1Σ+

u manifold. Since the present state-of-the-art ab inito methods cannot pre-dict the positions of the rovibrational levels with precision better than a few wavenumbers forsuch a heavy system like Sr2, the exact location of such levels requires a spectroscopic search.

53

54

CHAPTER 5

CONCLUSIONS

The presented thesis concerned theoretical studies on the interactions and dynamics inultracold collisions for a few selected molecular systems: LiH + Li, OH + N, (BaRb)+ andSr2. These systems have been important for the ongoing or planned experiments aiming atproduction of ultracold molecules by sympathetic cooling or photoassociation techniques.

The atomic/molecular interactions were studied with state-of-the-art electronic structuremethods, mainly the coupled cluster method for computations of the potential energy sur-faces, and multireference configuration interaction method for computations of the spin-orbitand nonadiabatic couplings between the surfaces. Particular attention was always paid to thecorrect treatment of the interaction in the long range, as it is a prerequisite for a proper theo-retical description of ultracold collisions and dynamics.

Quantum-dynamical calculations were performed by solving the close-coupling equa-tions. The employed rotational basis sets were always constructed in a way to maximallyexploit the symmetries existing in a given system and thus reduce the number of the coupledchannels.

The main achievements of the thesis may be summarized as follows:

1. Development of the theory of long-range interactions between a ground-state atom inan S state and a linear molecule in a degenerate state with nonzero electronic angularmomentum. Final formulas for the long-range coefficients were expressed in terms offirst and second order atomic and molecular properties.

2. Formulation of the effective Hamiltonian for collisions between an open-shell S-stateatom and a 2Π-state molecule in the presence of magnetic field. Mechanisms leading toinelasticity in such collisions were identified and analysed on the example of collisionsbetween OH radical and nitrogen atom.

3. Performing an accurate and comprehensive study of the interaction between the LiH

55

CHAPTER 5. CONCLUSIONS

molecule and a Li atom. The calculated cross sections for collisions between Li andLiH were employed for simulations of the sympathetic cooling process. The simulationsshowed that such cooling is feasible in a microwave trap but in the electrostatic oralternating gradient field traps it is likely to fail due to substantial molecular lossesfrom the trap.

4. Proving good prospects for the sympathetic cooling of the Ba+ ion by immersing inultracold Rb atomic gas. It follows from the calculated cross sections that the possiblecharge exchange process in this system should not hamper the thermalization.

5. Calculations of the excited states potential energy curves for Sr2 and rovibrational lev-els, which agree well with the newest experimental findings and disagree with mostpreviously published ab initio data.

6. Formulation of possible optical pathways for efficient photoassociative production ofdeeply bound ultracold Sr2 molecules, based on strong spin-orbit interaction betweenthe c3Πu(1S + 3P), A1Σ+

u (1S + 1D) and B1Σ+u (1S + 1P) electronic states.

Seven papers, published in international scientific journals, constitute the core of the thesisand contain a detailed account of the obtained results.

56

Bibliography

[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, Science,1995, 269, 198–201.

[2] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. VanDruten, D. S. Durfee, D. M. Kurn,W. Ketterle, Phys. Rev. Lett., 1995, 75, 3969–3973.

[3] J. P. Gordon, A. Ashkin, Phys. Rev. A, 1980, 21, 1606–1617.

[4] W. D. Phillips, H. Metcalf, Phys. Rev. Lett., 1982, 48, 596–599.

[5] A. L. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergeman, H. J. Metcalf, Phys. Rev.

Lett., 1985, 54, 2596–2599.

[6] L. Hollberg, S. Chu, J. E. Bjorkholm, A. Cable, A. Ashkin, J. Opt. Soc. Am. A – Opt.

Image Sci. Vis., 1985, 2, P50–P50.

[7] E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, Phys. Rev. Lett., 1987, 59,2631–2634.

[8] W. D. Phillips, Rev. Mod. Phys., 1998, 70, 721–741.

[9] K. B. Davis, M.-O. Mewes, M. A. Joffe, M. R. Andrews, W. Ketterle, Phys. Rev. Lett.,1995, 74, 5202–5205.

[10] R. V. Krems, Phys. Chem. Chem. Phys., 2008, 10, 4079–4092.

[11] E. P. Wigner, Phys. Rev., 1948, 73, 1002–1009.

[12] P. S. Julienne, F. H. Mies, J. Opt. Soc. Am. B, 1989, 6, 2257–2269.

[13] J. L. Bohn, P. S. Julienne, Phys. Rev. A, 1999, 60, 414–425.

[14] Z. Idziaszek, P. S. Julienne, Phys. Rev. Lett., 2010, 104, 113202.

57

BIBLIOGRAPHY

[15] J. M. Vogels, C. C. Tsai, R. S. Freeland, S. J. J. M. F. Kokkelmans, B. J. Verhaar, D. J.Heinzen, Phys. Rev. A, 1997, 56, R1067–R1070.

[16] P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, B. J. Verhaar, Phys. Rev.

Lett., 1998, 81, 69–72.

[17] S. Knoop, F. Ferlaino, M. Mark, M. Berninger, H. Schöbel, H.-C. Nägerl, R. Grimm,Nature Phys., 2009, 5, 227.

[18] E. S. Shuman, J. F. Barry, D. DeMille, Nature, 2010, 467, 820.

[19] J. F. Barry, E. S. Shuman, E. B. Norrgard, D. DeMille, Phys. Rev. Lett., 2012, 108,103002.

[20] J. D. Weinstein, R. deCarvalho, T. Guillet, B. Friedrich, J. M. Doyle, Nature, 1998,395, 148–150.

[21] S. C. Doret, C. B. Connolly, W. Ketterle, J. M. Doyle, Phys. Rev. Lett., 2009, 103,103005.

[22] H. L. Bethlem, G. Berden, G. Meijer, Phys. Rev. Lett., 1999, 83, 1558–1561.

[23] S. Y. T. van de Meerakker, H. L. Bethlem, G. Meijer, Nature Physics, 2008, 4, 595.

[24] S. A. Rangwala, T. Junglen, T. Rieger, P. W. H. Pinkse, G. Rempe, Phys. Rev. A, 2003,67, 043406.

[25] K. M. Jones, E. Tiesinga, P. D. Lett, P. S. Julienne, Rev. Mod. Phys., 2006, 78, 483–535.

[26] T. Köhler, K. Goral, P. S. Julienne, Rev. Mod. Phys., 2006, 78, 1311–1361.

[27] J. Deiglmayr, A. Grochola, M. Repp, K. Mörtlbauer, C. Glück, J. Lange, O. Dulieu,R. Wester, M. Weidemüller, Phys. Rev. Lett., 2008, 101, 133004.

[28] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel,S. Kotochigova, P. S. Julienne, D. S. Jin, J. Ye, Science, 2008, 322, 231.

[29] G. Reinaudi, C. B. Osborn, M. McDonald, S. Kotochigova, T. Zelevinsky, Phys. Rev.

Lett., 2012, 109, 115303.

[30] J. G. Danzl, E. Haller, M. Gustavsson, M. J. Mark, R. Hart, N. Bouloufa, O. Dulieu,H. Ritsch, H.-C. Nägerl, Science, 2008, 321, 1062–1066.

[31] H. L. Bethlem, G. Berden, F. M. H. Crompvoets, R. T. Jongma, A. J. A. van Roij,G. Meijer, Nature, 2000, 406, 491–494.

58

BIBLIOGRAPHY

[32] S. K. Tokunaga, J. M. Dyne, E. A. Hinds, M. R. Tarbutt, New J. Phys., 2009, 11,055038.

[33] S. Y. T. van de Meerakker, I. Labazan, S. Hoekstra, J. Küpper, G. Meijer, J. Phys. B,2006, 39, S1077.

[34] J. R. Bochinski, E. R. Hudson, H. J. Lewandowski, G. Meijer, J. Ye, Phys. Rev. Lett.,2003, 91, 243001.

[35] E. R. Hudson, C. Ticknor, B. C. Sawyer, C. A. Taatjes, H. J. Lewandowski, J. R.Bochinski, J. L. Bohn, J. Ye, Phys. Rev. A, 2006, 73, 063404.

[36] M. R. Tarbutt, H. L. Bethlem, J. J. Hudson, V. L. Ryabov, V. A. Ryzhov, B. E. Sauer,G. Meijer, E. A. Hinds, Phys. Rev. Lett., 2004, 92, 173002.

[37] D. Patterson, E. Tsikata, J. M. Doyle, Phys. Chem. Chem. Phys., 2010, 12, 9736–9741.

[38] K. Maussang, D. Egorov, J. S. Helton, S. V. Nguyen, J. M. Doyle, Phys. Rev. Lett.,2005, 94, 123002.

[39] D. Egorov, W. C. Campbell, B. Friedrich, S. E. Maxwell, E. Tsikata, L. D. van Buuren,J. M. Doyle, Eur. Phys. J. D, 2004, 31, 307–311.

[40] J. G. E. Harris, R. A. Michniak, S. V. Nguyen, N. Brahms, W. Ketterle, J. M. Doyle,EPL (Europhysics Letters), 2004, 67, 198.

[41] T. V. Tscherbul, J. Kłos, A. Dalgarno, B. Zygelman, Z. Pavlovic, M. T. Hummon, H.-I.Lu, E. Tsikata, J. M. Doyle, Phys. Rev. A, 2010, 82, 042718.

[42] T. Junglen, T. Rieger, S. A. Rangwala, P. W. H. Pinkse, G. Rempe, Eur. Phys. J. D,2004, 31, 365–373.

[43] E. Bodo, F. A. Gianturco, A. Dalgarno, J. Phys. B, 2002, 35, 2391–2396.

[44] E. Bodo, F. A. Gianturco, N. Balakrishnan, A. Dalgarno, J. Phys. B, 2004, 37, 3641–3648.

[45] P. Staanum, S. D. Kraft, J. Lange, R. Wester, M. Weidemüller, Phys. Rev. Lett., 2006,96, 023201.

[46] P. Soldán, M. T. Cvitaš, J. M. Hutson, P. Honvault, J. M. Launay, Phys. Rev. Lett., 2002,89, 153201.

[47] K.-K. Ni, S. Ospelkaus, D. Wang, G. Quéméner, B. N. M. H. G. de Miranda, , J. L.Bohn, J. Ye, D. S. Jin, Nature, 2010, 464, 1324.

[48] A. V. Avdeenkov, J. L. Bohn, Phys. Rev. A, 2002, 66, 052718.

59

BIBLIOGRAPHY

[49] G. Quéméner, J. L. Bohn, Phys. Rev. A, 2010, 81, 022702.

[50] M. H. G. de Miranda, A. Chotia, B. Neyenhuis, D. Wang, G. Quéméner, S. Ospelkaus,J. L. Bohn, J. Ye, D. S. Jin, Nature Physics, 2011, 7, 502.

[51] S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis, G. Quéméner,P. S. Julienne, J. L. Bohn, D. S. Jin, J. Ye, Science, 2010, 327, 853–857.

[52] T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye, R. Ciuryło, P. Naidon, P. S.Julienne, Phys. Rev. Lett., 2006, 96, 203201.

[53] Y. N. Martinez de Escobar, P. G. Mickelson, P. Pellegrini, S. B. Nagel, A. Traverso,M. Yan, R. Côté, T. C. Killian, Phys. Rev. A, 2008, 78, 062708.

[54] S. Tojo, M. Kitagawa, K. Enomoto, Y. Kato, Y. Takasu, M. Kumakura, Y. Takahashi,Phys. Rev. Lett., 2006, 96, 153201.

[55] A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, P. Pillet, Phys.

Rev. Lett., 1998, 80, 4402–4405.

[56] K. Enomoto, M. Kitagawa, S. Tojo, Y. Takahashi, Phys. Rev. Lett., 2008, 100, 123001.

[57] K. M. Jones, P. S. Julienne, P. D. Lett, W. D. Phillips, E. Tiesinga, C. J. Williams,Europhys. Lett., 1996, 35, 85.

[58] E. R. Hudson, H. J. Lewandowski, B. C. Sawyer, J. Ye, Phys. Rev. Lett., 2006, 96,143004.

[59] M. G. Kozlov, Phys. Rev. A, 2009, 80, 022118.

[60] H. L. Bethlem, W. Ubachs, Faraday Discuss., 2009, 142, 25–36.

[61] D. DeMille, S. Sainis, J. Sage, T. Bergeman, S. Kotochigova, E. Tiesinga, Phys. Rev.

Lett., 2008, 100, 043202.

[62] T. Zelevinsky, S. Kotochigova, J. Ye, Phys. Rev. Lett., 2008, 100, 043201.

[63] M. Kajita, Y. Moriwaki, J. Phys. B, 2009, 42, 154022.

[64] M. Kajita, Phys. Rev. A, 2008, 77, 012511.

[65] H. L. Bethlem, M. Kajita, B. Sartakov, G. Meijer, W. Ubachs, The European Physical

Journal Special Topics, 2008, 163, 55–69.

[66] A. Borschevsky, M. Iliaš, V. A. Dzuba, K. Beloy, V. V. Flambaum, P. Schwerdtfeger,Phys. Rev. A, 2012, 85, 052509.

60

BIBLIOGRAPHY

[67] J. J. Hudson, M. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, E. A. Hinds,Nature, 2011, 473, 493.

[68] L. D. Carr, D. DeMille, R. V. Krems, J. Ye, New J. Phys., 2009, 11, 055049.

[69] D. DeMille, Phys. Rev. Lett., 2002, 88, 067901.

[70] C. Lee, E. A. Ostrovskaya, Phys. Rev. A, 2005, 72, 062321.

[71] S. F. Yelin, K. Kirby, R. Côté, Phys. Rev. A, 2006, 74, 050301.

[72] A. Micheli, G. K. Brennen, P. Zoller, Nature Phys., 2006, 2, 341.

[73] R. Barnett, D. Petrov, M. Lukin, E. Demler, Phys. Rev. Lett., 2006, 96, 190401.

[74] H. P. Büchler, E. Demler, M. Lukin, A. Micheli, N. Prokof’ev, G. Pupillo, P. Zoller,Phys. Rev. Lett., 2007, 98, 060404.

[75] S. K. Tokunaga, J. O. Stack, J. J. Hudson, B. E. Sauer, E. A. Hinds, M. R. Tarbutt, J.

Chem. Phys., 2007, 126, 12431.

[76] J. J. Gilijamse, S. Hoekstra, S. Y. T. van de Meerakker, G. C. Groenenboom, G. Meijer,Science, 2006, 313, 1617–1620.

[77] B. C. Sawyer, B. K. Stuhl, D. Wang, M. Yeo, J. Ye, Phys. Rev. Lett., 2008, 101, 203203.

[78] L. Scharfenberg, J. Klos, P. J. Dagdigian, M. H. Alexander, G. Meijer, S. Y. T. van deMeerakker, Phys. Chem. Chem. Phys., 2010, 12, 10660–10670.

[79] M. Kirste, X. Wang, H. C. Schewe, G. Meijer, K. Liu, A. van der Avoird, L. M. C.Janssen, K. B. Gubbels, G. C. Groenenboom, S. Y. T. van de Meerakker, Science, 2012,338, 1060–1063.

[80] M. T. Hummon, W. C. Campbell, H.-I. Lu, E. Tsikata, Y. Wang, J. M. Doyle, Phys. Rev.

A, 2008, 78, 050702.

[81] C. Zipkes, S. Palzer, C. Sias, M. Köhl, Nature, 2010, 464, 388.

[82] C. Zipkes, S. Palzer, L. Ratschbacher, C. Sias, M. Köhl, Phys. Rev. Lett., 2010, 105,133201.

[83] S. Schmid, A. Härter, J. H. Denschlag, Phys. Rev. Lett., 2010, 105, 133202.

[84] F. H. J. Hall, M. Aymar, M. Raoult, O. Dulieu, S. Willitsch, ArXiv e-prints, 2013,1301.0724.

[85] G. C. Nielson, G. A. Parker, R. T Pack, J. Chem. Phys., 1976, 64, 2055–2061.

61

BIBLIOGRAPHY

[86] D. Spelsberg, J. Chem. Phys., 1999, 111, 9625–9633.

[87] B. Bussery-Honvault, F. Dayou, A. Zanchet, J. Chem. Phys., 2008, 129, 234302.

[88] J. Ulmanis, J. Deiglmayr, M. Repp, R. Wester, M. Weidemuller, Chem. Rev., 2012, 112,4890–4927.

[89] P. G. Mickelson, Y. N. Martinez, A. D. Saenz, S. B. Nagel, Y. C. Chen, T. C. Killian,P. Pellegrini, R. Côté, Phys. Rev. Lett., 2005, 95, 223002.

[90] T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye, R. Ciuryło, P. Naidon, P. S.Julienne, Phys. Rev. Lett., 2006, 96, 203201.

[91] A. Stein, H. Knöckel, E. Tiemann, Eur. Phys. J. D, 2011, 64, 227–238.

[92] E. Czuchaj, M. Krosnicki, H. Stoll, Chem. Phys. Lett., 2003, 371, 401 – 409.

[93] S. Kotochigova, J. Chem. Phys., 2008, 128, 024303.

[94] D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano, D. J. Wineland, Phys. Rev.

Lett., 1986, 57, 70–73.

[95] R. Drullinger, D. Wineland, J. Bergquist, Applied physics, 1980, 22, 365–368.

[96] P. Blythe, B. Roth, U. Fröhlich, H. Wenz, S. Schiller, Phys. Rev. Lett., 2005, 95, 183002.

[97] M. A. van Eijkelenborg, M. E. M. Storkey, D. M. Segal, R. C. Thompson, Phys. Rev.

A, 1999, 60, 3903–3910.

[98] F. Schreck, G. Ferrari, K. L. Corwin, J. Cubizolles, L. Khaykovich, M. O. Mewes,C. Salomon, Phys. Rev. A, 2001, 64, 011402.

[99] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, M. Inguscio, Science, 2001,294, 1320–1322.

[100] P. G. Mickelson, Y. N. Martinez de Escobar, M. Yan, B. J. DeSalvo, T. C. Killian, Phys.

Rev. A, 2010, 81, 051601.

[101] C. E. Heiner, H. L. Bethlem, G. Meijer, Phys. Chem. Chem. Phys., 2006, 8, 2666–2676.

[102] J. van Veldhoven, H. L. Bethlem, G. Meijer, Phys. Rev. Lett., 2005, 94, 083001.

[103] R. deCarvalho, J. Kim, J. D. Weinstein, J. M. Doyle, B. Friedrich, T. Guillet, D. Patter-son, Eur. Phys. J. D, 1999, 7, 289–309.

[104] P. Soldán, J. M. Hutson, Phys. Rev. Lett., 2004, 92, 163202.

62

BIBLIOGRAPHY

[105] M. Tacconi, L. Gonzalez-Sanchez, E. Bodo, F. A. Gianturco, Phys. Rev. A, 2007, 76,032702.

[106] M. Lara, J. L. Bohn, D. Potter, P. Soldán, J. M. Hutson, Phys. Rev. Lett., 2006, 97,183201.

[107] M. Lara, J. L. Bohn, D. E. Potter, P. Soldán, J. M. Hutson, Phys. Rev. A, 2007, 75,012704.

[108] A. O. G. Wallis, J. M. Hutson, Phys. Rev. Lett., 2009, 103, 183201.

[109] P. S. Zuchowski, J. M. Hutson, Phys. Rev. A, 2009, 79, 062708.

[110] P. E. S. Wormer, F. Mulder, A. Van der Avoird, Int. J. Quantum Chem., 1977, 11, 959–970.

[111] T. G. A. Heijmen, R. Moszynski, P. E. S. Wormer, A. van der Avoird, Mol. Phys., 1996,89, 81.

[112] T. Cwiok, B. Jeziorski, W. Kołos, R. Moszynski, J. Rychlewski, K. Szalewicz, Chem.

Phys. Lett., 1992, 195, 67 – 76.

[113] H. B. G. Casimir, D. Polder, Phys. Rev., 1948, 73, 360.

[114] M. H. Alexander, J. Chem. Phys., 1982, 76, 5974–5988.

[115] T. V. Tscherbul, G. C. Groenenboom, R. V. Krems, A. Dalgarno, Faraday Discuss.,2009, 142, 127–141.

[116] J. M. Brown, A. Carrington, Rotational Spectroscopy of Diatomic Molecules, Cam-bridge University Press, Cambridge, U.K., 2003.

[117] B. R. Johnson, J. Comput. Phys., 1973, 13, 445–449.

[118] M. H. Alexander, J. Chem. Phys., 1984, 81, 4510–4516.

[119] M. H. Alexander, D. E. Manolopoulos, J. Chem. Phys., 1987, 86, 2044.

[120] L. M. C. Janssen, G. C. Groenenboom, A. van der Avoird, P. S. Zuchowski,R. Podeszwa, J. Chem. Phys., 2009, 131, 224314.

[121] J. M. Hutson, S. Green, MOLSCAT computer program, version 14, distributed by Col-

laborative Computational Project No. 6 of the UK Engineering and Physical Sciences

Research Council, 2006.

[122] P. S. Zuchowski, J. M. Hutson, Phys. Chem. Chem. Phys., 2011, 13, 3669–3680.

63

BIBLIOGRAPHY

[123] L. Kong, F. A. Bischoff, E. F. Valeev, Chem. Rev., 2012, 112, 75–107.

[124] T. B. Adler, G. Knizia, H.-J. Werner, J. Chem. Phys., 2007, 127, 221106.

[125] G. Knizia, T. B. Adler, H.-J. Werner, J. Chem. Phys., 2009, 130, 054104.

[126] K. L. Bak, A. Halkier, P. Jørgensen, J. Olsen, T. Helgaker, W. Klopper, J. Mol. Struct.,2001, 567, 375–384.

[127] M. Jeziorska, R. Bukowski, W. Cencek, M. Jaszunski, B. Jeziorski, K. Szalewicz, Coll.

Czech. Chem. Commun., 2003, 68, 463.

[128] M. Jeziorska, W. Cencek, K. Patkowski, B. Jeziorski, K. Szalewicz, Int. J. Quantum

Chem., 2008, 108, 2053.

[129] H. J. Monkhorst, Int. J. Quantum Chem. Symp., 1977, 11, 421.

[130] H. Sekino, R. J. Bartlett, Int. J. Quantum Chem. Symp., 1984, 18, 255.

[131] J. F. Stanton, R. J. Bartlett, J. Chem. Phys., 1993, 98, 7029.

[132] Y. Shao, L. Fusti-Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. T. Brown, A. T. B.Gilbert, L. V. Slipchenko, S. V. Levchenko, D. P. O’Neill, R. A. D. Jr., R. C. Lochan,T. Wang, G. J. O. Beran, N. A. Besley, J. M., Herbert, C. Y. Lin, T. Van Voorhis, S. H.Chien, A. Sodt, R. P. Steele, V. A. Rassolov, P. E. Maslen, P. P. Korambath, R. D. Adam-son, B. Austin, J. Baker, E. F. C. Byrd, H. Dachsel, R. J. Doerksen, A. Dreuw, B. D.Dunietz, A. D. Dutoi, T. R. Furlani, S. R. Gwaltney, A. Heyden, S. Hirata, C.-P. Hsu,G. Kedziora, R. Z. Khalliulin, P. Klunzinger, A. M. Lee, M. S. Lee, W. Liang, I. Lotan,N. Nair, B. Peters, E. I. Proynov, P. A. Pieniazek, Y. M. Rhee, J. Ritchie, E. Rosta, C. D.Sherrill, A. C. Simmonett, J. E. Subotnik, H. L. Woodcock III, W. Zhang, A. T. Bell,A. K. Chakraborty, D. M. Chipman, F. J. Keil, A. Warshel, W. J. Hehre, H. F. SchaeferIII, J. Kong, A. I. Krylov, P. M. W. Gill, M. Head-Gordon, Phys. Chem. Chem. Phys.,2006, 8, 3172.

[133] M. Baer, Chem. Phys. Lett., 1975, 35, 112.

[134] M. Baer, Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and

Conical Intersections, Wiley, New York, 2006.

[135] H.-J. Werner, P. J. Knowles, J. Chem. Phys., 1988, 89, 5803–5814.

[136] P. J. Knowles, H.-J. Werner, Chem. Phys. Lett., 1988, 145, 514.

[137] H.-J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, M. Schütz, P. Celani, T. Korona,A. Mitrushenkov, G. Rauhut, T. B. Adler, R. D. Amos, A. Bernhardsson, A. Berning,

64

BIBLIOGRAPHY

D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, E. Goll, C. Hampel, G. Hetzer,T. Hrenar, G. Knizia, C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. McNi-cholas, W. Meyer, M. E. Mura, A. Nicklass, P. Palmieri, K. Pflüger, R. Pitzer, M. Rei-her, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson, M. Wang, A. Wolf, MOLPRO,

version 2008.3, a package of ab initio programs, 2008, see http://www.molpro.net.

[138] R. Abrol, A. Shaw, A. Kuppermann, D. R. Yarkony, J. Chem. Phys., 2001, 115, 4640.

[139] D. A. Brue, X. Li, G. A. Parker, J. Chem. Phys., 2005, 123, 091101.

[140] D. DeMille, D. Glenn, J. Petricka, Eur. Phys. J. D, 2004, 31, 375–384.

[141] H. Koch, P. Jorgensen, J. Chem. Phys., 1990, 93, 3333–3344.

[142] H. Koch, O. Christiansen, P. Jorgensen, A. M. S. de Meras, T. Helgaker, J. Chem. Phys.,1997, 106, 1808–1818.

[143] E. Timmermans, R. Côté, Phys. Rev. Lett., 1998, 80, 3419–3423.

[144] J. Tellinghuisen, P. S. Julienne, J. Chem. Phys., 1984, 81, 5779–5785.

[145] S. Knecht, L. K. Sorensen, H. J. A. Jensen, T. Fleig, C. M. Marian, J. Phys. B, 2010,43, 055101.

[146] P. D. Lett, K. Helmerson, W. D. Phillips, L. P. Ratliff, S. L. Rolston, M. E. Wagshul,Phys. Rev. Lett., 1993, 71, 2200–2203.

[147] J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. Lett., 1993, 71, 2204–2207.

[148] A. N. Nikolov, J. R. Ensher, E. E. Eyler, H. Wang, W. C. Stwalley, P. L. Gould, Phys.

Rev. Lett., 2000, 84, 246–249.

[149] K. Bergmann, H. Theuer, B. W. Shore, Rev. Mod. Phys., 1998, 70, 1003–1025.

[150] M. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, P. Pillet,Science, 2008, 321, 232.

[151] A. N. Nikolov, E. E. Eyler, X. T. Wang, J. Li, H. Wang, W. C. Stwalley, P. L. Gould,Phys. Rev. Lett., 1999, 82, 703–706.

[152] G. Zinner, T. Binnewies, F. Riehle, E. Tiemann, Phys. Rev. Lett., 2000, 85, 2292–2295.

[153] M. Kitagawa, K. Enomoto, K. Kasa, Y. Takahashi, R. Ciuryło, P. Naidon, P. S. Julienne,Phys. Rev. A, 2008, 77, 012719.

[154] C. P. Koch, Phys. Rev. A, 2008, 78, 063411.

65

BIBLIOGRAPHY

[155] C. P. Koch, R. Moszynski, Phys. Rev. A, 2008, 78, 043417.

[156] N. Boutassetta, A. R. Allouche, M. Aubert-Frécon, Phys. Rev. A, 1996, 53, 3845–3852.

[157] I. S. Lim, H. Stoll, P. Schwerdtfeger, J. Chem. Phys., 2006, 124, 034107.

[158] V. Kokoouline, O. Dulieu, R. Kosloff, F. Masnou-Seeuws, J. Chem. Phys., 1999, 110,9865.

[159] K. Willner, O. Dulieu, F. Masnou-Seeuws, J. Chem. Phys., 2004, 120, 548.

[160] T. Gonzalez-Lezana, E. J. Rackham, D. E. Manolopoulos, J. Chem. Phys., 2004, 120,2247–2254.

[161] T. Zelevinsky, S. Kotochigova, J. Ye, Phys. Rev. Lett., 2008, 100, 043201.

[162] S. Kotochigova, T. Zelevinsky, J. Ye, Phys. Rev. A, 2009, 79, 012504.

[163] A. Stein, H. Knöckel, E. Tiemann, Eur. Phys. J. D, 2010, 57, 171–177.

[164] H. A. Bethe, E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,Academic Press, New York, 1957, p. 170.

[165] R. Napolitano, J. Weiner, C. J. Williams, P. S. Julienne, Phys. Rev. Lett., 1994, 73,1352–1355.

[166] K. Sando, A. Dalgarno, Mol. Phys., 1971, 20, 103–112.

[167] C. M. Dion, C. Drag, O. Dulieu, B. Laburthe Tolra, F. Masnou-Seeuws, P. Pillet, Phys.

Rev. Lett., 2001, 86, 2253–2256.

[168] H. K. Pechkis, D. Wang, Y. Huang, E. E. Eyler, P. L. Gould, W. C. Stwalley, C. P. Koch,Phys. Rev. A, 2007, 76, 022504.

[169] M. Tomza, F. Pawłowski, M. Jeziorska, C. P. Koch, R. Moszynski, Phys. Chem. Chem.

Phys., 2011, 13, 18893.

66

APPENDIX A

PAPER I

“Long-range interactions between an atom in its ground S state

and an open-shell linear molecule”

W. Skomorowski and R. Moszynski

Journal of Chemical Physics 134, 124117 (2011)

67

THE JOURNAL OF CHEMICAL PHYSICS 134, 124117 (2011)

Long-range interactions between an atom in its ground S stateand an open-shell linear molecule

Wojciech Skomorowski and Robert Moszynskia)

Quantum Chemistry Laboratory, Department of Chemistry, University of Warsaw,Pasteura 1, 02-093 Warsaw, Poland

(Received 10 November 2010; accepted 25 February 2011; published online 31 March 2011)

Theory of long-range interactions between an atom in its ground S state and a linear molecule in adegenerate state with a nonzero projection of the electronic orbital angular momentum is presented.It is shown how the long-range coefficients can be related to the first and second-order molecularproperties. The expressions for the long-range coefficients are written in terms of all components ofthe static and dynamic multipole polarizability tensor, including the nondiagonal terms connectingstates with the opposite projection of the electronic orbital angular momentum. It is also shown thatfor the interactions of molecules in excited states that are connected to the ground state by multipolartransition moments additional terms in the long-range induction energy appear. All these theoreticaldevelopments are illustrated with the numerical results for systems of interest for the sympatheticcooling experiments: interactions of the ground state Rb(2S) atom with CO(3), OH(2), NH(1),and CH(2) and of the ground state Li(2S) atom with CH(2). © 2011 American Institute of Physics.[doi:10.1063/1.3567306]

I. INTRODUCTION

Recent developments in laser cooling and trapping tech-niques have opened the possibility of studying collisionaldynamics at ultralow temperatures. Atomic Bose–Einsteincondensates1 are of crucial importance in this respect sinceinvestigations of the collisions between ultracold atoms in thepresence of a weak laser field leads to precision measurementsof the atomic properties and interactions. Such collisions mayalso lead to the formation of ultracold molecules that can beused in high-resolution spectroscopic experiments to study in-elastic and reactive processes at very low temperatures, inter-atomic interactions at very large distances including the rela-tivistic and QED effects, or the thermodynamic properties ofthe quantum condensates of weakly interacting atoms.2

Experimental techniques based on the buffer gas cooling3

or Stark deceleration4 produced cold molecules with a tem-perature well below 1 K. Optical techniques, based on thelaser cooling of atoms to ultralow temperatures and photoas-sociation to create molecules,5 reached temperatures of theorder of a few μK or lower. Spectacular achievements werereported with the Bose–Einstein condensation of homonu-clear alkali molecules starting from fermionic atoms6–8 andrecently with ultracold dense gas of bialkali polar molecules.9

A major objective for the present day experiments oncold molecules is to achieve quantum degeneracy for polarmolecules. Two approaches to this problems are used: indi-rect methods, in which molecules are formed from pre-cooledatomic gases,9 and direct methods, in which molecules arecooled from room temperature. The Stark deceleration andtrapping methods pioneered by Meijer and collaborators10 are

a)Author to whom correspondence should be addressed. Electronic mail:[email protected].

the best developed of the direct methods and provide excitingpossibilities for progress toward quantum degeneracy.

Beam deceleration can achieve temperatures around 10mK. However, condensation requires submicro-Kelvin tem-peratures. Finding a second-stage cooling method to bridgethis gap is the biggest challenge facing the field. The mostpromising possibility is the so-called sympathetic cooling, inwhich cold molecules are introduced into an ultracold atomicgas and equilibrate with it. Sympathetic cooling has alreadybeen successfully used to achieve Fermi degeneracy in 6Li(Ref. 11) and Bose–Einstein condensation in 41K.12 However,it has not yet been attempted for molecular systems and thereare many challenges to overcome. In Berlin, Meijer’s grouphas now developed the capability to trap ultracold 87Rb atomsfor use in sympathetic cooling.13 In London, Tarbutt’s grouphas begun to set up experiments using an ultracold gas of 7Liatoms to cool molecules.14 Thus far, open shell moleculessuch as CO(3), OH(2), NH(1), and CH(2) could bedecelerated and are good candidates for sympathetic cooling.Theoretical studies on sympathetic cooling of polar moleculeswere first presented for the NH(3−) molecule with rubid-ium atoms used as a coolant.15 Later, other atoms, includingCs, Li, Mg, and N, were surveyed for sympathetic coolingof NH.16–19 Other simple molecular systems such as LiH,20

ND3 ,21, 22 and ions, Yb+ and Ba+,23–26 were also investigated,both experimentally and theoretically.

Very little is known about collisions between polarmolecules and alkali metal atoms, and results of theoreticalstudies on them are essential to guide the experimentsand later to interpret the results. There are two essentialingredients: good potential energy surfaces to describe theinteractions, and good methods for carrying out low-energycollision calculations. Hutson and collaborators has pioneeredthe study of potential energy surfaces for interactions between

0021-9606/2011/134(12)/124117/14/$30.00 © 2011 American Institute of Physics134, 124117-1

Downloaded 25 Oct 2011 to 212.87.2.134. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

124117-2 W. Skomorowski and R. Moszynski J. Chem. Phys. 134, 124117 (2011)

polar molecules and alkali metal atoms.21, 22, 27 Interactionsand ultracold collisions of alkali atoms with polar moleculeswere also explored theoretically by Tacconi et al.16, 28 How-ever, the theoretical methods available for calculating thesurfaces have some significant inadequacies, which need tobe addressed before quantitative predictions will be possible.

When dealing with collisions at ultralow temperatures theaccuracy of the potential in the long range is crucial. There-fore, the methods used in the calculations of the potential en-ergy surfaces should be size-consistent29 in order to ensurea proper dissociation of the electronic states, and a properlong-range asymptotics of the potential should be imposed.The latter task is highly nontrivial when a molecule is in anopen-shell degenerate state. To our knowledge, this problemhas only been addressed by Spelsberg30 for the CO+OH sys-tem, and by Nielson et al.31 and Bussery-Honvault et al.32, 33

for an atom interacting with open-shell molecule. The latterconsiderations were limited, however, to the C6 coefficientsat most. Standard approaches based on the symmetry-adaptedperturbation theory within the wave function34, 35 or densityfunctional formalisms36 are inapplicable in this case.

The significance of analytic representation of the long-range potential results from the fact that some important prop-erties concerning low-energy collisions, such as the meanscattering length and the heights of the centrifugal barriers,can be determined purely from the van der Waals constants.Other simplified scattering approaches such as multichannelquantum defect theory require the knowledge of the exact po-tential only in the long-range form −Cn/Rn , while the short-range part is confined to some adjustable parameters.37, 38 Onthe other side, precise theoretical determination of the scat-tering length require the inclusion of the asymptotic terms ofthe interaction potential up to and including C12/R12, as wasdemonstrated in Ref. 39 for spin-polarized helium. Therefore,the values of the asymptotic constants Cn form an importantinput for planned future studies of the quantum dynamics oflow-energy collisions and, by itself, they may provide qual-itative insight into possible interactions, especially in the ul-tralow temperature regime.

In this paper, we report a theoretical study of the long-range interactions between an atom in its ground S state and alinear molecule in a degenerate state with the projection of theelectronic orbital angular momentum . As we are interestedin the long-range interactions only, the resulting spin stateof the interacting system is of no importance. Thus, the pre-sented theory is applicable to all diatomic molecules interact-ing with any ground state S atom, including alkaline, alkaline-earth atoms, also He, N, Cr, Yb, and so on. The expressionsderived in this work are applied to systems of interest for thesympathetic cooling experiments, i.e., to interactions of theground state Rb(2S) atom with CO(3), OH(2), NH(1),and CH(2), and the Li(2S) atom with CH(2). The plan ofthis paper is as follows. In Sec. II, we present the derivationof the long-range asymptotics for the interaction of a groundS state atom with linear molecule in a degenerate state .We discuss which polarizability components of an open-shellspecies are needed to express the asymptotic interaction en-ergy and how the expression for the interaction energy de-pends on the adopted basis (spherical or Cartesian). In this

section, we also show that for molecules in an excited statethat is connected to the ground state by multipolar transitionmoments, a new term in the long-range expansion appears. InSec. III, we present the computational approach adopted in thethis paper, discuss our results, and compare them with the datafrom the supermolecule calculations and from the symmetry-adapted perturbation theory. Finally, Sec. IV concludes ourpaper.

II. THEORY

We consider the interaction of an atom A in the ground Sstate ψA(S) and a linear molecule B in a state ψB(), where is the projection of the electronic orbital angular momen-tum of the molecule on the molecular axis. All the quanti-ties relating to the atom and the molecule will be designatedby subscripts A and B, respectively. The resulting spin mul-tiplicity of the complex do not play any role in our furtherdevelopments and will be omitted. The electronic state of themolecule does not need to be its ground state. The Hamilto-nian H of the complex AB can be written as

H = H0 + V, (1)

where H0 is the sum of the Hamiltonians describing isolatedmonomers A and B, H0 = HA + HB, and V is the intermolec-ular interaction operator collecting all Coulombic interactionsbetween electrons and nuclei of the monomer A with the elec-trons and nuclei of the monomer B. Assuming that the elec-tron clouds of the monomers do not overlap, V can be repre-sented by the following multipole expansion:41–43

V =∞∑

lA,lB=0

ClA,lB R−lA−lB−1lA+lB∑

m=−lA−lB

(−1)mY −mlA+lB

(R)

× [QlA ⊗ QlB

]lA+lB

m , (2)

where the constant ClA,lB is given by

ClAlB = (−1)lB

[4π

2lA + 2lB + 1

]1/2 (2lA + 2lB

2lA

)1/2

, (3)

Y ml (R) is the normalized spherical harmonic depending on

the spherical angles R of the vector connecting the centersof mass of the monomers A and B in a space-fixed coordi-nate system, Qm

l denotes the multipole moment operator inthe space-fixed frame. We made also use of the coupled prod-uct of two spherical tensors,

[QlA ⊗ QlB

]L

M =∑

m A,m B

〈lA, m A; lB, m B |L , M〉 Qm AlA

Qm BlB

, (4)

where 〈lA, m A; lB, m B |L , M〉 is the Clebsch-Gordancoefficient.

A state ψB() of a linear molecule with = 0 is dou-bly degenerate, and so is the state of the complex AB at


124117-3 Long-range interactions in open-shell systems J. Chem. Phys. 134, 124117 (2011)

R = ∞, which is just a product ψA(S)ψB(). For the interac-tion of a ground state S atom with a molecule, the first-orderelectrostatic energy vanishes identically in the multipole ap-proximation, so the degeneracy is lifted in the second-order,leading to the splitting of the two states ψA(S)ψB(±||)into the A′ and A′′ states of the complex AB at finite R.Thus, to obtain the long-range behavior of the A′ and A′′

states, we have to diagonalize the second-order interaction

matrix:

V (2) =(

V (2), V (2)

,−

V (2)−, V (2)

−,−

). (5)

The elements V (2),′ of V (2) are given by the standard ex-

pressions of the polarization theory44 and can be decomposedinto the induction and dispersion parts:

V (2),′ = −

∞∑k

′ 〈ψA(S)ψB()|V |ψA(k)ψB(±||)〉〈ψA(k)ψB(±||)|V |ψA(S)ψB(′)〉ωS,k

−∞∑k

′∞∑n

′ 〈ψA(S)ψB()|V |ψA(k)ψB(n)〉〈ψA(k)ψB(n)|V |ψA(S)ψB(′)〉ωS,k + ω,n

, (6)

where ωS,k = Ek − ES is the excitation energy of the atomfrom the ground state ψA(S) to the excited state ψA(k)characterized by the set of quantum numbers k, and ω,n

= En − E is the excitation energy from the state ψB()to the excited state ψB(n) of the molecule with the set ofquantum numbers denoted by n. In the above equation, thesign prime on the summation symbol means that the groundstate of the atom or the states ψB(±||) of the molecule areexcluded from the summations.

It can easily be shown31 that in the case of interactionwith an atom in S state the matrix elements V (2)

, and V (2)−,−

are equal, and the same holds for off-diagonal elements V (2),−

and V (2)−,. The eigenfunctions and the corresponding eigen-

values of the matrix (5) can simply be constructed on the sym-metry basis only, as the proper zero-order wave function of thecomplex AB should have a defined behavior under the reflec-tion in the plane σ containing the three atoms: σψ

(+)AB = ψ

(+)AB

for A′ state and σψ(−)AB = (−1)ψ (−)

AB for A′′ state. In our ap-proach, the three atoms lie in the xz plane and the trans-formation between the space-fixed frame and the body-fixedframes is chosen in such a way that it leads to the coincidenceof the axis y in all relevant frames of reference, see Fig. 1.Thus, the symmetry (A′ or A′′) of the zero-order wave func-tion ψ

(±)AB characterizes its behavior under the reflection in the

plane σxz . It is determined by the symmetry properties of thewave functions of the two constituent monomers, i.e., ψA(S)and ψB(), with respect to the reflection in the plane σxz oftheir body-fixed frames, which happens to be the same. Thesymmetry relations for the monomer electronic wave func-tions are ( = 0) (Refs. 40 and 45)

σxzψB() = (−1)ψB(−), σxzψA(S) = (−1)pAψA(S),(7)

where pA defines the spatial parity of the atomic wavefunc-tion in the S state. Therefore, the properly adapted wave

functions read

ψ(±)AB = 1√

2

[ψA(S)ψB() + (−1)+pA+ f ψA(S)ψB(−)

].

(8)and the corresponding second-order energies (i.e., eigenval-ues of the matrix V (2)) are

E (2)(±) = V (2)

, + (−1)+pA+ f V (2),− (9)

with f = 0 for A′ and f = 1 for A′′ state. A standard phaseconvention of Condon and Shortley47 was adopted to defineψB() and ψB(−). More details on the symmetry proper-ties of the electronic wave functions of diatomic moleculescan be found in Refs. 40, 45, and 46. Here, we only want toconclude that in the case of the systems under study the A′

state corresponds to the combination of the two terms with theminus sign in Eqs. (8) and (9) for Rb+OH(2), Rb+CO(3),and Li+CH(2), while the A′ state has the plus sign forRb+NH(1). The opposite holds for the A′′ states.

The multipole expansion of V,′ is readily obtainedby inserting the multipole expansion (2) of the interaction

Y=y

x

X

z

Z

R

A

B

C

θ

FIG. 1. Relative orientation of the molecule-fixed frame (xyz) to the space-fixed frame (XY Z ) with its Z axis directed along the vector R joining thecenter of mass of the molecule BC and the atom A. The three atoms are inthe plane σxz = σX Z .



operator V into Eq. (6) and collecting terms, as it was done inRefs. 41–43. Specifically, the derivation is based on the wellknown transformation properties of the multipole operatorsfrom the space-fixed (with the index m X ) to the body-fixed(with the index kX ) frame of each monomer (X = A or B),

Qm XlX

=lX∑

kX =−lX

D(lX )

m X ,kX(RX )QkX

lX, (10)

where D(lX )

m X ,kX(RX ) is the Wigner D matrix, and on the addi-

tion theorems for the D functions and spherical harmonics,

D(lX )

m X ,kX(RX ) · D

(l ′X )

m ′X ,k ′

X(RX )

=∑L X

L X∑K X =−L X

L X∑MX =−L X

〈lX , kX ; l ′X , k ′X |L X , K X 〉

× 〈lX , m X ; l ′X , m ′X |L X , MX 〉D(L X )

MX ,K X(RX ), (11)

Y −mlA+lB

(R) · Y −m ′lA′ +lB′ (R)

=∑

L

L∑M=−L

[(2lA + 2lB + 1)(2lA′ + 2lB ′ + 1)

4π (2L + 1)

]1/2

×〈lA + lB,−m; lA′ + lB ′ ,−m ′|L , M〉×〈lA + lB, 0; l ′A + l ′B, 0|L , 0〉Y M

L (R). (12)

The sets of the Euler angles RA and RB describe rota-tions of the space-fixed frame to the appropriate body-fixedframes. For our convenience, we choose the Z axis along theintermolecular axis connecting the two centers of mass. Then,the angular factor Y M

L (R) reduces to

Y ML (R) = Y M

L (0, φ) =(

2L + 1

4π

)1/2

δM,0. (13)

For an atom in an S state the quantum numbers L A, MA, andK A are zero, so D(L A)

MA,K A(RA) = 1. For an open-shell linear

molecule, we have MB = 0, and the set of the three Eulerangles is RB = (0, θ, 0), so

D(L B )

0,K B(0, θ, 0) = (−1)K B

√(L B − K B)!

(L B + K B)!P K B

L B(cos θ ),

(14)where P K B

L B(cos θ ) denotes the associated Legendre polyno-

mial. The set of the Euler angles RB = (0, θ, 0) for a lin-ear molecule is consistent with the foregoing requirementof the coincidence of the axis y in all relevant frames ofreference, cf. Fig. 1. The other possible set would be RB

= (3π/2, θ, π/2) which leads to the coincidence of the axis xas it was adopted in Ref. 31.

It is useful to express the final equations for the elementsof the matrix V(2) in terms of the static and dynamic multipolepolarizabilities of the atom and molecule. For an S atom, we

use the standard definition

αll ′mm ′(ω) =

2∑k

′ ωS,k〈ψA(S)|Qm

l |ψA(k)〉〈ψA(k)|Qm ′l ′ |ψA(S)〉

ω2S,k − ω2

,

(15)

while for an open-shell linear molecule we introduce extrasuperscripts and ′ to distinguish between diagonal andoff-diagonal components,

,′αll ′

mm ′ (ω) =

2∑n

′ ω,n〈ψB()|Qm

l |ψB(n)〉〈ψB(n)|Qm ′l ′ |ψB(′)〉

ω2,n − ω2

.

(16)

The corresponding irreducible polarizabilities are ob-tained by Clebsch-Gordan coupling:

,′α

(ll ′)LK (ω) =

∑m,m ′

〈l, m; l ′, m ′|L , K 〉,′αll ′

mm ′ (ω). (17)

The only nonvanishing components of the irreducible polariz-ability for an atom in the S state are α

(ll)00 , while for an open-

shell linear molecule the nonzero components are ,′α

(ll ′)LK

with K = 0 for = ′ and with K = 2 if ′ = −.We are now ready to give final expressions for the long-

range coefficients expressed in terms of the irreducible com-ponents of the polarizabilities:

V (2),′ = −

∑lA,lB ,l ′B

∑L ,K

ξL ,KlAlBl ′B

R−(2+2lA+lB+l ′B )[,′

C L ,KlAlBl ′B

(ind)

+,′C L ,K

lAlBl ′B(disp)

]P |K |

L (cos θ ), (18)

where the constant ξL ,KlAlBl ′B

is given by

ξL ,KlAlBl ′B

=[

(2lA + 2lB + 1)!(2lA + 2lB ′ + 1)!

(2lA)!(2lB)!(2lA)!(2lB ′)!

]1/2

×(

lA + lB lA + l ′B L0 0 0

)

×(

2L + 1

2lA + 1

)1/2√

(L − K )!

(L + K )!

lB lA + lB lA

lA + l ′B l ′B L

,

(19)

and the expressions in the round and curly brackets are the 3 jand 6 j coefficients, respectively. Combining all terms withthe same power n = 2lA + lB + l ′B + 2 in the above expan-sion, we will get standard long-range coefficients C L ,K

n ,

C L ,Kn =

2lA+lB+l ′B+2=n∑lA,lB ,l ′B

ξL ,KlAlBl ′B

[,′

C L ,KlAlBl ′B

(ind)+,′C L ,K

lAlBl ′B(disp)

],

(20)



by means of which the asymptotic expansion of Eq. (9) simplyreads

E (2)(±) = −

∞∑n=6

∑L

[C L ,0

n

RnP0

L (cos θ ) + (−1)+pA+ f

×C L ,2n

RnP |2|

L (cos θ )

]. (21)

The dispersion part ,′C L ,K

lAlBl ′B(disp) is proportional to

the Casimir-Polder integral over the atomic and molecular

polarizabilities calculated at imaginary frequencies,

,′C L ,K

lAlBl ′B(disp) = 1

2π

∫ ∞

0α

(lAlA)00 (iω) ,′

α(lB l ′B )LK (iω)dω,

(22)

while the induction term ,′C L ,K

lAlBl ′B(ind) is the product of the

static polarizability of the atom and permanent multipole mo-ments of the open-shell molecule:

,′C L ,K

lAlBl ′B(ind) = α

(lAlA)00 (0)

[〈ψB()|QlB |ψB(±||)〉 ⊗ 〈ψB(±||)|Ql ′B |ψB(′)〉]L

K . (23)

A few comments are needed here. The expression forthe diagonal term V (2)

, is the same as for the interactionbetween atom and linear molecule in a spatially nondegen-erate state (). The additional term emerging when = 0is the off-diagonal V (2)

,−. It has both induction and disper-sion parts. As it was already stated, the only nonvanishingterm in V (2)

,− appears for K = 2. The lowest order long-range coefficient, to which the off-diagonal term contributesto, is C L ,K

n with n = 4 + 2|| for the dispersion part andn = 5 + 2|| for the induction part. The source of the induc-tion energy in V (2)

,− comes from the fact that open-shell lin-ear molecules have an additional independent component ofthe permanent multipole moments, 〈ψ()|Q2

l |ψ(−)〉, incontrast to the state molecules with only one independentcomponent, namely 〈ψ()|Q0

l |ψ()〉. It is obvious that thesecond independent component appears for l ≥ 2||; there-fore, in the case of the states there is no induction contribu-tion to the off-diagonal C2,2

6 coefficient. The dispersion termsin V (2)

,− results from the presence of additional componentsof the polarizability terms for the open-shell linear molecules.

In general, the number of independent diagonal terms,αll ′

kk ′ is equal to 2l< + 1 (where l< is the smaller of l andl ′). Each of the nonredundant components ,αll ′

kk ′ comes fromthe transitions in the sum (16) through the intermediate statesψ(n) with the projection of the electronic angular momen-tum ranging || − (2l< + 1) to || + (2l< + 1). Due to thetransformation properties, the number of independent irre-ducible components ,α

(ll ′)LK will also be 2l< + 1.

In addition, for open-shell linear molecules, there arenondiagonal components of the polarizability tensor ,−αll ′

kk ′

which do not vanish and are not related to the diagonal terms.The nondiagonal terms in the multipole polarizability ten-sor appear if the condition k ′ + k = 2 can be fulfilled fora given set of the quantum numbers (l, k) and (l ′, k ′). Theyresult from the fact that there are transitions through the sameoperator to states + and −, and then ,−αll ′

kk is the differ-ence between these two separate contributions (e.g., 1,1α11

11 forthe state). The other mechanism leading to the appearanceof the nondiagonal ,−αll ′

kk ′ terms is parallel with the sourceof the off-diagonal induction terms; namely, there are possi-ble two independent transition moments between the state in

question |ψ()〉 and the exited states through an operator ofthe same rank, e.g., 1,−1α12

02 for the state, to be comparedwith the diagonal 1,1α12

00.Let us discuss the dipole polarizabilities in more details.

For open-shell diatomic molecules there are three indepen-dent spherical components ,α11

kk ′ with (k, k ′) equal to (0, 0),(1,−1), and (−1, 1), the corresponding transitions in sum-over-states occur through the excited states with a projec-tion of electronic angular momentum equal to , − 1, and + 1, respectively. In case of the states, there is extra non-diagonal component 1,−1α11

11 with intermediate states + and− in the sum-over-state expression, which come with oppo-site sign, in contrast to 1,1α11

−11 in which states + and −

contribute with the same sign. The four corresponding irre-ducible dipole polarizability components would be ,α

(11)00 ,

,α(11)20 , ,α

(11)10 , and ,−α

(11)22 , the last one is nonvanish-

ing only for the states.From the analysis of Eq. (18) it turns out that not all non-

vanishing irreducible components ,′α

(ll ′)LK of the molecule

polarizability tensor are needed to express the interaction en-ergy of an open-shell linear molecule with an atom. Thedipole component ,α

(11)10 is redundant in this case. It fol-

lows from the expression (19) that all terms ,′α

(ll ′)LK with

l + l ′ + L odd do not contribute to the interaction energy inthe second order E (2)

(±). However, if we had an asymmetricmolecule or an other linear open-shell species instead of anatom then all nonzero irreducible components would be nec-essary. For a state, there are three independent dipole polar-izabilities, but again, in the case of interaction with an atom,2,2α

(11)10 is redundant. The first off-diagonal term for a state

will be 2,−2α(22)44 or equivalently 2,−2α22

22 .Sometimes it is more convenient to use the Cartesian ba-

sis both for the states |〉 and multipole moments. The trans-formation formulas between the two bases can be found inRef. 58. We focus again on the states. In the Cartesianbasis the two degenerate states |± 1〉 are usually referred toas |x〉 and |y〉. The four independent dipole polarizabilitytensor components would be x,xαxx, x,xαyy, x,xαzz, and x,yαxy,where we adopted the index (x,y) in place of (−1,1) to distin-guish between particular diagonal and off-diagonal compo-nents. It is possible to define the fifth Cartesian component,



namely x,yαyx, however, due to the relation x,xαxx − x,xαyy

= x,yαxy + x,yαyx, it will not be independent.30 To see betterwhy the Cartesian basis may be useful, let us mention how theirreducible spherical components are related to the Cartesianones for the states:

1,1α(1,1)00 =−1,−1 α

(1,1)00 = − 1√

3

(x,xαxx + x,xαyy + x,xαzz

),

1,1α(1,1)20 =−1,−1 α

(1,1)20 = 1√

6

(2 x,xαzz − x,xαxx − x,xαyy

),

1,1α(1,1)10 = −−1,−1α

(1,1)10 = − 1√

2

(x,yαxy − x,yαyx

),

1,−1α(1,1)22 =−1,1 α

(1,1)2−2 = x,xαyy − x,xαxx. (24)

An inspection of the above relations shows that in order toexpress the interaction energy of a molecule with an atom,only diagonal Cartesian components are needed, as the off-diagonal x,yαxy contributes only to 1,1α

(1,1)10 , i.e., the term not

present in expression for E (2)(±). The off-diagonal irreducible

component 1,−1α(1,1)22 is equal to the difference between two

perpendicular α’s calculated for one of the Cartesian states.This observation is more general and it turns out that if wedecide for a Cartesian representation of the |〉 states, thenall off-diagonal irreducible components ,−α

(ll ′)L2 can be re-

lated to some diagonal Cartesian terms. Still, we will have tocalculate some additional Cartesian components, however, di-agonal only, which are not present in the expression for the di-agonal ,α

(ll ′)L0 terms. Calculations of the off-diagonal polar-

izability components in the Cartesian basis are indispensableif the interacting system comprises of an open-shell moleculeand asymmetric species. It means that in such a case the in-teraction energy in the multipole approximation is expressedin terms of, for instance, x,yαxy component. It is worth notingthat x,yαxy is irrelevant in the description of the interaction ofthe open-shell molecule with the external electric field in theStark effect. Thus, the induction energy and, consequently,intermolecular forces explore properties of the open-shell di-atomic molecules which are not accessible otherwise.

Note that Eq. (22) is strictly valid only when the moleculeis in its ground electronic state. If the molecule is in an excitedstate that is connected to the ground state (or to any otherstate lower in energy) by multipolar transition moments thenthe Casimir–Polder integral is no longer valid and an extraterm has to be added to the energy. The reason behind thisis the property of the Casimir–Polder integral that if the twoelements in denominator, εA and εB , have opposite signs thenwe obtain (assuming that εB < 0)

2

π

∫ ∞

0

εA

ε2A + ω2

· εB

ε2B + ω2

dω = − 1

εA + |εB | , (25)

instead of the value of 1/(εA − |εB |), which we want todecompose into the product of two terms depending onmonomer properties only. Formally, we may write the follow-ing identity (again εB < 0):

1

εA − |εB | = 2

π

∫ ∞

0

εA

ε2A + ω2

· εB

ε2B + ω2

dω

+ 1

εA + |εB | + 1

εA − |εB | . (26)

For any molecular state with εB < 0, we have the summation(6) over all atomic states with positive excitation energy εA,this means that the two last factors in the above equation willadd up to yield dynamic polarizability of atom at frequencyω = |εB |. Therefore, if we want to express the whole interac-tion energy in the second order E (2)

(±) in terms of the monomersproperties only, then we are forced to add an extra term tothe dispersion part, Eq. (22), in order to compensate an er-ror introduced by the integral representation of the dispersionenergy. This additional term depends on the dynamic polar-izability of the atom calculated at frequency ω equal to theenergy of all possible deexcitations in the molecule and willbe denoted by C L ,K

lAlBl ′B(corr, deexc). Its form is slightly similar

to the induction part,

,′C L ,K

lAlBl ′B(corr, deexc) =

∑n−

α(lAlA)00 (ωn−,)

[〈ψB()|QlB |ψB(n−)〉 ⊗ 〈ψB(n−)|Ql ′B |ψB(′)〉]L

K. (27)

The summation in the above equation runs only over states ofthe molecule ψB(n−) with energy lower than the referenceone, i.e., if ωn−, = E − En− is positive, and hence ωn−,

corresponds to the possible deexcitations of the molecule.This term does not have a simple physical interpretation, butas shown in Ref. 48 it leads to a different QED retardation ofthe long-range potential than given by the classical Casimir–Polder formula.49 Note also that without this extra term, thesecond-order interaction energy in the long-range could notbe written correctly in terms of molecular properties of theisolated subsystems in the case when deexcitation may oc-cur. Obviously, a similar term will be needed if an atom isin an exited state and molecule in its ground state. Then

C L ,KlAlBl ′B

(corr, deexc) would depend on the the dynamic polariz-abilities of the molecule at frequency ω corresponding to pos-sible atomic deexcitation. However, this holds only for atomicexcited S states, as if the atomic state was either P, D, etc.,then the whole formalism presented here would not be longervalid due to nonvanishing first-order energy in the multipoleapproximation.

III. NUMERICAL RESULTS AND DISCUSSION

We have applied the theory exposed above to the in-teractions of the ground state rubidium atom Rb(2S) withCO(3), OH(2), NH(1), and CH(2), and of the ground



TABLE I. Diagonal Cartesian components of the static dipole polarizabil-ities (in a.u.) for CO(3), OH(2), NH(1), and CH(2).

CO(3) OH(2) NH(1) CH(2) Referencex,xαzz 17.97 8.29 11.23 15.80 present

8.75 3015.86 63

x,xαxx 12.75 5.99 9.15 14.32 present6.37 30

x,xαyy 9.91 7.31 9.15 11.81 present7.55 30

state lithium atom Li(2S) with CH(2). As discussed inSec. I, these molecules have been successfully deceleratedand are the best candidates for sympathetic cooling by col-lisions with ultracold rubidium atoms. At present no ab initiocode allows for the calculations of all components of the dy-namic polarizability tensor for open-shell linear molecules.Therefore, in our calculations we computed the polarizabili-ties appearing in Eqs. (22) and (23) from the sum-over-statesexpansion, Eq. (16). The appropriate transitions moments tothe excited states and excitation energies were calculated us-ing linear response formalism with reference wave functionobtained from the multireference self-consistent field method

–0.65

–0.60

–0.55

–0.50

–0.45

–0.40

0 20 40 60 80 100 120 140 160 180

Ein

t [cm

–1]

Θ [degrees]

FIG. 2. Comparison of the long-range anisotropy of the potential energy sur-faces for the A′ (solid line vs circles) and A′′ (dotted line vs crosses) statesof Rb–CH(2) computed from the mutlipole expansion up to and includingR−10 and from the supermolecule calculations, intermolecular distance R =25 bohr.

(MCSCF) with large active spaces. For some electronic states,the convergence of the sum in Eq. (16) was not very fast,and we had to include over 100 states in the expansion. TheDALTON program50 was used for linear response calculations.

TABLE II. Long-range coefficients (in a.u.) for Rb–CO(3). C L ,Kn is the sum C L ,K

n (ind) + C L ,Kn (disp). The

number in parentheses denotes the power of 10.

L → 0 1 2 3 4 5 6C L0

6 (ind) 1.187(2) 1.187(2)C L0

6 (disp) 3.797(2) 6.400(1)C L0

6 4.984(2) 1.827(2)

C L26 (ind) 0

C L26 (disp) –1.415(1)

C L26 –1.415(1)

C L07 (ind) 4.646(2) 3.103(2)

C L07 (disp) 1.045(3) –1.193(2)

C L07 1.470(3) 1.913(2)

C L27 (ind) 4.027(1)

C L27 (disp) –1.550(1)

C L27 2.477(1)

C L08 (ind) 7.406(3) 6.165(3) 1.423(3)

C L08 (disp) 3.144(4) 6.753(3) 2.401(2)

C L08 3.884(4) 1.292(4) 1.663(3)

C L28 (ind) 3.168(2) 1.136(2)

C L28 (disp) –4.200(2) –4.667(1)

C L28 –1.021(2) 6.695(1)

C L09 (ind) 4.166(4) 1.995(4) 7.102(3)

C L09 (disp) 1.226(5) –2.757(3) –2.947(3)

C L09 1.642(5) 1.719(4) 4.155(3)

C L29 (ind) 3.299(3) 8.970(1)

C L29 (disp) –6.668(2) –2.376(1)

C L29 2.632(3) 6.594(1)

C L010 (ind) 5.376(5) 4.472(5) 4.844(4) 1.609(4)

C L010 (disp) 9.360(5) 7.679(5) 4.014(4) –4.700(4)

C L010 1.474(6) 1.215(6) 8.858(4) –3.091(4)

C L210 (ind) 1.592(4) 2.822(4) 3.965(3)

C L210 (disp) –4.121(4) –6.491(3) –1.094(3)

C L210 1.180(4) 2.173(4) 2.862(3)



TABLE III. Long-range coefficients (in atomic units) for Rb–OH(2). C L ,Kn is the sum C L ,K

n (ind) + C L ,Kn (disp). The number in parentheses denotes the

power of 10. The values in square brackets are the results of Ref. 27.

L → 0 1 2 3 4 5 6C L0

6 (ind) 1.339(2) [1.33(2)] 1.339(2) [1.33(2)]C L0

6 (disp) 2.154(2) [1.92(2)] 1.654(1) [1.80(1)]C L0

6 3.494(2) [3.25(2)] 1.505(2) [1.51(2)]

C L26 (ind) 0

C L26 (disp) 3.010(0) [1.90(0)]

C L26 3.010(0) [1.90(0)]

C L07 (ind) 9.460(2) 6.306(2)

C L07 (disp) 2.679(2) 1.058(2)

C L07 1.214(3) [1.04(3)] 7.365(2) [6.30(2)]

C L27 (ind) –4.040(1)

C L27 (disp) 4.800(0)

C L27 –3.560(1) [–4.00(1)]

C L08 (ind) 8.518(3) 8.188(3) 2.117(3)

C L08 (disp) 1.626(4) 2.772(3) 0.274(0)

C L08 2.487(4) 1.096(4) 2.117(3)

C L28 (ind) 3.294(2) –4.442(1)

C L28 (disp) 3.856(2) 1.397(1)

C L28 7.712(2) –3.023(1)

C L09 (ind) 7.136(4) 4.188(4) 5.980(3)

C L09 (disp) 2.865(4) 1.211(4) 6.591(2)

C L09 1.000(5) 5.399(4) 6.641(3)

C L29 (ind) –1.811(3) 1.273(2)

C L29 (disp) 1.790(2) 3.633(1)

C L29 –1.632(3) 1.637(2)

C L010 (ind) 5.815(5) 5.738(5) 1.457(5) 8.853(3)

C L010 (disp) 3.085(5) 2.046(5) 2.241(4) 1.705(3)

C L010 8.900(5) 7.784(5) 1.681(5) 1.056(4)

C L210 (ind) 2.765(4) –1.320(2) 6.332(2)

C L210 (disp) 2.167(4) 3.751(2) 8.513(1)

C L210 4.932(4) 2.431(2) 7.183(2)

We have checked the convergence of the expansion (16) bycomparison of the static parallel components obtained fromthe MCSCF calculations with the results of finite-field unre-stricted open-shell coupled cluster calculations with single,double, and noniterative triple excitations, UCCSD(T). Thefinite field calculations were done with the MOLPRO code.51

Note parenthetically that finite field calculations can correctlyreproduce the parallel component of the polarizability tensor,but fail for the perpendicular component due to the symmetrybreaking. The nondiagonal components cannot be obtainedfrom finite-field calculations. We have attempted to use themulticonfiguration interaction method restricted to single anddouble excitations (MRCI) but due to the convergence prob-lems, we could get in this way only a few (up to ten) excitedstates. Multipole moments of diatomic molecules were cal-culated as expectation values from the MRCI wave function.The Li and Rb atoms polarizabilities at imaginary frequencywere taken from highly accurate relativistic calculations fromthe group of Derevianko.52

In order to judge the quality of the computed long-rangecoefficients, we have computed cuts through the potentialenergy surfaces of Rb–CH(2) at a fixed distance R = 25bohr from the atom to the center of mass of the molecule. The

zero of the angle θ corresponds to the rubidium atom on the Hside of CH. In these calculations, we have employed thesupermolecule method. The potential was computed as thedifference,

V2S+1||(R, θ ) = ESM

AB − ESMA − ESM

B , (28)

where ESMAB denotes the energy of the dimer computed

using the supermolecule method SM, and ESMX , X = A

or B, is the energy of the molecule X. For the high-spinstates [triplet for Rb(2S)–OH(2), Rb(2S)–CH(2), andLi(2S)–CH(2), quartet for Rb(2S)–CO(3), and doubletfor Rb(2S)–NH(1)], we used the unrestricted open-shellcoupled cluster method with single, double, and nonit-erative triple excitations [UCCSD(T)]. The unrestrictedversion of CCSD(T) method was chosen instead of therestricted formulation to circumvent problem with lack ofsize-consistency when the interaction between two open-shellsystems is considered.53 Since the low-spin states have thesame asymptotic behavior as the high-spin states, therewas no need to compute the ab initio points explicitly.The UCCSD(T) calculations were done with the MOLPRO

suite of codes.51 The distances in the diatomic molecules werefixed at their equilibrium values re corresponding to the elec-



TABLE IV. Long-range coefficients (in a.u.) for Rb–NH(1). C L ,Kn is the sum C L ,K

n (ind) + C L ,Kn (disp). The

number in parentheses denotes the power of 10.

L → 0 1 2 3 4 5 6C L0

6 (ind) 1.120(2) 1.120(2)C L0

6 (disp) 2.849(2) 2.094(1)C L0

6 3.969(2) 1.329(2)

C L07 (ind) 4.436(2) 2.958(2)

C L07 (disp) 2.456(2) 1.678(2)

C L07 6.892(2) 4.636(2)

C L08 (ind) 6.107(3) 6.830(3) 1.216(3)

C L08 (disp) 2.370(4) 2.611(3) 3.100(2)

C L08 2.981(4) 9.441(3) 1.526(3)

C L48 (ind) 0

C L48 (disp) 1.416(3)

C L48 1.416(3)

C L09 (ind) 3.324(4) 2.167(4) 4.089(3)

C L09 (disp) 2.808(4) 2.036(4) 9.281(2)

C L09 6.132(4) 4.204(4) 5.017(3)

C L49 (ind) –7.744(2)

C L49 (disp) 1.442(2)

C L49 –6.302(2)

C L010 (ind) 5.116(5) 4.777(5) 8.395(4) 4.889(3)

C L010 (disp) 6.581(5) 1.656(5) 5.146(4) 1.630(3)

C L010 1.170(6) 6.433(5) 1.354(5) 6.518(3)

C L410 (ind) 4.640(4) 1.128(3)

C L410 (disp) 4.258(4) 3.692(2)

C L410 8.898(4) 1.497(3)

tronic state considered. For CO(3) re was set equal to 2.279bohr, for OH(2) 1.834 bohr, for NH(1) 1.954 bohr, and2.116 bohr for CH(2).57 The angle θ = 0 corresponds tothe linear geometries CH–Rb, CH–Li, OH–Rb, NH–Rb, andOC–Rb. In order to mimic the scalar relativistic effects, someelectrons were described by pseudopotentials. For rubidium,we took the ECP28MDF pseudopotential from the Stuttgartlibrary,54 and the spd f g quality basis set suggested in Ref. 54.For the light atoms (hydrogen, carbon, nitrogen, and oxygen),we used the aug-cc-pVQZ bases and the cc-pVQZ basis setfor lithium.55 The full basis of the dimer was used in the su-permolecule calculations and the Boys and Bernardi schemewas used to correct for the basis-set superposition error.56

Before going on with the discussion of the long-rangeinteractions in the dimers, let us compare the diagonal staticpolarizabilities of CO(3), OH(2), NH(1), and CH(2)with the literature data. In fact, the data are very scarce. ForOH, the most recent calculations of Spelsberg30 date back to1999 (see also some older references such as Refs. 59–62).For CH, the only calculation we found in the literature is the2007 paper by Manohar and Pal.63 To our knowledge, no datafor the excited states of CO and NH were reported thus far. Aninspection of Table I shows a relatively good agreement withthe results of Spelsberg30 for OH. The differences are of theorder of a few percent, 5.5% for the parallel component, and6.3% and 3.3% for the perpendicular xx and yy components,respectively. For CH, Manohar and Pal63 reported only thevalue of the parallel component obtained from the analyticalsecond derivative calculations with the Fock space multiref-

erence coupled cluster theory restricted to single and doubleexcitations. The agreement of our result with the value ofRef. 63 is remarkably good: the two results agreewithin 0.4%.

The long-range coefficients C L ,Kn for the interactions

of the ground state rubidium atom Rb(2S) with CO(3),OH(2), NH(1), and CH(2), and of the ground statelithium atom Li(2S) with CH(2) are reported in Tables II–VI. Also reported in these tables are the values of the induc-tion and dispersion coefficients C L ,K

n (ind) and C L ,Kn (disp).

First, we note that for all systems and most of the coefficientsthe induction part is as important as the dispersion. This isnot very surprising since the Li and Rb atoms are highly po-larizable, and the molecules suitable for the Stark decelerationhave large dipole moments. Note parenthetically that the co-efficient C22

6 (ind) for the interactions of the state moleculesand C24

8 (ind) for the interactions of the state moleculesvanish for symmetry reasons. The leading contribution to theanisotropy of the potentials in the long range, as measured bythe ratio C20

6 /C006 , is quite substantial for all the systems. The

ratio C206 /C00

6 ranges between 0.26 for Li–CH and Rb–CH to0.43 for Rb–OH. The difference in the anisotropy due to thepresence of terms C L2

n /Rn is relatively modest, since the co-efficients C L2

n are at least one order of magnitude smaller thanC L0

n .Comparison of the long-range anisotropy of the poten-

tial energy surfaces for the singlet and triplet A′ and A′′ statesof Rb–CH(2) computed from the mutlipole expansion up toand including R−10 and from the supermolecule calculations



TABLE V. Long-range coefficients (in atomic units) for Rb–CH(2). C L ,Kn is the sum C L ,K

n (ind) + C L ,Kn (disp).

The number in parentheses denotes the power of 10.

L → 0 1 2 3 4 5 6C L0

6 (ind) 9.654(1) 9.654(1)C L0

6 (disp) 3.888(2) 2.844(1)C L0

6 4.853(2) 1.250(2)

C L26 (ind) 0

C L26 (disp) –7.650(0)

C L26 –7.650(0)

C L07 (ind) –2.974(2) –1.982(2)

C L07 (disp) –3.874(1) 3.254(2)

C L07 –3.361(2) 1.272(2)

C L27 (ind) 2.569(1)

C L27 (disp) –9.037(0)

C L27 1.665(1)

C L08 (ind) 6.731(3) 3.341(3) 8.140(2)

C L08 (disp) 3.151(4) 7.499(3) 6.711(2)

C L08 3.824(4) 1.084(4) 1.485(3)

C L28 (ind) 1.170(2) –7.964(0)

C L28 (disp) –8.110(2) –5.038(1)

C L28 –6.940(2) –5.834(1)

C L09 (ind) –2.130(4) –1.349(4) 3.089(3)

C L09 (disp) 1.180(3) 4.095(4) 1.888(3)

C L09 –2.012(4) 2.746(4) 4.977(3)

C L29 (ind) 1.108(3) –7.006(1)

C L29 (disp) –9.600(2) –2.901(1)

C L29 1.012(3) –9.907(1)

C L010 (ind) 3.591(5) 1.630(5) 5.587(4) 3.526(3)

C L010 (disp) 8.679(5) 8.055(5) 8.953(4) 8.714(3)

C L010 1.227(6) 9.685(5) 1.454(5) 1.224(4)

C L210 (ind) 1.170(3) 8.307(2) 7.917(2)

C L210 (disp) 1.238(3) 3.143(2) –2.617(2)

C L210 2.408(3) 1.145(3) 5.300(2)

is illustrated in Fig. 2. The agreement between the long-rangeand supermolecule results is relatively good, although smalldeviations can be observed. For the A′ state, the agreementis good at angles close to 180 and slightly deteriorates for θ

around 80. The same is true for the A′′ state, showing that our

computed coefficients are not the perfect representation of theasymptotic expansion of the UCCSD(T) potential for this sys-tem. It should be stressed here that the long-range coefficientsreported in the present paper do not describe the asymptoticsof any potential obtained from supermolecule calculations,

–0.24

–0.23

–0.22

–0.21

–0.20

–0.19

–0.18

–0.17

–0.16

–0.15

0 20 40 60 80 100 120 140 160 180

Ein

t [cm

–1]

Θ [degrees] Θ [degrees]

SAPT (disp), A’ –Cn/Rn (disp), A’

SAPT (disp), A’’–Cn/Rn (disp), A’’

SAPT (ind), A’ –Cn/Rn (ind), A’

SAPT (ind), A’’–Cn/Rn (ind), A’’

–0.14

–0.12

–0.10

–0.08

–0.06

–0.04

–0.02

0.00

0 20 40 60 80 100 120 140 160 180

Ein

t [cm

–1]

FIG. 3. Comparison of the long-range dispersion (left-hand panel) and induction (right-hand panel) energies for the A′ and A′′ states of Li–CH(2) computedfrom the mutlipole expansion and from the symmetry-adapted perturbation theory (SAPT), intermolecular distance R = 25 bohr.



TABLE VI. Long-range coefficients (in atomic units) for Li–CH(2). C L ,Kn is the sum C L ,K

n (ind) +C L ,K

n (disp). The number in parentheses denotes the power of 10.

L → 0 1 2 3 4 6 5C L0

6 (ind) 4.969(1) 4.969(1)C L0

6 (disp) 2.034(2) 1.521(1)C L0

6 2.537(2) 6.491(1)

C L26 (ind) 0

C L26 (disp) –4.227(0)

C L26 –4.227(0)

C L07 (ind) –1.531(2) –1.020(2)

C L07 (disp) –2.863(1) 1.771(2)

C L07 –1.817(2) 7.502(2)

C L27 (ind) 1.322(1)

C L27 (disp) –5.003(0)

C L27 8.220(0)

C L08 (ind) 1.944(3) 5.030(2) 6.634(2)

C L08 (disp) 1.125(4) 3.670(3) 3.547(2)

C L08 1.319(4) 4.173(3) 1.018(3)

C L28 (ind) 6.025(1) –4.100(0)

C L28 (disp) –3.344(2) –2.741(1)

C L28 –2.742(2) –3.151(1)

C L09 (ind) –4.721(3) –4.619(2) 1.590(3)

C L09 (disp) 2.573(3) 1.663(4) 1.019(3)

C L09 –2.147(3) 1.619(4) 2.609(3)

C L29 (ind) 1.604(2) –3.607(1)

C L29 (disp) –5.096(1) –1.568(1)

C L29 1.095(2) –5.175(1)

C L010 (ind) 6.133(4) 1.983(4) 1.337(4) 1.815(3)

C L010 (disp) 9.074(5) 3.187(5) 3.728(4) 4.704(3)

C L010 9.687(5) 3.385(5) 5.065(4) 6.519(3)

C L210 (ind) 1.424(3) 5.150(2) 5.621(2)

C L210 (disp) 3.582(2) 2.407(2) –1.328(2)

C L210 1.782(3) 7.557(2) 4.293(2)

since for most of the supermolecule methods the long-rangeasymptotics is not known. See, e.g., Ref. 64 for a more de-tailed discussion of this point. However, the data reportedin the present paper can be used in the fits of the poten-tials, or in the case of lack of ab initio points at large dis-tances, to fix the long-range asymptotics with some switchingfunction.53

Two separate components of the long-range interactionpotential, induction, and dispersion energies, can be com-pared independently with calculations utilizing symmetry-adapted perturbation theory (SAPT).34 We have performedthe SAPT calculations using spin-restricted time-dependentHartree–Fock (TDHF) response function,65 implemented inthe SAPT programme package.66 The SAPT induction anddispersion energies for the Li-CH(2) system at a fixeddistance R = 25 bohr, and the results from the multipole ex-pansion, are compared in Fig. 3. The dispersion componentsof the SAPT interaction energy and from the multipole ex-pansion agree very well for all angles. This clearly reflectsthe fact that TDHF approach gives very accurate dynamicpolarizabilities of Li and CH. Comparison of the inductionenergies from the two approaches is less satisfactory, as theSAPT overshoots the results from multipole expansion by

5–20%. Overestimation of the induction energy in the SAPT(TDHF) can be easily attributed to the error in the dipolemoment of CH molecule obtained from the SCF method.Dipole moment of CH from the SCF calculation (0.618 a.u.) isover 10% larger than the CCSD(T) finite-field value of 0.555a.u. Difference of 10% in dipole moment of CH results inapproximately 20% increase of the SAPT induction energiesfor linear geometries. This is exactly what is observed in theright-hand panel of Fig. 3. Taking into consideration impre-cision in the SCF dipole moment, we may safely concludethat angular dependence of the dispersion and induction com-ponents of the long-range potential obtained from asymptoticexpansion coincides reasonably well with the SAPT results,which confirms correctness of the presented analytic repre-sentation of the long-range interaction energy.

To illustrate the importance of the long-range coefficientswith n > 6 in Fig. 4, we report cuts through the potentialenergy surfaces of the least (Rb–CH) and most (Rb–OH)anisotropic systems for a fixed distance R = 25 bohr. Aninspection of this figure shows that the contribution of thecoefficients beyond n = 6 is very important. For Rb–CH, theR−6 terms qualitatively reproduce the anisotropy of the poten-tial. This is not the case for Rb–OH. The inclusion of all terms



–0.65

–0.60

–0.55

–0.50

–0.45

–0.40

–0.35

–0.30

0 20 40 60 80 100 120 140 160 180

Ein

t [cm

–1]

Ein

t [cm

–1]

Rb–CH, A’’

–0.60

–0.55

–0.50

–0.45

–0.40

–0.35

–0.30

–0.25

–0.20

0 20 40 60 80 100 120 140 160 180

Rb–OH, A’’

–0.65

–0.60

–0.55

–0.50

–0.45

–0.40

–0.35

0 20 40 60 80 100 120 140 160 180

Ein

t [cm

–1]

Ein

t [cm

–1]



Rb–CH, A’ Up to C6Up to C8Up to C10

Up to C6Up to C8Up to C10



–0.60

–0.55

–0.50

–0.45

–0.40

–0.35

–0.30

–0.25

–0.20

0 20 40 60 80 100 120 140 160 180

Rb–OH, A’

FIG. 4. Importance of the R−6, R−8, and R−10 contributions to the long-range anisotropy of the potential energy surfaces for the A′ and A′′ states of Rb–CH(2) and Rb–OH(2) computed from the mutlipole expansion, intermolecular distance R = 25 bohr.

up to n = 8 gives the correct picture of the anisotropy, and theR−9 and R−10 contributions are of minor importance at thisdistance. It follows from the comparison of the UCCSD(T) re-sults with the data computed from the asymptotic expansion,cf. Fig. 2, that the short-range exchange-repulsion effects arenegligible at this distance. Thus, our illustration of Fig. 4 trulydemonstrates the importance of the R−8 and higher terms inthe multipole expansion of the interaction energy. Obviously,the importance of the contributions beyond the C6 depends onthe distance R, but our plot clearly shows that in the region ofnegligible exchange and overlap the contributions beyond C6

are important.No literature data are available for comparison, except

for the long-range coefficients for Rb–OH obtained by Laraet al.27 by fitting the CCSD(T) potential energy surfaces inthe A′ and A′′ symmetries at large distances to the functionalform of Eq. (18). The values of the long-range coefficientstaken from Ref. 27 are included in Table III. The agreementbetween the two sets of the results is very reasonable. Theisotropic C00

6 coefficients agree within 7%. The discrepanciesof the anisotropic coefficients are of the order of 10–15%.Such an agreement is satisfactory given the fact that the fit-ted values effectively account for the higher coefficients thatcould not be obtained from the fit. The only significant differ-

ence is in C226 . Here, the difference is as large as 37%, but this

coefficient is small, and most probably could not be correctlyreproduced from the fitting procedure. By contrast, the valuesof C32

7 agree relatively well within 10%.Hutson67 estimated the lowest dispersion coefficients

C006 , C20

6 , and C226 for Rb–OH by using the best available

data for the static polarizability of Rb and OH, and the Slater-Kirkwood rules. He obtained an isotropic C00

6 coefficient of149.4 a.u., 30% off our value. Such an agreement is reason-able given all the simplifications of the Slater–Kirkwood ap-proach. The values of the anisotropic coefficients C20

6 andC22

6 , 4.3 and 0.9 a.u., respectively, are four and three timessmaller than those reported in the present paper and thus unre-alistic. This shows that the applicability of simple semiempir-ical rules to anisotropic interactions in open-shell complexesis of limited utility. We have performed a similar analysis forother complexes considered in our paper and came to similarconclusions.

IV. SUMMARY AND CONCLUSIONS

In the present paper, we have formulated the theory oflong-range interactions between a ground state atom in anS state and a linear molecule in a degenerate state with a



nonzero projection of the electronic orbital angular momen-tum. We have shown that the long-range coefficients describ-ing the induction and dispersion interactions at large atom–diatom distances can be related to the first and second-ordermolecular properties. The final expressions for the long-rangecoefficients were written in terms of all components of thestatic and dynamic multipole polarizability tensor, includ-ing the nondiagonal terms connecting states with the op-posite projection of the electronic orbital angular momen-tum. It was also shown that for the interactions of moleculesin excited states that are connected to the ground state bymultipolar transition moments additional terms in the long-range induction energy appears. All these theoretical develop-ments were illustrated with the numerical results for systemsof interest for the sympathetic cooling experiments: interac-tions of the ground state Rb(2S) atom with CO(3), OH(2),NH(1), and CH(2), and of the ground state Li(2S) atomwith CH(2). Our results for the static polarizabilities of theOH and CH molecules are in a good agreement with the abinitio data from other authors.30, 63 For all systems consid-ered in the present paper, the induction contribution to thelong-range potential was found to be important. Also theanisotropy of the long-range interaction, as measured bythe ratio C20

6 /C006 , is substantial, while the anisotropy due to

the C226 is of modest importance. Relatively good agreement

between the multipole-expanded and ab initio UCCSD(T) re-sults was found. Comparison of the multipole expansion withSAPT results have shown highly satisfactory agreement fordispersion energy and relatively good for induction energy. Inthe asymptotic region, where the exchange effects are negli-gible, terms R−n with n ≥ 7 are very important and cannotbe neglected. For Rb–OH, we could compare our results withthe fit of ab initio RCCSD(T) points.27 In general, relativelygood agreement was found, except for the small C22

6 coeffi-cient. It was also found that the Slater–Kirkwood rules for theanisotropic long-range coefficients fail in the case of open-shell monomers with spatial degeneracy.

ACKNOWLEDGMENTS

We thank Bogumił Jeziorski for reading and commentingon the manuscript and Piotr S. Zuchowski for providing uswith the SAPT results.

This work was supported by the Polish Ministry of Sci-ence and Higher Education (grant 1165/ESF/2007/03).

1F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys.71, 463 (1999).

2J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod. Phys. 71,1 (1999).

3J. D. Weinstein, R. de Carvalho, T. Guillet, B. Friedrich, and J. M. Doyle,Nature (London) 395, 148 (1998).

4H. L. Bethlem, G. Berden, and G. Meijer, Phys. Rev. Lett. 83, 1558(1999).

5H. R. Thorsheim, J. Weiner, and P. S. Julienne, Phys. Rev. Lett. 58, 2420(1987).

6J. Hrbig, T. Krämer, M. Mark, T. Weber, C. Chin, H. C. Nägerl, andR. Grimm, Science 301, 1510 (2003).

7M. Greiner, C. A. Regal, and D. S. Lin, Nature (London) 426, 537 (2003).8M. W. Zwierlein, C. A. Stan, C. H. Schunk, S. M.F. Raupach, S. Gupta,Z. Hadzibabic, and W. Ketterle, Phys. Rev. Lett. 91, 250401 (2003).

9K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Peer, B. Neyenhuis,J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Science322, 231 (2008).

10H. L. Bethlem, G. Meijer, Int. Rev. Phys. Chem. 22, 73 (2003).11B. DeMarco and D. S. Jin, Science 285, 1703 (1999).12G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M. Ingus-

cio, Science 294, 1320 (2001).13S. Schlunk, A. Marian, P. Geng, A. P. Mosk, G. Meijer, and W. Schøllkopf,

Phys. Rev. Lett. 98, 233002 (2007).14S. K. Tokunaga, J. M. Dyne, E. A. Hinds, and M. R. Tarbutt, New J. Phys.

11 055038 (2009).15P. Soldan, and J. M. Hutson, Phys. Rev. Lett. 92, 163202 (2004).16M. Tacconi, E. Bodo, and F. A. Gianturco, Phys. Rev. A 75, 012708

(2007).17A. O. G. Wallis, and J. M. Hutson, Phys. Rev. Lett. 103, 183201 (2009).18P. S. Zuchowski, and J. M. Hutson, Phys. Chem. Chem. Phys. 13, 3669

(2011).19M. T. Hummon, T. V. Tscherbul, J. Kłos, H.-I. Lu, E. Tsikata, W. C.

Campbell, A. Dalgarno, and J. M. Doyle, Phys. Rev. Lett. 106, 053201(2011).

20S. K. Tokunaga, W. Skomorowski, P. S. Zuchowski, R. Moszynski, J. M.Hutson, E. A. Hinds, and M. R. Tarbutt, arXiv:1012.5029, Eur. Phys. J. D(2011), in press.

21P. S. Zuchowski and J. M. Hutson, Phys. Rev. A 78, 022701 (2008).22P. S. Zuchowski and J. M. Hutson, Phys. Rev. A 79, 062708 (2009).23C. Zipkes, S. Palzer, C. Sias, and M. Köhl, Nature (London) 464, 388

(2010).24C. Zipkes, S. Palzer, L. Ratschbacher, C. Sias, and M. Köhl, Phys. Rev.

Lett. 105, 133201 (2010).25S. Schmid, A. Härter, and J. H. Denschlag, Phys. Rev. Lett. 105, 133202

(2010).26M. Krych, W. Skomorowski, F. Pawłowski, R. Moszynski, and Z.

Idziaszek, arXiv:1008.0840, Phys. Rev. A (2011), in press.27M. Lara, J. L. Bohn, D. E. Potter, P. Soldan, J. M. Hutson, Phys. Rev. A 75,

012704 (2007).28M. Tacconi, L. Gonzalez-Sanchez, E. Bodo, and F. A. Gianturco, Phys.

Rev. A 76, 032702 (2007).29R. J. Bartlett, Ann. Rev. Phys. Chem. 32, 359 (1981).30D. Spelsberg, J. Chem. Phys. 111, 9625 (1999).31G. C. Nielson, G. A. Parker and R. T. Pack, J. Chem. Phys. 64, 2055

(1976).32B. Bussery-Honvault, F. Dayou, and A. Zanchet, J. Chem. Phys. 129,

234302 (2008).33B. Bussery-Honvault, and F. Dayou, J. Phys. Chem. A 113, 14961 (2009).34B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev. 94, 1887

(1994).35R. Moszynski, in Molecular Materials with Specific Interactions – Mod-

eling and Design, edited by W. A. Sokalski (Springer, New York, 2007),pp. 1–157.

36P. S. Zuchowski, R. Podeszwa, R. Moszynski, B. Jeziorski, and K.Szalewicz, J. Chem. Phys. 129, 084101 (2008).

37Z. Idziaszek, and P. S. Julienne, Phys. Rev. Lett. 104, 113202 (2010).38G. Quèmèner, and J. L. Bohn, Phys. Rev. A 83, 012705 (2011).39M. Przybytek, and B. Jeziorski, J. Chem. Phys. 123, 134315 (2005).40R. N. Zare, Angular momentum. Understanding Spatial Aspects in Chem-

istry and Physics (Wiley, New York, 1988).41P. E.S. Wormer, F. Mulder, and A. van der Avoird, Int. J. Quantum Chem.

11, 959 (1977).42A. van der Avoird, P. E.S. Wormer, F. Mulder, and R. M. Berns, Top. Curr.

Chem. 93, 1 (1980).43T. G. A. Heijmen, R. Moszynski, P. E. S. Wormer, and A. van der Avoird,

Mol. Phys. 89, 81 (1996).44T. Cwiok, B. Jeziorski, W. Kolos, R. Moszynski, J. Rychlewski, and

K. Szalewicz, Chem. Phys. Lett. 195, 67 (1992).45I. Røeggen, Theoret. Chim. Acta 21, 398 (1971).46R. T. Pack and J. O. Hirschfelder, J. Chem. Phys. 52, 521 (1970).47E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cam-

bridge University Press, Cambridge, 1963).48W. Skomorowski and R. Moszynski (to be published).49H. B. G. Casimir, and D. Polder, Phys. Rev. 73, 360 (1948).50T. Helgaker, H. J. A. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H.

Ågren, A. A. Auer, K. L. Bak, V. Bakken, O. Christiansen, et al.,DALTON, an ab initio electronic structure program, Release 2.0 (2005), Seehttp://www.kjemi.uio.no/software/dalton/dalton.html



51MOLPRO is a package of ab initio programs written by H.-J. Werner andP. J. Knowles, with contributions from R. D. Amos, A. Bernhardsson, A.Berning, et al.

52A. Derevianko, S. G. Porsev, and J. F. Babb, At. Data Nucl. Data Tables96, 323 (2010).

53L. M. C. Janssen, G. C. Groenenboom, A. van der Avoird, P. S. Zuchowski,and R. Podeszwa, J. Chem. Phys. 131, 224314 (2009).

54I. S. Lim, P. Schwerdtfeger, B. Metz, H. Stoll, J. Chem. Phys. 122, 104103(2005).

55D. E. Woon and T. H. Dunning, Jr, J. Chem. Phys. 94, 2975(1994).

56S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 (1970).57See http://www.nist.gov/pml/data/msd-di.58C. G. Gray and B. W. N. Lo, Chem. Phys. 14, 73 (1976).59L. Adamowicz, J. Chem. Phys. 89, 6305 (1988).

60U. Dinur, J. Mol. Struct. (Theochem) 303, 227 (1994).61S. P. Karna, J. Chem. Phys. 104, 6590 (1996).62A. D. Esposti and H. J. Werner, J. Chem. Phys. 93, 3351 (1990).63P. U. Manohar and S. Pal, Chem. Phys. Lett. 438, 321 (2007).64M. Rode, J. Sadlej, R. Moszynski, P. E.S. Wormer, and A. van der Avoird,

Chem. Phys. Lett. 314, 326 (1999).65P. S. Zuchowski, B. Bussery-Honvault, R. Moszynski, and B. Jeziorski, J.

Chem. Phys. 119, 10497 (2003).66R. Bukowski, W. Cencek, P. Jankowski, M. Jeziorska, B. Jeziorski,

S. A. Kucharski, V. F. Lotrich, A. J. Misquitta, R. Moszynski, K.Patkowski, et al., SAPT2008: An ab initio program for many-bodysymmetry-adapted perturbation theory calculations of intermolecular inter-action energies, University of Delaware and University of Warsaw (2008),http://www.physics.udel.edu/szalewic/SAPT/SAPT.html.

67J. M. Hutson, private communication (2007).


82

APPENDIX B

PAPER II

“Cold collisions of an open-shell S-state atom with a 2Π molecule:

N(4S) colliding with OH in a magnetic field”

W. Skomorowski, M. L. González-Martínez, R. Moszynski

and J.M. Hutson

Physical Chemistry Chemical Physics 13, 19077 (2011)

83

This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 19077–19088 19077

Cite this: Phys. Chem. Chem. Phys., 2011, 13, 19077–19088

Cold collisions of an open-shell S-state atom with a 2P molecule:

N(4S) colliding with OH in a magnetic field

Wojciech Skomorowski,aMaykel L. Gonzalez-Martınez,

bRobert Moszynski*

aand

Jeremy M. Hutson*b

Received 15th April 2011, Accepted 19th July 2011

DOI: 10.1039/c1cp21200a

We present quantum-theoretical studies of collisions between an open-shell S-state atom and

a 2P-state molecule in the presence of a magnetic field. We analyze the collisional Hamiltonian

and discuss possible mechanisms for inelastic collisions in such systems. The theory is applied to

the collisions of the nitrogen atom (4S) with the OH molecule, with both collision partners

initially in fully spin-stretched (magnetically trappable) states, assuming that the interaction

takes place exclusively on the two high-spin (quintet) potential energy surfaces. The surfaces

for the quintet states are obtained from spin-unrestricted coupled-cluster calculations with single,

double, and noniterative triple excitations. We find substantial inelasticity, arising from strong

couplings due to the anisotropy of the interaction potential and the anisotropic spin–spin dipolar

interaction. The mechanism involving the dipolar interaction dominates for small magnetic

field strengths and ultralow collision energies, while the mechanism involving the potential

anisotropy prevails when the field strength is larger (above 100 G) or the collision energy is

higher (above 1 mK). The numerical results suggest that sympathetic cooling of magnetically

trapped OH by collisions with ultracold N atoms will not be successful at higher

temperatures.

1 Introduction

The first experimental realization of Bose–Einstein condensation

in a dilute gas in 19951 opened up a novel and fast-growing

field of research on cold and ultracold matter. At temperatures

below about 106 K, novel properties emerge in which the

quantum nature of atoms and molecules is crucial. Although

the original experiments involved quantum-degenerate states in

atomic systems, it was soon realised that molecules, especially

those with a permanent dipole moment, offer an additional

range of applications in physics and chemistry. These include

development of new frequency standards, tests of fundamental

physical concepts such as parity and time-reversal violation,2,3

spectroscopic measurements of unprecedented accuracy,4,5

quantum information processing,6,7 and control of chemical

reactions with state-selected reagents and products.8–10

In contrast to atoms, which nowadays can be cooled

relatively easily by laser Doppler cooling and evaporative

cooling,11 molecules are incomparably more challenging

because of their complicated internal structure. Two main classes

of methods have been established to produce cold molecules:

direct methods, in which molecules are cooled from high

temperature by means of a buffer gas or external fields, and

indirect methods, in which cold molecules are formed from

precooled atoms by photoassociation or magnetoassociation.

Indirect methods can now produce ground-state molecules

at temperatures below 1 mK.12–14 It has been shown recently

that, for KRb, the rates of chemical reactions change spectacularly

between different nuclear spin states and can be dramatically

affected by applied electric fields.10 However, indirect methods

are so far restricted to alkali-metal dimers and it will be challen-

ging to extend them to other regions of the periodic table.15

Direct cooling methods can be applied to a much larger

variety of chemically interesting molecules, including OH,

NH3, CO and LiH.16–19 Stark deceleration, pioneered by

Meijer and coworkers,16 can be applied to polar molecules

with large Stark effects, while helium buffer-gas cooling20 has

been particularly successful for paramagnetic species. How-

ever, the temperatures so far achieved with direct methods are

limited to tens of millikelvin, which is not cold enough to

achieve quantum degeneracy. The development of a second-

stage cooling method for such molecules is the biggest current

challenge in the field. One of the most promising proposals is

to use sympathetic cooling, which is based on the conceptually

simple idea of bringing cold molecules into thermal contact

with a bath containing ultracold atoms. So far sympathetic

cooling has been successfully realized for ions21,22 and some

neutral atoms,23,24 but not for neutral molecules.

a Faculty of Chemistry, University of Warsaw, Pasteura 1,02-093 Warsaw, Poland. E-mail: [email protected]

bDepartment of Chemistry, Durham University, South Road, DurhamDH1 3LE, UK. E-mail: [email protected]

PCCP Dynamic Article Links

www.rsc.org/pccp PAPER

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online / Journal Homepage / Table of Contents for this issue

19078 Phys. Chem. Chem. Phys., 2011, 13, 19077–19088 This journal is c the Owner Societies 2011

Linear molecules in spatially degenerate electronic states

(P, D, etc.) are particularly attractive for Stark deceleration, as

they exhibit first-order Stark effects at moderate electric fields

(in contrast to molecules in S states, which exhibit only

second-order Stark effects). After deceleration, the molecules

can be loaded into traps where they are confined by static

electric or magnetic fields. Such static traps are not the only

way to confine cold molecular species,25 but they are experi-

mentally the most accessible. In addition, atoms in open-shell

S states (such as alkali-metal atoms, H(2S), N(4S), He(3S), and

Cr(7S)) can be held in magnetic traps and may be suitable as

coolants.

Trapping with a static field is possible only if the atom or

molecule is in a low-field-seeking state. However, the absolute

ground state is always high-field-seeking. Thus, in addition to

the elastic collisions that lead to thermalization of the sample,

there is always the possibility of inelastic collisions that

transfer the colliding partners to a lower state and release

kinetic energy. Inelastic collisions eject molecules from the

trap and may lead to the heating of the sample. The success of

sympathetic cooling therefore depends on the ratio of elastic

to inelastic events, which should preferably be as large as

possible.

Molecular sympathetic cooling was first suggested for Rb+

NH(3S).26 Subsequently, potential energy surfaces and the

appropriate collision cross sections have been calculated for a

variety of candidate systems, including Mg + NH(3S),27

Li + LiH(1S+),28,29 Rb +NH330 and He + CH2(

3B1).31 Rb +

ND3 has also been explored experimentally,32 though the

inelastic collision rate in an electric field turned out to be

too high for cooling. Studies of cold collisions with linear

molecules in a P state in the presence of external fields have

mostly been limited to cases when the second colliding partner

is closed-shell. In particular, Tscherbul et al.33 have investigated

OH + He collisions and have shown how the inelastic cross

sections can be reduced by combining electric and magnetic

fields to eliminate certain inelastic channels. Collisions of

rotationally excited OH with He in the presence of electro-

magnetic field were analyzed by Pavlovic et al.,34 while

Bohn and coworkers35 studied cold collisions between two

OH molecules with long-range dipole–dipole interactions and

concluded that the evaporative cooling of OH would be

challenging. Lara et al.36 carried out theoretical studies of

cold collisions of OH with Rb, taking account of multiple

potential energy surfaces and including the hyperfine structure

of OH. However, they did not include external field effects.

There is thus a need for rigorous quantum studies of

collisions between a P-state molecule and an open-shell

S-state atom in the presence of external fields. In this paper,

we extend the theory presented in ref. 36 and 37, to handle

this case. This theory will be applicable to a broad set of

experimentally important systems, including interactions of

molecules such as OH, NO, ClO, and CH with alkali-metal

and other magnetically trappable atoms. As an example, we

present numerical results for collisions between OH(2P) and

N(4S) in a magnetic field, with both colliding species initially in

their fully spin-stretched low-field-seeking states. OH was one

of the first molecules to be decelerated and trapped,19,38 and

many pioneering experiments with it have been reported.

Gilijamse et al.19 carried out a crossed-beam experiment,

colliding velocity-controlled OH molecules with Xe atoms;

they were able to resolve state-to-state inelastic cross sections

as a function of the collision energy. Similar experiments with

improved sensitivity have recently been performed for OH

colliding with Ar, He, and D2.39–41 An experiment to collide two

velocity-controlled beams, of OH and NO, is in preparation.42

Sawyer et al.38 have measured energy-dependent cross sections

for collisions between magnetically trapped OH and slow D2

molecules.

Tscherbul et al.43 have recently suggested that spin-polarized

nitrogen atoms are a promising coolant for sympathetic cooling

experiments. N atoms at T4 1 mK are stable against collisional

relaxation between different Zeeman levels for a wide range

of magnetic field strengths. Moreover, the low polarizability

of the N atom leads to potential energy surfaces with an

anisotropy much smaller than is usually encountered for inter-

actions with alkali-metal atoms. Theoretical and experimental

studies for collisions of magnetically trapped N(4S) and

NH(3S) have been reported,44,45 showing that the trap loss

in this system is fairly small and is caused mostly by the

anisotropic magnetic dipole–dipole interaction between the

atomic and molecular spins.

This paper is organized as follows. In Section 2 we describe

calculations of the high-spin (quintet) potential energy surfaces

resulting from interaction of the N(4S) atom with the OH(2P)

molecule. In Section 3 we describe the effective Hamiltonian

used in the dynamical calculations and give the expressions for

the matrix elements of the Hamiltonian. In Section 4 we

discuss the results of the scattering calculations and their

implications for sympathetic cooling of OH by N atoms.

Finally, Section 5 summarizes and concludes the paper.

2 Potential energy surfaces

The interaction between the N(4S) atom and the OH(2P)

molecule occurs on four adiabatic surfaces: 3A0, 3A0 0, 5A0,

and 5A0 0. The triplet surfaces have been studied extensively

to investigate the reaction N + OH - NO + H that can

take place on the 3A0 0 surface.46–51 This reaction is the major

source of the NO radical in the interstellar medium and is

one of the key elementary processes in nitrogen chemistry.

Formation of NO is barrierless, via a stable intermediate

complex NOH, and is highly exothermic with 1.83 eV energy

release. The other possible reaction channel N + OH -

NH + O is energetically forbidden for low-energy collisions.

If we neglect minor spin–orbit coupling effects between the

triplet and quintet states, the quintet surfaces are non-reactive.

To our knowledge, the quintet surfaces of N + OH have not

been reported in the literature thus far.

We have carried out calculations of the quintet surfaces

using the unrestricted version of the coupled-cluster method

with single, double, and noniterative triple excitations

[UCCSD(T)]. The unrestricted version was chosen to circumvent

the problem of the lack of size-consistency for the interaction

between two open-shell systems in spin-restricted coupled-

cluster calculations.52 The highly accurate aug-cc-pV5Z

basis set of Dunning53 was employed for all atoms and the

counterpoise procedure54 was used to correct the computed

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


interaction energies for basis-set superposition error. The

MOLPRO suite of codes55 was used in the electronic structure

calculations.

Both the 5A0 and 5A0 0 potential energy surfaces were

computed on a grid of points in Jacobi coordinates (R, y),where R is the intermolecular distance measured from the

centre of mass of 16OH to the 14N atom and y is the angle

between the vector pointing from O to H in the OH molecule

and the vector pointing from the centre of mass of OH to the

N atom. The angle y = 01 thus corresponds to the linear

O–H–N arrangement. The distance R was varied from 4.0 to

12.0a0 with an interval of 0.5a0 and from 12.0 to 20.0a0 with an

interval of 1.0a0. The angular grid points were chosen as the

set of points for 11-point Gauss–Lobatto quadrature, which

include points at y = 01 and 1801. The OH bond length was

kept fixed at the monomer equilibrium value of 1.834a0.

Contour plots of the 5A0 and 5A0 0 potential energy surfaces

are shown in Fig. 1. The shapes of the two quintet potentials

are quite similar. The global minima appear for the linear

geometry O–H–N and have a depth of 120.9 cm1. There are

also local minima 71.5 cm1 deep, which occur at the linear

N–O–H configuration. Note that for linear geometries the5A0 and 5A0 0 states are degenerate, so these minima are

common to the two surfaces. The set of stationary points of

the potentials is completed by saddle points between the two

minima, which are located at slightly different positions for

the 5A0 and 5A0 0 states. Table 1 gives the positions of the

stationary points on the surfaces and the corresponding inter-

action energies. The shapes of the quintet potential energy

surfaces for N + OH closely resemble the high-spin (quartet)

surface for N + NH reported by Zuchowski and Hutson,44

although the global minimum for N + OH is about 30 cm1

deeper than for N + NH.

To perform quantum scattering calculations, it is necessary

to expand the 5A0 and 5A0 0 surfaces in terms of angular

functions. We adopt the convention of Alexander37 and use

spherical harmonics in the Racah normalization Ck,q(y, f)(with angle f = 0) for angular representation of the potential.

For the interaction of an S-state atom with aP-state molecule,

there are nonvanishing terms with q = 0 and q = 2. The sum

of the 5A0 and 5A0 0 potentials is expanded in terms of functions

with q = 0,

1

2½VA0 ðR; yÞ þ VA00 ðR; yÞ ¼

X1k¼0

Ck;0ðy; 0ÞVk0ðRÞ; ð1Þ

while the difference between the 5A0 and 5A0 0 potentials is

expanded in terms of functions with q = 2,

1

2½VA0 ðR; yÞ VA00 ðR; yÞ ¼

X1k¼2

Ck;2ðy; 0ÞVk2ðRÞ: ð2Þ

Note that the definition of the difference potential, either

VA0 VA0 0 or VA0 0 VA0, depends in general on the symmetry

of the electronic wave functions of the interacting

subsystems.56,57 The radial functions Vkq(R) are obtained by

projecting the sum or difference onto the appropriate angular

function, using Gauss–Lobatto quadrature to perform the

numerical integration. Prior to this projection, the interpolation

to obtain VA0(R, y) and VA0 0 (R, y) at an arbitrary value of R is

done for each value of y using the reproducing kernel Hilbert

space (RKHS) procedure.58 For the quintet states of N(4S) +

OH(2P), the dominant anisotropic term in expansion (1) is

V20(R) with a well depth of approximately 28 cm1, while the

dominant term in expansion (2) comes from V22(R).

To improve the description of the potential at large R, we

use an analytic representation in this region. Each radial

component Vkq(R) is expanded at long range in terms of van

der Waals coefficients,

VkqðRÞ ¼ Xn¼6

Ckqn =R

n: ð3Þ

Fig. 1 Contour plots of the quintet interaction potentials for N + OH: 5A0 (left-hand panel) and 5A0 0 (right-hand panel). Energies are in cm1.

Table 1 Characteristic points of the interaction potentials for thequintet states of N(4S) + OH(2P)

R/a0 y/1 V/cm1 Surface

Global minimum 6.55 0.0 120.9 5A0, 5A0 0

Local minimum 6.36 180.0 71.5 5A0, 5A0 0

Saddle point 6.56 97.2 61.0 5A0

Saddle point 6.66 100.1 45.8 5A0 0

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


The expressions for the Ckqn coefficients have been given

by Skomorowski and Moszynski,57 though with a different

normalisation for q4 0 from the one used here. We calculated

the van der Waals constants up to and including n = 8, using

the method described in ref. 57. The results are listed in

Table 2. For a weakly polarizable system such as N + OH,

the neglect of higher-order coefficients with n 4 8 is fully

justified. We used the switching function of Janssen et al.,52

with parameters a=15a0 and b=25a0, to join the asymptotic

form based on the long-range coefficients and the RKHS

interpolation of the ab initio points.

3 Collision Hamiltonian

3.1 Effective Hamiltonian

We consider the case of an atom A(2s1+1S), interacting with a

diatomic molecule BC(2P), in the presence of an external

magnetic field B. The direction of the field defines the laboratory

(space-fixed) Z-axis. The system A–BC is described in Jacobi

coordinates, with the r vector connecting the heavier and

lighter of the atoms B and C, and R connecting the centre of

mass of BC and the atom A. By convention, lower-case and

capital letters are used to represent the quantum numbers of

the monomers and of the complex as a whole, respectively.

The subscripts 1 and 2 refer to the monomers A and BC,

respectively. For simplicity, the diatom will be treated as a

rigid rotor in vibrational state v, although generalization to

include its vibrations is straightforward.

The Hamiltonian describing the nuclear motions of A+ BC

in the presence of magnetic field B can be written

H ¼ h2

2mR1

d2

dR2Rþ L

2

2mR2þ Hmon þ H12; ð4Þ

where L is the space-fixed angular momentum operator

describing the end-over-end rotation of A and BC about one

another and m is the reduced mass of the complex.

Hmon contains all terms describing the isolated monomers,

i.e. Hmon = H1 + H2. H12 describes the interaction between

the monomers:

H12 = Hs1s2+ V (R,y). (5)

Here, Hs1s2accounts for the direct dipolar interaction between

the magnetic moments due to the unpaired electrons of the

monomers, and V is the intermolecular interaction potential.

Eqn (5) neglects the small dipolar interaction between the spin

of A and the orbital magnetic moment of BC.

If s1 a 0 and hyperfine terms are neglected, the Hamiltonian

for an isolated atom in the state 2s1+1S is fully determined by

the Zeeman interaction between the electron spin and the

external magnetic field,

H1 = gSmBs1B, (6)

where gS is the electron g-factor, mB is the electron Bohr

magneton, and s1 is the spin operator.

The analogous Hamiltonian for a 2P molecule can be

written59

H2 = Hrso + HZ,2 + Hl, (7)

where the rotational and spin–orbit contributions within thePstate are collapsed into the first term,

Hrso Bvn2 + Avls2. (8)

Bv and Av are the molecular rotational and spin–orbit

constants, respectively, and n is the operator of the mechanical

rotation of BC, which can be expressed as jˆ lˆ s2, where

j, lˆand s2 are the operators for the rotational, electronic orbital

and spin angular momenta, respectively. Hrso can be rewritten

Hrso = (Av + 2Bv)lzs2z + Bv[j2 + l2 + s22 2 js2 + l2z

2jzlz]. (9)

The terms l2, s22 and l2z simply shift all the levels by a constant

amount and are omitted below. The term Hl, responsible for

the L-doubling of the rotational levels of BC, is represented

by the effective Hamiltonian

Hl ¼Xq¼1

e2iqfr ½qvT22qðj; jÞ þ ðpv þ 2qvÞT2

2qðj; s2Þ; ð10Þ

where fr is the azimuthal angle associated with the electron

orbital angular momentum about the molecular axis defined

by r, while pv and qv are empirical parameters. In eqn (10), the

second-rank tensor T2q that couples two vectors k1 and k2 is

defined as

T2q ðk1; k2Þ ¼

Xq1;q2

h1; q1; 1; q2j2; qiT1q1ðk1ÞT1

q2ðk2Þ; ð11Þ

where h1, q1; 1, q2|2, qi is a Clebsch–Gordan coefficient and the

first-rank tensor components are T10(k) = kz and

T 11ðkÞ ¼ ðkx ikyÞ=

ffiffiffi2p

. If only the electron spin and orbital

contributions are taken into account, the Zeeman term is

HZ;2 ¼ gSmBs2 Bþ g0LmB l B; ð12Þ

where g0L is the orbital g-factor. For diatomic molecules of

multiplicity higher than 2 (for example 3P), an additional term

describing the intramolecular spin–spin interaction must be

included in the monomer Hamiltonian (7).

The spin–spin dipolar interaction can conveniently be

written:59

Hs1s2 ¼ g2Sm2Bðm0=4pÞffiffiffi6p X

q

ð1ÞqT2q ðs1; s2ÞT2

qðCÞ; ð13Þ

with T2q(C) = C2,q(y, f)R

3, where C2,q(y, f) is a spherical

harmonic function in the Racah normalization and (R, y, f)is the set of relative spherical coordinates of the ‘composite’

atomic and diatomic electronic spins in the space-fixed frame.

m0 is the magnetic permeability of the vacuum.

Table 2 Long-range coefficients (in atomic units) for N(4S) + OH(2P)

k - 0 1 2 3 4

Ck06 27.84 4.92

Ck26 1.23

Ck07 51.60 24.61

Ck27 6.38

Ck08 583.34 312.00 48.29

Ck28 159.09 31.42

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


3.2 Basis sets and matrix elements

The state of the BC molecule can conveniently be described

using Hund’s case (a) basis functions |l; s2s2; jomji, where s2 isthe electron spin with projection s2 onto the molecular axis

(body-fixed z axis), l is the (signed) projection of the electronic

orbital angular momentum onto the molecular axis, and j

is the angular momentum of BC with projections o onto

the molecular axis and mj onto the space-fixed Z-axis. For the

body-fixed projections we have o = l + s. The state of the

atom is characterized by the electronic spin function |s1ms1i.

The (primitive) basis set used here for the A–BC collision system

is constructed as |s1ms1i|l; s2s2; jomji|LMLi, where |LMLi are

functions describing the relative motion of A and BC in the

space-fixed reference frame.

In the presence of a magnetic field, the conserved quantities

are the projection Mtot of the total angular momentum,

Mtot = ms1+ mj + ML, and the total parity P. An electric

field would mix states of different total parities. In the absence

of an electric field it is most efficient to use a parity-adapted

basis set, |s1ms1i|s2; jomjei|LMLi, with

js2; jomjei 1ffiffiffi2p ½j1; s2s2; j omji

þ eð1Þjs2 j 1; s2 s2; j omji;ð14Þ

where o |o|, s2 = o 1 and e = 1. In this basis set,

the parity of BC is p2 = e(1)js2, and that of the triatomic

system is P = p1p2(1)L. The matrix elements of L2 and

H1 are diagonal, and given by h2L(L + 1) and gSmBms1B,

respectively.

We next give the matrix elements of all terms in the

Hamiltonian of eqn (4), although only those involving the

atomic spin are new in the present work. The terms that

do not involve atomic spin are the same as for collisions with

a closed-shell atom and were previously given by Tscherbul

et al.33 However, the published version of ref. 33 contains a

number of typographical errors, so we report the correct

expressions here.

The matrix elements of the molecular rotation/spin–orbit

operator are

hs2; jomje|Hrso|s2; jo0mjei= doo0(Av + 2Bv)(o 1) + Bv[j(j + 1) 2o2]

Bv[doo01a(j,o0)a(s2, o0 1)

+ doo0+1a+(j,o0)a+(s2, o0 1)] (15)

where we use aðj;mÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijðj þ 1Þ mðm 1Þ

pboth to

simplify the equations and to ease comparison with ref. 33.

The off-diagonal terms on the right-hand side connect different

spin–orbit manifolds related by o0 = o 1.

The L-doubling matrix elements are

hs2; jomje|Hl|s2; jo0mjei= 1

2e(1)js2a(j,o0)[do2o0qva(j,o0 1)

do1o0(pv + 2qv)a+(s2,o0 1)] (16)

and also couple states with different o. For a 2P molecule,

the first factor inside the square brackets mixes the 1/2

and 3/2 states, while the second is non-zero only for

o = o0 = 1/2.

The matrix elements of the Zeeman interaction for the

molecule BC are

hs2; jomjejHZ;2js2; j0 o0mjei

¼ mBBð1Þmjo0 ½ð2j þ 1Þð2j0 þ 1Þ1=2j 1 j0

mj 0 mj

!

gSaþðs2; o0 1Þffiffiffi

2p

j 1 j0

o 1 o0

! gS

aðs2; o0 1Þffiffiffi2p

"

j 1 j0

o 1 o0

!þ ½gSðo 1Þ þ g0L

j 1 j0

o 0 o0

!#;

ð17Þ

and mix both different rotational and different spin–orbit

states.

To determine the matrix elements of the spin–spin dipolar

interaction it is natural to expand the second-rank tensor

T2(s1,s2) as a linear combination of the products of the

space-fixed components of first-rank tensors T 1p1ðs1Þ and

T 1p2ðs2Þ. The matrix elements of T 1

p1ðs1Þ can be calculated

directly in our basis set, while for T 1p2ðs2Þ we first need to

transform from the space- to the body-fixed frame,

T 1p2ðs2Þ ¼

Xq

Dð1Þp2qðXÞT 1

qðs2Þ; ð18Þ

where DJKM is a Wigner rotation matrix and X represents the

Euler angles for the transformation. The matrix elements in

the primitive basis set are

hLML; l; s2s2; jomj ; s1ms1 jHs1s2 js1m0s1 ; l; s2s02; j0o0m0j ;L

0M0Li

¼ ffiffiffiffiffi30p

ls1s2ðRÞð1Þs1ms1

þs2s2þmjoML

½s1ðs1 þ 1Þðs2 þ 1Þs2ð2s1 þ 1Þð2s2 þ 1Þð2j þ 1Þð2j0 þ 1Þ1=2

½ð2Lþ 1Þð2L0 þ 1Þ1=2L 2 L0

0 0 0

!

Xp1;p2 ;q

1 1 2

p1 p2 p

!s1 1 s1

ms1 p1 m0s1

0@

1A

s2 1 s2

s2 q s02

!j 1 j0

mj p2 m0j

0@

1A

j 1 j0

o q o0

!L 2 L0

ML p ML0

!;

ð19Þ

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


and the corresponding matrix elements in the parity-adapted

basis set are

hLML; s2; jomje; s1ms1 jHs1s2 js1m0s1 ; s2; j0 o0m0je;L

0M0Li

¼ffiffiffiffiffi30p

ls1s2ðRÞð1Þs1ms1

þs2þmjþ2oML

½s1ðs1 þ 1Þð2s1 þ 1Þs2ðs2 þ 1Þð2s2 þ 1Þð2j þ 1Þð2j0 þ 1Þ1=2

½ð2Lþ 1Þð2L0 þ 1Þ1=2L 2 L0

0 0 0

!

Xp1;p2;q

1 1 2

p1 p2 p

!s1 1 s1

ms1 p1 m0s1

0@

1A

s2 1 s2

oþ 1 q o0 1

!j 1 j0

mj p2 m0j

0@

1A

j 1 j0

o q o0

!L 2 L0

ML p M0L

!;

ð20Þ

where p p1 + p2, ls1s2(R) = Eha30a

2/R3 is the R-dependent

spin–spin dipolar coupling constant and a E 1/137 is the fine-

structure constant.

Finally, the matrix elements of the interaction potential are

hLML; s2; jomjejVjs2; j0 o0m0je0;L0M0Li

¼ ð1Þmjo0ML ½ð2j þ 1Þð2j0 þ 1Þð2Lþ 1Þð2L0 þ 1Þ1=2

Xk;mk

1

2½1þ ee0ð1Þkð1Þmk

j k j0

mj mk m0j

0@

1A

L k L0

0 0 0

!L k L0

ML mk M0L

!

j k j0

o 0 o0

!Vk0ðRÞ

"

ð1 doo0 Þe0j k j0

o 2 o0

!Vk2ðRÞ

#;

ð21Þ

where Vk0(R) and Vk2(R) are the radial strength functions of

eqn (1) and (2). It is readily seen that states belonging to the

same spin–orbit manifold are coupled through the ‘average’ of

the A0 and A0 0 potential surfaces, while those of different

manifolds are connected through their difference. In addition,

the factor 12[1 + ee0(1)k] guarantees that states of the same

monomer parity are connected by terms Vkq(R) with even k,

while those with odd k couple rotational levels of opposite

parity. It follows from this that a strong parity-conserving

propensity rule for transitions involving different spin–orbit

manifolds can be expected.

4 Dynamical calculations

4.1 Computational details

Expanding the Schrodinger equation with the Hamiltonian of

eqn (4) in the parity-adapted basis set (14) yields a set of

coupled differential equations. We have written a plug-in for

the MOLSCAT general-purpose quantum molecular scattering

package60 to implement the matrix elements described above

for collisions between an open-shell S-state atom and a 2P-state

molecule in a magnetic field. We solved the coupled equations

numerically using the hybrid propagator of Alexander and

Manolopoulos,61 propagating from Rmin = 4a0 to Rmid =

25a0 using a fixed-step log-derivative propagator with interval

size 0.02a0 and from Rmid to Rmax = 800a0 using a variable-step

log-derivative propagator based on Airy functions. MOLSCAT

applies scattering boundary conditions at Rmax to extract

scattering S-matrices, which are then used to calculate elastic

and inelastic cross sections.

Values of the OH molecular constants in the monomer

Hamiltonian were taken from ref. 62 and 63. After performing

numerous test calculations, we decided to include basis functions

with j r 9/2 and L r 8, which gives convergence of the cross

sections to within approximately 1%.

4.2 Results

The lowest rotational state of OH in its ground X2P state at

zero field is a L doublet with j = 3/2. The doublet consists of

two states, referred to as e and f, which have opposite parity

and are separated by 0.059 cm1, with |j = 3/2, ei being the

ground state. A magnetic field splits each component of the

doublet into four states differing by the projection of the

angular momentum mj on the field axis (mj = 3/2, 1/2, 1/2,3/2). For the N atom in its 4S ground state, a magnetic field

produces four Zeeman levels, with spin projections ms1= 3/2,

1/2, 1/2 and 3/2. The combination of 8 Zeeman levels

of OH with 4 of the N atom yields 32 asymptotic levels

(thresholds), as shown in Fig. 2. In principle, even at zero

field each of the levels is further split due to hyperfine inter-

actions, although in the present work hyperfine effects are

neglected for simplicity.

N and OH can both be magnetically trapped in their spin-

stretched states, with ms1= 3/2 and mj = 3/2, respectively.

There are two such states for OH, originating from the e and f

components of the L doublet. We choose the initial state to be

|ms1= 3/2i|mj =3/2, ei, shown with a red line in Fig. 2. This is

likely to be more favourable for sympathetic cooling than

|ms1= 3/2i|mj = 3/2, fi (shown with a solid blue line in Fig. 2),

because there are fewer inelastic channels open for Zeeman

relaxation at low collision energies. In particular, there are no

transitions between the two fully spin-stretched states, from

|ms1= 3/2i|mj = 3/2, fi to |ms1

= 3/2i|mj = 3/2, ei, at

collision energies below about 85 mK. The only centrifugal

suppression in such a process, even for an incoming s wave

(Li = 0, ML,i = 0) is due to a p-wave barrier in the outgoing

channel (Lf = 1, ML,f = 0) with a height of only 11 mK,

necessitated by the change in OH monomer parity.

The interaction between collision partners that are initially

in fully spin-stretched states takes place almost entirely on the

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


quintet (high-spin) potential energy surfaces. A full description

of exit channels in which ms1+ mj has changed requires triplet

surfaces, but including these explicitly would be computation-

ally prohibitively expensive. In the present work, we effectively

approximate the triplet potential surfaces with the corresponding

quintet ones. This approximation is closely analogous to that

used for N + NH in ref. 44.

In a low-energy inelastic collision, the quantum state of

at least one of the colliding species changes and kinetic energy

is released. There are two main mechanisms that produce

inelasticity in ultracold collisions of an open-shell S-state atom

with a molecule in a 2P state. The first is direct coupling

through the anisotropy of the interaction potential, which

drives transitions to states with the molecular quantum

number mj reduced by at least 1 and the atomic spin projection

ms1unchanged. This mechanism is also present in collisions

between a closed-shell atom and a 2P molecule and has been

described by Tscherbul et al.33 The second mechanism arises

from coupling by the spin–spin dipolar interaction Hs1s2. Here,

the final Zeeman state may have quantum numbers mj and

ms1reduced by at most one. Such processes are also present

in collisions of an open-shell S-state atom with 2S or 3Smolecules, or indeed between two alkali-metal atoms. Collisions

of spin-polarized S-state atoms with 2Pmolecules thus combine

two direct mechanisms for coupling between different Zeeman

levels.

The most important contribution to coupling by the inter-

action potential comes from the anisotropic term V20(R),

which induces direct transitions from the OH state |mj =

3/2, ei to |mj = 1/2, ei and |mj = 1/2, ei. This occurs evenin the s-wave regime (Li = 0). An s-wave collision in which

ms1+ mj decreases requires ML,f Z 1 to conserve Mtot. If the

monomer parity is unchanged, conservation of total parity

then requires Lf Z 2. There is thus a centrifugal barrier in the

outgoing channel, which suppresses the inelastic cross sections

for low collision energies and low fields. For N + OH, the

centrifugal barriers are relatively high due to the low reduced

mass and small C006 coefficient: the height of the d-wave barrier

is 71 mK.

Fig. 3 shows the cross sections for Zeeman relaxation in

collisions of OH(X2P,|mj = 3/2, ei) with N(4S, |ms1= 3/2i)

for magnetic field strengths B = 10, 100, 500 and 1000 G. At

low collision energies (below 0.1 mK), the cross sections

behave according to the Wigner threshold laws:64 the elastic

cross section is constant, while the total inelastic cross section

grows with decreasing energy as E1/2. The elastic cross

section is almost unaffected by the magnetic field. At ultralow

collision energies, the inelastic cross sections are suppressed

due to centrifugal barriers in the outgoing channels, and the

total inelastic cross section grows with increasing field because

the increasing kinetic energy release helps overcome these

barriers. For example, the energy released by relaxation to

the state |ms1= 3/2i|mj = 1/2, ei at a field of 560 G is sufficient

to overcome the d-wave barrier, and Fig. 3 shows how the

inelastic cross section is enhanced for fields of 500 G and

higher in the s-wave regime. For collision energies between

4 mK and 80 mK, both the elastic and inelastic cross sections

feature numerous resonances, mostly Feshbach resonances

due to coupling with higher-energy closed channels.

As discussed above, there are two mechanisms driving

transitions between different Zeeman levels, one driven by

the spin–spin dipolar term Hs1s2and the other driven by the

anisotropy of the intermolecular potential Vkq(R). The

mechanism involving Hs1s2dominates for low fields (10 G

and below) and in the s-wave regime. For higher fields

(100 G and above), the opposite is true and the relaxation is

driven by Vkq(R). Fig. 4 shows the integral cross sections and

the s-wave contribution for the two lowest fields (10 and

100 G), with the Hs1s2term in the Hamiltonian included or

neglected. Fig. 5 compares the s-wave contributions for the

Fig. 2 Energy levels of noninteracting N(4S) + OH(X2P, j = 3/2)

in a magnetic field. The solid red and blue lines indicate the

spin-stretched low-field-seeking states |ms1= 3/2i|mj = 3/2, ei (red)

and |ms1= 3/2i|mj = 3/2, fi (blue). The dotted blue line shows state

|ms1= 3/2i|mj = 1/2, fi.

Fig. 3 Elastic and total inelastic cross sections for N + OH scatter-

ing at different magnetic field strengths B. The elastic cross section

is almost unaffected by the field strength and is shown only for

B = 10 G.

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


same two fields with those obtained by neglecting either the

spin–spin dipolar term Hs1s2or all the anisotropic terms

Vkq(R). At 10 G, Hs1s2greatly enhances inelastic processes in

the ultracold regime: at 105 K, the enhancement is almost two

orders of magnitude. However, for B = 100 G, Vkq(R) is

dominant over the whole range of energies.

The way in which the spin–spin dipolar interaction induces

transitions between different Zeeman levels is exactly parallel

to that described by Janssen et al.65,66 It is a purely long-range

effect caused by narrowly avoided crossings between the

potential adiabats at very long range, which enable transitions

between Zeeman levels without the need to penetrate

centrifugal barriers. In the present case, avoided crossings due to

the dipolar term are present between the adiabat asymptotically

correlating with the incident s-wave channel |ms1= 3/2i|mj =

3/2, ei and other adiabats correlating with the states

|ms1= 3/2i|mj = 1/2, ei, |ms1

= 1/2i|mj = 3/2, ei, and |ms1=

1/2i|mj = 1/2, ei. The p-wave and higher-L contributions to

the total inelastic cross sections are almost unaffected by the

inclusion of Hs1s2for any field and collision energy. This arises

because the long-range avoided crossings for incident channels

with centrifugal barriers are energetically inaccessible at low

energies.

Channels corresponding to different Zeeman levels are also

directly coupled by the anisotropy of the intermolecular

potential Vkq(R). Fig. 6 shows a schematic illustration of the

first-order couplings by Hs1s2and Vkq(R) for collisions

involving an incoming s wave and outgoing d waves. Because of

this, long-range avoided crossings are present even if we

neglect Hs1s2. However, the effect of the avoided crossings on

collision outcomes is much more pronounced for crossings due

Fig. 4 Comparison of the total inelastic cross sections (upper panel)

and the s-wave contributions to them (lower panel) for N + OH,

obtained with the spin–spin dipolar interaction included or neglected

in the Hamiltonian, for magnetic fields B = 10 and 100 G.

Fig. 5 Comparison of the s wave total inelastic cross sections for N + OH with those obtained with either the spin–spin dipolar term or the

anisotropy of the interaction potential neglected. Left-hand panel: B = 10 G; right-hand panel: B = 100 G.

Fig. 6 Pattern of first-order couplings between different Zeeman

levels of N(4S) + OH(X2P, j = 3/2) through the spin–spin dipolar

interaction and the anisotropy of the interaction potential, for incoming

s wave (Li = 0) and outgoing d wave (Lf = 2).

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


to Hs1s2than for those due to Vkq(R). The latter dies off much

faster with R than Hs1s2, and is one or two orders of magnitude

weaker at the positions of the long-range avoided crossings.

The ratio of the coupling strengths is approximately Hs1s2(R)/

V20(R) = Eha2R3a30/C

206 . The avoided crossings for B = 10 G

occur at distances ranging from 159 to 342a0, corresponding to

a ratio Hs1s2(R)/V20(R) between 10 and 100. It follows from an

approximate Landau–Zener model67 that the probability of

ending in a different asymptotic level after a nonadiabatic

transition is proportional to the square of the coupling

between the diabats if the coupling is relatively small.

The interplay between the spin–spin dipolar term and the

intermolecular potential terms is also manifested in the state-

to-state cross sections. Fig. 7 shows state-to-state cross

sections (s-wave contributions only) for B = 10 G and 100 G.

At B = 10 G, in the region where the Hs1s2term dominates

(below 1 mK), the most important transitions are to states

with mj orms1quantum numbers reduced by 1, which are those

coupled to the incident channel |ms1= 3/2i|mj = 3/2, ei by

Hs1s2, while for collision energies above 1 mK the dominant

inelastic channels become |ms1= 3/2i|mj = 1/2, ei and |ms1

=

3/2i|mj = 1/2, ei, which are those coupled by Vkq(R). At B

= 100 G, only channels coupled by Vkq(R) are important.

The s-wave cross sections at B = 10 G exhibit two distinct

resonant structures: a strong feature near 15 mK and a weaker

one around 41 mK. Both are Feshbach resonances caused by

coupling to closed channels arising from the f component of

the L doublet of OH. The coupling arises almost exclusively

from the V10(R) term in the intermolecular potential, which

couples states of different monomer parities. The Feshbach

resonance near 15 mK can be attributed to a bound state on

the p-wave adiabat correlating with the |ms1= 3/2i|mj = 1/2, fi

threshold, as shown in Fig. 8. This resonance moves to smaller

energies with increasing field, because the energy difference

between the |mj = 3/2, ei and |mj = 1/2, fi states (red and

dotted blue lines in Fig. 2, respectively) decreases as the field

increases. For sufficiently large field (B 4 1200 G), this

resonance will disappear as the |mj = 1/2, fi level drops below|mj = 3/2, ei. The second Feshbach resonance near 41 mK

can be attributed to a bound state on the p-wave adiabat

correlating with the |ms1= 3/2i|mj = 3/2, fi threshold. The

position of this resonance is almost unaffected by the field

strength since the energy difference between the two spin-

stretched states, |mj = 3/2, ei and |mj = 3/2, fi, is independentof magnetic field.

The fact that these are Feshbach (rather than shape)

resonances is confirmed by several observations. Firstly, the

d-wave contributions to the inelastic cross sections show

resonant structures at exactly the same energies as the s-wave

contribution. Secondly, the positions and shapes of the

Feshbach resonances can be reproduced using even the

smallest possible basis set that allows inelastic transitions,

with j r 3/2, L r 2, and potential terms Vkq(R), k r 2.

Thirdly, the presence of the V10(R) term, which does not

couple the incident and outgoing channels directly, is crucial

for the existence of the resonances. It is worth noting that no

such structure due to Feshbach resonances would be present

for collisions involving the initial state |ms1= 3/2i|mj = 3/2, fi,

with OH in the upper component of its L doublet, since no

low-lying closed channels are present in that case. However,

molecules in the f state are likely to undergo fast relaxation to

the e state in collisions driven directly by V10(R).

Fig. 9 shows the ratio of the elastic to total inelastic cross

sections as a function of collision energy. The ratio is notFig. 7 State-to-state inelastic cross sections (s-wave contribution

only) for fields of 10 G (upper panel) and 100 G (lower panel).

Fig. 8 The lowest adiabatic potential energy curves forMtot = 3 and

Lmax = 2 correlating with thresholds with the state of the N atom

unchanged (|ms1= 3/2i), at B = 10 G. Two solid horizontal lines

indicate the position of the bound states responsible for the two sharp

Feshbach resonances in the s and d-wave contribution to the inelastic

cross sections.

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


favourable for sympathetic cooling of OH by collision with

ultracold N atoms, except at fields below 10 G and collision

energies below 1 mK. The cross sections presented here may be

compared to those for N(4S) + NH(3S) by Zuchowski and

Hutson.44 The ratio of the elastic to inelastic cross sections is

at least an order of magnitude lower for N + OH than for

N + NH. Two main reasons for this can be identified. Firstly,

the spin-stretched component of the rotational ground state of

NH (3S, n = 0) is not directly coupled by the potential

anisotropy to any other accessible Zeeman level, whereas such

a coupling does exist for the ground state of OH(2P, j = 3/2)

(or any other molecule with j Z 1). Secondly, there are low-

lying states arising from the f component of the L doublet in

the OH radical that create many Feshbach resonances and

increase the inelasticity. Both effects are particularly strong for

collision energies above 10 mK, where the contributions from

p and d incoming waves to the inelastic cross sections are

dominant. For all field strengths, the ratio of elastic to inelastic

cross sections at collision energies above 1 mK is more than

10 times larger for N + NH than for N + OH.

4.3 Potential dependence

The results of scattering calculations at ultralow collision

energies are in general very sensitive to the details of inter-

action potentials. To estimate the accuracy of the calculated

potential energy surfaces for N + OH, we have carried out

additional electronic structure calculations for the geometry

corresponding to the global minimum of the potentials at the

linear N–OH geometry. In the aug-cc-pV5Z basis set, the

global minimum has a well depth of 120.9 cm1, while in

the aug-cc-pV6Z basis set this shifts to 121.8 cm1. Based on these

two results, we can estimate the complete basis set limit of the

CCSD(T) method to be 122.9 cm1, using the extrapolation

formula for correlation energy as given in ref. 68. This

corresponds to an error estimate of 1.7% for our full potential

surfaces using the aug-cc-pV5Z basis set. To estimate the error

in the correlation energy obtained from the CCSD(T) method,

we have performed full configuration-interaction (FCI) calcu-

lations with eight electrons correlated in the cc-pVDZ basis

set. The relative contribution of the FCI correction to the

CCSD(T) result should only be weakly dependent on the basis

set used, so even in the small cc-pVDZ basis set we should

obtain a reliable estimate of the FCI valence–valence correla-

tion correction. The FCI correction to the CCSD(T) result

accounts for approximately 1.5% of the interaction energy at

the global minimum. We can thus estimate the uncertainty of

our potential energy surfaces to be 4% at worst.

To assess the sensitivity of the scattering results to the

uncertainty in the interaction potential, we have carried out

calculations with the interaction potential scaled by a constant

factor l in the range 0.96 r l r 1.04, corresponding to the

estimated error bounds in the calculated potential energy

surfaces. The results at B = 10 G are shown in Fig. 10 for

collision energies of 1 mK and 10 mK. The weak dependence of

the cross sections on the potential scaling is disturbed by the

presence of sharp resonances, which occur when bound states

of the N–OH complex cross the incoming threshold (or more

precisely the collision energy) as a function of l. Two of

the peaks in the inelastic cross sections, near l = 1.010 and

l = 1.026, can be attributed to the Feshbach resonances in

the s and d partial-wave contributions discussed above. The

two additional peaks at l = 0.97 and l = 1.017, which

broaden substantially with collision energy, are due to shape

resonances in the p-wave partial cross section. If the true

potential happens to bring one of these resonances close to

zero energy, it may change the ratio of elastic to inelastic cross

sections quite dramatically. However, Fig. 10 shows that the

resonances occur in quite narrow ranges of l, so that there is a

Fig. 9 Ratio of elastic to total inelastic cross sections for N + OH at

different magnetic fields.

Fig. 10 Cross sections obtained with the interaction potential scaled

by a constant factor, V(R) - lV(R), for magnetic field B = 10 G and

collision energies of 10 mK (upper panel) and 1 mK (lower panel).

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


low probability that the true potential will be such that the

ratio of elastic to inelastic cross sections is seriously affected by

resonances for collision energies below 1 mK. It may also be

noted that the numerical results obtained with the unscaled

potential (l = 1) are fairly typical of the range expected for

N+OH on plausible interaction potentials, in the sense that the

low-energy elastic cross section (around 1000 A2) is close to

the value s = 4pa2 = 712 A2 obtained from the mean

scattering length a defined by Gribakin and Flambaum.69

5 Summary and conclusions

We have presented a theoretical study of the relaxation processes

in collisions between an atom in an open-shell S state and a

molecule in a 2P state, in a magnetic field, using the example

of N(4S) + OH(2P). The transitions between different Zeeman

levels in such collisions are driven by two mechanisms: coupling

through the spin–spin dipolar term and through the anisotropy

of the interaction potential. Both mechanisms are present in

first order. The spin–spin dipolar term dominates when both

the collision energy and the magnetic field are low, while the

anisotropy of the interaction potential dominates at higher

energies or fields. In the latter regime, the spin–spin dipolar term

can be neglected. Neglecting the dipolar interaction is equivalent

to treating the atom as closed-shell, which dramatically reduces

the cost of the scattering calculations.

An important general point is that spin relaxation collisions can

be driven directly by the anisotropy of the interaction potential

for anymolecule that has rotational angular momentum. Since the

anisotropies of atom–molecule and molecule–molecule interaction

potentials are typically quite large, this will often provide an

important trap loss mechanism for such states. For molecules

in 2P states, this is true even for the molecular ground state.

For the case of N + OH, the spin–spin dipolar term

dominates at collision energies below about 1 mK and magnetic

fields of 10 G or less. In this regime, the ratio of elastic to

inelastic cross sections is greater than 100 and thus favourable

for sympathetic cooling. However, if either the collision energy

or the magnetic field is significantly above this, inelastic processes

due to the potential anisotropy dominate and the ratio of elastic

to inelastic cross sections falls. This suggests that sympathetic

cooling of OH by collisions with N atoms is unlikely to be

successful except at collision energies below 1 mK.

Acknowledgements

The authors are grateful to the Polish Ministry of Science and

Higher Education (project N N204 215539) and to the UK

Engineering and Physical Sciences Research Council for

financial support. The collaboration was supported by the

EuroQUAM Programme of the European Science Foundation.

References

1 M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemanand E. A. Cornell, Science, 1995, 269, 198–201.

2 E. R. Hudson, H. J. Lewandowski, B. C. Sawyer and J. Ye, Phys.Rev. Lett., 2006, 96, 143004.

3 M. R. Tarbutt, J. J. Hudson, B. E. Sauer and E. A. Hinds, FaradayDiscuss., 2009, 142, 37–56.

4 S. Tojo, M. Kitagawa, K. Enomoto, Y. Kato, Y. Takasu,M. Kumakura and Y. Takahashi, Phys. Rev. Lett., 2006,96, 153201.

5 K. Enomoto, M. Kitagawa, S. Tojo and Y. Takahashi, Phys. Rev.Lett., 2008, 100, 123001.

6 P. Rabl, D. DeMille, J. M. Doyle, M. D. Lukin, R. J. Schoelkopfand P. Zoller, Phys. Rev. Lett., 2006, 97, 033003.

7 D. DeMille, Phys. Rev. Lett., 2002, 88, 067901.8 R. V. Krems, Phys. Chem. Chem. Phys., 2008, 10, 4079–4092.9 J. M. Hutson, Science, 2010, 327, 788–789.10 S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda,

B. Neyenhuis, G. Quemener, P. S. Julienne, J. L. Bohn, D. S. Jinand J. Ye, Science, 2010, 327, 853–857.

11 C. N. Cohen-Tannoudji, Rev. Mod. Phys., 1998, 70, 707–719.12 K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er,

B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne,D. S. Jin and J. Ye, Science, 2008, 322, 231–235.

13 J. Deiglmayr, A. Grochola, M. Repp, K. Mortlbauer, C. Gluck,J. Lange, O. Dulieu, R. Wester and M. Weidemuller, Phys. Rev.Lett., 2008, 101, 133004.

14 J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, R. Hart,J. Aldegunde, J. M. Hutson and H.-C. Nagerl, Nat. Phys., 2010, 6,265–270.

15 P. S. Zuchowski, J. Aldegunde and J. M. Hutson, Phys. Rev. Lett.,2010, 105, 153201.

16 H. L. Bethlem, G. Berden and G. Meijer, Phys. Rev. Lett., 1999,83, 1558–1561.

17 H. L. Bethlem, G. Berden, F. M. H. Crompvoets, R. T. Jongma,A. J. A. van Roij and G. Meijer, Nature, 2000, 406, 491.

18 S. K. Tokunaga, J. M. Dyne, E. A. Hinds and M. R. Tarbutt, NewJ. Phys., 2009, 11, 055038.

19 J. J. Gilijamse, S. Hoekstra, S. Y. T. van de Meerakker,G. C. Groenenboom and G. Meijer, Science, 2006, 313, 1617–1620.

20 J. D. Weinstein, R. deCarvalho, T. Guillet, B. Friedrich andJ. M. Doyle, Nature, 1998, 395, 148–150.

21 D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano andD. J. Wineland, Phys. Rev. Lett., 1986, 57, 70–73.

22 A. Ostendorf, C. B. Zhang, M. A. Wilson, D. Offenberg, B. Rothand S. Schiller, Phys. Rev. Lett., 2006, 97, 243005.

23 C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell andC. E. Wieman, Phys. Rev. Lett., 1997, 78, 586–589.

24 G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni andM. Inguscio, Science, 2001, 294, 1320–1322.

25 J. van Veldhoven, H. L. Bethlem and G. Meijer, Phys. Rev. Lett.,2005, 94, 083001.

26 P. Soldan and J. M. Hutson, Phys. Rev. Lett., 2004, 92, 163202.27 A. O. G. Wallis and J. M. Hutson, Phys. Rev. Lett., 2009,

103, 183201.28 S. K. Tokunaga, W. Skomorowski, P. S. Zuchowski,

R. Moszynski, J. M. Hutson, E. A. Hinds and M. R. Tarbutt,Eur. Phys. J. D, 2011, DOI: 10.1140/epjd/e2011-10719-x.

29 W. Skomorowski, F. Pawlowski, T. Korona, R. Moszynski,P. S. Zuchowski and J. M. Hutson, J. Chem. Phys., 2011,134, 114109.

30 P. S. Zuchowski and J. M. Hutson, Phys. Rev. A: At., Mol., Opt.Phys., 2009, 79, 062708.

31 T. V. Tscherbul, H.-G. Yu and A. Dalgarno, Phys. Rev. Lett.,2011, 106, 073201.

32 L. P. Parazzoli, N. J. Fitch, P. S. Zuchowski, J. M. Hutson andH. J. Lewandowski, Phys. Rev. Lett., 2011, 106, 193201.

33 T. V. Tscherbul, G. C. Groenenboom, R. V. Krems andA. Dalgarno, Faraday Discuss., 2009, 142, 127–141.

34 Z. Pavlovic, T. V. Tscherbul, H. R. Sadeghpour,G. C. Groenenboom and A. Dalgarno, J. Phys. Chem. A, 2009,113, 14670–14680.

35 A. V. Avdeenkov and J. L. Bohn, Phys. Rev. A: At., Mol., Opt.Phys., 2002, 66, 052718.

36 M. Lara, J. L. Bohn, D. E. Potter, P. Soldan and J. M. Hutson,Phys. Rev. A: At., Mol., Opt. Phys., 2007, 75, 012704.

37 M. H. Alexander, J. Chem. Phys., 1982, 76, 5974–5988.38 B. C. Sawyer, B. K. Stuhl, D. Wang, M. Yeo and J. Ye, Phys. Rev.

Lett., 2008, 101, 203203.39 M. Kirste, L. Scharfenberg, J. K"os, F. Lique, M. H. Alexander,

G. Meijer and S. Y. T. van de Meerakker, Phys. Rev. A: At., Mol.,Opt. Phys., 2010, 82, 042717.

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online


40 L. Scharfenberg, J. K"os, P. J. Dagdigian, M. H. Alexander,G. Meijer and S. Y. T. van de Meerakker, Phys. Chem. Chem.Phys., 2010, 12, 10660–10670.

41 L. Scharfenberg, S. Y. T. van de Meerakker and G. Meijer, Phys.Chem. Chem. Phys., 2011, 13, 8448–8456.

42 S. Y. T. van de Meerakker and G. Meijer, Faraday Discuss., 2009,142, 113–126.

43 T. V. Tscherbul, J. K"os, A. Dalgarno, B. Zygelman, Z. Pavlovic,M. T. Hummon, H.-I. Lu, E. Tsikata and J. M. Doyle, Phys. Rev.A: At., Mol., Opt. Phys., 2010, 82, 042718.

44 P. S. Zuchowski and J. M. Hutson, Phys. Chem. Chem. Phys.,2011, 13, 3669–3680.

45 M. T. Hummon, T. V. Tscherbul, J. K"os, H.-I. Lu, E. Tsikata,W. C. Campbell, A. Dalgarno and J. M. Doyle, Phys. Rev. Lett.,2011, 106, 053201.

46 F. Pauzat, Y. Ellinger, G. Berthier, M. Grin and Y. Viala, Chem.Phys., 1993, 174, 71–79.

47 D. Sengupta and A. K. Chandra, J. Chem. Phys., 1994, 101,3906–3915.

48 R. Guadagnini, G. C. Schatz and S. P. Walch, J. Chem. Phys.,1995, 102, 774–783.

49 R. Guadagnini, G. C. Schatz and S. P. Walch, J. Chem. Phys.,1995, 102, 784–791.

50 M. Jorfi, P. Honvault and P. Halvick, J. Chem. Phys., 2009,131, 094302.

51 D. Edvardsson, C. F. Williams and D. C. Clary, Chem. Phys. Lett.,2006, 431, 261–266.

52 L. M. C. Janssen, G. C. Groenenboom, A. van der Avoird,P. S. Zuchowski and R. Podeszwa, J. Chem. Phys., 2009,131, 224314.

53 T. H. Dunning, Jr., J. Chem. Phys., 1989, 90, 1007–1023.54 S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, 553.55 H.-J. Werner, P. J. Knowles, F. R. Manby, R. Lindh, M. Schutz,

P. Celani, T. Korona, A. Mitrushenkov, G. Rauhut, T. B. Adler,R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper,M. J. O. Deegan, A. J. Dobbyn, E. Goll, F. Eckert, C. Hampel,

G. Hetzer, T. Hrenar, G. Knizia, C. Koppl, Y. Liu, A. W. Lloyd,R. A. Mata, A. J. May, S. J. McNicholas, W. Meyer, M. E. Mura,A. Nicklass, P. Palmieri, K. Pfluger, R. Pitzer, M. Reiher,U. Schumann, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson,M. Wang and A. Wolf, MOLPRO, version 2008.1, a package ofab initio programs, 2008.

56 G. C. Nielson, G. A. Parker and R. T Pack, J. Chem. Phys., 1976,64, 2055–2061.

57 W. Skomorowski and R. Moszynski, J. Chem. Phys., 2011,134, 124117.

58 T.-S. Ho and H. Rabitz, J. Chem. Phys., 1996, 104, 2584–2597.59 J. M. Brown and A. Carrington, Rotational Spectroscopy of

Diatomic Molecules, Cambridge University Press, Cambridge,U.K., 2003.

60 J. M. Hutson and S. Green,MOLSCAT computer program, version14, distributed by Collaborative Computational Project No. 6 ofthe UK Engineering and Physical Sciences Research Council, 2006.

61 M. H. Alexander and D. E. Manolopoulos, J. Chem. Phys., 1987,86, 2044–2050.

62 C. Ticknor and J. L. Bohn, Phys. Rev. A: At., Mol., Opt. Phys.,2005, 71, 022709.

63 J. M. Brown, K. Kaise, C. M. L. Kerr and D. J. Milton, Mol.Phys., 1978, 36, 553.

64 E. P. Wigner, Phys. Rev., 1948, 73, 1002–1009.65 L. M. C. Janssen, P. S. Zuchowski, A. van der Avoird,

G. C. Groenenboom and J. M. Hutson, Phys. Rev. A: At., Mol.,Opt. Phys., 2011, 83, 022713.

66 L. M. C. Janssen, A. van der Avoird and G. C. Groenenboom,Eur. Phys. J. D, 2011, DOI: 10.1140/epjd/e2011-20093-4.

67 E. E. Nikitin, in Handbook of Atomic, Molecular and OpticalPhysics, ed. G. W. F. Drake, Springer Science + Business Media,New York, USA, 2006, pp. 741–752.

68 K. L. Bak, A. Halkier, P. Jørgensen, J. Olsen, T. Helgaker andW. Klopper, J. Mol. Struct., 2001, 567, 375–384.

69 G. F. Gribakin and V. V. Flambaum, Phys. Rev. A: At., Mol., Opt.Phys., 1993, 48, 546.

Dow

nloa

ded

by U

nive

rsity

of

War

saw

on

25 O

ctob

er 2

011

Publ

ishe

d on

15

Aug

ust 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C1C

P212

00A

View Online

96

APPENDIX C

PAPER III

“Interaction between LiH molecule and Li atom from state-of-the-art

electronic structure calculations”

W. Skomorowski, F. Pawłowski, T. Korona, R. Moszynski,

P.S. Zuchowski and J.M. Hutson


97


Interaction between LiH molecule and Li atom from state-of-the-artelectronic structure calculations

Wojciech Skomorowski,1 Filip Pawłowski,1 Tatiana Korona,1 Robert Moszynski,1,a)

Piotr S. Zuchowski,2 and Jeremy M. Hutson2

1Faculty of Chemistry, University of Warsaw, Pasteura 1, Warsaw 02-093, Poland2Department of Chemistry, Durham University, South Road, Durham DH1 3LE, United Kingdom

(Received 22 September 2010; accepted 18 February 2011; published online 18 March 2011)

State-of-the-art ab initio techniques have been applied to compute the potential energy surface for thelithium atom interacting with the lithium hydride molecule in the Born–Oppenheimer approximation.The interaction potential was obtained using a combination of the explicitly correlated unrestrictedcoupled-cluster method with single, double, and noniterative triple excitations [UCCSD(T)-F12] forthe core–core and core–valence correlation and full configuration interaction for the valence–valencecorrelation. The potential energy surface has a global minimum 8743 cm−1 deep if the Li–H bondlength is held fixed at the monomer equilibrium distance or 8825 cm−1 deep if it is allowed tovary. In order to evaluate the performance of the conventional CCSD(T) approach, calculations werecarried out using correlation-consistent polarized valence X -tuple-zeta basis sets, with X rangingfrom 2 to 5, and a very large set of bond functions. Using simple two-point extrapolations basedon the single-power laws X−2 and X−3 for the orbital basis sets, we were able to reproduce theCCSD(T)–F12 results for the characteristic points of the potential with an error of 0.49% at worst.The contribution beyond the CCSD(T)–F12 model, obtained from full configuration interaction cal-culations for the valence–valence correlation, was shown to be very small, and the error bars on thepotential were estimated. At linear LiH–Li geometries, the ground-state potential shows an avoidedcrossing with an ion-pair potential. The energy difference between the ground-state and excited-state potentials at the avoided crossing is only 94 cm−1. Using both adiabatic and diabatic pictures,we analyze the interaction between the two potential energy surfaces and its possible impact onthe collisional dynamics. When the Li–H bond is allowed to vary, a seam of conical intersectionsappears at C2v geometries. At the linear LiH–Li geometry, the conical intersection is at a Li–Hdistance which is only slightly larger than the monomer equilibrium distance, but for nonlinear ge-ometries it quickly shifts to Li–H distances that are well outside the classical turning points of theground-state potential of LiH. This suggests that the conical intersection will have little impact onthe dynamics of Li–LiH collisions at ultralow temperatures. Finally, the reaction channels for theexchange and insertion reactions are also analyzed and found to be unimportant for the dynamics.© 2011 American Institute of Physics. [doi:10.1063/1.3563613]

I. INTRODUCTION

Ultracold molecules offer new opportunities for scientificexploration, including studies of molecular Bose–Einsteincondensates, novel quantum phases, and ultracold chemistry.For molecular interactions that take place at micro-Kelvintemperatures, even the smallest activation energy exceedsthe available thermal energy. This opens up new possibilitiesfor controlling the pathways of chemical reactions (see, e.g.,Ref. 1).

A major objective of current experiments on coldmolecules is to achieve quantum degeneracy, particularlyfor polar molecules. Two approaches are being pursued: in-direct methods in which molecules are formed from pre-cooled atomic gases, and direct methods in which moleculesare cooled from room temperature. There have been very


substantial recent advances, particularly in indirect methods.In particular, the JILA (Ref. 2) and Innsbruck3 groups haveformed deeply bound ground-state molecules at temperaturesbelow 1 μK by magnetoassociation of pairs of ultracold atomsfollowed by coherent state transfer with lasers. Methods thatform ultracold molecules from ultracold atoms are howeverrestricted at present to species formed from heavy alkali-metalatoms.

Direct methods, such as buffer-gas cooling,4 Starkdeceleration,5 crossed-beam collisional cooling,6 and Max-well extraction,7 are applicable to a much larger variety ofchemically interesting molecules. However, these methodscannot yet reach temperatures below 10–100 mK. Findinga way to cool these molecules further, below 1 mK, is oneof the biggest challenges facing the field. The most promis-ing possibility is so-called sympathetic cooling in which coldmolecules are introduced into an ultracold atomic gas andthermalize with it. Sympathetic cooling has been success-fully used to achieve Fermi degeneracy in 6Li,8 Bose–Einstein



114109-2 Skomorowski et al. J. Chem. Phys. 134, 114109 (2011)

condensation in 41K,9 and for producing ultracold ions.10–12

However, it has not yet been achieved for molecular systems,although there are theoretical proposals for experiments inwhich ultracold NH or ND3 molecules are obtained by col-lisions with a bath of colder atoms such as Rb, Mg, or N.13–15

The group at Imperial College London recently suc-ceeded in producing samples of cold LiH molecules in thefirst rotationally excited state16, 17 using Stark deceleration.LiH is an attractive molecule for cooling, since it has largedipole moment and light mass, so that it can be controlledeasily with fields. It has a relatively large rotational con-stant (7.52 cm−1), which opens up the possibility of produc-ing cold molecules in a single excited rotational state. There isproposal to produce ultracold LiH molecules by sympatheticcooling with Li.18 However, sympathetic cooling can be suc-cessful only if the rate of elastic (thermalization) collisionsis large compared to the rate of inelastic (deexcitation) colli-sions, which causes trap loss. The main objects of the presentpaper are to explore the interaction between Li atoms and LiHmolecules, to understand the nature of the interaction betweenthese two species, and to obtain a detailed and accurate poten-tial energy surface for the Li–LiH system.

The results of scattering calculations at ultralow tem-perature are very sensitive to the details of the interactionpotential.13, 15 For systems containing heavy atoms, the meth-ods of quantum chemistry currently available cannot generateinteraction potentials with accuracy better than a few percent.This limitation is caused by approximate treatments of corre-lation effects and relativistic contributions. With potential en-ergy surfaces of moderate precision, it is usually possible toextract only qualitative information from low-energy collisioncalculations. By contrast, Li–LiH is a light system containingonly seven electrons, and state-of-the-art ab initio electronicstructure calculations can be performed with no significantapproximations. It therefore offers a unique possibility to pro-duce a very precise interaction potential, which will allow aquantitative description of Li–LiH collision dynamics even inthe ultralow temperature regime.

In electronic structure calculations, one aims at approach-ing the exact solution of the Schrödinger equation as closelyas possible within the algebraic approximation. In practice,this is accomplished by combining hierarchies of one-electronand N -electron expansions. The accuracy increases across thehierarchies in a systematic manner, allowing the errors in thecalculations to be controlled and a systematic approach tothe exact solution to be achieved. The standard N -electronhierarchy employed in electronic structure calculations con-sists of the Hartree–Fock (HF), second-order Møller–Plessetperturbation theory (MP2), coupled-cluster with single anddouble excitations (CCSD), and coupled-cluster with sin-gle, double, and approximate noniterative triple excitations[CCSD(T)] models, with the latter recovering most of the cor-relation energy. Thus, CCSD(T) constitutes a robust and accu-rate computational tool nowadays. All these models are size-consistent, which means that the interaction potential showsthe correct dissociation behavior at large intermolecular dis-tances. In contrast, methods based on the configuration in-teraction approach with a restricted excitation space such asmultireference configuration interaction limited to single and

double excitations (MRCISD) are not size-consistent; there-fore, they are not well suited for calculations of the interactionenergy.

The most popular example of a one-electron hierar-chy is the family of Dunning correlation-consistent polar-ized valence basis sets, cc-pVXZ (Ref. 19) with the cardi-nal number X going from D (double-zeta), through T indi-cating triple-zeta, and so on. These have successfully beencombined with the HF, MP2, CCSD, and CCSD(T) hierar-chy of wave function models for the calculation of variousmolecular properties.20–22 The basis-set limit, correspondingto X → ∞, may be approached either by extrapolating the re-sults obtained with finite cardinal numbers toward infinite X(Refs. 23 and 24) or by replacing the standard one-electron hi-erarchy by explicitly-correlated methods, such as CCSD–F12and CCSD(T)–F12,25–30 in which the interelectron distancer12 is explicitly introduced into the wave function.31–33 TheF12 methods have recently been implemented efficiently34–37

and shown to accelerate the convergence toward the basis-setlimit for a number of properties.38–40

In the present paper, we combine all-electron spin-unrestricted CCSD(T)–F12 calculations with frozen-core fullconfiguration interaction (FCI) calculations to yield a highlyaccurate best estimate of the Li–LiH interaction potential. Wealso compare the F12 interaction energies with results ob-tained from standard (not explicitly correlated) CCSD(T) cal-culations. We then characterize the ground-state potential, an-alyze possible interactions with excited states, and investigatechannels for reactive collisions.

II. COMPUTATIONAL DETAILS

We have calculated the interaction energies between thelithium atom and the lithium hydride molecule in Jacobi co-ordinates (R, r, θ ), defined for the isotopic combination 7Li–7Li1H. Calculations were performed for states of 2 A′ symme-try in the Cs point group. The Li–H bond distance, r , wasinitially kept frozen at the LiH monomer equilibrium distanceof 3.014 bohr.41 The distance R between Li and the centerof mass of LiH ranged from 3.0 to 10.0 bohr with an in-terval of 0.5 bohr and then from 11.0 to 20.0 bohr with aninterval of 1.0 bohr. Additional distances of 30.0, 40.0, and50.0 bohr were also used. The angle θ , between the vectorpointing from Li to H in the LiH molecule and the vectorpointing from the center of mass of the molecule to the Liatom, was varied from 0 to 180 with an interval of 15;θ = 0 corresponds to Li–H–Li configurations. We thus useda total of 28 intermonomer distances, R, which combinedwith the 13 values of θ yielded 364 grid points on the two-dimensional interaction energy surface.

Calculations with uncorrelated basis functions were car-ried out using the unrestricted version of the coupled-clustermodel CCSD(T) with Dunning’s cc-pVXZ(-mid) basis setswith X = D, T, Q, and 5, where mid indicates the inclu-sion of an additional set of basis functions, the so-calledmidbond-95 set,42 placed at the middle of the Li–LiH distanceR. All electrons were correlated in these calculations. Addi-tionally, for the purpose of comparison with the FCI results


114109-3 State-of-the-art ab initio PES for Li-LiH J. Chem. Phys. 134, 114109 (2011)

(see below), the frozen-core approximation (1σLiH and 1sLi

orbitals kept frozen) was used for the cc-pVQZ basis.All these calculations were carried out using the MOLPRO

package.43 The full basis set of the dimer was used inthe supermolecular calculations and the Boys and Bernardischeme44 was used to correct for basis-set superpositionerror.

The explicitly correlated spin-unrestricted CCSD–F12and CCSD(T)–F12 (Refs. 34, 35, 45, and 46) calculationswere carried out with the MOLPRO code43 to establish theCCSD and CCSD(T) basis-set limits for the LiH–Li inter-action. We chose to use the F12b variant35, 46 of the ex-plicitly correlated spin-unrestricted energy implemented inthe MOLPRO code. Employing the fixed-amplitude ansatz forthe F12 wave function ensured the orbital invariance andsize-consistency of the CCSD-F12 and CCSD(T)-F12 results.The QZVPP basis set47 was employed as the orbital basisin the F12 calculations. The corresponding QZVPP-jk basisset48 was used as the auxiliary basis for the density-fittingapproximation45, 49 for many-electron integrals, while the un-contracted version of the QZVPP-jk basis was used to approx-imate the resolution-of-identity in the F12 integrals.50, 51 Inaddition, the valence correlation in the dimer was describedwith the full configuration interaction method (FCI). The FCIand standard CCSD(T) calculations in the frozen-core ap-proximation were carried out using the cc-pVQZ basis. TheDALTON package52 and the LUCIA program53 were combinedto yield the FCI results.

To calculate potential energy surface V (R, θ ) with theLi–H bond length kept fixed at its equilibrium value, we usedcomputational scheme which was previously applied in theo-retical studies of the ground and excited states of the calciumdimer.54–58 The potential V (R, θ ) was constructed accordingto the following expression:

V (R, θ ) = V CCSD(T)−F12(R, θ ) + δV FCIv−v(R, θ ), (1)

where V CCSD(T)−F12(R, θ ) contribution was obtained fromall-electron CCSD(T)-F12 calculations, while the correctionfor the valence-valance correlation beyond the CCSD(T)-F12level, δV FCI

v−v(R, θ ), was calculated in an orbital cc-pVQZ ba-sis set. Both terms, V CCSD(T)−F12(R, θ ) and δV FCI

v−v(R, θ ), wereobtained from the standard expressions for the supermoleculeinteraction energy, as given in Ref. 57.

The long-range asymptotic form of the potentials is ofprimary importance for cold collisions. We have thereforecomputed the leading long-range coefficients that describethe induction and dispersion interactions up to and includingR−10 and l = 4 terms,

V (R, θ ) = −10∑

n=6

n−4∑l=0

Cln

RnPl (cos θ ), (2)

where l is even/odd for n even/odd, and Cln = Cl

n(ind)+ Cl

n(disp). The long-range coefficients Cln(ind) and Cl

n(disp)are given by the standard expressions (see, e.g., Refs. 59and 60). The multipole moments and polarizabilities of LiHwere computed with the recently introduced explicitly con-nected representation of the expectation value and polar-ization propagator within the coupled-cluster method,61–63

while the Li polarizabilities (both static and at imaginaryfrequencies) were taken from highly accurate relativistic cal-culations from Derevianko et al.64

The interaction potentials were interpolated between cal-culated points using the reproducing kernel Hilbert spacemethod (RKHS),65 with the asymptotics fixed using theab initio long-range Van der Waals coefficients. The switchingfunction of Ref. 66 was used to join the RKHS interpolationsmoothly with the Van der Waals part in the interval betweenRa = 18 and Rb = 26 bohr.

III. CONVERGENCE OF THE LI–LIH INTERACTIONPOTENTIAL TOWARD THE EXACT SOLUTION

In Sec. III A, we analyze the convergence of the Li–LiH interaction potential with respect to the one-electron andN -electron hierarchies. Based on the analysis, we give inSec. III B our best estimate for the ground-state interactionpotential with the Li–H bond length fixed at its monomerequilibrium value. The features of the potential are presentedin Sec. III C.

A. Convergence of the one-electron andN-electron hierarchies

In order to investigate the saturation of the Li–LiH in-teraction energy in the one-electron space, we have ana-lyzed four characteristic points of the Li–LiH potential (theglobal minimum, the saddle point, the local minimum, andone point very close to the avoided crossing: R = 5.5 bohrand θ = 0.0). The characteristic points were obtained fromthe potentials calculated at the CCSD(T)/cc-pVXZ-mid levelof theory, for X = D, T, Q, and 5. The interaction en-ergies were then compared to the corresponding energiesof the spin-unrestricted CCSD(T)-F12/QZVPP potential (ap-proximation F12b), which serves as the basis-set limit. Toevaluate the accuracy of the pure one-electron basis (notexplicitly correlated), the relative percentage errors, F12b

= (V cc−pVXZ − V F12b)/|V F12b| × 100%, were determinedfor each X at every characteristic point. The results are givenin Table I.

We have also evaluated the characteristic points from theextrapolated interaction energy surfaces, which were gener-ated as follows: at each grid point, the extrapolated total en-ergies for Li, LiH, and Li–LiH were obtained by adding theHartree–Fock energy calculated with cardinal number X tothe extrapolated correlation energy, Ecorr

(X−1)X , obtained fromthe two-point extrapolation formula,23, 24

Ecorr(X−1)X = Ecorr

X + EcorrX − Ecorr

(X−1)

[1 − (X )−1]−α − 1, (3)

where Ecorr(X−1) and Ecorr

X are the correlation energies obtainedfor two consecutive cardinal numbers, (X − 1) and X , re-spectively. The final extrapolated interaction energy at a sin-gle grid point is obtained by subtracting the Li and LiH ex-trapolated total energies from the Li–LiH extrapolated to-tal energy. We used the values α = 2 and α = 3, whichwere recommended by Jeziorska et al. in their helium dimerstudy67, 68 as the ones most suited for extrapolating all the



TABLE I. Performance of various orbital basis sets and extrapolation schemes compared to the CCSD(T)-F12a and CCSD(T)-F12b results at the charac-teristic points of the ground state potential energy surface of Li–LiH. The notation VXZ with X=D, T, Q, and 5 denotes the result of the orbital CCSD(T)calculations in the cc-pVXZ basis with midbond-95 set, XYα with X and Y = D, T, Q, and 5, and α = 2 or 3 denote the extrapolated result according toEq. (3), while F12a and F12b stand for the explicitly correlated CCSD(T) results with the a and b approximation schemes. F12a and F12b are the percenterror of given result with respect to the CCSD(T)-F12a and CCSD(T)-F12b results, respectively.

V (cm−1) F12a F12b V (cm−1) F12a F12b V (cm−1) F12a F12b V (cm−1) F12a F12b

Global minimum Saddle point Local minimum Avoided crossing

VDZ –8547.77 1.87 1.84 –1548.64 –0.39 –0.61 –1616.54 –0.65 –0.77 –4771.50 4.59 4.65VTZ –8652.38 0.67 0.64 –1566.01 –1.52 –1.73 –1629.62 –1.46 –1.59 –4892.57 2.17 2.23VQZ –8683.85 0.31 0.27 –1555.93 –0.86 –1.08 –1618.22 –0.75 –0.88 –4955.39 0.91 0.97V5Z –8698.84 0.14 0.10 –1551.91 –0.60 –0.82 –1612.70 –0.41 –0.53 –4974.35 0.53 0.60

DT2 –8683.31 0.32 0.28 –1575.59 –2.14 –2.36 –1637.22 –1.94 –2.06 –4934.30 1.33 1.40TQ2 –8704.14 0.08 0.04 –1541.85 0.05 –0.17 –1601.47 0.29 0.17 –4983.15 0.36 0.42Q52 –8714.77 –0.05 –0.08 –1543.21 –0.04 –0.25 –1601.76 0.27 0.15 –4991.70 0.19 0.25

DT3 –8668.66 0.48 0.45 –1571.05 –1.84 –2.06 –1633.62 –1.71 –1.84 –4914.53 1.73 1.79TQ3 –8695.37 0.18 0.14 –1547.94 –0.35 –0.56 –1608.71 –0.16 –0.28 –4971.14 0.60 0.66Q53 –8707.24 0.03 0.01 –1546.78 –0.27 –0.49 –1606.24 –0.01 –0.13 –4984.59 0.33 0.39

F12a –8710.85 0.00 –0.04 –1542.60 0.00 –0.21 –1606.14 0.00 –0.12 –5001.10 0.00 0.06F12b –8707.77 0.04 0.00 –1539.31 0.21 0.00 –1604.17 0.12 0.00 –5004.15 –0.06 0.00

components of the interaction energy. The energies of thecharacteristic points obtained in this way were compared withthe CCSD(T)-F12/QZVPP results and the corresponding val-ues of F12b are included in Table I.

The relative percentage errors, F12b, are plotted inFig. 1 for both plain (nonextrapolated) and extrapolated char-acteristic points. For the global minimum, the plain cc-pVXZ results approach the basis-set limit from above andthe convergence is smooth and fast: the error is reduced bya factor of 2–3 for each increment in X . The extrapolationaccelerates the convergence: the (X − 1)X extrapolated in-teraction energies have a quality at least that of the plaincc-pV(X + 1)Z results. Though the extrapolation with α = 2seems to be more efficient than that with α = 3 for the DTand TQ cases, it actually overshoots the basis-set limit whenthe Q and 5 cardinal numbers are used. More importantly,using α = 2 leads to irregular behavior: the Q5 extrapola-tion results in a lower quality than the TQ extrapolation. Incontrast, extrapolation with α = 3, though slightly less effi-cient for low cardinal numbers, exhibits highly systematic be-havior and leads to an error as small as 0.01% for the Q5extrapolation.

Similar behavior of the extrapolation schemes is ob-served for the point near the avoided crossing. Both extrapo-lations, with α = 2 and α = 3, converge smoothly toward thebasis-set limit, but the convergence is not as fast as in the caseof the global minimum. In contrast to the global minimum,there is no problem here with overshooting the basis-set limit.For each pair of cardinal numbers (X − 1)X , the extrapola-tion with α = 2 gives results slightly more favourable thanusing α = 3, with the smallest error of 0.19% for the Q5 ex-trapolation.

For the saddle point and local minimum, the convergenceof the relative errors is not as smooth as for the global mini-mum: the relative error for X = D is surprisingly small. Thisis obviously accidental and does not reflect particularly highquality of the cc-pVDZ basis set. Indeed, when the cc-pVDZ

results are employed in Eq. (3), the extrapolation worsens theaccuracy: the errors for the DT extrapolation are much largerthan the errors for both the X = D and X = T plain results,independent of the value of the α extrapolation parameter.Starting from X = T, the plain results smoothly approach thebasis-set limit, though the convergence is clearly slower thanin the case of the global minimum. The extrapolation withα = 2 is unsystematic and unpredictable, as in the case of theglobal minimum, while that with α = 3 smoothly approachesthe basis-set limit. The errors of the Q5 extrapolation withα = 3 are −0.49% for the saddle point and −0.13% for thelocal minimum.

Patkowski and Szalewicz69 recently investigated Ar2

with the CCSD(T)-F12 method. They found that the F12a andF12b variants35 gave significantly different results. They alsoconcluded that, for Ar2, calculations with explicitly correlatedfunctions cannot yet compete with calculations employing ex-trapolation based on conventional orbital basis sets. Indeed,while their orbital results converged smoothly toward the ex-trapolated results, the CCSD(T)-F12a and CCSD(T)-F12b re-sults behaved erratically with respect to both the orbital andthe extrapolated results. Table I shows that this is not the casefor the Li–LiH system. In our case the CCSD(T)-F12a andCCSD(T)-F12b results are quite similar and are fully con-sistent with the plain and extrapolated results with conven-tional basis sets. It should be stressed, however, that Ar2 isbound mostly by dispersion forces, while the main sourceof the bonding in Li–LiH is the induction energy, which isless sensitive to the basis-set quality. This may at least partlyexplain the success of the CCSD(T)-F12 calculations forLi–LiH.

Finally, it is important to note here that, while the inter-action energy at the characteristic points varies considerablywith the basis set and extrapolation method, the positions ofthe points (i.e., the distance R and angle θ at which the char-acteristic points occur) remain practically unaffected by thechoice of the basis set and extrapolation scheme.



-0.5

0

0.5

1

1.5

Rel

ativ

e er

ror

[%]

Global minimum

D

T

Q5

DTTQ

Q5

DT

TQ Q5

cc-pVX Z extrapolationusing α = 2

extrapolation using α = 3

-2

-1.5

-1

-0.5

0

0.5Local Minimum

D

T

Q

5

DT

TQ Q5

DT

TQQ5

cc-pVX Z extrapolation using α = 2


-1

0

1

2

3

4

5

Rel

ativ

e er

ror

[%]

Near Avoided Crossing

D

T

Q5

DT

TQ Q5

DT

TQ

Q5

cc-pVxZ extrapolationusing α=2

e xtrapolationusing α=3

-2

-1.5

-1

-0.5

0

0.5Saddle Point

D

T

Q

5

DT

TQQ5

DT

TQ Q5

cc-pVX Z extrapolation using α = 2


FIG. 1. Relative percentage errors of the interaction energy at the characteristic points (global minimum, saddle point, local minimum, and near the avoidedcrossing: R = 5.5 bohr and θ = 0) of the LiH–Li potential calculated at the CCSD(T)/cc-pVXZ-mid level, where X = D, T, Q, 5, and mid stands for themidbond-95 set. The errors for the characteristic points obtained by extrapolating the plain basis-set results with the two-point extrapolation formula are alsoshown for α = 2 and α = 3. The errors were obtained by comparison with the CCSD(T)-F12b/QZVPP results.



0

20

40

60

80

100

120

140

N-e

lect

ron

err

or

[cm

-1]

CCSD

CCSD(T)

CCSD

CCSD(T)

CCSD

CCSD(T)

CCSD

CCSD(T)

Global Minimum

Saddle Point

Local Minimum

Near AvoidedCrossing

FIG. 2. The N -electron error of the characteristic points of the LiH–Li interaction potential calculated at the CCSD/cc-pVQZ and CCSD(T)/cc-pVQZ levelsof theory. The error was determined by comparison with the FCI/cc-pVQZ interaction potential. The 1σLiH1sLi frozen-core approximation was used.

To analyze the convergence of the CCSD and CCSD(T)models in the N -electron space, Fig. 2 compares thecharacteristic points (global minimum, saddle point, localminimum, and near the avoided crossing) of the Li–LiH po-tential calculated at the CCSD/cc-pVQZ and CCSD(T)/cc-pVQZ levels of theory with the characteristic points obtainedat the FCI/cc-pVQZ level. The 1σLiH and 1sLi orbitals werekept frozen in the calculations. As expected, the N -electronerror is reduced by a factor of 3–4 when the approximatetriples correction is included in the calculations. It can alsobe seen from the figure that the global minimum is the mostsensitive and the local minimum is the least sensitive to thedescription of the electron correlation.

B. The best estimate of the ground-state Li–LiHpotential energy surface

Because of the negligible one-electron error in theCCSD(T)–F12 calculations and to the rather large basis setused in the FCI/cc-pVQZ calculations, and assuming that theone-electron and N -electron errors are approximately inde-pendent, the best estimate of the ground-state interaction en-ergy surface for the LiH-Li is

V best = V CCSD(T)−F12 + δV FCIv−v + δV FCI, (4)

where V CCSD(T)−F12 is the CCSD(T) basis-set limit energy(i.e., the CCSD(T)-F12 result) and the FCI correction, δV FCI

v−v,

is obtained by subtracting the CCSD(T)/cc-pVQZ energyfrom the FCI/cc-pVQZ energy, both calculated in the frozen-core approximation. The quantity δV FCI accounts for the lastremaining correction (in the nonrelativistic limit); namely,the effects of core-core and core-valence correlation in theFCI/cc-pVQZ calculations,

δV FCI = δV FCIall−all − δV FCI

v−v, (5)

where the subscript “all” refers to all electrons correlated.The quantity δV FCI is a measure of the uncertainty in our

best estimate V best. To estimate this, we may safely assumethat δV FCI is at most as large as the corresponding δV (T),

δV FCI ≤ δV (T) = δV (T)all−all − δV (T)

v−v, (6)

where

δV (T)all−all = V CCSD(T)

all−all − V CCSDall−all , (7)

δV (T)v−v = V CCSD(T)

v−v − V CCSDv−v , (8)

with V CCSD(T)all−all , V CCSD

all−all , V CCSD(T)v−v , and V CCSD

v−v denoting in-teraction energies calculated at the CCSD(T)/cc-pVQZ orCCSD/cc-pVQZ level, correlating all electrons or using thefrozen-core approximation, as appropriate. As can be seenfrom Fig. 2, the differences between CCSD(T) and CCSD are,for the characteristic points of the potential, 2–3 times larger(and for the rest of the potential at least 1.5 times larger) than



TABLE II. Performance of the valence–valence FCI correction δV FCIv−v

against exact FCI results for the characteristic points of the LiH–Li poten-tial (in cm−1). Calculations were done in the cc-pVDZ basis set. Subscriptall–all refers to all electrons correlated, while v–v denotes frozen-core re-sults. is the percentage error of the δV FCI

v−v approximation with respect tothe δV FCI

all−all: = (δV FCIv−v − δV FCI

all−all)/|δV FCIall−all| · 100%.

GM SP LM AC

V CCSD(T)v−v –7553.16 –1407.50 –1502.25 –3785.91

V FCIv−v –7587.86 –1437.27 –1522.05 –3806.93

δV FCIv−v –34.70 –29.77 –19.80 –21.02

V CCSD(T)all−all –7590.40 –1415.56 –1509.04 –3842.01

V FCIall−all –7625.07 –1445.32 –1528.78 –3863.46

δV FCIall−all –34.67 –29.76 –19.74 –21.45

[%] –0.09 –0.04 –0.30 0.76

the differences between FCI and CCSD(T). Equation (6) istherefore actually a conservative estimate for δV FCI. The rootmean square error for δV (T)/δV (T)

all−all over the whole potentialis 4.1%. We thus consider that our best estimate of the ground-state interaction energy for LiH–Li, Eq. (4), has a (conserva-tive) total uncertainty of 5% of the FCI correction (δV FCI

v−v).The analysis of the Li–LiH potential in the remainder ofthis paper is based on the interaction energies obtained usingEq. (4), unless otherwise stated.

To justify our error estimation, we have performed calcu-lations with all electron correlated at the FCI level for the setof characteristic points of the potential. Due to the immensememory requirements of the FCI calculations with seven elec-trons, we were able to apply the cc-pVDZ basis set only. TheFCI/cc-pVDZ results together with the CCSD(T)/cc-pVDZ,both with and without the frozen-core approximation, are pre-sented in Table II. The error in the FCI correction calculatedwith frozen core is as small as 0.76 % for the examined points.We may see that the approximation with the FCI valence cor-rection added to CCSD(T), Eq. (1), reproduces the exact FCIresults with accuracy better than 1% of the FCI correction(δV FCI

v−v). This confirms our estimate of 5% uncertainty in theFCI correction δV FCI

v−v.

C. Features of the ground-state potentialenergy surface

In Table III, we have listed the characteristic points ofthe potential energy surfaces of the ground state, which cor-relates at long range with Li(2S) + LiH (X 1+), and the firstexcited state, which correlates at the long range with Li(2P)+ LiH (X 1+). Both these states are of 2 A′ symmetry in theCs point group. The latter is included in Table III since, as willbe discussed in Sec. IV, it shows an avoided crossing withthe ground-state potential for the linear LiH–Li geometry.Table III shows that the interaction potential for the groundstate of Li–LiH is deeply bound, with a binding energy of8743 cm−1 at the global minimum. The global minimum islocated at a skew geometry with Re = 4.40 bohr, θe = 46.5

and is separated by a barrier around R = 6.3 bohr, θ = 136.0

from a shallow local minimum at the linear Li–LiH

TABLE III. Characteristic points of the interaction potentials for theground state Li(2S) + LiH (X 1+) and the first excited state, which cor-relates asymptotically with Li(2P) + LiH (X 1+).

R (bohr) θ (degrees) V (cm–1)Ground state

Global minimum 4.40 46.5 –8743Local minimum 6.56 180.0 –1623Saddle point 6.28 136.0 –1565

Excited state

Global minimum 5.66 0.0 –4743

geometry. The local minimum is at R = 6.56 bohr, with a welldepth of only 1623 cm−1. The excited-state potential showsonly one minimum, at R = 5.66 bohr, with a binding energy of4743 cm−1.

A contour plot of the ground-state potential is shown inthe left-hand panel of Fig. 3, while the full-CI correction to theCCSD(T) potential, δV FCI

v−v, is shown in the right-hand panel.The correction is very small compared to the best potential.It amounts to 0.4% around the global minimum and approx-imately 1% at the local minimum. Thus, our estimated errorof the calculation, 5% of the full-CI correction, translates into0.05% error in the potential itself. We would like to reiter-ate here that such a small error was achieved not only be-cause the interelectron distance was included explicitly in theab initio CCSD(T)–F12 calculations, but also because of thevery small valence-valence correlation beyond the CCSD(T)level. The smallness of the valence–valence correlation be-yond the CCSD(T) level is not so surprising, since Li–LiHhas only three valence electrons, and the exact model for athree-electron system would be CCSDT, coupled-cluster withsingle, double, and exact triple excitations.70 Our results showthat the triples contribution to the correlation energy beyondthe CCSD(T) model for the valence electrons is very small.

The potential for the ground state of Li–LiH is verystrongly anisotropic. This is easily seen in the left-hand panelof Fig. 3, and in Fig. 4, which shows the expansion coef-ficients of the potential in terms of Legendre polynomialsPl (cos θ ),

V (R, θ ) =∞∑

l=0

Vl (R) Pl(cos θ ). (9)

Here, V0(R) is the isotropic part of the potential andVl(R)∞l=1 is the set of anisotropic coefficients. Figure 4shows that, around the radial position of the global minimum,R = 4.4 bohr, the first anisotropic contribution to the poten-tial, V1(R), is far larger than the isotropic term, V0(R). Thehigher anisotropic components, with l = 2, 3, etc., contributemuch less to the potential.

As mentioned above, calculations of collision dynam-ics at ultralow temperatures require accurate values of thelong-range potential coefficients, Eq. (2). Some importantscattering properties, such as the mean scattering length andthe heights of centrifugal barriers, are determined purely bythe Van der Waals coefficients. The calculated coefficientsfor Li–LiH are presented in Table IV. Because of the largedipole moment of lithium hydride and the relatively high



Θ [degrees]

R [b

ohr]

−50−50−100

−100−300

−300

−500

−500

−300

−500

−1000

−1000

−1000−1600

−1600−1600

−1000

−2000

−3000

−3000−4000

−7000−8500 0

00

0 20 40 60 80 100 120 140 160 180

4

5

6

7

8

9

10

11

12

13

14

15

Θ [degrees]

R [b

ohr] –10

–10

–5

–5

–2

–2

1

–1

–20

–20–30

–30–40

–40–50–40

–50

–30–40–50

0 20 40 60 80 100 120 140 160 180

4

5

6

7

8

9

10

11

12

13

14

15

–

FIG. 3. Contour plots of the best ab initio potential for the ground state of Li–LiH (left-hand panel), and of the full-CI correction to the CCSD(T)–F12 potential(right-hand panel). Energies are in cm−1.

polarizability of the lithium atom, the lowest-order, most im-portant, coefficients are dominated by the induction contribu-tion. For example, the induction part of C0

6 and C26 is 887 a.u.,

which accounts for 71% of C06 and 98% of C2

6 .

IV. INTERACTION BETWEEN THE GROUNDAND EXCITED STATES

A. Low-lying excited state potential, nonadiabaticcoupling matrix elements, and diabatic potentials

We encountered convergence problems with CCSD(T)calculations at the linear LiH–Li geometry around R = 5.6bohr, due to the presence of a low-lying excited state. Theexcited state correlates with the Li(2P) + LiH(X1+) disso-ciation limit, but closer investigation revealed that, at linearLi–HLi geometries near the crossing with the ground state,it has ion-pair character, Li+(1S) + LiH−(2). The ion-pairstate itself has a crossing near R = 9 bohr with the lowest2 A′ state correlating with Li(2P) + LiH(X1). This is shownschematically in Fig. 5. Away from linear Li–HLi geometries,

–6000

–4000

–2000

0

2000

4000

6000

3 4 5 6 7 8 9 10

Pot

enti

al e

nerg

y [

cm–1

]

R [bohr]

V0

V1

V2

V3

V4

FIG. 4. The Legendre components Vl (R) (l = 0, 1, 2, 3, 4) of the ground-state Li(2S) + LiH (X 1+) interaction potential, see Eq. (9).

the excited state has covalent character and remains below theion-pair state all the way to dissociation. The avoided cross-ing between the ground state and the first excited state is atR = 5.66 bohr, which is near the minimum of the ground-statepotential at the linear geometry, and the energetic distance be-tween the two states at the avoided crossing is only 94 cm−1.

In order to investigate how far the excited state may af-fect the scattering dynamics, we computed the full potentialenergy surface for the excited state in question by means ofequation-of-motion coupled-cluster method with single anddouble excitations71–73 implemented in the QCHEM code,74

using the orbital cc-pVQZ basis set. Cuts through the ground-state and excited-state potential energy surfaces at selectedvalues of the angle θ are shown in Fig. 6. It may be seenthat it is only near the linear LiH–Li geometry that the twostates come very close together. If we distort the systemfrom the linear geometry, the excited state goes up in en-ergy very rapidly, and around the global minimum energy,θ ≈ 45, it is almost 6000 cm−1 above the ground state. Theimportance of the possible interaction between the groundand excited states can be measured by analyzing the (vecto-rial) nonadiabatic coupling matrix elements τ 12, defined asτ 12 = 〈1|∇2〉, where ∇ is the gradient operator of theposition vector and 1 and 2 are the wave functions ofthe two lowest states. On the two-dimensional surface, wemay define radial τ12,R = 〈1|∂2/∂ R〉 and angular τ12,θ

= 〈1|∂2/∂θ〉 components of the vector τ 12. We evaluatedτ 12 for all (R, θ ) geometries by means of the multireference

TABLE IV. Long-range coefficients (in atomic units) for Li–LiH, from per-turbation theory. Numbers in parentheses indicate powers of 10.

l → 0 1 2 3 4Cl

6 1247.8 869.7

Cl7 8304.2 2902.3

Cl8 4.83(4) 4.24(4) 8923.0

Cl9 3.94(5) 6.42(4)

Cl10 2.03(6) 2.02(6) 2.44(5)



–5000

0

5000

10000

15000

20000

25000

30000

35000

4 6 8 10 12 14 16 18

Pot

enti

al e

nerg

y [

cm–1

]

R [bohr]

Li+(1S) + LiH–(2Σ)

Li(2P) + LiH(1Σ)

Li(2S) + LiH(1Σ)

FIG. 5. Ground-state and excited-state potentials at the linear geometryLiH–Li: demonstration of the ion-pair nature of the excited-state potential.

configuration interaction method limited to single and dou-ble excitations (MRCI),75, 76 using the MOLPRO code.43 Thenonadiabatic coupling is largest when the two states are veryclose in energy, as it can be seen in Fig. 6. While at θ = 0

the radial component of the nonadiabatic coupling approachesthe Dirac delta form near the crossing point Rac, with in-creasing angle it becomes a broad function of approximatelyLorentzian shape. [Note the different scales on the verticalaxes of the different panels.]

The transformation from the adiabatic representation to adiabatic representation may be expressed in terms of a mixingangle γ ,

H1 = V2 sin2 γ + V1 cos2 γ, H2 = V1 sin2 γ + V2 cos2 γ,

H12 = (V2 − V1) sin γ cos γ, (10)

where V1 and V2 are the ground-state and excited-state adi-abatic potentials, H1 and H2 are the diabatic potentials, andH12 is the diabatic coupling potential. In principle, the mix-ing angle γ may be obtained by performing line integrationof the nonadiabatic coupling τ 12,

γ (R) = γ (R0) +∫ R

R0

τ 12 · d l, (11)

where R0 is the starting point of the integration. For poly-atomic molecules, however, the mixing angle γ obtained byintegrating this equation is nonunique due to the contributionsfrom higher states. To circumvent the problem of path depen-dence, one may assume that we deal with an ideal two-statemodel.

In our case, however, the ion-pair surfaceLi+(1S) + LiH−(2) shows another crossing at smallangles and large distances, θ ≤ 15 and R ≈ 9 bohr, withanother excited-state potential that correlates with theLi(2P) + LiH(1) dissociation limit. Thus, a third state 3

comes into play and a two-state model is not strictly valid.The energy of the first excited state goes up very rapidlywith the angle θ , and at the same time the contribution of theion-pair configuration to the wave function of the first excitedstate, 2, diminishes rapidly. Fortunately, the nonadiabatic

coupling matrix elements between the two lower states τ 12

and between the two higher states τ 23 are well isolated. Themaximum of τ 12 is separated from the maximum of τ 23 bymore than 4 bohr; the locations of the crossing points betweenthe surfaces for θ = 0 are shown in Fig. 5. Moreover, thecoupling τ 13 between the ground state and the third stateis negligible over the whole configurational space. Thus,following the discussion of Baer et al.77 on the application ofthe two-state model, we conclude that the necessary condi-tions are fulfilled for the Li–LiH system. Due to the spatialseparation of the nonadiabatic couplings τ 12 and τ 23, usingthe diabatization procedure based on the two-state model isjustified. It is worth noting that in our particular case, wecould not use the so-called quasi-diabatization procedure,78

since it is not possible to assign a single-reference wavefunction. This is due to the fact that the excited state showsadmixture from the ion-pair state.

As the starting point of the integration in Eq. (11), wechose R = 20 bohr and θ = 0 and followed a radial pathalong θ = 0 and subsequently angular paths at constant R.The diabatic potentials were then generated according toEq. (10). Contour plots of the adiabatic, diabatic, and couplingpotentials, and of the mixing angle γ , are presented as func-tions of R and θ in Fig. 7. We consider first the mixing angleγ , which is plotted in the bottom right-hand panel of Fig. 7.As expected, the mixing angle shows an accumulation pointat θ = 0 at a distance R corresponding to the closely avoidedcrossing between the ground and excited states. For θ = 180,the mixing angle is non-negligible, even at large distances.The coupling potential H12 vanishes quite slowly with dis-tance R, as R−3. For the coupling between the ground andion-pair states, this long-range decay is exponential becauseof the different dissociation limits of the two surfaces. As ex-pected, at large distances the two diabatic surfaces approachthe respective adiabatic surfaces. The diabatic surface thatcorrelates asymptotically with the excited-state Li(2P) + LiHsurface has an important contribution from the ground-stateadiabatic potential only inside the avoided crossing and atsmall angles θ . The diabatic surface that correlates asymp-totically with the ground state resembles the ground-state adi-abatic surface rather less closely, especially at large values ofθ . The coupling between the diabatic states is small over asignificant region of θ and Li–H bond length r in the vicinityof the crossing. Physically, this means that the dynamics willbe strongly nonadiabatic in this region, and to take this rig-orously into account would require a full two-state treatmentof the dynamics. However, there are no open channels that in-volve the second surface, and any collisions that cross onto itmust eventually return to the original surface. Its effect in col-lision calculations will therefore be at most to cause a phasechange in the outgoing wavefunction.

B. Conical intersection

It is well known that potential energy surfaces forhomonuclear triatomic systems composed of hydrogen79 orlithium atoms80 show conical intersections at equilateral tri-angular geometries. Analogous behavior may be expectedfor Li2H, at geometries where the two lithium atoms are



–10000

–5000

0

5000

10000

15000

4 5 6 7 8 9 10 11 12R [bohr]

Θ = 15.°

Li+LiH

–10000

–5000

0

5000

10000

15000

4 5 6 7 8 9 10 11 12R [bohr]

Θ = 45.°

Li+LiH

–10000

–5000

0

5000

10000

15000

4 5 6 7 8 9 10 11 12

Pot

enti

al e

nerg

y [c

m–1

]

R [bohr]

Θ = 0.°

Li+LiH

–10000

–5000

0

5000

10000

15000

4 5 6 7 8 9 10 11 12

Pot

enti

al e

nerg

y [c

m–1

]

R [bohr]

Θ = 30.°

Li+LiH

–1

–0.5

0

0.5

1

1.5

2

4 4.5 5 5.5 6 6.5 7 7.5 8R [bohr]

Θ = 15.°

–1

–0.5

0

0.5

1

1.5

2

4 4.5 5 5.5 6 6.5 7 7.5 8R [bohr]

Θ = 45.°

0

5

10

15

20

4 4.5 5 5.5 6 6.5 7 7.5 8

Non

adia

bati

c ra

dial

cou

plin

g

R [bohr]

Θ = 0.°

–1

–0.5

0

0.5

1

1.5

2

4 4.5 5 5.5 6 6.5 7 7.5 8

Non

adia

bati

c ra

dial

cou

plin

g

R [bohr]

Θ = 30.°

FIG. 6. Cuts through the potential energy surfaces for the ground and the first excited states of 2 A′ symmetry for selected values of the angle θ , and thecorresponding radial nonadiabatic coupling matrix elements as functions of the distance R.



Θ [degrees]

R [b

ohr]

−50 −50−100

−100−300

−300

−500

−500

−300

−500

−1000

−1000

−1000−1600

−1600−1600

−1000

−2000

−3000

−3000−4000

−7000

−8500

0

00

0 20 40 60 80 100 120 140 160 180

4

5

6

7

8

9

10

11

12

13

14

15

Θ [degrees]

R [b

ohr]

14500

14500

14000

14000

1325013250

1275012000

100008000400020000−2000

−3000

−4000−4500

1275

0 13250

14000

14500

0 20 40 60 80 100 120 140 160 180

4

5

6

7

8

9

10

11

12

13

14

15

R [b

ohr]

Θ [degrees]

−5000

−4000

−3000

−2000

−1000

−500

−300

−50−1000

100

300

500

1000

2000

3000

5000

10000

20000

8000

1500020000

0 20 40 60 80 100 120 140 160 180

4

5

6

7

8

9

10

11

12

13

14

15

R [b

ohr]

Θ [degrees]

1450

0

14000

14000

13000

13000

12000

12000

11000

11000

1000010000

8000 8000

−7000−6000

−4000 −20000 2000400050006000 7000

700060005000

5000

4000

600070008000 10000

14500

0 20 40 60 80 100 120 140 160 180

4

5

6

7

8

9

10

11

12

13

14

15

Θ [degrees]

R [b

ohr]

−8000−7000

−6000−5000

−4000−3

000

−200

0

−100

0

−500−2

50

−100

−700

0

−600

0

−500

0

−400

0

−300

0

−200

0

−10

00

001−052−005−

−7000−6000

−5000

0 20 40 60 80 100 120 140 160 180

4

5

6

7

8

9

10

11

12

13

14

15

Θ [degrees]

R [b

ohr]

70

60

50

40

30

20

15

1052

2

5

1520

40

50

7080

85

10

30

60

0 20 40 60 80 100 120 140 160 180

4

5

6

7

8

9

10

11

12

13

14

15

FIG. 7. Adiabatic potentials (top two panels), diabatic potentials (middle two panels), and coupling potential (bottom left-hand panel), and the mixing angle γ

(bottom right-hand panel), as functions of the geometrical parameters R and θ . Energies are in cm−1.

equivalent, i.e., C2v geometries. Thus far, our discussion of thepotential for Li–LiH has been restricted to two dimensionswith the bond length of the LiH molecule fixed at its equilib-rium value, and no conical intersection was observed. How-ever, if we start to vary the bond length of the LiH molecule,conical intersections show up immediately.

At C2v geometries, with the two Li–H bond lengths equal,there are two low-lying electronic states, of 2A1 and 2B2 sym-metries, that cross each other as a function of the internuclearcoordinates. Figure 8 shows contour plots of the two poten-tial energy surfaces and of the difference between them, and

the top panel of Fig. 9 summarizes some key features of thesurfaces. The 2A1 state has a minimum energy of −8825 cm−1

at r (LiH) = 3.22 bohr and an Li-H-Li angle of 95. MRCI cal-culations with all coordinates free to vary confirm that this isindeed the absolute minimum geometry. There is also a sad-dle point on the 2A1 surface at a linear H–Li–H geometry withr (LiH) = 3.04 bohr and an energy of −4992 cm−1, whichis a minimum in D∞h symmetry. The 2B2 state has a mini-mum energy of −5136 cm−1 at r (LiH) = 3.17 bohr at a linearLi–H–Li geometry. The 2A1 saddle point and 2B2 linear min-imum have symmetries 2+

g and 2+u respectively in D∞h



α [degrees]

r [b

ohr]

10000

100005000

5000

2000

2000

0

0

0

−2000

−2000

−4000

−4000

−5000

−5000

−6000

−6000

−7000

−8000

−7000

−4000

−4000

−20000

20005000

10000

10000

5000

−2000−2000

60 80 100 120 140 160 1802

2.5

3

3.5

4

4.5

5

5.5

6

α [degrees]

r [b

ohr]

3000

2000

1000

0

−1000−2000

−3000−4000

−5000−8000

−10000

2000

1000

0

−1000

−2000

−3000

−4000

−5000

−8000

−10000

60 80 100 120 140 160 1802

2.5

3

3.5

4

4.5

5

5.5

6

α [degrees]

r [b

ohr]

1000010000

5000

5000

2000

2000

0

0

−1000−1000

−2000

−3000

−4000

−5000

−4000−3000−2000

10000

50002000

0

5000

20000

20000

−4000

60 80 100 120 140 160 1802

2.5

3

3.5

4

4.5

5

5.5

6

H

Li Li

r rα

FIG. 8. Potential energy surfaces for the 2A1 (top left-hand panel) and 2B2 (bottom left-hand panel) states of Li2H in C2v symmetry and the difference betweenthem (top right-hand panel), which is zero along the seam of conical intersections. Also shown is the coordinate system used for C2v geometries. Energies arein cm−1.

symmetry, but mix and distort if the constraint on the Li–Hbond lengths is relaxed, to form a 2+ state in C∞v symmetrywith a minimum at a linear geometry with r (LiH) distances of3.00 and 3.33 bohr and an energy of −5323 cm−1. Even thisis a saddle point with respect to bending on the full potentialsurface in Cs symmetry.

The 2A1 and 2B2 states are of different symmetries at C2v

geometries, but both are of 2A′ symmetry when the geometryis distorted from C2v to Cs symmetry. The two states thereforemix and repel one another at geometries where the two Li–Hbond lengths are different, but a seam of conical intersectionsruns along the line where the energy difference is zero at C2v

geometries.The fixed LiH distance used in Secs. I–IV A (r = 3.014

bohr, shown as a dashed line on the figure) keeps the 2A1 sur-face just below the 2B2 surface. However, if we allow for thevibrations of LiH, the seam of conical intersections becomesaccessible at near-linear LiH–Li geometries, where the zeroof the energy difference appears for an Li–H distance onlyslightly larger than 3.014 bohr. At nonlinear geometries theseam quickly moves to Li–H distances far outside the classi-cal turning points of the ground vibrational level of free LiH,which are 2.72 and 3.35 bohr.

It is interesting to compare the features of the conical in-tersections in Li2H with those in other triatomic moleculesformed from Li and H atoms: LiH2, Li3, and H3. In the caseof LiH2, the seam of intersections occurs at highly bent C2v

geometries with an angle between the two Li–H bonds of ap-proximately 30 and arises from degeneracy between the sur-faces of A1 and B2 symmetry. The global minimum of B2

symmetry is located at r (LiH) = 3.23 bohr and a bond angleH–Li–H of 28.81 This contrasts with Li2H, where the min-imum of B2 symmetry is at a linear Li–H–Li configuration.The energy of the lowest point on the seam of intersections isabout 9000 cm−1 above the Li(2S) + H2(X1g) threshold, sothat it is irrelevant for low and medium-energy collisions be-tween H2 and Li in their ground states, though it is importantfor quenching of Li(2P) by H2.81, 82

The conical intersections for the doublet states of Li3and H3 occur at equilateral triangular geometries, wherethe ground state is doubly degenerate and has symmetry2E′ in the D3h point group. In the case of H3, the lowest-energy point on the seam is located at an energy more than20000 cm−1 above the H(2S) + H2(X1g) threshold, so thatnonadiabatic effects are negligible in H + H2 collisions,83 al-though the conical intersection also produces geometric phase



–8825

(a)

(b)

–5323–5136

0

12021

22539

Li(2S)+LiH(X1Σ)

Li2(X1Σg)+H(2S)

LiH(X1Σ)+Li(2S)

Li2(a3Σu)+H(2S)

Li2H( A1)Li--H-Li(2Σ)

Li-H-Li(2Σu)

Li(1)−H distance [bohr]

Li(2

)−H

dis

tanc

e [b

ohr]

5000

8000

10000

−10000

2000

0−1

000

−200

0 −2000

−300

0−4

000

−4000

−3000

−2000

−1000

0

2000

5000

8000

0

0−1000

−2000−3000

−1000−2000

−5000

00

2.5 3 3.5 4 4.5 5 5.5 6 6.5 72.5

3

3.5

4

4.5

5

5.5

6

6.5

7

Li−Li distance [bohr]

Li−

H d

ista

nce

[boh

r]

18000

18000

12000

0004

1 16000

1600

014

000

1200

0

1800

0

10000

8000

5000

20000

−2000−3000−5000−7000−8000

0 0

00

20005000

20005000

800010000

1400012000

16000

12000100008000

5000

3 4 5 6 7 8 9 10 11 122

4

6

8

10

12

14

2

FIG. 9. Schematic representation of the possible reaction pathways in collisions of the lithium atom with the lithium hydride molecule (upper panel),and two-dimensional cuts of the reactive potential energy surfaces for the exchange (left-hand panel) and insertion (right-hand panel) reactions. Energiesare in cm−1.

effects.84 For Li3, the energetics are essentially different. Thelowest-energy point on the seam is around 4000 cm−1 belowthe Li(2S) + Li2(X1g) threshold and only 500 cm−1 abovethe C2v global minimum.85 This is likely to produce consid-erable nonadiabacity in collisions of Li2 with Li.

To conclude, in all the triatomic molecules formed fromH and Li there are seams of crossings that occur at configu-rations of the highest possible symmetry, either C2v or D3h .For Li3 and Li2H the conical intersections are accessible dur-ing atom-molecule collisions, while for H3 and LiH2 nonadi-abatic processes are unimportant if the colliding partners arein their ground states and have relatively low kinetic energy.

V. REACTION CHANNELS

Several reaction channels exist that might affect sympa-thetic cooling86 in Li + LiH. These are the exchange reaction,

LiH + Li → Li + HLi, (12)

and two insertion reactions,

LiH + Li → Li2 + H, (13)

producing Li2(X1+g ) and Li2(a3+

u ) plus a ground-state Hatom. The energetic location of the entrance and exit channels



of these reactions, as well as those of the potential minima forlinear and C2v geometries, are shown in the upper panel ofFig. 9. The insertion reactions are highly endothermic, withan energy difference between the entrance and exit channelsof the order of 12000 and 22500 cm−1 for Li2(X1+

g ) + Hand Li2(a3+

u ) + H, respectively.To make the discussion more quantitative, Fig. 9 also

shows two-dimensional plots of the energy as functions ofthe internal coordinates. For the exchange reaction, we heldLi–H–Li at linear geometries and varied the distances fromthe two lithium atoms to the hydrogen atom. For the insertionreaction, Li–Li–H was kept bent, with the angle (HLi1Li2)held constant at the C2v equilibrium value 42.5, while theLi–Li and Li–H distances were varied. To make the plotsconsistent with the correlation diagram shown on the upperpanel, the zero of energy was fixed at that of Li–LiH sepa-rated to infinite distance with the Li–H bond length fixed atthe monomer equilibrium value.

Let us consider the exchange reaction first. The two-dimensional cut through the potential energy surface is pre-sented in the left-hand panel of Fig. 9. The potential energysurface of linear Li2H has two equivalent minima with an en-ergy of −5323 cm−1, separated by a small barrier 187 cm−1

high. The linear minima are in any case substantially abovethe absolute minimum (8825 cm−1), so this small barrier willhave no important effect on the collision dynamics. The ex-change reaction produces products that are indistinguishablefrom the reactants, so reactive collisions cannot be distin-guished from inelastic collisions experimentally (unless thetwo Li atoms are different isotopes).

An analogous two-dimensional cut through the potentialenergy surface corresponding to Li2(X1g) + H products ispresented in the right-hand panel of Fig. 9. The plot illus-trating the reaction to form Li2(a3u) + H products is not re-ported, as the reaction is even more endothermic. The surfaceincludes the absolute minimum at an energy of −8825 cm−1.The entrance channel for this reaction corresponds to anLi–H distance of 3.014 bohr at large Li–Li distance, while inthe exit channel the Li–Li distance is approximately 5.05 bohrwhen the Li–H distance is very large. However, this reactioncannot occur at low collision energies.

VI. SUMMARY AND CONCLUSIONS

In the present paper, state-of-the-art ab initio techniqueshave been applied to compute the ground-state potential en-ergy surface for Li–LiH in the Born–Oppenheimer approxi-mation. The interaction potential was obtained using a combi-nation of the explicitly-correlated unrestricted coupled-clustermethod with single, double, and approximate noniterativetriple excitations [UCCSD(T)–F12] for the core–core andcore–valence correlation, with full configuration interactionfor the valence–valence correlation. The main results of thispaper can be summarized as follows:

(1) The Li–LiH system is strongly bound: if the Li–H bondlength is held fixed at the monomer equilibrium distanceof 3.014 bohr, the potential energy surface has a global

minimum 8743 cm−1 deep at a distance R = 4.40 bohrfrom the lithium atom to the center of mass of LiH, anda Jacobi angle θ = 46.5. It also shows a weak localminimum 1623 cm−1 deep at the linear Li–LiH geom-etry for R = 6.56 bohr, separated from the global mini-mum by a barrier at R = 6.28 bohr and θ = 136. If theLi–H bond length is allowed to vary, the potential mini-mum is at a depth of 8825 cm−1, at a C2v geometry withLi–H bond length of 3.22 bohr and an Li–H–Li angleof 95.

(2) The full-CI correction for the valence-valence corre-lation to the explicitly correlated CCSD(T)–F12 po-tential is very small. The remaining error in our cal-culations is due to the neglect of the core–core andcore–valence contributions, and is estimated to be of theorder of 0.05% of the total potential.

(3) To evaluate the performance of the conventional orbitalelectron-correlated methods, CCSD and CCSD(T), cal-culations were carried out using correlation-consistentpolarized valence X -tuple zeta basis sets, with X rangingfrom D to 5, and a very large set of midbond functions.Simple two-point extrapolations based on the single-power laws X−2 and X−3 for the basis-set truncationerror reproduce the CCSD(T)–F12 results for the char-acteristic points of the potential with an error of 0.49%at worst.

(4) The potential for the ground state of Li–LiH is stronglyanisotropic. Around the distance of the global minimum,the isotropic potential V0(R) is almost two times smallerthan the first anisotropic contribution V1(R). Higheranisotropic components, with l = 2, 3, etc., do not con-tribute much to the potential.

(5) At the linear LiH–Li geometry, the ground-state poten-tial shows a close avoided crossing with the first excited-state potential, which has ion-pair character around theavoided crossing point. The full potential energy surfacefor the excited state was obtained with the equation-of-motion method within the framework of coupled-clustertheory with single and double excitations. The excited-state potential has a single minimum 4743 cm−1 deep forthe linear LiH–Li geometry at R = 5.66 bohr. The en-ergy difference between the ground and excited states atthe avoided crossing is only 94 cm−1. An analysis of thenonadiabatic coupling matrix elements suggests that dy-namics in the vicinity of the avoided crossing will havenonadiabatic character.

(6) When stretching the Li–H bond in the Li–LiH system,a seam of conical intersections appears for C2v geome-tries, between the ground state of 2A1 symmetry andan excited state of 2B2 symmetry. At the linear LiH–Li geometry, the conical intersection occurs for an Li–Hdistance which is only slightly larger than the equilib-rium distance of the LiH monomer, but for significantlynonlinear geometries it moves to Li–H distances far out-side the classical turning points of LiH.

(7) The Li–LiH system has several possible reaction chan-nels: an exchange reaction to form products identicalto the reactants, and two insertion reactions that pro-duce Li2(a3+

u ) and Li2(X1+g ) plus a ground-state



hydrogen atom. The insertion reactions are highly en-dothermic, with the energy difference between the en-trance and exit channels of the order of 12000 and 22500cm−1 for Li2(X1+

g ) + H and Li2(a3+u ) +H, respec-

tively.

In a subsequent paper,87 we will analyze the dynamics ofLi–LiH collisions at ultralow temperatures, based on our bestab initio potential, and discuss the prospects of sympatheticcooling of lithium hydride by collisions with ultracold lithiumatoms.

ACKNOWLEDGMENTS

We would like to thank Dr. Michał Przybytek for hisinvaluable technical help with the FCI calculations. We ac-knowledge the financial support from the Polish Ministryof Science and Higher Education (grant 1165/ESF/2007/03)and from the Foundation for Polish Science (FNP) via Hom-ing program (grant HOM/2008/10B) within EEA FinancialMechanism. We also thank Engineering and Physical Sci-ences Research Consul (U.K.) (EPSRC) for support undercollaborative project CoPoMol of the ESF EUROCORESProgramme EuroQUAM.

1R. V. Krems, Phys. Chem. Chem. Phys. 10, 4079 (2008).2K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis,J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Science322, 231 (2008).

3J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, R. Hart, J. Aldegunde,J. M. Hutson, and H.-C. Nägerl, Nature Physics 6, 265 (2010).

4J. D. Weinstein, R. de Carvalho, T. Guillet, B. Friedrich, and J. M. Doyle,Nature (London) 395, 148 (1998).

5H. L. Bethlem, G. Berden, and G. Meijer, Phys. Rev. Lett. 83, 1558(1999).

6M. S. Elioff, J. J. Valentini, and D. W. Chandler, Science 302, 1940(2003).

7L. D. van Buuren, C. Sommer, M. Motsch, M. S. S. Pohle, J. Bayerl, P. W.H. Pinkse, and G. Rempe, Phys. Rev. Lett. 102, 033001 (2009).

8B. DeMarco and D. S. Jin, Science 285, 1703 (1999).9G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M.Inguscio, Science 294, 1320 (2001).

10C. Zipkes, S. Palzer, C. Sias, and M. Köhl, Nature (London) 464, 388(2010).

11C. Zipkes, S. Palzer, L. Ratschbacher, C. Sias, and M. Köhl, Phys. Rev.Lett. 105, 133201 (2010).

12S. Schmid, A. Härter, and J. H. Denschlag, Phys. Rev. Lett. 105, 133202(2010).

13P. S. Zuchowski and J. M. Hutson, Phys. Rev. A 79, 062708 (2009).14P. Soldán, P. S. Zuchowski, and J. M. Hutson, Faraday Discuss. 142, 191

(2009).15A. O. G. Wallis and J. M. Hutson, Phys. Rev. Lett. 103, 183201

(2009).16S. K. Tokunaga, J. O. Stack, J. J. Hudson, B. E. Sauer, E. A. Hinds, and

M. R. Tarbutt, J. Chem. Phys. 126, 12431 (2007).17S. K. Tokunaga, J. M. Dyne, E. A. Hinds, and M. R. Tarbutt, New J. Phys.

11, 055038 (2009).18M. Tarbutt, private communication (2008).19T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989).20W. Klopper, K. L. Bak, P. Jørgensen, J. Olsen, and T. Helgaker, J. Phys. B

32, R103 (1999).21D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 100, 2975 (1994).22K. L. Bak, J. Gauss, T. Helgaker, P. Jørgensen, and J. Olsen, Chem. Phys.

Lett. 319, 563 (2000).23K. L. Bak, A. Halkier, P. Jørgensen, J. Olsen, T. Helgaker, and W. Klopper,

J. Mol. Struct. 567, 375 (2001).24T. Helgaker, W. Klopper, H. Koch, and J. Noga, J. Chem. Phys. 106, 9639

(1997).

25T. Shiozaki, M. Kamiya, S. Hirata, and E. F. Valeev, J. Chem. Phys. 129,071101 (2008).

26T. Shiozaki, M. Kamiya, S. Hirata, and E. F. Valeev, Phys. Chem. Chem.Phys. 10, 3358 (2008).

27A. Köhn, G. W. Richings, and D. P. Tew, J. Chem. Phys. 129, 201103(2008).

28D. P. Tew, W. Klopper, C. Neiss, and C. Hättig, Phys. Chem. Chem. Phys.9, 1921 (2007).

29D. Bokhan, S. Ten-no, and J. Noga, Phys. Chem. Chem. Phys. 10, 3320(2008).

30D. P. Tew, W. Klopper, and C. Hättig, Chem. Phys. Lett. 452, 326 (2008).31W. Kutzelnigg, Theor. Chim. Acta 68, 445 (1985).32W. Klopper and W. Kutzelnigg, Chem. Phys. Lett. 134, 17 (1987).33S. Ten-no, Chem. Phys. Lett. 398, 56 (2004).34C. Hättig, D. P. Tew, and A. Kohn, J. Chem. Phys. 132, 231102 (2010).35T. B. Adler, G. Knizia, and H.-J. Werner, J. Chem. Phys. 127, 221106

(2007).36H.-J. Werner, T. B. Adler, G. Knizia, and F. R. Manby, in Recent Progress

In Coupled Cluster Method, edited by P. Cársky, J. Paldus, and J. Pittner(Springer, Heidelberg, 2010).

37D. P. Tew, C. Hättig, R. A. Bachorz, and W. Klopper, in Recent ProgressIn Coupled Cluster Method, edited by P. Cársky, J. Paldus, and J. Pittner(Springer, Heidelberg, 2010).

38J. Yang and C. Hättig, J. Chem. Phys. 131, 074102 (2009).39C. Neiss and C. Hättig, J. Chem. Phys. 126, 154101 (2007).40J. Yang and C. Hättig, J. Chem. Phys. 130, 124101 (2009).41G. Plummer, E. Herbst, and F. DeLucia, J. Chem. Phys. 81, 4893 (1984).42H. Partridge and C. W. Bauschlicher Jr., Mol. Phys. 96, 705 (1999).43H.-J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, M. Schütz, P.

Celani, T. Korona, A. Mitrushenkov, G. Rauhut, T. B. Adler, et al.,MOLPRO, version 2008.3, a package of ab initio programs (2008), seehttp://www.molpro.net.

44S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 (1970).45G. Knizia and H.-J. Werner, J. Chem. Phys. 128, 154103 (2008).46G. Knizia, T. B. Adler, and H.-J. Werner, J. Chem. Phys. 130, 054104

(2009).47F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys. 7, 3297 (2005).48F. Weigend, J. Comput. Chem. 29, 167 (2008).49H. J. Werner, T. B. Adler, and F. R. Manby, J. Chem. Phys. 126, 164102

(2007).50W. Klopper and C. C. M. Samson, J. Chem. Phys. 116, 6397 (2002).51E. F. Valeev, Chem. Phys. Lett. 395, 190 (2004).52T. Helgaker, H. J. A. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H.

Ågren, A. A. Auer, K. L. Bak, V. Bakken, O. Christiansen, et al.,DALTON, an ab initio electronic structure program, Release 2.0 (2005), seehttp://www.kjemi.uio.no/software/dalton/dalton.html.

53LUCIA, a general active space program by J. Olsen, with contributions fromH. Larsen and M. Fülscher.

54B. Bussery-Honvault, J.-M. Launay, and R. Moszynski, Phys. Rev. A 68,032718 (2003).

55B. Bussery-Honvault, J.-M. Launay, and R. Moszynski, Phys. Rev. A 72,012702 (2005).

56B. Bussery-Honvault, J.-M. Launay, T. Korona, and R. Moszynski, J.Chem. Phys. 125, 114315 (2006).

57B. Bussery-Honvault and R. Moszynski, Mol. Phys. 104, 2387 (2006).58C. P. Koch and R. Moszynski, Phys. Rev. A 78, 043417 (2008).59B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev. 94, 1887

(1994).60R. Moszynski, in Molecular Materials with Specific Interactions—

Modeling and Design, edited by W. A. Sokalski (Springer, New York,2007), pp. 1–157.

61B. Jeziorski and R. Moszynski, Int. J. Quantum Chem. 48, 161 (1993).62R. Moszynski, P. S. Zuchowski, and B. Jeziorski, Coll. Czech. Chem.

Commun. 70, 1109 (2005).63T. Korona, M. Przybytek, and B. Jeziorski, Mol. Phys. 104, 2303 (2006).64A. Derevianko, S. G. Porsev, and J. F. Babb, At. Data Nucl. Data Tables

96, 323 (2010).65T.-S. Ho and H. Rabitz, J. Chem. Phys. 104, 2584 (1996).66L. M.C. Janssen, G. C. Groenenboom, A. van der Avoird, P. S. Zuchowski,

and R. Podeszwa, J. Chem. Phys. 131, 224314 (2009).67M. Jeziorska, R. Bukowski, W. Cencek, M. Jaszunski, B. Jeziorski, and K.

Szalewicz, Coll. Czech. Chem. Commun. 68, 463 (2003).68M. Jeziorska, W. Cencek, K. Patkowski, B. Jeziorski, and K. Szalewicz,

Int. J. Quantum Chem. 108, 2053 (2008).



69K. Patkowski and K. Szalewicz, J. Chem. Phys. 133, 094304 (2010).70J. Noga and R. J. Bartlett, J. Chem. Phys. 86, 7041 (1987).71H. J. Monkhorst, Int. J. Quantum Chem. Symp. 11, 421 (1977).72H. Sekino and R. J. Bartlett, Int. J. Quantum Chem., Symp. 18, 255

(1984).73J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993).74Y. Shao, L. Fusti-Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. T.

Brown, A. T. B. Gilbert, L. V. Slipchenko, S. V. Levchenko, D. P. O’Neill,R. A. Distasio Jr., R. C. Lochan, T. Wang, G. J. O. Beran, N. A. Besley, J.M. Herbert, C. Y. Lin, T. Van Voorhis, S. H. Chien, A. Sodt, R. P. Steele,V. A. Rassolov, P. E. Maslen, P. P. Korambath, R. D. Adamson, B. Austin,J. Baker, E. F. C. Byrd, H. Dachsel, R. J. Doerksen, A. Dreuw, B. D. Duni-etz, A. D. Dutoi, T. R. Furlani, S. R. Gwaltney, A. Heyden, S. Hirata, C.-P.Hsu, G. Kedziora, R. Z. Khalliulin, P. Klunzinger, A. M. Lee, M. S. Lee,W. Liang, I. Lotan, N. Nair, B. Peters, E. I. Proynov, P. A. Pieniazek, Y. M.Rhee, J. Ritchie, E. Rosta, C. D. Sherrill, A. C. Simmonett, J. E. Subotnik,H. L. Woodcock III, W. Zhang, A. T. Bell, A. K. chakraborty, D. M. Chip-man, F. J. Keil, A. Warshel, W. J. Hehre, H. F. Schaefer III, J. Kong, A. I.Krylov, P. M. W. Gill, and M. Head-Gordon, Phys. Chem. Chem. Phys. 8,3172 (2006).

75H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 (1988).76P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. 145, 514 (1988).77M. Baer, A. M. Mebel, and G. D. Billing, J. Phys. Chem. A 106, 6499

(2002).78D. Simah, B. Hartke, and H.-J. Werner, J. Chem. Phys. 111, 4523 (1999).79R. Abrol, A. Shaw, A. Kuppermann, and D. R. Yarkony, J. Chem. Phys.

115, 4640 (2001).80D. A. Brue, X. Li, and G. A. Parker, J. Chem. Phys. 123, 091101 (2005).81E. S. Kryachko and D. R. Yarkony, Theor. Chem. Acc. 100, 154 (1998).82T. J. Martinez, Chem. Phys. Lett. 272, 139 (1997).83T.-S. Chu, K.-L. Han, M. Hankel, G. G. Balint-Kurti, A. Kuppermann, and

R. Abrol, J. Chem. Phys. 130, 144301 (2009).84J. C. Juanes-Marcos and S. C. Althorpe, J. Chem. Phys. 122, 204324

(2005).85A. I. Voronin, J. M. C. Marques, and A. J. C. Varandas, J. Phys. Chem. A

102, 6057 (1998).86P. S. Zuchowski and J. M. Hutson, Phys. Rev. A 81, 060703 (2010).87S. K. Tokunaga, W. Skomorowski, P. S. Zuchowski, R. Moszynski, J. M.

Hutson, E. A. Hinds, and M. R. Tarbutt, arXiv:1012.5029, Eur. Phys. J. D(2011) in press.


114

APPENDIX D

PAPER IV

“Prospects for sympathetic cooling of molecules in electrostatic,

ac and microwave traps”

S.K. Tokunaga, W. Skomorowski, P.S. Zuchowski, R. Moszynski,

J.M. Hutson, E.A. Hinds and M.R. Tarbutt

European Physical Journal D 65, 141 (2011)

115

Eur. Phys. J. D 65, 141–149 (2011)DOI: 10.1140/epjd/e2011-10719-x

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Prospects for sympathetic cooling of moleculesin electrostatic, ac and microwave traps

S.K. Tokunaga1, W. Skomorowski2, P.S. Zuchowski3, R. Moszynski2, J.M. Hutson3,E.A. Hinds1, and M.R. Tarbutt1,a

1 Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, UK2 Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland3 Department of Chemistry, Durham University, South Road, DH1 3LE Durham, UK

Received 22 December 2010 / Received in final form 18 February 2011Published online 15 April 2011 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2011

Abstract. We consider how trapped molecules can be sympathetically cooled by ultracold atoms. As aprototypical system, we study LiH molecules co-trapped with ultracold Li atoms. We calculate the elasticand inelastic collision cross sections of 7LiH + 7Li with the molecules initially in the ground state and inthe first rotationally excited state. We then use these cross sections to simulate sympathetic cooling in astatic electric trap, an ac electric trap, and a microwave trap. In the static trap we find that inelastic lossesare too great for cooling to be feasible for this system. The ac and microwave traps confine ground-statemolecules, and so inelastic losses are suppressed. However, collisions in the ac trap can take moleculesfrom stable trajectories to unstable ones and so sympathetic cooling is accompanied by trap loss. In themicrowave trap there are no such losses and sympathetic cooling should be possible.

1 Introduction

There has been rapid progress in the field of cold and ul-tracold molecular gases over the last decade, driven by adiverse range of applications in physics and chemistry [1].Polar molecules are of particular interest because they in-teract strongly with applied electric fields, and interactwith one another through dipole-dipole interactions thatare long-range, anisotropic, and tuneable. These proper-ties, along with the exceptional control that is possible atlow temperatures over all the degrees of freedom, make anultracold gas of polar molecules an ideal tool for simulat-ing strongly interacting condensed-matter systems and theremarkable quantum phenomena they exhibit [2]. In low-temperature molecular gases it becomes possible to con-trol chemical reactions using electric and magnetic fieldsand to study the role of quantum effects in determin-ing chemical reactivity [3]. Cold molecules are also use-ful for testing fundamental symmetries, for example bymeasuring the value of the electron’s electric dipole mo-ment [4], searching for a time-variation of fundamentalconstants [5–7], or measuring parity violation in nuclei [8]or in chiral molecules [9]. For these applications, a greatleap in sensitivity could be obtained by cooling the rele-vant molecules to low temperatures so that, for example,the experiment could be done in a trap or a fountain [10].

a e-mail: [email protected]

The bialkali molecules can be produced at very lowtemperatures by binding together ultracold alkali atoms,by either photoassociation [11,12] or magnetoassocia-tion [13–16]. A few specific species of other moleculesare amenable to direct laser cooling to ultralow temper-atures [17]. A large variety of useful molecules can beproduced with temperatures in the range 10mK to 1Kby decelerating supersonic beams [18–20] or capturingthe lowest-energy molecules formed in a cold buffer-gassource [21,22]. For many applications it is desirable to coolthese molecules to lower temperatures, and this could bedone by mixing the molecules with ultracold atoms andencouraging the two to thermalize. This sympathetic cool-ing method has not yet been demonstrated for neutralmolecules, but is often used to cool neutral atoms [23–25],atomic ions [26] and molecular ions [27,28].

For sympathetic cooling to yield ultracold molecules,the rate of atom-molecule elastic collisions, which are re-sponsible for the cooling, must be sufficiently high that themolecules cool in the available time. In practice this re-quires that both atoms and molecules be trapped, so thatthey are held at high density and interact for a long time.The easiest way to trap molecules is in a static electric ormagnetic trap. However, static traps can confine moleculesonly in weak-field-seeking states; since the ground stateis always strong-field-seeking, inelastic collisions can ejectmolecules from the trap by de-exciting them to lower-lying strong-field-seeking states. These traps are therefore

142 The European Physical Journal D

unsuitable for sympathetic cooling unless the ratio of theelastic to inelastic cross section happens to be particularlylarge. Inelastic losses can be avoided by trapping ground-state molecules, but such traps are more difficult to realize.

In this paper, we consider the sympathetic cooling ofLiH molecules with ultracold Li atoms. Due to its largedipole moment of 5.88D, its low mass, and its simplestructure, LiH is an attractive molecule for studying thephysics of dipolar gases and the electric field control ofcollisions and chemical reactions. A supersonic beam ofcold LiH molecules has been produced [29] and deceler-ated to low speed using a Stark decelerator [30]. Ultra-cold Li is likely to be a good coolant for LiH because theclosely matching masses ensure that energy is transferredefficiently in an elastic collision. Also, the low mass of Liensures that inelastic collisions with non-zero angular mo-mentum are suppressed by a centrifugal barrier, even atrelatively high collision energies [31]. We have prepared amagneto-optical trap of 1010 Li atoms for the purpose ofsympathetic cooling.

We begin by calculating the elastic and inelastic crosssections for LiH+ Li collisions. Then we calculate the tra-jectories of a set of trapped LiH molecules that have oc-casional collisions with a co-trapped cloud of ultracold Li.Our aims are to calculate how the molecular temperatureevolves with time, to investigate loss mechanisms in dif-ferent kinds of traps, and to establish how the ultracoldatoms should be distributed so that the cooling is mostefficient.

2 Scattering calculations

We have carried out quantum-mechanical scattering cal-culations on 7Li+ 7LiH collisions on the potential energysurface of reference [32]. The calculations are carried outusing the MOLSCAT program [33]. We use full close-coupling calculations for the energy range of importancefor sympathetic cooling, up to collision energies of 1 K,and coupled states (CS) calculations over an extendedrange up to 100 cm−1. The calculations are carried outtreating LiH as a rigid rotor, with rotational constantbLiH/hc = 7.5202 cm−1. Because of the deep potentialwell (8743 cm−1) and strong anisotropy, a large rota-tional basis set is needed. The present calculations includeall functions with LiH rotational quantum number j upto jmax = 37. The coupled equations are solved usingthe hybrid log-derivative/Airy propagator of Alexanderand Manolopoulos [34] with the propagation continuedto 500 A.

The collision calculations treat the Li atom as struc-tureless. This is justified because there are almost no termsin the collision Hamiltonian that can cause a change in theLi hyperfine state or magnetic projection quantum num-ber [35,36]. The (very small) hyperfine structure of theLiH molecule is also neglected.

The results of close-coupling calculations for LiHmolecules initially in j = 1 are shown in Figure 1. Asexpected from the Wigner threshold laws [37], the elasticcross section becomes constant at very low energy (below

102

103

104

105

10−5 10−4 10−3 10−2 10−1 100

σ (Å

2 )

Collision energy (K)

Elastic j=0Elastic j=1

Inelastic j=1 → j=0

Fig. 1. (Color online) Elastic (red) and inelastic (blue) crosssections from close-coupling calculations on 7Li-7LiH with 7LiHinitially in j = 1. Also shown (black) are elastic cross sectionsfor 7LiH initially in j = 0.

about 1 mK) and the inelastic cross section is approxi-mately proportional to E−1 in this region. Above about10 mK the ratio of elastic to inelastic cross sections stabi-lizes at a factor of 5 to 10.

Close-coupling calculations are carried out for fixedvalues of the total angular momentum J , and the result-ing partial wave contributions are summed to form crosssections. The results in Figure 1 include contributions upto Jmax = 10, and Figure 2 shows the individual partialwave contributions for 0 < J ≤ 6. There is no signifi-cant resonance structure in the inelastic cross sections forJ < 6, although shape resonances appear in the elasticcross sections for J ≥ 2 and are particularly prominentfor J = 4 and 6.

The cross sections obtained from quantum scatteringcalculations on an individual potential energy surface arein general quite sensitive to small potential scalings be-cause of variations in the scattering length a. However,this sensitivity is relatively small in Li +LiH because ofthe low reduced mass. The low-energy limit of the elasticcross sections shown in Figure 1 may be compared withthe value σ = 4πa2 = 3200 A2 obtained from the meanscattering length a = 16.01 A, as defined by Gribakin andFlambaum [38].

The higher-energy results from CS calculations areshown in Figure 3. It is interesting to compare the inelas-tic cross section with the Langevin limit, which assumesthat all collisions that cross the centrifugal barrier lead toinelastic events. The Langevin limit is shown as a dashedline in Figure 3, and it may be seen that the inelastic crosssection remains below this limit even at collision energiesaround 100 cm−1.

We note that the reaction LiH + Li → Li2+H is highlyendothermic and so cannot occur at the low collision en-ergies of interest here.

S.K. Tokunaga et al.: Prospects for sympathetic cooling of molecules in electrostatic, ac and microwave traps 143

101

102

103

104

10−5 10−4 10−3 10−2 10−1 100

σ (Å

2 )


J=0J=1J=2J=3J=4J=5J=6

100

101

102

103

104

105

10−5 10−4 10−3 10−2 10−1 100

σ (Å

2 )


J=0J=1J=2J=3J=4J=5J=6

Fig. 2. (Color online) Partial-wave contributions to elastic(upper panel) and inelastic (lower panel) cross sections fromclose-coupling calculations.

3 Sympathetic cooling simulations

Using the cross sections calculated in Section 2, we simu-late the sympathetic cooling of LiH molecules co-trappedwith ultracold Li atoms. We consider three types of trap.The first is a static electric trap for molecules in the weak-field-seeking state (j, m) = (1, 0). Here, elastic collisionswith the Li atoms cool the molecules, whereas an inelasticcollision transfers a molecule to a lower-lying high-field-seeking state causing it to be lost from the trap. The othertwo traps we consider are a microwave trap and an ac elec-tric trap, both of which can trap ground-state moleculesso that inelastic losses are avoided. We note that sympa-thetic cooling in an optical dipole trap has been studiedpreviously [39].

3.1 Cooling in a static electric trap

We first consider the sympathetic cooling of LiH moleculesin the weak-field-seeking (j, m) = (1, 0) state. The simu-lation starts with a large set of molecules with a velocitydistribution that fills the trap. Later, when we considerac and microwave traps, we will track individual molecu-lar trajectories in these traps, but we do not need do this

100

101

102

103

104

105

0 20 40 60 80 100

σ (Å

2 )

Collision energy (cm−1)

Elastic j=1Total inelastic

Inelastic j=1 → j=2Inelastic j=1 → j=3

Inelastic from Langevin formula

Fig. 3. (Color online) Elastic (red) and state-to-state inelas-tic cross sections from coupled states calculations on 7Li-7LiHwith 7LiH initially in j = 1. The dashed black line shows theLangevin limit for the total inelastic cross section.

for the electrostatic trap. Our aim is only to calculate thefraction of all the molecules that cool to low temperaturewithout being lost from the trap and this is determinedentirely by the ratio of elastic to inelastic cross sections asa function of the collision energy.

For each collision we find the kinetic energy in thecentre of mass frame, look up the relative probability forelastic and inelastic collisions as given in Figure 1, andthen make a random choice between these two processesaccording to this probability. If the collision is inelastic,the molecule is lost from the trap. If the collision is elastic,the molecule’s velocity is transformed by the collision intoa new velocity using a hard-sphere collision model. Thevelocity vector of the atom is selected at random froman isotropic Gaussian velocity distribution whose width isfixed by the temperature of the atom cloud, chosen hereto be 140μK. The atom and molecule velocities are trans-formed into the centre of momentum frame, where themolecular momenta before and after the collision, p andp′, are related by

p′ = p − 2 (p · e) e, (1)

where e is a unit vector along the line joining the centresof the spheres. It is given by

e =√

1 − |b|2 p

|p| + b (2)

where b is a vector that lies in the plane perpendicularto p and whose magnitude is the impact parameter of thecollision normalized to the sum of the radii of the twospheres. For each collision, b is chosen at random from auniform distribution subject to the constraints b · p = 0and |b| ≤ 1. The new momentum of the molecule, p′,is finally transformed back into the lab frame and thismomentum is used in the next collision.

Figure 4 gives the results of these simulations, showinghow the fraction of molecules remaining in the trap and


10-1

1

10-2

10-3

10-4

Tem

pera

ture

(K) \

Fra

ctio

n R

emai

ning

Number of collisions5 10 15 20 25 30

Fig. 4. (Color online) Cooling and loss in a trap for weak-field-seeking molecules. The temperature of the molecules (bluecircles) and the fraction remaining in the trap (red squares) areplotted against the number of collisions.

the temperature of their distribution depends on the num-ber of collisions that have occurred. The molecules havean initial temperature near 100mK, and for every 10 colli-sions their temperature falls by about a factor of 10. Afterabout 20 collisions, they have reached a temperature of1mK, but only 0.7% of them remain. After 28 collisions,the molecules have thermalized to the temperature of theatoms, but now the fraction that remains is only 4×10−4.We see that the ratio of elastic to inelastic cross sectionsis too small in this case for sympathetic cooling in a statictrap to be feasible.

Our simulations use collision cross sections calculatedin zero field, even though the molecules are electrostati-cally trapped. An electric field can have a large effect onatom-molecule collisions, as recently demonstrated for col-lisions between Rb and ND3 in an electrostatic trap [40].Here, it was found that the trapping field increases theinelastic cross section. If a similar effect occurred for Li-LiH, it would strengthen our conclusion that sympatheticcooling is not feasible in the static trap for this system.

3.2 Cooling in an ac electric trap

To eliminate trap loss due to inelastic collisions, it is de-sirable to trap the molecules in their ground state. Theground state of every molecule is strong-field-seeking, andstrong-field-seeking molecules cannot be trapped usingstatic fields. One solution is to use an ac electric trapwhere the molecules move on a saddle-shaped potentialthat focusses them towards the centre of the trap in onedirection, but defocusses them in another direction. Byalternating the focussing and defocussing directions at asuitable rate, molecules are confined near the trap cen-tre. Such ac electric traps have already been used to trappolar molecules in strong-field-seeking states [41–43], andalso to trap ground-state atoms [44,45], and sympatheticcooling of molecules in ac traps has been proposed [44,45].

Motion in an ac trap consists of a small-amplitudemicromotion at the switching frequency of the trap,superimposed on a larger-amplitude, lower-frequency

Position

Spee

d

Position

PositionPosition

Spee

d

Spee

d

Spee

d

(a) (b)

(c) (d)

Fig. 5. Phase-space acceptance of an ac trap in one dimensionfor 4 phases of the switching cycle: (a) start of focussing period,(b) centre of focussing period, (c) start of defocussing period,(d) centre of defocussing period. The shaded areas indicateregions of phase space that are unstable in an idealised head-on collision.

macromotion. The stability of the molecules derives fromthe micromotion, and a collision which interrupts the mi-cromotion may put the molecule onto an unstable tra-jectory. This means that a molecule initially confined inthe trap may be ejected by a collision even though thecollision reduces its energy. Figure 5 gives a simple pic-ture of how this can happen. In each dimension, the setof all the stable molecules forms an ellipse in phase space,and this ellipse evolves periodically with the phase of theswitching cycle [42]. Figure 5a shows the ellipse at thestart of a focussing phase, showing that the positions andspeeds of the molecules are positively correlated at thisphase. To illustrate what may happen in a collision, con-sider the case where molecules collide head-on (b = 0)with stationary atoms of the same mass. In this idealisedcase, a collision reduces the speed of a molecule to zero,leaving its position unchanged, as indicated by the arrowin Figure 5a. The molecule will remain trapped only if itis still inside the ellipse, so the shaded regions of the el-lipse are unstable against collisions. The same argumentsapply at the start of the defocussing phase, as indicatedin Figure 5c. Half-way through the focussing and defo-cussing phases (Figs. 5b, 5d), the molecules remain insidethe stable region when their speeds are reduced to zeroand so there will be no collisional loss at these phases.The phase-space plots show what happens in only one di-mension. In a cylindrically symmetric ac trap, the sameplots can be made for the radial and longitudinal direc-tions separately, one being half a period out of phase withthe other. Thus when the focussing phase begins in theradial direction, the defocussing phase is beginning in thelongitudinal direction, and a molecule must not be inside


any of the shaded regions if it is to remain trapped aftera collision.

This picture is, of course, a highly simplified one. Thecollisions do not reduce the speed to zero, and they cancouple energy from one direction to another, which tendsto increase the opportunities for loss. Nevertheless, we ex-pect the same general conclusions to hold – a large portionof the trap’s stable phase-space volume becomes unstablewhen sympathetic cooling collisions are introduced, andcollisions are more likely to result in loss at the start of afocus/defocus phase than half-way through.

Turning now to a complete simulation, we considerLiH molecules in a cylindrical ac trap consisting of tworing electrodes and two cylindrically symmetric end caps,as used in references [41,42]. The square of the electricfield magnitude in this trap is well approximated by theexpression

E2(z, ρ) = E20

(1 + 2a3

(z2 − 12ρ2)

z20

+ a23

(z4 + 14ρ4)

z40

+2a5

(z4 − 3z2ρ2 + 38ρ4)

z40

), (3)

where z0 is the characteristic size of the trap, E0 is theelectric field magnitude at the trap centre, and a3 and a5

are the coefficients in a multipole expansion of the elec-trostatic potential. Considering the same trap as used inreference [42], we set a3 = −1.29 and a5 = 0.63 for thelongitudinal focussing phase, a3 = 1.29 and a5 = 0.44for the radial focussing phase, z0 = 4.55mm, and E0 =50kV/cm. We switch the trap between the two configura-tions at a frequency of 5 kHz with a 50:50 duty cycle.

An ensemble of initially warm molecules evolves withinthe trap, each molecule having occasional collisions witha distribution of 1010 ultracold Li atoms. We calculate thetrajectories of many molecules moving in the trap by solv-ing the equations of motion numerically using a Runge-Kutta method with a fixed time step. The force acting onthe molecules is F = −∇W where W is the Stark shift.For the electric field magnitudes considered here, the Starkshift of ground-state LiH is small compared to the rota-tional spacing, and is given to a good approximation bysecond-order perturbation theory:

W = −μ2eE

2

6bLiH. (4)

Here, μe is the electric dipole moment of the molecule andbLiH is the rotational constant (in energy units).

We suppose that the atoms are trapped independentlyfrom the molecules, for example in a magnetic trap, andwe give them a spherically symmetric Gaussian spatialdistribution with a 1/e half-width wa = 3 mm, and aMaxwell-Boltzmann velocity distribution with a temper-ature Ta = 50 μK.

We first simulate trajectories without any collisions,starting with an initial phase-space distribution that islarger than the trap acceptance, so as to obtain a set ofmolecules that, in the absence of collisions, survive in thetrap for 10 s. This set of molecules defines the phase-space

acceptance of the trap and is then used for the full simula-tion including the collisions. This ensures that moleculesare lost from the trap only as a result of collisions. Aftereach interval of time Δt, the calculation of the molec-ular trajectory is stopped and the probability, P , of themolecule having a collision during this time interval is cal-culated. The value of Δt is chosen such that P 1 andwe take P = n σ(K) v Δt, where n is the local density ofatoms, v is the relative velocity of the LiH molecule andLi atom at the time of collision, and σ(K) is the elasticcollision cross section at collision energy K. A randomnumber, r, is chosen from a uniform distribution in theinterval from 0 to 1, and a collision occurs only if P > r.When a collision does occur, the velocity vector obtainedfrom the molecular trajectory is transformed using thehard-sphere collision model outlined above. The numeri-cal integration of the trajectory then continues using thistransformed velocity. Since the number of trapped atomsis many orders of magnitude larger than the number oftrapped molecules, we assume that the atom distributionis unaffected by the presence of the molecules. We also ne-glect collisions between the molecules, since their densityis so low.

The dashed line in Figure 6i shows the fraction ofmolecules that survive in the ac trap as a function of time.We see that most of the molecules are lost due to collisionsand that this loss occurs on two separate time scales. Dur-ing the first 1 s, 94% of all the molecules are lost fromthe trap. Between 1 s and 10 s the number of trappedmolecules continues to fall, so that after 10 s only 1%remain in the trap. It is surprising to find that the losscontinues at long times since we would expect there to bea small region close to the origin of phase space that is sta-ble against collisions. It appears that even these moleculesare eventually being destabilized by the collisions. To in-vestigate why this happens, we repeated the simulationwith the atom temperature reduced to zero. The result isshown by the dotted line in Figure 6i. Here, the loss atearly times is the same as before, but after a few secondsof cooling the fraction remaining in the trap stabilizes ataround 5%. When the atoms have non-zero temperature,the collisions cause molecules near the origin of phase-space to diffuse away from the origin, eventually endingup on an unstable trajectory. The atoms cool the hottermolecules, but they also tend to heat the coolest ones, andeven when the atom temperature is only 50 μK the heat-ing results in significant additional losses from the trap ona 10 s timescale.

Next, to shed some light on why there is so much col-lisional loss in the ac trap, we simplified the simulationsby neglecting terms in the electric field beyond the sec-ond term in equation (3). In this harmonic approximationthe phase-space acceptance of the trap is maximized, andalthough this field cannot be realized in practice [42] itis helpful to make this approximation since the dynamicsin such an ideal ac trap are well understood. The higher-order terms complicate the dynamics by introducing non-linear forces into the trap and coupling the axial and radialmotions, and this greatly reduces the trap acceptance.


(i)

(ii)

0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

Phase of switching cycle

Num

ber l

ost (

arbi

trary

uni

ts) (b) (c) (d) (a)

Time (s)

Frac

tion

rem

aini

ng

0 2 4 6 8 100.01

0.02

0.05

0.10

0.20

0.50

1.00

Fig. 6. (Color online) Simulated molecule loss in an ac trap.(i) Fraction of molecules surviving as a function of time.Dashed (purple) line: real trap, Ta = 50 µK. Dotted (blue)line: real trap, Ta = 0. Solid (red) line: ideal trap, Ta = 50 µK.(ii) Number of collisions that result in molecule loss as a func-tion of the phase of the switching cycle, in an ideal trap. Thezero of phase corresponds to the centre of the radial focussingperiod and the labels (a)−(d) refer to the phases depicted bythe same labels in Figure 5.

Our simulations show that neglect of these higher-orderterms increases the acceptance by a factor of 4 in bothposition and velocity. The solid line in Figure 6i showsthe fraction of molecules that survive in the ideal ac trapas a function of time. In this case almost all the loss oc-curs in the first 1 s of cooling. The loss occurs from theouter regions of the trap and there is a ‘safe’ region aroundthe phase-space origin where a molecule will remain insidethe trap’s acceptance for all possible outcomes of a singlecollision. Once molecules have been cooled into this regionthere are no losses. The fraction of all the initial moleculesthat remain in the trap after 3 s is 38%, and there are nofurther losses between 3 s and 10 s. These results conformto the intuitive expectations obtained from our discussionof Figure 5. Comparing the results obtained for the idealtrap with those of the real trap, we see that it is primar-ily the large non-linear forces that make the ac trap anunsuitable environment for sympathetic cooling.

Figure 6ii shows how the number of collisions result-ing in trap loss depends on the phase of the switchingcycle, in the ideal ac trap. As expected from the discus-sion of Figure 5, the simulations confirm that collisions

50

1510

-5-10-15

-4 -2 0 2 4

-4 -2 0 2 4 -4 -2 0 2 4

-4 -2 0 2 4

50

1510

-5-10-15

50

1510

-5-10-15

50

1510

-5-10-15

Radial position (mm)

Rad

ial s

peed

(m/s

)

(i) (ii)

(iii) (iv)

Fig. 7. (Color online) Simulated time evolution of the radialphase-space distributions of molecules in an ideal ac trap over-lapped with a 50 µK atom cloud with a width parameter ofwa = 3 mm. Only the 1st and 2nd terms in equation (3) areincluded in the expression for the electric field. The phase ofthe switching cycle is the same in each plot and, to within oneswitching cycle, the times are (i) 0.002 s, (ii) 0.5 s, (iii) 3 s and(iv) 10 s.

occurring at the start of a focussing/defocussing periodare far more likely to cause loss than those occurring half-way through these periods. Similar results are obtainedwhen the higher-order terms are included, except that themodulation observed in the figure is then not so deep.

Figure 7 shows how the radial phase-space distributionin the ideal ac trap evolves with time. At early times themolecules fill the available trap acceptance, but as timegoes on they congregate near the origin of phase space.After 0.5 s (Fig. 7ii) the ellipse has become dense near thecentre and sparse elsewhere. Most molecules have had oneor more collisions by this time and these collisions tendto remove molecules from the outer regions of the distri-bution, either by cooling them towards the centre or, asdiscussed above, kicking them out of the trap. As time goeson, almost all the molecules in the outer regions of phasespace disappear and the molecules that remain are cooledinto a small region near the origin. After 10 s (Fig. 7iv)this cold distribution has a full width at half maximum of0.3mm in radial position and 0.9m/s in radial speed. Thetime evolution of the longitudinal phase-space distributionis similar.

3.3 Cooling in a microwave trap

An alternative way to trap ground-state molecules is touse a microwave trap, as discussed in reference [46]. Theground-state molecules are attracted to the electric fieldmaximum of the standing-wave microwave field inside aresonant cavity. The trap depth is particularly large whenthe detuning of the microwave frequency from the rota-tional transition frequency is small, although this placesa stringent requirement that the microwave field be circu-larly polarised in order to avoid multi-photon excitation torotationally excited states [46]. Collision-induced absorp-tion of microwave photons may also occur in the trap, and


again this unwanted process is far more probable when thedetuning is small [47]. Here, we consider a far-detunedmicrowave trap for ground-state LiH molecules, operat-ing at a frequency of 15GHz. Since this frequency is verysmall compared to the rotational frequency (2bLiH/h =445GHz), and since the Stark shift will also be smallcompared to bLiH for all attainable electric field strengths,the Stark shift is given to a good approximation by equa-tion (4), where E2 is now the time-averaged squared elec-tric field. We take the microwave field to be the fundamen-tal Gaussian mode of a symmetrical Fabry-Perot cavity,having a beam waist of 15mm and an rms electric field atthe trap centre of E0 = 40kV/cm. This is the field pro-duced by coupling 2.6 kW of power into a cavity whoseQ-factor is set by the reflectivity of room-temperature cop-per mirrors. The trap has a depth of 500mK and the sim-ulation begins with a trap whose phase-space acceptanceis completely filled. We simulate individual molecular tra-jectories in the microwave trap with collisions modelled inexactly the same way as outlined above for the ac trap.We use the field-free elastic cross section shown in Fig-ure 1 since this is insensitive to the microwave field [47],and we neglect inelastic relaxation between field-dressedstates because, for our trap conditions, the rate of thisinelastic process is expected to be very low compared tothe elastic collision rate [47].

Each time a molecule has a collision with an ultracoldatom, its energy is reduced. Nevertheless, it is possiblefor a collision to transfer energy between axial and ra-dial motions so that it has enough energy in one directionto leave the trap. By running simulations both with andwithout collisions, we find that there is no additional traploss as a result of the collisions. This is because the axialand radial motions in the trap are weakly coupled, so thateven in the absence of collisions a molecule whose energyis greater than the trap depth eventually leaves the trap.

Figure 8 shows how the distribution of molecules inthe trap evolves with time as they cool to the tempera-ture of the atom cloud whose width is wa = 3mm. Eachplot shows the radial position and speed of each moleculein the trap. After 0.1 s very little cooling has occurredand the molecules have the full range of speeds and po-sitions that the trap can accept. After 0.5 s it is clearthat the molecules are accumulating near the phase-spaceorigin as expected. They have small speeds and are con-fined near the centre of the trap. As time goes on, theaccumulation of cold molecules continues. After 2 s themajority of the molecules have been cooled into the smallarea of phase space near the origin, but some molecules re-main distributed throughout the phase-space acceptance.The distribution has separated into two components, onecold and one hot. After 10 s, 90% of the molecules arein the cold component, the remaining hot molecules forma halo in phase space around the cold ones, and there isa region in between where there are no molecules at all.The molecules that are slow to cool are the ones that ini-tially have large angular momentum about the trap centre.In the absence of collisions, these molecules cannot reachthe centre, and if they cannot reach the centre they are

-0.02 -0.01 0.00 0.01 0.02-30-20-10

0102030

-0.02 -0.01 0.00 0.01 0.02-30-20-10

0102030

-0.02 -0.01 0.00 0.01 0.02-30-20-10

0102030

-0.02 -0.01 0.00 0.01 0.02-30-20-10

0102030

Radial position (m)

Rad

ial s

peed

(m/s

)

(a) 0.1 s (b) 0.5 s

(c) 2 s (d) 10 s

Fig. 8. (Color online) Simulated time evolution of the radialphase-space distributions of molecules in the microwave trapoverlapped with a 140 µK atom cloud with a width of wa =3 mm. The cooling times are (a) 0.1 s, (b) 0.5 s (c) 2 s and(d) 10 s.

0 2 4 6 8 10

0.5

1.0

5.0

10.0

50.0

100.0

Time (s)

Tem

pera

ture

(mK

)

Fig. 9. Temperature versus time for molecules thermalizingwith cold atoms in a microwave trap. The atom distributionhas a temperature of Ta = 140 µK and a width of wa = 5 mm.To obtain the molecule temperature, we take the kinetic energydistribution, remove high energy outliers, take the mean anddivide by 3

2k.

unlikely to have any collisions. The molecules in the haloin Figure 8d have particularly large angular momenta andso spend all their time in the far wings of the atomic distri-bution; they are unlikely to have collisions even after 10 s.

The two-component speed distribution is even moreevident when the atom cloud width is reduced to 1mm.In this case, a cold distribution develops rapidly at thecentre of the trap but the rest of the trap phase-space ac-ceptance is filled with hot molecules, apart from a thinempty region separating the hot and cold distributions.When the atom cloud width is wa = 5mm the atom-

molecule overlap is sufficient that the molecules form asingle-component speed distribution. In this case it is pos-sible to give a sensible measure of the temperature. Themean kinetic energy is not a good measure because a fewremaining outliers with high kinetic energy have a dis-proportionate effect on the mean. Instead, we trim thedistribution by removing the 5% that have the highestkinetic energy, take the mean, and then divide by 3

2k toobtain a temperature. Figure 9 shows the result. After10 s the molecules have cooled from an initial temperature


0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Time (s)

Frac

tion

of c

old

mol

ecul

es

Fig. 10. (Color online) Fraction of cold molecules in the sim-ulated microwave trap as a function of time, for three differentatom cloud sizes: 1 mm (red solid line), 3 mm (green dashedline) and 5 mm (blue dotted line). A molecule is classified ascold if its kinetic energy is less than 3

2kT , where T = 1 mK.

of 100mK to a final temperature of 200μK, close to thetemperature of the atom cloud. The mean number of col-lisions per molecule required to reach this temperatureis 30. As the molecules cool they move into the densestpart of the atomic distribution and, as shown in Figure 1,the collision cross section tends to increase. As a result, thecooling rate tends to increase gradually between 100mKand 1mK, despite the fact that the collision rate is pro-portional to the decreasing speed.

Figure 10 shows the number of cold molecules as afunction of time for three different atomic cloud sizes. Thenumber of molecules is normalized to the total number inthe trap, and a molecule is taken to be cold once its ki-netic energy is less than 3

2kT , where T = 1mK. Whenwa = 5 mm, almost all the molecules in the trap cool tolow temperature, but the cooling is slow because the atomdensity is low. After 1 s, only 2% of the molecules are cold,but after 10 s, 96% of them are cold. As the atomic densityis increased by reducing the size of the cloud, the coolingrate at early times increases. However, the number of coldmolecules obtained after a long period of cooling is lowerwith these smaller atom clouds. For both the 1mm and3mm atom clouds, about 25% of the molecules are coldafter 1 s. After 10 s, 90% of the molecules are cold whenwa = 3 mm, but only 42% when wa = 1mm. As discussedabove, the molecules that fail to cool are those that havelarge angular momentum about the centre of the trap.These results show that the most suitable choice of atomcloud size depends on the trap lifetime. If the lifetime islong enough, it is best to use a large atom cloud to maxi-mize the number of cold molecules obtained. If the lifetimeis short, it is better to use a small atom cloud to maxi-mize the cooling rate and then to remove the moleculesthat remain hot, for example by lowering the trap depth.

4 Conclusions

Molecules are most easily trapped when they are preparedin weak-field-seeking states, but then sympathetic cooling

is feasible only if the ratio of elastic to inelastic cross sec-tions is high. In the Li+LiH system with LiH in its ro-tationally excited state (j = 1), we find this ratio to beapproximately 5 at a collision energy of 100mK, graduallyfalling as the collision energy decreases and reaching 1 atabout 30μK. This ratio is too small for sympathetic cool-ing to be effective, since cooling from 100mK to 100μKresults in 4 orders of magnitude of trap loss. To avoidinelastic losses in this system, the molecules need to betrapped in the ground state. This can be done using an acelectric trap, but in this trap collisions can transfer stablemolecules onto unstable trajectories. This occurs becausestability of motion in an ac trap relies on a specific cor-relation between position and speed, and this necessarycorrelation tends to be upset by collisions. Our simula-tions suggest that the resulting trap losses are too greatfor sympathetic cooling to be feasible in a realistic ac trap.

Alternatively, ground-state molecules can be trappedin the electric field maximum of a standing wave mi-crowave field formed inside a microwave cavity. The mi-crowave trap appears to be suitable for sympathetic cool-ing. We find that both the cooling rate and the fractionof molecules that cool depend on the degree of overlapbetween the atom and molecule distributions. When theatoms are compressed to a small volume, the cooling rateis high at the trap centre, but a large fraction of themolecules do not cool because their angular momentumprevents them from reaching the centre of the trap. Whenthe atom cloud is larger, more of the molecules cool butthe cooling is slower. The typical time required for a largefraction of the molecules to reach ultracold temperaturesis a few seconds. For ground-state LiH, in a good vac-uum, the trap lifetime will be limited by black-body heat-ing of the rotational motion [48]. This black-body-limitedlifetime is 2.1 s at room temperature, rising to 9.1 s at77K [49]. This suggests that liquid nitrogen cooling ofthe microwave cavity may be necessary in order to coola large fraction of the molecules in the time available.Cooling of the cavity mirrors would also allow for a factorof 10 increase in the cavity Q-factor, and a correspondingdecrease in the power required to obtain the same trapdepth. Note that the black-body heating rate is consider-ably slower for many other molecules of interest [48,49],and then trap lifetimes of several tens of seconds shouldbe attainable under good vacuum conditions.

Our use of a hard-sphere scattering model may under-estimate the degree of forward scattering, thereby overes-timating the cooling rate since scattering in the forwarddirection does little to cool the molecules. We will inves-tigate this in future work. We have used an unchangingatomic distribution in our simulations, but it is clear thatthe atoms could be used more efficiently. It would be bet-ter to compress the atom cloud gradually so that the sizeof the atom distribution matches that of the moleculesas they cool towards the trap centre. This optimizes thecollision rate by optimizing the atom density and the over-lap between the two clouds at all times. Compressing theatom cloud will raise its temperature, but at high den-sities the atoms can be evaporatively cooled which will


in turn sympathetically cool the molecules to even lowertemperatures. We have focussed on LiH molecules sympa-thetically cooled with Li atoms, but we expect our generalconclusions to apply to a wide range of other systems.

We are grateful to Isabel Llorente-Garcia and Benoit Darquiefor helpful discussions regarding the collision simulations. Weacknowledge financial support from the Polish Ministry of Sci-ence and Higher Education (grant 1165/ESF/2007/03), fromEPSRC under collaborative project CoPoMol of the ESF EU-ROCORES Programme EuroQUAM, and from the Alexandervon Humboldt Foundation.

References1. L.D. Carr, D. DeMille, R.V. Krems, J. Ye, New. J. Phys.

11, 055049 (2009)2. A. Micheli, G.K. Brennen, P. Zoller, Nature Phys. 2, 341

(2006)

3. R.V. Krems, Int. Rev. Phys. Chem. 24, 99 (2005)

4. J.J. Hudson, B.E. Sauer, M.R. Tarbutt, E.A. Hinds, Phys.Rev. Lett. 89, 023003 (2002)

5. E.R. Hudson, H.J. Lewandowski, B.C. Sawyer, J. Ye, Phys.Rev. Lett. 96, 143004 (2006)

6. H.L. Bethlem, M. Kajita, B. Sartakov, G. Meijer,W. Ubachs, Eur. Phys. J. Special Topics 163, 55 (2008)

7. H.L. Bethlem, W. Ubachs, Faraday Discuss. 142, 25 (2009)

8. D. DeMille, S.B. Cahn, D. Murphree, D.A. Rahmlow, M.G.Kozlov, Phys. Rev. Lett. 100, 023003 (2008)

9. B. Darquie, C. Stoeffler, A. Shelkovnikov, C. Daussy,A. Amy-Klein, C. Chardonnet, S. Zrig, L. Guy,J. Crassous, P. Soulard et al., Chirality 22, 870 (2010)

10. M.R. Tarbutt, J.J. Hudson, B.E. Sauer, E.A. Hinds,Faraday Discuss. 142, 370 (2009)

11. J.M. Sage, S. Sainis, T. Bergeman, D. DeMille, Phys. Rev.Lett. 94, 203001 (2005)

12. J. Deiglmayr, A. Grochola, M. Repp, K. Mortlbauer, C.Gluck, J. Lange, O. Dulieu, R. Wester, M. Weidemuller,Phys. Rev. Lett. 101, 133004 (2008)

13. S. Ospelkaus, A. Pe’er, K.-K. Ni, J.J. Zirbel, B. Neyenhuis,S. Kotochigova, P.S. Julienne, J. Ye, D.S. Jin, Nature Phys.4, 622 (2008)

14. F. Lang, K. Winkler, C. Strauss, R. Grimm, J. HeckerDenschlag, Phys. Rev. Lett. 101, 133005 (2008)

15. J.G. Danzl, E. Haller, M. Gustavsson, M.J. Mark, R. Hart,N. Bouloufa, O. Dulieu, H. Ritsch, H.C. Nagerl, Science321, 1062 (2008)

16. J.G. Danzl, M.J. Mark, E. Haller, M. Gustavsson, R. Hart,J. Aldegunde, J.M. Hutson, H.C. Nagerl, Nature Phys. 6,265 (2010)

17. E.S. Shuman, J.F. Barry, D. DeMille, Nature 467, 820(2010)

18. H.L. Bethlem, G. Berden, G. Meijer, Phys. Rev. Lett. 83,1558 (1999)

19. E. Narevicius, A. Libson, C.G. Parthey, I. Chavez, J.Narevicius, U. Even, M. Raizen, Phys. Rev. A 77, R051401(2008)

20. R. Fulton, A.I. Bishop, M.N. Schneider, P.F. Barker,Nature Phys. 2, 465 (2006)

21. J.D. Weinstein, R. de Carvalho, T. Guillet, B. Friedrich,J.M. Doyle, Nature 395, 148 (1998)

22. L.D. van Buuren, C. Sommer, M. Motsch, S. Pohle, M.Schenk, J. Bayerl, P.W.H. Pinkse, G. Rempe, Phys. Rev.Lett. 102, 033001 (2009)

23. C.J. Myatt, E.A. Burt, R.W. Ghrist, E.A. Cornell, C.E.Wieman, Phys. Rev. Lett. 78, 586 (1997)

24. A.G. Truscott, K.E. Strecker, W.I. McAlexander, G.B.Partridge, R.G. Hulet, Science 291, 2570 (2001)

25. G. Modugno, G. Ferrari, G. Roati, R.J. Brecha, A. Simoni,M. Inguscio, Science 294, 1320 (2001)

26. D.J. Larson, J.C. Bergquist, J. Bollinger, W.M. Itano, D.J.Wineland, Phys. Rev. Lett. 57, 70 (1986)

27. M. Drewsen, A. Mortensen, R. Martinussen, P. Staanum,J.L. Sørensen, Phys. Rev. Lett. 93, 243201 (2004)

28. A. Ostendorf, C.B. Zhang, M.A. Wilson, D. Offenberg, B.Roth, S. Schiller, Phys. Rev. Lett. 97, 243005 (2006)

29. S.K. Tokunaga, J.J. Hudson, B.E. Sauer, E.A. Hinds, M.R.Tarbutt, J. Chem. Phys. 126, 124314 (2007)

30. S.K. Tokunaga, J.M. Dyne, E.A. Hinds, M.R. Tarbutt,New J. Phys. 11, 055038 (2009)

31. M. Lara, J.L. Bohn, D. Potter, P. Soldan, J.M. Hutson,Phys. Rev. Lett. 97, 183201 (2006)

32. W. Skomorowski, F. Pawlowski, T. Korona, R. Moszynski,P.S. Zuchowski, J.M. Hutson, J. Chem. Phys. 134, 114109(2011)

33. J.M. Hutson, S. Green, MOLSCAT computer program,version 14, distributed by Collaborative ComputationalProject No. 6 of the UK Engineering and Physical SciencesResearch Council (1994)

34. M.H. Alexander, D.E. Manolopoulos, J. Chem. Phys. 86,2044 (1987)

35. P. Soldan, P.S. Zuchowski, J.M. Hutson, Faraday Discuss.142, 191 (2009)

36. P.S. Zuchowski, J.M. Hutson, Phys. Rev. A 79, 062708(2009)

37. E.P. Wigner, Phys. Rev. 73, 1002 (1948)38. G.F. Gribakin, V.V. Flambaum, Phys. Rev. A 48, 546

(1993)39. P. Barletta, J. Tennyson, P.F. Barker, New J. Phys. 12,

113002 (2010)40. L.P. Parazzoli, N.J. Fitch, P.S. Zuchowski, J.M. Hutson,

H.J. Lewandowski (2011), arXiv:1101.288641. J. van Veldhoven, H.L. Bethlem, G. Meijer, Phys. Rev.

Lett. 94, 083001 (2005)42. H.L. Bethlem, J. van Veldhoven, M. Schnell, G. Meijer,

Phys. Rev. A 74, 063403 (2006)43. M. Schnell, P. Lutzow, J. van Veldhoven, H.L. Bethlem,

J. Kupper, B. Friedrich, M. Schleier-Smith, H. Haak, G.Meijer, J. Phys. Chem. A 111, 7411 (2007)

44. S. Schlunk, A. Marian, P. Geng, A.P. Mosk, G. Meijer, W.Schollkopf, Phys. Rev. Lett. 98, 223002 (2007)

45. T. Rieger, P. Windpassinger, S.A. Rangwala, G. Rempe,P.W.H. Pinkse, Phys. Rev. Lett. 99, 063001 (2007)

46. D. DeMille, D.R. Glenn, J. Petricka, Eur. Phys. J. D 31,375 (2004)

47. S.V. Alyabyshev, T.V. Tscherbul, R.V. Krems, Phys. Rev.A 79, R060703 (2009)

48. S. Hoekstra, J.J. Gilijamse, B. Sartakov, N. Vanhaecke, L.Scharfenberg, S.Y.T. van de Meerakker, G. Meijer, Phys.Rev. Lett. 98, 133001 (2007)

49. S.Y. Buhmann, M.R. Tarbutt, S. Scheel, E.A. Hinds, Phys.Rev. A 78, 052901 (2008)

APPENDIX E

PAPER V

“Sympathetic cooling of the Ba+ ion by collisions with ultracold Rb atoms:

Theoretical prospects”

M. Krych, W. Skomorowski, F. Pawłowski, R. Moszynski

and Z. Idziaszek

Physical Review A 83, 032723 (2011)

125

PHYSICAL REVIEW A 83, 032723 (2011)

Sympathetic cooling of the Ba+ ion by collisions with ultracold Rb atoms: Theoretical prospects

Michał KrychInstitute of Theoretical Physics, Department of Physics, University of Warsaw, Hoza 69, 00-681 Warsaw, Poland and Quantum Chemistry

Laboratory, Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland

Wojciech Skomorowski, Filip Pawłowski, and Robert Moszynski*

Quantum Chemistry Laboratory, Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland

Zbigniew IdziaszekInstitute of Theoretical Physics, Department of Physics, University of Warsaw, Hoza 69, 00-681 Warsaw, Poland and Institut de Physique de

Rennes, UMR 6251 du CNRS and Universite de Rennes I, F-35042 Rennes Cedex, France(Received 4 August 2010; published 31 March 2011)

State-of-the-art ab initio techniques have been applied to compute the potential energy curves of the (BaRb)+

molecular ion in the Born-Oppenheimer approximation for the singlet and triplet states dissociating into theground-state 1S Rb+ ion and the Ba atom in the ground 1S state or the lowest singlet or triplet D excited states,and for the singlet and triplet states dissociating into the ground-state 2S Rb atom and the ground-state 2S Ba+ ion.The ground-state potential energy was obtained with the coupled-cluster method restricted to single, double, andnonperturbative triple excitations. The first triplet states in the , , and symmetries were computed with therestricted open-shell coupled-cluster method restricted to single, double, and nonperturbative triple excitations.All other excited-state potential energy curves were computed using the equation of motion approach withinthe coupled-cluster singles, doubles, and linear triples framework. The long-range coefficients describing theelectrostatic, induction, and dispersion interactions at large interatomic distances are also reported. The electrictransition dipole moments governing the X 1 →1 ,1 transitions have been obtained as the first residue of thepolarization propagator computed with the linear response coupled-cluster method restricted to single and doubleexcitations. Nonadiabatic radial and angular coupling matrix elements as well as the spin-orbit coupling matrixelements have been evaluated using the multireference configuration-interaction method restricted to single anddouble excitations with a large active space. With these couplings, the spin-orbit-coupled (relativistic) potentialenergy curves for the 0+ and 1 states relevant for the running experiments have been obtained. Finally, relativistictransition moments and nonadiabatic coupling matrix elements were obtained from the nonrelativistic resultsand spin-orbit eigenvectors. The electronic structure input has been employed in the single-channel scatteringcalculations of the collisional cross sections between the Ba+ ion and Rb atom. Both nonrelativistic and relativisticpotentials were used in these calculations. Our results show that the inelastic cross section corresponding to thecharge transfer from the Rb atom to the Ba+ ion is much smaller than the elastic one over a wide range of energiesup to 1 mK. This suggests that sympathetic cooling of the Ba+ ion by collisions with ultracold Rb atoms shouldbe possible.

DOI: 10.1103/PhysRevA.83.032723 PACS number(s): 34.20.Cf, 34.70.+e, 34.50.Cx

I. INTRODUCTION

Nowadays an increasing number of experimental groupsworldwide have started to work with hybrid systems involvingcold or ultracold trapped atoms and ions [1–5]. Apart from thefundamental interest in the physics of atom-ion collisions inthe quantum regime [6–8], these systems are very attractivefrom the point of view of quantum-information processing[9,10], study of many-body effects of ion impurities [11],or creation of molecular ions [12]. One type of ongoingexperiments is focused on studying the collisional processesin cold clouds of atoms and ions, like Yb with Yb+ and Nawith Ca+ [1,2], stored in dual hybrid charged-neutral traps atmillikelvin temperatures. Other experiments, in contrast, studythe dynamics of single ions like Ba+ or Yb+ trapped in rfpotentials and immersed in Bose-Einstein condensates [3–5].In such experiments the ultracold cloud of atoms is prepared

*Corresponding author; [email protected]

using standard cooling and trapping techniques, while the ionis laser cooled separately in the rf trap and later overlapped withthe Bose-Einstein-condensed atoms. The atom-ion collisionscan lead to further sympathetic cooling of the ion, but thenet cooling effect depends on the interplay of (i) the two-bodycollisional properties, (ii) the micromotion of the ion due to thetime-dependent rf potential, and (iii) the collective phenomenaresulting from the coherent properties of the condensateadditionally modified by the presence of an ionic impurity.In this paper we perform a first step toward understanding thesympathetic cooling process of Ba+ ions with Rb atoms, bycalculating highly accurate molecular potentials and determingthe single-channel elastic, spin-exchange, and charge-transfercollision rates.

In a recent paper Makarov et al. [13] performed ab initioand dynamical calculations on the (CaNa)+ molecular ion.The results of these calculations suggest that the millikelvinregime of collisional cooling of calcium ions by sodium atomsis very favorable, with the rate coefficient for charge transferfrom the Na atom to the Ca+ ion several orders of magnitude

032723-11050-2947/2011/83(3)/032723(16) ©2011 American Physical Society

MICHAŁ KRYCH et al. PHYSICAL REVIEW A 83, 032723 (2011)

smaller than the rate for elastic and spin-exchange collisions.This strongly suggests that sympathetic cooling of ions bycollisions with ultracold atoms should be possible. This systemwas further studied in the ultracold regime in Ref. [8] withinthe multichannel quantum defect formalism. In the presentpaper we investigate the possibility of sympathetic coolingfor yet another system of experimental interest [5]: Ba+ ionscooled by collisions with ultracold Rb atoms.

Theoretical modeling of collisions in the ultracold regimerequires a lot of care [14–18]. First of all, the electronicstates involved in the dynamics must be computed withstate-of-the-art methods of quantum chemistry. In particular,these methods should be size consistent in order to ensure aproper dissociation of the molecular state, and must sufficientlyaccount for the electronic correlation. Moreover, any informa-tion on the long-range asymptotics of the potentials is veryimportant. Finally, all couplings between the molecular states,those resulting both from the spin-orbit interaction and fromthe nonadiabatic effects, should be considered. Having theelectronic structure results at hand, exact quantum-dynamicalcalculations should be performed to get the cross sections andcollisional rates.

In a recent series of papers Knecht et al. [19,20] re-ported nonrelativistic, scalar relativistic, and fully relativisticab initio potential energy curves for the (BaRb)+ molecularion. Unfortunately, the approach adopted in these papers is notsize consistent, so the results at large internuclear distancesmay not be accurate enough. Moreover, excitations in thewave function beyond the doubles levels were included onlyfor the simplest single-reference closed-shell and high-spinstates. Also, the transition moments between states, necessaryto model the radiative charge-transfer process from the Rbatom to the Ba+ ion, were not computed. Finally, the nonadi-abatic coupling matrix elements between different molecularstates were not considered. Therefore, in the present paperwe report state-of-the-art ab initio potential energy curvesfor the (BaRb)+ molecular ion in the Born-Oppenheimerapproximation for the singlet and triplet states dissociatinginto the ground-state 1S Rb+ ion and the Ba atom in the ground1S state or the lowest singlet or triplet D excited states, and forthe singlet and triplet states dissociating into the ground-state2S Rb atom and the ground-state 2S Ba+ ion, electric transitiondipole moments and nonadiabatic and spin-orbit-couplingmatrix elements. Except for the spin-orbit-coupling and somenonadiabatic coupling matrix elements, all the results wereobtained with size-consistent methods based on the coupled-cluster ansatz including triple excitations. The nonrelativisticresults are transformed to the relativistic basis, thus allowingus to judge the importance of the relativistic effects, and touse our data in quantum-dynamical calculations within boththe Hund’s cases (a) and (c). Finally, we report single-channelcalculations of the elastic, spin-flip, and charge-transfer crosssections. This will allow us to obtain the ratio of inelastic toelastic cross sections at various temperatures, and thus givean estimate of the efficiency of the sympathetic cooling of thebarium ion by collisions with ultracold rubidium atoms.

The plan of this paper is as follows. In Sec. II weintroduce the theoretical models used in our calculations. Westart this section with a description of the methods used inab initio calculations of the Born-Oppenheimer potential

energy curves and electric transition dipole moments for(BaRb)+. We continue with the calculations of the nonadi-abatic and spin-orbit-coupling matrix elements, and of therelativistic potentials. The choice of fixing the long-rangecoefficients at their ab initio values is also addressed. Theremaining part of this section is devoted to the second stepof the Born-Oppenheimer approximation, i.e., to dynamicalcalculations of the elastic, spin-flip, and charge-transferscattering cross sections. Numerical results are presented anddiscussed in Sec. III. We start this section with the discussionof the ground- and excited-state potentials. Next we turn to thenonadiabatic coupling matrix elements and electric transitiondipole moments. The effects of the spin-orbit coupling onthe potentials, transition moments, and nonadiabatic couplingmatrix elements are also addressed. Whenever possible, ourtheoretical results are compared with the available ab initiodata [19,20]. Once the results of the electronic structurecalculations are presented and discussed we turn to the problemof producing cold barium ions. We present the results for theelastic, spin-flip, and charge-transfer cross sections for boththe nonrelativistic and relativistic potentials, and discuss theefficiency of the sympathetic cooling leading to cold bariumions. Finally, in Sec. IV we conclude our paper.


A. Electronic structure calculations

When dealing with collisions at ultralow temperatures theaccuracy of the potential in the long range is very important.Therefore, the methods used in the calculations should besize consistent in order to ensure a proper dissociation of theelectronic state, and a proper long-range asymptotics of thepotential. In the present paper we adopt the computationalscheme successfully applied to the ground and excited statesof the calcium dimer [14–18]. The potential energy curves forthe ground and excited states of the (BaRb)+ molecular ionhave been obtained by a supermolecule method:

V2S+1||(R) = ESM

AB − ESMA − ESM

B (1)

where ESMAB denotes the energy of the dimer computed using

the supermolecule (SM) method, and ESMX , X = A or B, is

the energy of the atom X. For the ground-state potential weused the coupled-cluster method restricted to single, double,and noniterative triple excitations, CCSD(T). For the firsttriplet states of the , , and symmetries we employedthe restricted open-shell coupled-cluster method restricted tosingle, double, and noniterative triple excitations, RCCSD(T).Calculations on all other excited states employed the linearresponse theory (equation of motion) within the coupled-cluster singles, doubles, and linear triples (LRCC3) framework[21,22]. Note that the second and higher triplet states and allexcited singlet states are open-shell systems that cannot bedescribed with a single high-spin reference function, so one hasto resort to methods especially designed to describe open-shellsituations [23]. The CCSD(T) and LRCC3 calculations wereperformed with the DALTON program [24], while RCCSD(T)calculations were done with the MOLPRO suite of codes [25].It is interesting to note at this point that even though (BaRb)+is effectively a two-electron system, triple excitations are

032723-2

SYMPATHETIC COOLING OF THE Ba+ ION BY . . . PHYSICAL REVIEW A 83, 032723 (2011)

very important. A more detailed discussion of this pointwill be presented in Sec. III. In principle we could use theLRCC3 method to obtain all the excited-state potentials in anysymmetry. However, the LRCC3 method is computationallymore expensive than the RCCSD(T) approach, so we decidedto use the latter when possible. However, we have checked fora few points that the RCCSD(T) and LRCC3 results for the(1) 3, (1) 3, and (1) 3 are very close.

The long-range asymptotics of the potentials is of primaryimportance for cold collisions. Therefore, for each state wehave computed the leading long-range coefficients describingthe electrostatic, induction, and dispersion interactions. Forthe ground X 1 state dissociating into Rb+(1S) + Ba(1S) andfor the first excited singlet and triplet states dissociating intoBa+(2S) + Rb(2S) the leading long-range asymptotics at largeinternuclear distances R reads

V2S+1||(R) = −C ind

4

R4− C ind

6

R6− C

disp6

R6+ · · · , (2)

where the long-range coefficients C ind4 , C ind

6 , and Cdisp6 are

given by the standard expressions (see, e.g. Refs. [26,27])that can be derived from the multipole expansion of theinteratomic interaction operator. The long-range inductionand dispersion coefficients were computed with the recentlyintroduced explicitly connected representation of the expec-tation value and polarization propagator within the coupledcluster method [28,29], and the best approximation XCCSD4proposed by Korona and collaborators [30]. For the singletand triplet states dissociating into Ba+(2S) ion and Rb(2S)atom the induction coefficients were obtained from finite-fieldRCCSD(T) calculations, while the dispersion coefficient fromthe sum-over-state expression with the transition momentsand excitation energies computed with the multireferenceconfiguration-interaction method limited to single and doubleexcitations (MRCI). Specifically, the transition moments andexcitation energies of the Ba+(2S) ion were obtained in thisway, while the Rb polarizability at imaginary frequency wastaken from highly accurate relativistic calculations from thegroup of Derevianko [31].

For the molecular states of the (BaRb)+ ion dissociatinginto Ba(1/3D) + Rb+(1S) the long-range asymptotics of thepotentials is slightly more complicated and reads

V2S+1||(R) = Celst

3

R3− C ind

4

R4+ Celst

5

R5− C ind

6

R6− C

disp6

R6+ · · · ,

(3)

where the new terms appearing in the expression abovedescribe the long-range charge-quadrupole (Celst

3 ) and charge-hexadecapole (Celst

5 ) interactions. The mathematical expres-sions for the coefficients of Eq. (3) are given by

Celst3 = (−1)2+

(2 2 2

− 0

)〈1/3D||Q2||1/3D〉, (4)

C ind4 = 1

2

(α0 + 32 − 6

6α2

), (5)

Celst5 = (−1)2+

(2 4 2

− 0

)〈1/3D||Q4||1/3D〉, (6)

C ind6 = 1

2Czz,zz, (7)

Cdisp6 = 1

2π

∫ ∞

0αRb+

0 (iω)

(6α0(iω) + 3

32 − 6

6α2(iω)

)dω.

(8)

In these equations the expression in parentheses is a 3j

symbol, 〈1/3D||Ql||1/3D〉 (l = 2 or 4) denotes the reducedmatrix element of the quadrupole and hexadecapole moment,respectively, α0 and α2 are the scalar and tensor componentsof the electric dipole polarizability tensor of the Ba atom inthe 1/3D state [32], while Czz,zz is the z component of thequadrupole polarizability of Ba(1/3D). Finally, αRb+

0 is thepolarizability of the rubidium ion Rb+(1S). The electrostaticcoefficients Celst

3 and Celst5 for the , , and states are

not independent, and are connected one to the other by thefollowing relations:

Celst3 () = −Celst

3 () = 2Celst3 (),

(9)

Celst5 () = 6Celst

5 () = −3

2Celst

5 ().

Note parenthetically that all coefficients that lead to attractiveinteractions (induction and dispersion terms) are assumedto be positive, while the electrostatic constants may resultin both attractive and repulsive interactions, so they enterwith their true sign. The values of the quadrupole moments,and scalar and tensor dipole polarizabilities, as well as thecomponents of the quadrupole polarizability, were obtainedfrom finite-field LRCC3 calculations on the 1/3D state ofthe atom. The hexadecapole moment of the singlet state wasobtained as an excited-state expectation value within the linearresponse CCSD formalism of Christiansen et al. [33], whilefor the triplet state from the MRCI calculations. The vectorcomponent of the dipole polarizability cannot be obtainedfrom finite-field calculations, so it was obtained from thesum-over-state expression with the transition moments andexcitation energies computed with the MRCI method. Allcalculations of the long-range coefficients employed bothDALTON [24] and MOLPRO [25] suites of codes.

The transitions from the ground X 1 state to the 1 and1 states are electric dipole allowed. The transition dipolemoments for the electric, µi , transitions were computed fromthe following expression [34]:

µi = ⟨X

AB

∣∣ri

∣∣excAB

⟩, (10)

where ri denotes the ith component of the position vector,while X

AB and excAB are the wave functions for the ground

and excited states, respectively. Note that in Eq. (10) i = x ory corresponds to transitions to 1 states, while i = z corre-sponds to transitions to 1 states. In the present calculationsthe electric transition dipole moments were computed as thefirst residue of the LRCCSD linear response function with twoelectric, r , operators [21]. In these calculations we have usedthe DALTON program [24]. We have evaluated the dependenceof the transition dipole moments on the internuclear distancefor the same set of distances as the excited-state potentialenergy curves.

032723-3


As will be shown in the next section the electronic statesof the low-lying excited states of the (BaRb)+ molecular ionshow strong nonadiabatic couplings. Therefore, in this workwe have computed the most important radial

R(R) =⟨(n)2S+1||

∣∣∣∣ ∂

∂R

∣∣∣∣ (n′)2S+1||⟩, (11)

and angular

A(R) = 〈(n)2S+1|||L+|(n′)2S+1|′|〉, (12)

coupling matrix elements. In the above equations ∂/∂R andL+ denote differentiation with respect to the internucleardistance and the electronic angular momentum operator,respectively. Note that the radial operator couples states of thesame multiplicity and symmetry, while the electronic angularmomentum operator couples states with differing by 1. In thepresent calculations the angular coupling between the singletstates was computed as the first residue of the LRCCSD linearresponse function with two angular momentum operators L[21]. In these calculations we have used the DALTON program

[24]. All other nonadiabatic couplings were obtained with theMRCI method and the MOLPRO code [25]. We have evaluatedthe dependence of the nonadiabatic coupling matrix elementson the internuclear distance for the same set of distances asthe excited-state potential energy curves.

Barium and rubidium are heavy systems, so the electronicstates of the (BaRb)+ molecular ion are strongly mixed by thespin-orbit (SO) interactions. Therefore, in any analysis of thecollisional cross sections between Ba+ and Rb the SO couplingand its dependence on the internuclear distance R must betaken into account. We have evaluated the spin-orbit-couplingmatrix elements for the lowest dimer states that couple tothe 0+ and 1 states of (BaRb)+, with the spin-orbit-couplingoperator HSO defined within the Breit-Pauli approximation[35]. The spin-orbit-coupling matrix elements have beencomputed within the MRCI framework with the MOLPRO

code [25]. Having the spin-orbit-coupling matrix elementsat hand, we can build up the matrices that will generate thepotential energies of the spin-orbit states that couple to 0+ and1 symmetry. The matrix for the 1 states is written as

⎛⎜⎜⎜⎜⎜⎜⎝

V (1) 1 〈(1) 1|HSO|(1) 3〉〈(1) 1|HSO|(1) 3〉〈(1) 1|HSO|(2) 3〉〈(1) 1|HSO|(1) 3〉〈(1) 3|HSO|(1) 1〉 V (1) 3 0 0 〈(1) 3|HSO|(1) 3〉〈(1) 3|HSO|(1) 1〉 0 V (1) 3 − 〈(1) 3|HSO|(1) 3〉 0 〈(1) 3|HSO|(1) 3〉〈(2) 3|HSO|(1) 1〉 0 0 V (2) 3 〈(2) 3|HSO|(1) 3〉〈(1) 3|HSO|(1) 1〉〈(1) 3|HSO|(1) 3〉〈(1) 3||HSO|(1) 3〉〈(1) 3|HSO|(2) 3〉 V (1) 3

⎞⎟⎟⎟⎟⎟⎟⎠

(13)

while the matrix for the 0+ states is given by⎛⎜⎜⎜⎜⎝

V (1) 3 − 〈(1) 3|HSO|(1) 3〉〈(1) 3|HSO|X 1〉〈(1) 3|HSO|(2) 1〉〈(1) 3|HSO|(1) 1〉〈X 1|HSO|(1) 3〉 V X 1 0 0

〈(2) 1|HSO|(1) 3〉 0 V (2) 1 0

〈(3) 1|HSO|(1) 3〉 0 0 V (3) 1

⎞⎟⎟⎟⎟⎠. (14)

Diagonalization of these matrices gives the spin-orbit-coupledpotential energy curves for the 1 and 0+ states, respectively.Note that all potentials in the matrices (13) and (14) weretaken from CCSD(T), RCCSD(T), and LRCC3 calculations.Only the diagonal and nondiagonal spin-orbit-coupling matrixelements were obtained with the MRCI method. Once theeigenvectors of these matrices are available, one can easily getthe electric dipole transition moments and the nonadiabaticcoupling matrix elements between the relativistic states.

Finally, the long-range coefficients corresponding to therelativistic potentials were obtained by diagonalizing thematrices of Eqs. (13) and (14) with the potentials expandedaccording to Eqs. (2) and (3) and the spin-orbit-coupling matrixelements fixed at the atomic values. Note that unlike in thecase of resonant interactions between like atoms [36], the SOcoupling does not change the leading power in the asymptoticexpansion of the interaction energy, but only changes thenumerical values of the coefficients. One should also note

that the atomic SO coupling does not change in our model thelong-range coefficients for the (1) 1 and (2) 1 states due tothe different dissociation limits: Ba(1/3D) + Rb+(1S) versusBa+(2S) + Rb(2S).

In order to mimic the scalar and spin-orbit relativistic effectssome electrons were described by pseudopotentials. For Baand Rb we took the ECP46MDF [37] and ECP28MDF [38]pseudopotentials, respectively, from the Stuttgart library. Forboth barium and rubidium we used the spdfg quality basissets suggested in Refs. [37,38]. As shown in Refs. [37,38] thepseudopotentials ECP46MDF and ECP28MDF and the basissets associated with them work remarkably well for the atomicproperties; cf. the second paragraph of Sec. III A. The full basisof the dimer was used in the supermolecule calculations andthe Boys and Bernardi scheme was used to correct for thebasis-set superposition error [39].

It should be stressed at this point that the ab initio resultsreported in the present paper obtained by the ab initio methods

032723-4


described above will allow one to perform dynamical calcu-lations of the cross sections in the nonadiabatic, multichannelregime, in both the Hund’s cases (a) (nonrelativistic states, SOand nonadiabatic couplings, and transition moments) and (c)(relativistic potentials, nonadiabatic couplings, and transitionmoments); cf. Ref. [40].

Finally, to conclude this section we would like to emphasizethat almost all ab initio results were obtained with themost advanced size-consistent methods of quantum chemistry:CCSD(T), RCCSD(T), and LRCC3. Only the SO-couplingmatrix elements and nonadiabatic matrix elements wereobtained with the MRCI method which is not size consistent.Fortunately enough, all of the couplings are important in theregion of the curve crossings or avoided crossings and vanishat large distances, so the effect of the size inconsistency ofMRCI on our results should not be dramatic.

B. Dynamical calculations

In collisions of the Ba+(2S) ion with Rb(2S) atom wehave basically three types of processes: elastic scattering inthe singlet and triplet potentials, spin-flip (spin-exchange)processes, and the inelastic radiative charge transfer fromthe singlet and triplet manifolds of Ba+Rb to the groundstate of BaRb+. In the present paper we restrict ourselvesto single-channel calculations. A more detailed multichanneltreatment will be presented elsewhere.

To compute the elastic cross sections we need to solve theradial Schrodinger equation for the relative motion of the Ba+ion and Rb atom at an energy E:

(d2

dR2− 2µ

h2 V (R) − J (J + 1)

R2+ 2µE

h2

)EJ (R) = 0,

(15)

subject to the following normalization conditions:∫ ∞

0

E1J(R)E2J (R)dR = δ(E1 − E2), (16)

where EJ (R) is the scattering wave function, µ is the reducedmass of the (BaRb)+ ion, and V (R) stands for the interactionpotential of Ba+(2S) with Rb(2S) in the singlet or tripletmanifold. Note that the normalization condition of Eq. (16)is equivalent to the following large-R behavior of the wavefunction EJ (R):

EJ (R) ∼(

2µ

πh2k

)1/2

sin

(kR − Jπ

2+ δJ (E)

), (17)

where δJ (E) denotes the phase shift corresponding to the J

partial wave, and the wave vector k is given by the standard ex-pression, E = h2k2

2µ. Equation (15) subject to the normalization

condition (16) allows us to compute the cross sections for theelastic and spin-flip collisions from the standard expressions:

σ sel(E) = 4π

k2

∞∑J=0

(2J + 1) sin2 δsJ (E),

(18)

σ tel(E) = 4π

k2

∞∑J=0

(2J + 1) sin2 δtJ (E),

σsf(E) = π

k2

∞∑J=0

(2J + 1) sin2[δsJ (E) − δt

J (E)], (19)

where the superscripts “s” and “t” on σel and δJ pertain to thesinglet and triplet potentials, respectively. Note that an exactdescription of the spin-flip process would require at least twocoupled channels, so the expression (19) is only approximate[41]. It is derived under assumption that the hyperfine splittingsare much smaller than the collision energy. However, it wasshown to work relatively well, even at low energies [42,43].

Theoretical description of the charge-transfer process be-tween the atom and the ion is somewhat more elaborate. To thefirst order of perturbation theory the radiative charge transfercan be described by the following Fermi golden rule type ofexpression [44–46]:

σct(E) = 4π2h

k2A(E), (20)

where the Einstein coefficient A(E) is given by [40]

A(E′) = 4α3

3e4h2

∞∑J ′=0

∑J ′′=J ′±1

(2J ′ + 1)

×[ ∫ ∞

0ε3HJ ′ |〈E′J ′ |µ(R)|E′′J ′′ 〉|2dε

+∑v′′

HJ ′ (Ev′′J ′′ −E′)3|〈E′J ′ |µ(R)|v′′J ′′ 〉|2],

(21)

where the primed and double-primed quantities pertain to theexcited- and ground-state potentials, respectively, µ(R) is thetransition moment from the ground to the excited electronicstate, α = 1/137.035 999 679 (94) is the fine structure con-stant, e is the electron charge, and the Hohn-London factor HJ ′

is equal to (J ′ + 1)/(2J ′ + 1) for the P branch (J ′ = J ′′ − 1),and to J ′/(2J ′ + 1) for the R branch (J ′ = J ′′ + 1). Here, ε

stands for the energy difference

ε = E′′ − E′ + IP, (22)

where IP is the difference of the ionization potentials. Thescattering wave functions appearing in the expression (21) aresolutions of Eq. (15), while the bound-state wave functionsfulfill the following Schrodinger equation:(

d2

dR2− 2µ

h2 V (R) − J ′′(J ′′ + 1)

R2+ 2µEv′′J ′′

h2

)v′′J ′′ (R) = 0,

(23)

subject to the following normalization conditions:∫ ∞

0

v1J(R)v2J (R)dR = δv1v2 , (24)

where V (R) stands for the ground-state potential of the(BaRb)+ molecular ion.

A significantly simpler approach proposed in Ref. [46]approximates the sum over all continuum and bound statesin Eq. (21) with a simple average of a space-varying Einsteincoefficient over the initial scattering wave function E′J ′ :

A(E′) =∞∑

J ′=0

(2J ′ + 1)〈E′J ′ |A(R)|E′J ′ 〉, (25)

032723-5


FIG. 1. (Color online) Nonrelativistic potential energy curves forthe excited states of the (BaRb)+ molecular ion.

where

A(R) = α3

3he6(δV )3(R)µ2(R), (26)

and δV (R) is the difference between the excited- and ground-state potentials.


A. Nonrelativistic potential energy curves and spectroscopiccharacteristics of the ground and excited states

Calculations were done for the ground state and first eight(four singlet and four triplet) excited states of (BaRb)+.Two states dissociate into Ba+(2S) + Rb(2S), three states intoBa(3D) + Rb+(1S), and three states into Ba(1D) + Rb+(1S).The potential energies were calculated for 20 interatomicdistances R ranging from 4 to 50 bohrs for each potentialcurve. The potential curves are plotted in Fig. 1, while thespectroscopic characteristics of these states are reported in

Table I. The ground state is absent on the figures due to itsregular behavior and single minimum, and in order to increasethe visibility of the other states. The separated-atom energy foreach state was set equal to the experimental value. Numericalvalues of the potentials are available from the authors onrequest.

Before going on with the discussion of the potentials letus note that the atomic excitation energies obtained from theLRCC3 calculations are very accurate. Our predicted positionof the nonrelativistic 3D state of barium is 9422 cm−1, tobe compared with the experimental value of 9357 cm−1 [47]deduced from the positions of the states in the D multiplet andthe Lande rule. For the 1D state of Ba we obtain 11 907 cm−1,in a relatively good agreement with the experimental value of11 395 cm−1 [47]. It is worth noting that the present results forthe atomic excitation energies are as accurate as the results offully relativistic atomic calculations of Kozlov and Porsev [48],and more accurate than the data obtained from fully relativisticDirac-Coulomb calculations [20]. To further assess the qualityof the methods, basis sets, and pseudopotentials employed inthe present paper we have computed the static polarizabilitiesof the ground state of the Ba atom and of the ground stateof the Ba+ ion, and the scalar and tensor components ofthe polarizability of Ba(1D). The present polarizabiity of theground state of the barium atom is 272.5 a.u. The experimentalvalue is 268 ± 22 [49], while the best theoretical result ofKozlov and Porsev [48] is 264 a.u. Also the static polarizabilityof the Ba+ ion, 132 a.u., is in a fairly good agreement with theresult of fully relativistic calculations of Ref. [50], 124.15 a.u.,of Ref. [51], 124.26 a.u., and with the most recent experimentalresult, 123.88(5) a.u. [52]. For the 1D state we get the scalar andtensor polarizabilities of 289 and 73 a.u., in a fair agreementwith the results of Ref. [48], 266 and 81 a.u., respectively.The methods employed in the present paper do not allow fora consistent calculation of the dissociation limit IP of the(2) 1 and (1) 3 states corresponding to Ba+(2S) + Rb(2S)separated atoms, since the ground-state calculations employed

TABLE I. Spectroscopic characteristics of the nonrelativistic electronic states of the (BaRb)+ molecular ion.

State Re (bohr) De (cm−1) Te (cm−1) Reference Dissociation

X 1 8.67 5136 0 Present Ba(1S) + Rb+(1S)8.85 0 Ref. [19]

(1) 3 9.27 6587 6893 Present Ba+(2S) + Rb(2S)9.38 6808 Ref. [19]

(2) 1 (primary minimum) 9.02 911 12569 Present Ba+(2S) + Rb(2S)(2) 1 (secondary minimum) 15.19 576 12904 Present Ba+(2S) + Rb(2S)(2) 3 (primary minimum) 9.82 1874 12683 Present Ba(3D) + Rb+(1S)

9.58 13478 Ref. [19](2) 3 (secondary minimum) 16.78 697 13861 Present Ba(3D) + Rb+(1S)(1) 3 8.19 6301 8257 Present Ba(3D) + Rb+(1S)

8.17 8832 Ref. [19](1) 3 9.11 4380 10178 Present Ba(3D) + Rb+(1S)

9.19 10776 Ref. [19](3) 1 12.00 1858 15146 Present Ba(1D) + Rb+(1S)(1) 1 9.04 4403 12601 Present Ba(1D) + Rb+(1S)

8.44 15906 Ref. [19](1) 1 9.08 5769 11236 Present Ba(1D) + Rb+(1S)

9.19 11276 Ref. [19]

032723-6


CCSD(T) and the calculations on the singlet and triplet excitedstates the LRCC3 and RCCSD(T) methods, respectively.However, we can estimate IP from the energies of the (2) 1

and (1) 3 states at R = 50 bohrs. Both the LRCC3 calculationon the singlet state and RCCSD(T) calculation on the tripletstate give IP = 8097 cm−1 in a good agreement with theexperimental value of 8344 cm−1 [47]. The fully relativisticresult of Ref. [20], 8065 cm−1, is very close to our value.

The ground X 1 state of the (BaRb)+ molecular ion is astrongly bound state with the binding energy of 5136 cm−1.The minimum on the potential energy curve for this stateappears at a relatively large distance Re = 8.67 bohrs. Theorigins of the binding can be explained by using the symmetry-adapted perturbation theory of intermolecular forces (SAPT)[26,27]. As could be guessed, the interaction energy at theminimum results from a subtle balance of the inductionattraction and exchange repulsion. The induction energy ishuge, –14 499 cm−1, but is strongly quenched by the exchange-repulsion term, 10 068 cm−1. The electrostatic contributiondue to the charge overlap of the unperturbed electron cloudsof Ba and Rb+, –554 cm−1, and dispersion term, –598 cm−1

is of minor importance, again in agreement with our intuition.An inspection of Fig. 1 shows that the potential energy

curves for the excited states of the (BaRb)+ molecular ionare smooth with well-defined minima. The potential energycurves of the (2) and (3) 1 states show an avoided crossing.The potential energy curves of the (2) 1 and (2) 3 statesexhibit a double-minimum structure. The double minimumon the potential energy curve of the (2) 1 state is due tothe interaction with the (3) 1 state. Other double-minimum

structures can be explained from the long-range theoryand will be discussed below. Some potential energy curvesshow maxima. These are due to the first-order electrostaticinteractions in the long range, and will also be discussedin more detail in Sec. III B. Except for the shallow double-minima structure of the (2) 1 and (2) 3 states, and ashallow (3) 1 state, all other excited states of the (BaRb)+molecular ion are strongly bound with binding energiesDe ranging from approximately 4380 cm−1 for the (1) 3

state up to 6301 cm−1 for the (1) 3 state. The (2) 1

state, important from the experimental point of view, hastwo minima at Re = 9.02 and 15.19 bohrs. The depths ofthese minima are 911 and 576 cm−1, while the barrierseparating them is 30 cm−1 suggesting that the tunnelingbetween the two wells will be very fast. The (2) 3 statealso shows two minima at Re 9.82 and 16.78 bohrs of1874 and 697 cm−1, respectively, separated by a barrier of681 cm−1.

Before comparing our results with the ab initio datareported in Refs. [19,20] let us stress the importance of thetriple excitations in the wave functions for some states. Thispoint is illustrated in Fig. 2, where LRCCSD versus LRCC3and CCSD versus CCSD(T) potential energy curves are plottedfor selected states. An inspection of these figures shows thatthe contribution of the triple excitations is important for theexcited (2) 1 and (2) 3 states, and relatively unimportantfor the (1) 3 and for the (1) 3 states. Not shown onthese figures are the (1) 3, (1) 1, and (1) 1 states. Forthese states we find that the potential for the (1) 3 stateis relatively unaffected by the triple excitations, while the

FIG. 2. (Color online) Comparison of the LRCC3 with CCSD, and RCCSD(T) with LRCCSD potential energy curves for selected statesof the (BaRb)+ molecular ion.

032723-7


two other potential energy curves show large differencesdepending on whether the T3 cluster operator is includedor not in the wave functions. These results strongly suggestthat the CCSD or LRCCSD method works for those statesthat can be described by a single reference determinant. Foropen-shell states, i.e., all singlet states and the (2) 3 state,the effect of the triple excitations is large and changes boththe well depths and the barriers. In principle, the differencescould be due to CCSD(T) vs LRCC3 methods used in thecalculations. However, as we already stated in Sec. II, wehave checked for a few points that the RCCSD(T) and LRCC3results for the (1) 3, (1) 3, and (1) 3 are very close. Toget a better understanding of the importance of the tripleexcitations we analyzed the energy gaps between the lowestelectron configuration for a given state and the lowest triplyexcited configuration. It turned out that for the effectivelyhigh-spin states this energy difference was high, while forthe manifestly states the opposite was true. Using simpleperturbation theory open-shell arguments one can deduce thatthe correlation due to the triple excitations will be importantfor states with a small energy gap between the lowest and triplyexcited configurations, and much less important for states withlarge energy difference.

Let us compare our results with other available ab initiodata [19]. The spectroscopic constants are listed in Table Iand compared with the results of Ref. [19]. Unfortunately,Knecht et al. [19] did not report the binding energies De ofthe molecular states, but only the values of the electronic termvalues Te, taking the minimum of the ground state as zero ofenergy. An inspection of Table I shows that the agreement withthe data of Knecht et al. [19] is relatively good, given the factthat their results were obtained using the internally contractedmultireference configuration singles and doubles methodbased on a CASSCF reference function. For most of the statesthe computed electronic terms agree within a few hundredcm−1. For the high-spin states the differences in the positionsof the minima are 0.1 bohr at worst, while the electronic termsdiffer by 300 to 600 cm−1. For the open-shell states, wherethe triple excitations in the wave function are important, thedifferences in the positions and well depths are more important.The most striking difference between the present results andthe data of Ref. [19] is in the (1) 1 state. Here, the differencein the position of the minimum is 0.6 bohr, and the differencein Te is as much as 3300 cm−1. The double-minimum structureof the (2) 1 and (2) 3 states was not reproduced by MRCIcalculations, but the barriers and the avoided crossing betweenthe (2) 1 and (3) 1 states are reproduced. In general, theagreement is only qualitative. To end this paragraph we wouldlike to say that the results of fully relativistic Dirac-Coulombcalculations published by the same authors in Ref. [20] are ina much better agreement with the present data; cf. Sec. III C.

B. Nonadiabatic coupling matrix elements and electrictransition dipole moments from the ground X 1 state

The most important nonadiabatic coupling matrix elementsbetween the excited states of the (BaRb)+ molecular ion arereported on Fig. 3. The regions where these couplings couldpossibly be important are indicated on Fig. 1. The left panelon this figure shows the radial coupling between the (2) 1

FIG. 3. (Color online) Nonadiabatic coupling matrix elements ofthe nonrelativistic electronic states of the (BaRb)+ molecular ion. Theleft and right panels correspond to the radial and angular couplings,respectively.

and (3) 1 states. The potential energy curves for these statesreveal an avoided crossing shown on Fig. 1 at R ≈ 12 bohrs.An inspection of Fig. 3 shows that the maximum of theradial coupling between these two states corresponds to thisdistance. In general, the radial coupling as a function of thedistance R is small, and rather localized around the point ofthe avoided crossing. The angular coupling matrix elementsreported on the right panel of Fig. 4 show more variationswith R. The angular coupling between the (1) 1 and (2) 1

states has a broad maximum around R ≈ 11 bohrs, and thisdistance roughly corresponds to the crossing of the (1) 1

and (2) 1 potential energy curves. The angular couplingbetween the (1) 3 and (1) 3 states shows a broad minimumat R ≈ 7 bohrs, and again this distance roughly corresponds tothe crossing of the (1)3 and (1) 3 potential energy curves.The last angular coupling that may influence the dynamicsof the (BaRb)+ molecular ion corresponds to the (2) 3 and(1) 3 states. Here, the R dependence of the angular couplingis quite different, but the largest variations correspond again

FIG. 4. (Color online) Electric transition dipole moments fromthe ground X 1 state to the 1 and 1 states of (BaRb)+.

032723-8


to the point where the two curves cross, R ≈ 12 bohrs. Noteparenthetically that radial coupling tends to zero as R−7; theangular coupling between the (2) 3 and (1) 3 states tendsto a constant value, 〈3D(ML = 1)|L+|3D(ML = 0)〉 = √

3,while the coupling between the (1) 1 and (2) 1 statesdecays exponentially at large R. This unusual exponentialdecay is due to the different dissociation limits of the groundX 1 and excited (2) 1 states: Ba(1S) + Rb+(1S) versusBa+(2S) + Rb(2S). It is gratifying to note that the MRCImethod used in the calculations of the radial coupling andangular couplings between the triplet states quite preciselylocated the regions of the avoided crossing and curve crossingdespite the fact that the latter were determined from theRCCSD(T) and LRCC3 calculations. This suggests that thecomputed nonadiabatic coupling matrix elements are reliable,at least around the crossings.

The electric dipole transition moments between the groundstate and the three excited electric-dipole-allowed states, two1 and one 1, are plotted in Fig. 4 as functions of theinteratomic distance R. The calculated electric transitionmoments show a strong dependence on the internucleardistance R. For the transition moments to the excited statesof the symmetry the curves show broad maxima around thepositions of the depths on the potential energy curves. Thetransition moment to the state is very small, suggestingthat this state will be of minor importance in the dynamics ofthe (BaRb)+ ion. At large interatomic distances the transitionmoments to the (3) 1 and (1) 1 states tend to zero asµ4R

−4, while the transition moment to the (2) 1 state decaysexponentially with the internuclear distance R. This R−4

dependence can be derived fron the multipole expansion of

the wave functions of the (3) 1, (1) 1, and X 1 states. Theexpressions for the leading long-range coefficient µ4 of theelectric transition dipole moments read

µ4 = A||αRb+0 〈1S||Q2||1D〉, (27)

where A0 = 3/2√

5 and A1 = A0/√

3.

C. Spin-orbit coupling and relativistic potential energycurves, nonadiabatc coupling, and electric transition

dipole moments

A large number of spin-orbit interactions couple the dimerstates of the (BaRb)+ molecular ion. In Fig. 5 we reportthe R dependence of the spin-orbit-coupling matrix elementsthat couple to the 1 and 0+ states. These states are mostinteresting for the collisional dynamics of (BaRb)+. Similarresults can easily be obtained for the 0−, 2, and 3 states.An inspection of Fig. 5 shows that the SO-coupling matrixelements have relatively large variations at small internucleardistances and tend to zero or to the atomic values at large R.All diagonal matrix elements of the spin-orbit Hamiltonianare lower than the atomic 1 3D spin-orbit constant of barium(≈150 cm−1), while the largest nondiagonal couplings areobserved for the pairs of states associated with the crossingof the corresponding potential energy curves. The accuracyof the atomic SO couplings can be judged by comparison ofthe computed and observed positions of the energy levels inthe 3D multiplet. The calculated energies of the 3D1, 3D2,and 3D3 states are 9035, 9254, and 9680 cm−1, and are in avery good agreement with the experimental values of 9034,9216, and 9597 cm−1 [47]. It is worth noting that some of

FIG. 5. (Color online) Matrix elements of the spin-orbit coupling for the electronic states of (BaRb)+. The left and right panels correspondto 0+ and 1 states, respectively.

032723-9


the couplings vanish at large distances due to the differentdissociation limits: Ba + Rb+ versus Ba+ + Rb. This meansthat the neglect of the R dependence of the spin-orbit matrixelements would lead to wrong relativistic potentials since someimportant couplings would be neglected. As an example wecan cite the coupling of the (1) 3 and (1) 3 states. Theasymptotic value is zero due to the different dissociation limitsof these states: Ba(3D) + Rb+(1S) versus Ba+(2S) + Rb(2S).Therefore, when approximating in Eq. (13) all SO couplingsby the corresponding atomic values one would obtain acompletely wrong matrix of the spin-orbit Hamiltonian for the1 states since in the atomic limit 〈(1) 3|HSO|(1) 3〉 = 0. Inparticular, in the atomic approximation the (1) 3 state wouldremain unchanged, and as will be shown below it changesquite a lot at small internuclear distances. The same is true forthe (2) 1 state.

The diagonalization of the spin-orbit Hamiltonian matrices,Eqs. (13) and (14), gives the potential energy curves of thestates that couple to 1 or 0+ symmetry. The correspondingcurves for the 0+ and 1 symmetries are reported on Fig. 6.An inspection of Figs. 1 and 6 shows that the crossingsof the diabatic (nonrelativistic) states are transformed intoavoided crossings on the spin-orbit-coupled relativistic curves.Obviously, the inclusion of the spin-orbit interaction results indifferent dissociation pathways. Due to the presence of manyclosely located molecular states in the 3D–1D energy rangethat couple to the 0+ and 1 symmetries, the effect of thespin-orbit coupling is very pronounced. Indeed, comparisonof Figs. 1 and 6 shows that the behavior of some relativisticcurves is drastically modified compared to the nonrelativisticcase.

In Table II we report the spectroscopic constants ofthe relativistic states and compare them with the availableab initio data of Ref. [20]. An inspection of this table showsthat the agreement between the present results and the data ofRef. [20] is excellent for the dissociations Ba(1S0) + Rb+(1S0)and Ba+(2S1/2) + Rb(2S1/2). Indeed, our result for the welldepth of the ground state overestimates the data of Ref. [20]

FIG. 6. (Color online) Relativistic potential energy curves forthe excited states of the (BaRb)+ molecular ion. Red dashed linescorrespond to the 0+ states and the blue dotted lines to the 1 states.

by only 1.6%. For the first excited states of 0+ and 1 symmetry,our results underestimate the values of Knecht et al. [20] by3.1% and 3.7%, respectively. For all states mentioned above,the positions of the minima in the two calculations agree within0.1 bohr or better. It is gratifying to observe such an excellentagreement between two different sets of ab initio calculationsperformed with different methods, CCSD(T) and RCCSD(T)in the present work vs MRCI in Ref. [20]. Such a goodagreement was expected from the analysis of the nonrelativisticresults, since the triple excitations are relatively unimportantfor these states. It is also worth noting that the pseudopotentialsand basis sets used in our calculations [37,38] do a verygood job, as compared to the fully relativistic Dirac-Coulombcalculations [20]. The comparison for higher excited states isless favorable. For the (3)1 state the agreement of De is within4%, and Re is shifted by 0.1 bohr. This good agreement is againnot fortuitous, since the (3)1 relativistic state is dominatedby the nonrelativistic (1)3 component, and the latter is ahigh-spin state not very sensitive to triple excitations in the

TABLE II. Spectroscopic characteristics of the relativistic electronic states of the (BaRb)+ molecular ion.

State Re (bohr) De (cm−1) Te (cm−1) Reference Dissociation

(1)0+ 8.67 5136 0 Present Ba(1S0)+Rb+(1S0)8.72 5055 0 Ref. [20]

(1)1 9.25 6609 6872 Present Ba+(2S1/2)+Rb(2S1/2)9.22 6871 6638 Ref. [20]

(2)0+ 8.17 5403 8077 Present Ba+(2S1/2)+Rb(2S1/2)8.28 5899 7775 Ref. [20]

(2)1 8.27 5878 8293 Present Ba(3D1)+Rb+(1S0)8.28 5742 7932 Ref. [20]


(4)1 9.41 2353 12464 present Ba(3D3)+Rb+(1S0)9.22 2258 11963 Ref. [20]

(3)0+ 9.01 1801 12589 Present Ba(3D2)+Rb+(1S0)9.03 12005 Ref. [20]


(4)0+ 12.01 1890 15153 Present Ba(1D2)+Rb+(1S0)

032723-10


FIG. 7. (Color online) Nonadiabatic coupling matrix elements of the relativistic electronic states and relativistic electric transition dipolemoments of the (BaRb)+ molecular ion. The left and right panels correspond to the nonadiabatic couplings and transition moments, respectively.

wave function. For other states the differences in the welldepths are of the order of 8% to 13%, and mostly reflect thelack of triple excitations in the calculations of Ref. [20]. Wewould like to stress, however, that the overall pictures of therelativistic states in the present paper and in Ref. [20] agreequite well.

As in the nonrelativistic case, in the relativistic picturestates of the same symmetry do not cross, while statesof different symmetries can cross. Given the complicatedpattern of the molecular states (cf. Fig. 6), the knowledgeof the nonadiabatic couplings is essential for the multichanneldynamics. The nonadiabatic couplings between the relativisticstates as functions of the internuclear distance R are presentedon the left panel of Fig. 7. An inspection of this figure showsthat some couplings are rather localized in space with sharpmaxima or minima, and some others show broad structure.All this can be rationalized by looking at the predominantsinglet or triplet character of the states involved. Since allthese structures can be explained in such a way, we takethe coupling 〈(5)1|L+|(3)0+〉 as an example. The (5)1 and(3)0+ show a crossing around R ≈ 12 bohrs, and in thisregion the nonadiabatic coupling has a broad maximum. Atthese distances both states are primarily singlets with onlya small admixture of some triplet states. At R ≈ 15 bohrsthe (3)0+ state shows an avoided crossing with the (2)0+state. In the nonrelativistic picture the (2)0+ state is mostlydominated by the (1) 3 state, and the (3)0+ state by the (2) 1

state. Thus, at distances larger than the avoided crossing thematrix element of L+ between the (1) 3 and (1) 1 states iszero.

In the relativistic case many transitions that were forbiddenat the nonrelativistic level due to the different multiplicities ofthe states involved become allowed due to admixtures of sin-

glets to triplet and vice versa; cf. the right panel of Fig. 7. Theseadditional transition moments are very small, showing that therelativistic states obtained by admixture of the singlet states totriplets are almost pure triplet states. The transition moment〈(1)0+|z|(4)0+〉 resembles very much 〈(1) 1|z|(3)1〉, withlittle differences only at small internuclear distances. Moreinteresting are the transition moments 〈(1)0+|z|(2)0+〉 and〈(1)0+|z|(3)0+〉. If one were to take the sum of these twocurves, the resulting curve would strongly resemble the graphfor the 〈(1) 1|z|(2)1〉 transition moment; cf. Fig. 4. Theexplanation of this fact is very simple. At distances smallerthan R ≈ 15 bohrs the (3)0+ is predominantly a singlet state,while (4)0+ has mostly triplet character. The situation isopposite at R larger than 15 bohrs. This explains why weget two curves with a sharp decay to zero around 15 bohrs.

D. Long-range behavior of the nonrelativisticand relativistic potentials

When describing cold collisions between atoms it iscrucial to ensure the proper long-range asymptotics of theinteraction potential. The long-range coefficients describingthe asymptotics of the nonrelativistic potentials are reported inTable III. An inspection of this table shows that the dispersioncontribution C

disp6 is modest, but not negligible, of the order

of 5% to 20%. The induction and dispersion coefficientsare always positive, so they describe the attractive parts ofthe potentials. The electrostatic coefficients differ in signdepending on the state considered and are responsible for theappearance of barriers and long-range minima on the potentialenergy curves. This is illustrated on the left panel of Fig. 8,where it is shown how the long-range asymptotics nicely

032723-11


TABLE III. Long-range coefficients (in atomic units) for the nonrelativistic electronic states of the (BaRb)+ molecular ion. C6 is the sumC ind

6 + Cdisp6 .

State Celst3 C ind

4 Celst5 C ind

6 Cdisp6 C6 Dissociation

X 1 136.8 4450 368 4818 Ba(1S)+Rb+(1S)(1) 3 159.3 3260 2510 5770 Ba+(2S)+Rb(2S)(2) 1 159.3 3260 2510 5770 Ba+(2S)+Rb(2S)(2)3 −4.55 324.1 106.3 1873 392 2265 Ba(3D)+Rb+(1S)(1)3 −2.27 272.0 −71.0 2718 372 3090 Ba(3D)+Rb+(1S)(1)3 4.55 115.1 18.1 3060 322 3382 Ba(3D)+Rb+(1S)(3)1 −1.16 108.1 154.3 2844 661 3505 Ba(1D)+Rb+(1S)(1)1 −0.58 127.0 −102.9 2878 244 3122 Ba(1D)+Rb+(1S)(1)1 1.16 180.9 25.7 3802 273 4075 Ba(1D)+Rb+(1S)

fits the ab initio points. It is worth noting that the quadrupolemoments are extremely sensitive to the electronic correlation.For instance, the value of Celst

3 for the (3) 1 state is 1.16 atthe LRCC3 level and 2.05 with the LRCCSD method. Thismeans that the triple excitations diminish Celst

3 by as much as44%. Surprisingly enough, such an effect is not observed forthe hexadecapole moment. Here, the Celst

5 at the LRCC3 andLRCCSD differ by only 6%.

The long-range coefficients describing the large-R asymp-totics of the relativistic states are presented in Table IV. Asdiscussed in Sec. III C, in the atomic limit there is no spin-orbitcoupling between the ground X 1, (1) 3, and (2) 1 statesand other states, so the long-range coefficients of the (1)0+,(1)1, and (2)0+ states remain unchanged in this approximation.An inspection of Table IV shows that the SO coupling of the 3Dand 1D states modestly affects the long-range coefficients inthe Ba(1D2) + Rb+(1S0) dissociation limit. The SO couplingin the 3D multiplet introduces larger changes. For instance, theCelst

5 coefficient for the (2)1 state is zero. This is not fortuitous,but only reflects the fact that the hexadecapole moment of anatom in a 3D1 state is identically zero. Again, the signs of the

electrostatic coefficients are responsible for the barriers andlong-range minima; cf. the right panel of Fig. 8.

E. Elastic cross sections, spin exchange, and radiativecharge transfer

Thus far we have discussed the results of the electronicstructure calculations. Now, we turn to the problem of sym-pathetic cooling of cold barium ions, and present the resultsfor the elastic, spin-flip, and charge-transfer cross sections forboth the nonrelativistic and relativistic potentials. In Fig. 9 wereport the elastic cross sections in the singlet and triplet mani-folds, the spin-flip cross section, and the charge-transfer crosssection from the (2) 1 to groundX 1 state, all calculatedfrom the nonrelativistic potentials. An inspection of this figureshows that in the ultracold regime the elastic cross sections be-have according to Wigner’s threshold law, and the value of thecross section extrapolated to zero energy agrees very well withthe one determined from the s wave scattering length. At ener-gies around 100 nK the energy dependence of the cross sectionsstarts to exhibit some structures related to shape resonances

FIG. 8. (Color online) Long-range nonrelativistic (left panel) and relativistic (right panel) potential energy curves of the (BaRb)+ molecularion. Stars are based on long-range coefficients like in Eq. (3).

032723-12


TABLE IV. Long-range coefficients (in atomic units) for the relativistic electronic states of the (BaRb)+ molecular ion.

State Celst3 C ind

4 Celst5 C6 Dissociation

(1)0+ 136.8 4818 Ba(1S0)+Rb+(1S0)(1)1 159.3 5770 Ba+(2S1/2)+Rb(2S1/2)(2)0+ 159.3 5770 Ba+(2S1/2)+Rb(2S1/2)(2)1 1.60 183.0 0.0 3183 Ba(3D1)+Rb+(1S0)(3)1 −1.13 244.0 46.0 2780 Ba(3D2)+Rb+(1S0)(4)1 −2.73 282.0 6.0 3382 Ba(3D3)+Rb+(1S0)(3)0+ −2.25 270.0 −66.7 3096 Ba(3D2)+Rb+(1S0)(5)1 −0.59 129.0 −101.0 3117 Ba(1D2)+Rb+(1S0)(4)0+ −1.18 110.0 151.0 3493 Ba(1D2)+Rb+(1S0)

appearing due to the contributions of higher partial waves inthe expansions (18) and to glory interference effects. Note thatin the range of energies up to 1 mK the curves representingσ s

el(E) and σ tel(E) are hardly distinguishable, despite the fact

that the potential energy curves for the (2) 1 and (1) 3 statesare quite different. This behavior is purely fortuitous, and is dueto the fact that these two states have the same asymptotics andvery close scattering lengths, as and at, equal to −3.53 × 105

and −4.26 × 105 A, respectively. The superelastic spin-flipcross section shows a qualitatively similar behavior. Overall,all the elastic cross sections are very large, from around 1010 A2

at ultralow temperatures to ≈106 A2 in the millikelvin region.In the ultacold regime the charge-transfer cross section,

which is an inelastic state-changing cross section, decaysas E−1/2, in accordance with Wigner’s threshold law. Atnanokelvin temperatures this cross section, of the order of104 A2, is five orders of magnitude smaller than the elastic

cross sections. When we go up to millikelvin temperaturesthis ratio is even slightly more favorable, about five ordersof magnitude of difference. Thus we can conclude that at thenonrelativistic level and with the single-channel descriptionof the collisional dynamics, cooling of the barium ion bycollisions with ultracold rubidium atoms should be veryefficient.

The results of dynamical calculations on the relativistic(2)0+ and (1)1 potentials are presented on Fig. 10. The elasticcross section from the (1)1 state is almost identical to the crosssection obtained with the nonrelativistic (1) 3 potential. Thisis not surprising since the spin-orbit coupling has a very smalleffect on the potential, slightly moving the repulsive wall;cf. Figs. 1 and 6. The relativistic scattering length is lowercompared to the nonrelativistic (1) 3 value, but most of thefeatures, resonance structure, and glory interference effectsare almost the same. By contrast, the energy dependence of

FIG. 9. Elastic, spin-flip, and charge-transfer cross sections for collisions of 138Ba+(2S) and 87Rb(2S) as functions of the collision energyfrom nonrelativistic potentials.

032723-13


FIG. 10. Elastic, spin-flip, and charge-transfer cross sections for collisions of 138Ba+(2S) and 87Rb(2S) as functions of the collision energyfrom relativistic potentials.

the elastic cross section in the (2)0+ potential is very differentfrom that presented in Fig. 9 for the nonrelativistic potential.We note that Wigner’s limit is very different, and the resonantstructure, glory undulation, and all details at higher energieschanged drastically. For instance, the scattering lengths for the(2)0+ and (1)1 states are −1.58 × 103 and −2.56 × 105 A,to be compared with the values of as and at, −4.26 × 105

and −3.53 × 105 A, quoted above. All these difference arenot surprising, however, since the spin-orbit mixing has aprofound effect in this case. In fact, up to R ≈ 15 bohr, thepoint of the avoided crossing between the potential energycurves of the (2)0+ and (3)0+ states, the (2)0+ potential haspredominantly the character of the (1) 3 state. Only after theavoided crossing and mostly in the long range does it becomean almost pure (2) 1 state. Also the spin-flip cross sectioncomputed from the relativistic potentials is quite different fromthe nonrelativistic one. It shows a sharp resonant structure inthe microkelvin region.

The stricking difference between the nonrelativistic andrelativistic pictures is the fact that the charge-transfer processis now allowed from both the (2)0+ and (1)1 states. Thecorresponding cross sections as functions of the energy are alsoshown on Fig. 9. Again, we observe that Wigner’s thresholdlaw with the E−1/2 decay of the cross section in the ultracoldregime is preserved. An inspection of this figure shows thatthe charge transfer process from the (1)1 state to the groundstate will be very slow, unimportant at temperatures interestingfrom the experimental point of view. This is not a surprise,since the transition dipole moment from the (1)1 state to theground state is very small; cf. Fig. 7. The charge-transfer cross

section from the (2)0+ state is significantly larger, but againis several orders of magnitude smaller than the elastic oneover all the range of temperatures considered in our work. Inparticular, in the millikelvin regime, the inelastic events aresix orders of magnitude less probable than the elastic. Thus,the relativistic calculations confirm the conclusions from thenonrelativistic case that the sympathetic cooling of the bariumion by collisions with ultracold rubidium atoms will be veryefficient.

We would like to conclude this section by saying thatthe present single-channel calculations strongly suggest thatthe sympathetic cooling of the barium atom by collisionswith ultracold rubidium atoms should be very efficient in thetemperature range up to millikelvins. The analysis presentedhere neglects the effect of the hyperfine splittings which areimportant at low collision energies. Thus, our single-channelanalysis performed in terms of singlet and triplet propertiesis only approximate at ultracold temperatures, and the actualcollision rates will depend on the hyperfine spin states of theatom and ion. Moreover, other inelastic processes due to thepresence of other channels could enter the game, and finalconclusions can be presented only when full multichannelcalculations are performed.


In this paper we have reported theoretical prospects forthe sympathetic cooling of the barium ion by collisions withan ultracold buffer gas of rubidium atoms. Potential energycurves for the ground X 1 state of the (BaRb)+ molecular

032723-14


ion corresponding to the Rb+(1S) + Ba(1S) dissociation andfor the excited states, (1) 3 and (2) 1, corresponding to theRb(2S) + Ba+(2S) dissociation were computed by means ofsize-consistent coupled-cluster methods with single, double,and triple excitations in the wave function. The inclusionof the triple excitations in the wave function was shown tobe essential for the open-shell (2) 1 state. The (BaRb)+molecular ion shows a lot of low-lying molecular states, and thecorresponding potential energy curves were computed as well.Using the molecular spin-orbit-coupling matrix elements rela-tivistic potential energy curves were obtained. The asymptoticsof the nonrelativistic and relativistic potentials was fixed withlong-range coefficients calculated from C3 up to and includingC6. A good understanding of the dynamics in the (BaRb)+system requires the knowledge of the nonadiabatic (radial andangular) coupling matrix elements, which were calculated aswell. Finally, the electric dipole transition moments from theground state were computed. Based on the above ab initioelectronic structure calculations, single-channel dynamical

calculations of the elastic, spin-exchange, and inelastic crosssections for the collisions of the Ba+(2S) ion with Rb(2S) in theenergy range from 0 to 1 mK were performed, using both thenonrelativistic and relativistic potentials. It was found that theelastic processes are a few orders of magnitude more favorablethan the inelastic ones. Thus, we can conclude our paper bysaying that the sympathetic cooling of the barium ion by abuffer gas of ultracold rubidium atoms should be very efficient,taking into account the two-body collisional properties of Rband Ba+.

ACKNOWLEDGMENTS

We would like to thank Agnieszka Witkowska for fruit-ful discussions, and for reading and commenting on themanuscript. This work was supported by the Polish Ministry ofScience and Higher Education (Grants No. 1165/ESF/2007/03and No. PBZ/MNiSW/07/2006/41), and the Foundation forPolish Science (FNP) via the Homing program (Grant No.HOM/2008/10B) within the EEA Financial Mechanism.

[1] A. T. Grier, M. Cetina, F. Oruceevicæ, and V. Vuleticæ, Phys.Rev. Lett. 102, 223201 (2009).

[2] W. W. Smith, O. P. Makarov, and J. Lin, J. Mod. Opt. 52, 2253(2005).

[3] C. Zipkes, S. Palzer, C. Sias, and M. Kohl, Nature (London)464, 388 (2010).

[4] C. Zipkes, S. Palzer, L. Ratschbacher, C. Sias, and M. Kohl,Phys. Rev. Lett. 105, 133201 (2010).

[5] S. Schmid, A. Harter, and J. H. Denschlag, Phys. Rev. Lett. 105,133202 (2010).

[6] R. Cote and A. Dalgarno, Phys. Rev. A 62, 012709(2000).

[7] E. Bodo, P. Zhang, and A. Dalgarno, New J. Phys. 10, 033024(2008).

[8] Z. Idziaszek, T. Calarco, P. S. Julienne, and A. Simoni, Phys.Rev. A 79, 010702(R) (2009).

[9] Z. Idziaszek, T. Calarco, and P. Zoller, Phys. Rev. A 76, 033409(2007).

[10] H. Doerk, Z. Idziaszek, and T. Calarco, Phys. Rev. A 81, 012708(2010).

[11] P. Massignan, C. J. Pethick, and H. Smith, Phys. Rev. A 71,023606 (2005).

[12] R. Cote, V. Kharchenko, and M. D. Lukin, Phys. Rev. Lett. 89,093001 (2002).

[13] O. P. Makarov, R. Cote, H. Michels, and W. W. Smith, Phys.Rev. A 67, 042705 (2003).

[14] B. Bussery-Honvault, J.-M. Launay, and R. Moszynski, Phys.Rev. A 68, 032718 (2003).

[15] B. Bussery-Honvault, J.-M. Launay, and R. Moszynski, Phys.Rev. A 72, 012702 (2005).

[16] B. Bussery-Honvault, J.-M. Launay, T. Korona, and R.Moszynski, J. Chem. Phys. 125, 114315 (2006).

[17] B. Bussery-Honvault and R. Moszynski, Mol. Phys. 104, 2387(2006).

[18] C. P. Koch and R. Moszynski, Phys. Rev. A 78, 043417 (2008).[19] S. Knecht, H. J. Aa. Jensen, and T. Fleig, J. Chem. Phys. 128,

014108 (2008).

[20] S. Knecht, L. K. Sørensen, H. J. Aa. Jensen, T. Fleig, and C. M.Marian, J. Phys. B 43, 055101 (2010).

[21] H. Koch and P. Jørgensen, J. Chem. Phys. 93, 3333 (1990).[22] H. Koch, O. Christiansen, P. Jørgensen, A. Sanchez de Meras,

and T. Helgaker, J. Chem. Phys. 106, 1808 (1997).[23] B. Jeziorski, Mol. Phys. 108, 3043 (2010).[24] T. Helgaker et al., DALTON, an ab initio electronic structure

program, Release 1.2 (2001).[25] MOLPRO is a package of ab initio programs written by H.-J.

Werner and P. J. Knowles, with contributions from R. D. Amoset al.

[26] B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev. 94,1887 (1994).

[27] R. Moszynski, in Molecular Materials with SpecificInteractions—Modeling and Design, edited by W. A. Sokalski(Springer, New York, 2007), pp. 1–157.

[28] B. Jeziorski and R. Moszynski, Int. J. Quantum Chem. 48, 161(1993).

[29] R. Moszynski and P. S. Zuchowski, and B. Jeziorski, Collect.Czech. Chem. Comun. 70, 1109 (2005).

[30] T. Korona, M. Przybytek, and B. Jeziorski, Mol. Phys. 104, 2303(2006).

[31] A. Derevianko, S. G. Porsev, and J. F. Babb, At. Data Nucl. DataTables 96, 323 (2010).

[32] K. D. Bonin and V. V. Kresnin, Electric Dipole Polarizabilitiesof Atoms, Molecules, and Clusters (Eorld Scientific, New York,1997), pp. 36–38.

[33] O. Christiansen, A. Halkier, H. Koch, P. Jørgensen, andT. Helgaker, J. Chem. Phys. 108, 2801 (1998).

[34] P. R. Bunker and P. Jensen, Molecular Symmetry andSpectroscopy (NRC Press, Ottawa, 1998).

[35] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Academic Press, New York, 1957),p. 170.

[36] W. Skomorowski and R. Moszynski (unpublished).[37] I. S. Lim, H. Stoll, and P. Schwerdtfeger, J. Chem. Phys. 124,

034107 (2006).

032723-15


[38] I. S. Lim, P. Schwerdtfeger, B. Metz, and H. Stoll, J. Chem.Phys. 122, 104103 (2005).

[39] S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 (1970).[40] J. Brown and A. Carrington, Rotational Spectroscopy of Di-

atomic Molecules (Cambridge University Press, Cambridge,2003).

[41] A. Dalgarno and M. R. H. Rudge, Proc. R. Soc. London, Ser. A286, 519 (1965).

[42] H. T. C. Stoof, J. M. V. A. Koelman, and B. J. Verhaar, Phys.Rev. B 38, 4688 (1988).

[43] E. Timmermans and R. Cote, Phys. Rev. Lett. 80, 3419 (1998).[44] B. Zygelman and A. Dalgarno, Phys. Rev. A 38, 1877 (1988).[45] P. S. Julienne, J. Chem. Phys. 68, 32 (1978).

[46] J. Tellinghuisen and P. S. Julienne, J. Chem. Phys. 81, 5779(1984).

[47] [http://physics.nist.gov/PhysRefData/Handbook/Tables/bariumtable5.htm].

[48] M. G. Kozlov and S. G. Porsev, Eur. Phys. J. D 5, 59 (1999).[49] H. L. Schwartz, T. M. Miller, and B. Bederson, Phys. Rev. A 10,

1924 (1974).[50] E. Iskrenova-Tchoukova and M. S. Safronova, Phys. Rev. A 78,

012508 (2008).[51] B. K. Sahoo, R. G. E. Timmermans, B. P. Das, and D. Mukherjee,

Phys. Rev. A 80, 062506 (2009).[52] E. L. Snow and S. R. Lundeen, Phys. Rev. A 76, 052505

(2007).

032723-16

142

APPENDIX F

PAPER VI

“Rovibrational dynamics of the strontium molecule in the A1Σ+u , c3Πu, and

a3Σ+u manifold from state-of-the-art ab initio calculations”

W. Skomorowski, F. Pawłowski, C.P. Koch and R. Moszynski


143


Rovibrational dynamics of the strontium molecule in the A 1

+u , c3u,

and a 3

+u manifold from state-of-the-art ab initio calculations

Wojciech Skomorowski,1 Filip Pawłowski,1,2 Christiane P. Koch,3 and Robert Moszynski1,a)

1Quantum Chemistry Laboratory, Department of Chemistry, University of Warsaw, Pasteura 1,02-093 Warsaw, Poland2Physics Institute, Kazimierz Wielki University, pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland3Theoretische Physik, Universität Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany

(Received 16 March 2012; accepted 25 April 2012; published online 18 May 2012)

State-of-the-art ab initio techniques have been applied to compute the potential energy curves forthe electronic states in the A 1

+u , c3u, and a 3

+u manifold of the strontium dimer, the spin-orbit

and nonadiabatic coupling matrix elements between the states in the manifold, and the electric transi-tion dipole moment from the ground X 1

+g to the nonrelativistic and relativistic states in the A+c+a

manifold. The potential energy curves and transition moments were obtained with the linear response(equation of motion) coupled cluster method limited to single, double, and linear triple excitations forthe potentials and limited to single and double excitations for the transition moments. The spin-orbitand nonadiabatic coupling matrix elements were computed with the multireference configuration in-teraction method limited to single and double excitations. Our results for the nonrelativistic and rel-ativistic (spin-orbit coupled) potentials deviate substantially from recent ab initio calculations. Thepotential energy curve for the spectroscopically active (1)0+

u state is in quantitative agreement withthe empirical potential fitted to high-resolution Fourier transform spectra [A. Stein, H. Knöckel, andE. Tiemann, Eur. Phys. J. D 64, 227 (2011)]. The computed ab initio points were fitted to physicallysound analytical expressions, and used in converged coupled channel calculations of the rovibrationalenergy levels in the A+c+a manifold and line strengths for the A 1

+u ← X 1

+g transitions. Posi-

tions and lifetimes of quasi-bound Feshbach resonances lying above the 1S0 + 3P1 dissociation limitwere also obtained. Our results reproduce (semi)quantitatively the experimental data observed thusfar. Predictions for on-going and future experiments are also reported. © 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.4713939]

I. INTRODUCTION

In recent years the strontium diatomic molecule, Sr2, hasattracted the interest of theoreticians and experimentalists.Similar to the calcium dimer, Sr2 in its ground X 1

+g state

does not form a chemical bond. Indeed, the binding energy of1081.6 cm−1(Ref. 1) is typical for weakly bound complexesrather than for chemically bound molecules. However, excitedstates of Sr2 are strongly bound, and have been observed inmany experiments. As a matter of fact, the strontium moleculewas the subject of numerous high-resolution spectroscopicstudies in the gas phase2, 3 and in rare gas matrices.4, 5 Thedissociation energy of the ground state was first estimatedfrom the Rydberg-Klein-Rees (RKR) inversion of the spec-troscopic data for the B 1

+u ← X 1

+g transitions.3 Recently,

a more elaborate study of Sr2 was reported,1 in which mea-surements of the B 1

+u ← X 1

+g transitions covering large

parts of the ground state well were recorded. The spectrumcorresponding to the B 1

+u ← X 1

+g transitions could eas-

ily be assigned using standard spectroscopic techniques be-cause the B state dissociating into 1S + 1P atoms is rela-tively well isolated, thus not perturbed by any other electronicstate. While the same is true for the A′1u state reported in


Ref. 6, this is not the case for the A 1

+u state dissociating into

1S + 1D atoms. Here the potential energy curve of the A statecrosses the curve of the c3u state dissociating into 1S + 3Pstates, and the corresponding spectrum cannot easily be as-signed. Experimental investigation of the A state is limited toRef. 6. Tiemann and collaborators measured the spectrumcorresponding to the A 1

+u ← X 1

+g transition by high-

resolution Fourier transform spectroscopy, and showed thatthe rovibrational levels of the A state get strongly perturbedby the c3u state.6 Unfortunately, the amount of the measureddata was not sufficient to apply a deperturbation procedurethat could be used with trust to determine the spectroscopicconstants of the A 1

+u and c3u states. In particular, the key

information on the spin-orbit coupling between the A and cstates could not be determined from the analysis of the spec-tra, and only an effective potential was obtained.

Most of the ab initio calculations on the Sr2 molecule re-ported in the literature thus far are concerned with the groundstate potential energy curve7, 8 and the van der Waals con-stants governing the long-range behavior of the ground statepotential.9–13 To the best of our knowledge only three the-oretical papers considered the excited states of the stron-tium dimer.14–16 However, in view of the recent findingsof Tiemann and collaborators6 the quality of these data isquestionable.


Downloaded 15 Mar 2013 to 212.87.13.78. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions


It should be stressed that alkaline-earth atoms andmolecules are not only interesting for conventional spec-troscopy, but are also intensely investigated in experiments atultralow temperatures. Closed-shell atoms such as alkaline-earth metal atoms are much more challenging to cool and trapthan open-shell atoms such as the alkali atoms. They do nothave magnetic moments in the ground state that would enablemagnetic trapping. Moreover, the short lifetime of the first ex-cited 1P1 state implies rather high Doppler temperatures, re-quiring a dual-stage cooling with the second stage operatingnear the 3P1 intercombination line. Despite these challenges,cooling of calcium, strontium, and ytterbium atoms to micro-Kelvin temperatures has been realized, and Bose-Einsteincondensates of 40Ca,17 84Sr,18, 19 86Sr,20 88Sr,21 170Yb,22 and174Yb (Ref. 23) have been obtained.

On the other hand, the closed-shell structure of thealkaline-earth metal atoms leads to very simple molecularpotentials with low radiative losses and weak coupling tothe environment. This opens new areas of possible applica-tions, such as manipulation of the scattering properties withlow-loss optical Feshbach resonances,24 high-resolution pho-toassociation spectroscopy at the intercombination line,25, 26

precision measurements to test for a time variation of theproton-to-electron mass ratio27, 28 and of the fine struc-ture constant,29 quantum computation with trapped polarmolecules,30 and ultracold chemistry.31

Of particular interest for the present work are the ex-perimental investigations of the photoassociation spectra nearthe intercombination line25, 26, 32 and proposed precision mea-surements of fundamental constants.27, 28 In particular, thelatter require a precise route for the production of ultracoldmolecules in predefined rovibrational states. This, in turn, re-quires an accurate knowledge of the potential energy curvesand the various couplings that may occur in the A 1

+u , c3u,

and a 3

+u manifold of electronic states. Also, the assignment

of the photoassociation spectrum requires a detailed knowl-edge of the rovibrational levels close to the dissociation limit.At present, the experimental data on the A state recorded todate6 is far from complete. Therefore, in the present paper wereport a theoretical study of the spectroscopy of the strontiumdimer in the A 1

+u , c3u, and a 3

+u manifold by state-of-

the-art ab initio methods. The plan of this paper is as follows.In Sec. II we describe the ab initio electronic structure andquantum dynamical calculations. We present the numericalresults in Sec. III and discuss at length the accuracy of thepresent results, compare with the available experimental data,and report predictions for the on-going experiments.33 Finally,in Sec. IV we conclude our paper.


A. Ab initio electronic structure calculations

In the present study, we adopt the computational schemesuccessfully applied to the ground and excited states of thecalcium dimer,34–38 magnesium dimer,39, 40 (BaRb)+ molecu-lar ion,41 and SrYb heteronuclear molecule.42 The potentialenergy curves for the lowest singlet and triplet excited unger-ade states of the Sr2 molecule corresponding to the 1S + 3P,

1S + 3D, and 1S + 1D dissociation limits have been obtainedby a supermolecule method,

V2S+1||u (R) = ESM

AB − ESMA − ESM

B , (1)

where ESMAB denotes the energy of the dimer computed using

the supermolecule method (SM), and ESMX , X=A or B, is the

energy of the atom X. Here, the molecular electronic termis denoted by 2S + 1||u, where S is the total electronic spinquantum number, and the projection of the electronic or-bital angular momentum on the molecular axis. The excitedstates were calculated employing the linear response theory(equation of motion) within the coupled-cluster singles, dou-bles, and linear triples (LRCC3) framework.43–45 Note that inRefs. 34 and 35 the full configuration interaction correction(FCI) for the four-electron valence-valence correlation wasadded on top of the linear response result. However, the re-sults for Ca2 (Refs. 34 and 35) and more recently on Mg2

show that for the states of interest this correction is so smallthat it can safely be neglected. We assume that this is also thecase for Sr2.

Transitions from the ground X 1

+g state to the 1

+u

and 1u states are electric dipole allowed. The correspond-ing transition dipole moments are given by the followingexpression:46

μi(n ← X) = 〈X 1

+g |ri |(n)1||u〉, (2)

where n numbers the consecutive nonrelativistic dissociationlimits of the 1||u states, and ri, i = x, y, or z, denotes theith component of the position vector. Note that in Eq. (2), i= x or y corresponds to transitions to 1u states, while i = zcorresponds to transitions to 1

+u states. In the present calcu-

lations, the electric transition dipole moments were computedas the first residue of the coupled-cluster linear response func-tion restricted to single and double excitations (LRCCSD)with two electric dipole operators.44 Note that in principlethe more advanced LRCC3 method could be used to calcu-late the transition moments. However, since the intensities inthe spectra cannot be measured to a high precision, we haveemployed a less accurate method limited to single and dou-ble excitations. Comparison of the computed and measuredatomic lifetimes will show below that such an approximationis sufficient for the purpose of the present study.

The rovibrational energy levels of the electronically ex-cited states of Sr2 are expected to show some perturba-tions due to the nonadiabatic coupling between the electronicstates. Analysis of the potential energy curves, cf. Sec. III, re-veals angular couplings of the a 3

+u and b 3

+u states with

the c3u state. Therefore, in this work we have computed themost important angular coupling matrix elements defined bythe expression

L(n ↔ n′) = 〈(n)2S+1||u|L±|(n′)2S+1|′|u〉, (3)

with L± the ladder operator of the electronic angular mo-mentum and n ↔ n′ denoting the coupling between electronicstates n and n′ (here n will stand for the a 3

+u or b 3

+u

states, n′ for the c3u state). Note that the electronic angularmomentum operator couples states with the projection of theelectronic orbital angular momentum on the molecular axis



differing by one. In the present calculations, the angular cou-pling between the triplet states was computed directly fromthe multireference configuration interaction wave functionslimited to single and double excitations (MRCI).

Strontium is a heavy atom, and the electronic states ofthe Sr2 molecule are expected to be strongly mixed by thespin-orbit interaction. Therefore, the spin-orbit coupling andits dependence on the internuclear distance R must be takeninto account in our analysis of the spectra in the mixed sin-glet/triplet A 1

+u , c3u, and a 3

+u manifold of electronic

states. We have evaluated the spin-orbit coupling matrix ele-ments for the lowest dimer states that couple to the 0+

u and 1u

states of Sr2 within the MRCI framework. In our case the non-relativistic states that are coupled through the spin-orbit inter-action to the 0+

u symmetry are c3u, A 1

+u , and B 1

+u .37, 47

Other electronic states of Sr2 can be omitted from the presentanalysis due to their very weak couplings with the A+c+amanifold and significant energetic gaps as compared to theseelectronic states. Thus, the most important spin-orbit couplingmatrix elements for states of the 0+

u symmetry are given by

A(R)

= 〈c3u(= ±1, = ∓1)|HSO|c3u( = ±1, = ∓1)〉,(4)

ξ1(R) = 〈c3u( = ±1, = ∓1)|HSO|A 1

+u 〉, (5)

ξ2(R) = 〈c3u( = ±1, = ∓1)|HSO|B 1

+u 〉, (6)

where HSO is the spin-orbit Hamiltonian in the Breit-Pauliapproximation48 and denotes the projection of the elec-tron spin angular momentum on the molecular axis. For the1u symmetry the most important couplings occur betweenthe c3u, a 3

+u , b 3

+u , and B′1u states,37, 47 and the cor-

responding spin-orbit matrix elements read

ϕ1(R)=〈c3u(=0,=±1)|HSO|a 3

+u (=±1, = 0)〉,

(7)

ϕ2(R)=〈c3u(=0,=±1)|HSO|b 3

+u (=±1, = 0)〉,

(8)

ϕ3(R) = 〈c3u( = 0, = ±1)|HSO|B′1u〉, (9)

ζ1(R) = 〈a 3

+u ( = ±1, = 0)|HSO|B′1u〉, (10)

ζ2(R) = 〈b 3

+u ( = ±1, = 0)|HSO|B′1u〉. (11)

With the spin-orbit coupling matrix elements at hand, wecan build up the matrices that will generate the potential ener-gies of the spin-orbit states that couple to 0+

u and 1u symmetry.

The matrices for the 0+u and 1u states are given by

V 0+u =

⎛⎜⎜⎝

V c3u(R) − A(R) ξ1(R) ξ2(R)

ξ1(R) V A1

+u (R) 0

ξ2(R) 0 V B1

+u (R)

⎞⎟⎟⎠(12)

and

V 1u =

⎛⎜⎜⎜⎜⎜⎝

V a3

+u 0 ϕ1(R) ζ1(R)

0 V b3

+u ϕ2(R) ζ2(R)

ϕ1(R) ϕ2(R) V c3u(R) ϕ3(R)

ζ1(R) ζ2(R) ϕ3(R) V B′1u(R)

⎞⎟⎟⎟⎟⎟⎠ ,

(13)respectively. Diagonalization of these matrices yields thespin-orbit coupled potential energy curves for the 0+

u and 1u

states. Note that all the potentials in the matrices (12) and (13)are taken from the LRCC3 calculations. Only the diagonaland nondiagonal spin-orbit coupling matrix elements wereobtained with the MRCI method. Once the eigenvectors ofthese matrices are available, one can easily get the electricdipole transition moments and the nonadiabatic coupling ma-trix elements between the relativistic states. It is worth notingthat here, unlike in the case of Ca2, the B 1

+u and B′1u

states are included in the model. This is due to the fact thatfor Sr2 the long-range spin-orbit interactions of the c3u statewith the B and B′ states have some significance since they areresponsible for the existence of very weakly bound states lo-cated just below the 1S0 + 3P1 threshold that were observedin the photoassociation experiment by Zelevinsky et al.26 Thelong-range character of these couplings makes the photoasso-ciation of the ultracold strontium atoms possible, yielding anon-negligible resonant δCres

3 R−3 interaction for the 0+u and

1u potentials at the 1S0 + 3P1 asymptote, and a non-negligiblerelativistic transition moment from the X 1

+g ground state.

By contrast, the coupling with the A′1u state which wasreported in Ref. 6 was neglected in the present calculationssince it is asymptotically zero for the 1u states of interest andinfluences the R−6 and higher asymptotics of the (1)1u state,as opposed to the δCres

3 R−3 asymptotics due to spin-orbit cou-pling with the B′1u state, cf. Ref. 49 for a simple atomicmodel and Ref. 50 for a rigorous explanation. Note, finallythat the spin-orbit coupling between the c3u, a 3

+u , b 3

+u ,

and B′1u states also leads to states of 0−u and 2u symmetry.

In the absence of strong nonadiabatic effects these states arenot optically active and we do not discuss them here.

In order to mimic the scalar relativistic effects, some elec-trons were described by the ECP28MDF pseudopotential51

from the Stuttgart library. Thus, in the present study the Sr2

molecule was treated as a system of effectively 20 electrons.In all calculations the [8s8p5d4f1g] basis set suggested inRef. 51 was used, augmented by a set of [1s1p1d1f3g] dif-fuse functions. In the calculations of the potentials this basiswas supplemented by a set of spdfg bond functions.52 Thefull basis of the dimer was employed in the supermoleculecalculations and the Boys and Bernardi scheme was utilizedto correct for the basis-set superposition error.53 Ab initio cal-culations were performed for a set of 22 interatomic distances



TABLE I. Parameters of the analytical fits of the a 3

+u , b 3

+u , c3u, and A 1

+u potentials (in atomic units) for Sr2. Numbers in parentheses denote the

power of 10.

Parameter a 3

+u b 3

+u c3u A 1

+u

A0 6.763580846(2) 3.812370308(5) 5.806031460(6) 1.0713571607(3)A1 − 2.7286531831(2) − 2.7785352110(5) − 3.433599894(6) − 3.1369577995(2)A2 4.4648343035(1) 7.599745922(4) 7.677597973(5) 4.259633532(1)A3 − 3.4042105677 − 9.26259308(3) − 7.75564134(4) − 2.828788402A4 0.1048331384 4.2607133921(2) 3.0330564923(3) 0.08207453705α 0.03722198867 0.7652092533 2.102245892 0.2810137665β 3.14301181557 0.7516213273 1.03238202 1.195331858γ 0.07613828056 0.148486204 1.349678173(−3) 0.0362537402C12 − 5.318418476(9) 1.1998965806(11) − 1.06415514(10) 7.278665842(11)C5 − 8.649(2)C6 4.488(3) 6.750(3) 3.951(3) 2.72(3)C8 1.426(6) 2.044(4) 3.521(5) 2.285(5)C10 2.321(8) 1.010(6) 3.296(7) 9.223(7)

ranging from R = 5 to 50 bohrs. In the calculations of thepotential energy curves and transition dipole moments theDALTON code54 was used. All MRCI calculations weredone with the MOLPRO code.55 We would like to empha-size that almost all ab initio results were obtained with themost advanced size-consistent methods of quantum chem-istry, LRCC3 and LRCCSD. Only the spin-orbit coupling ma-trix elements and nonadiabatic matrix elements were obtainedwith the MRCI method which is not size consistent. Fortu-nately, all of the couplings are important in the region of thecurve crossings or avoided crossings at short interatomic dis-tances, so the effect of the size-inconsistency of MRCI on ourresults should not be dramatic.

B. Analytical fits

The computed points of the potential energy curves werefitted to the following analytical expression:

V2S+1||u (R) = e−αR−γR2

4∑i=0

AiRi + Cres

k

Rkfk(β,R)

−6∑

n=3

C2n

R2nf2n(β,R), (14)

where Ai4i=0, α, β, γ , and C12 were adjusted to the com-

puted points. The damping function fn(β, R) was employedin the form proposed by Tang and Toennies.56 The long-range coefficients C2n5

n=3 were not fitted, but fixed at theab initio values taken from Ref. 13. In our case the leadinglong-range coefficient Cres

k describes the first-order resonantinteraction57 between the Sr(1S) and Sr(1D) atoms, and wasalso fixed at the ab initio value.13 For the triplet states theresonant interaction term vanishes identically in the nonrela-tivistic approximation, so Cres

k is equal to zero. The param-eters of the analytical fits of the potentials are reported inTable I.

The transition moments μn←X0 (R), spin-orbit coupling

matrix elements, A(R), ξi(R)2i=1, ζ i(R)i = 1, 2, ϕ(R)3

i=1,and angular coupling matrix elements, L(a ↔ c) and L(b ↔ c),were fitted to the following generic expression:

X(R)=X∞ + BXe−αX1 R+(AX

0 + AX1 R + AX

2 R2)e−αX2 R−γ XR2

+6∑

n=n0

Xn

Rnfn(βX,R), (15)

where X stands for μn←X0 (R), A(R), ξi(R)2

i=1, ζi(R)2i=1,

ϕ(R)3i=1, L(a↔c), and L(b↔c). The leading power in the in-

verse power expansion of Eq. (15) depends on the asymptoticmultipole expansion of the wave functions in the polarizationapproximation58–60 and of the appropriate operator and variesbetween 3 and 6 for the different quantities X. The atomicvalues, X∞, were fixed as follows:

A∞ = ϕ∞1 = 193.68 cm−1,

ϕ∞3 = −ζ∞

1 = ξ∞2 /

√2 = −153.02 cm−1, (16)

ξ∞1 = ϕ∞

2 = ζ∞2 = 0, (17)

μ∞0 (A ← X) = 0, μ∞

0 (B ← X) =√

2 · 3.07 a.u.,

(18)

L∞(a ↔ c) =√

2, L∞(b ↔ c) = 0, (19)

and the remaining parameters were adjusted to the ab initiopoints. The value for A∞ was derived from the experimentalpositions of the states in the 3P multiplet assuming pure LScoupling, while the values of ξ∞

2 and μ∞0 (B ← X) are based

on the present ab initio atomic calculations. The parametersof the analytical fits of the most important spin-orbit cou-pling matrix elements A(R), ξ 1(R), and ξ 2(R) are reported inTable II. All parameters for other fitted quantities can be ob-tained from the authors upon request.

Note that fixing our fits at their proper asymptotic valuesis crucial for a proper description of the rovibrational transi-tions near the dissociation threshold. This is in a sharp con-trast with some potentials fitted to the experimental data thatmay not be sufficiently sensitive to the long-range tail of thepotential and spin-orbit couplings.



C. Quantum-dynamical calculations

In the present paper, we consider the homonuclearbosonic 88Sr2 molecule. The rovibrational energy levels andwave functions for the ground X 1

+g state were obtained

by diagonalizing the Hamiltonian for the nuclear motion inthe Born-Oppenheimer approximation with the variable step-size Fourier grid representation.61–63 For the ground state, anaccurate potential fitted to the experimental high-resolution

Fourier transform spectra is available1 and is used in our cal-culations.

Due to the spin-statistical weights for the bosonic 88Sr2

molecule, we can limit ourselves to odd values of the rota-tional quantum number J and e parity levels.46 Rovibrationalenergy levels for the excited electronic states in the A 1

+u ,

c3u, and a 3

+u manifold were obtained by diagonalizing the

following Hamiltonian:

H =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Hc3u

diag − A(R) ξ1 ξ2 −CJL −CJL CJS 0

ξ1 HA

1

+u

diag 0 0 0 0√

2CJL

ξ2 0 HB

1

+u

diag 0 0 0√

2CJL

−CJL 0 0 Ha

3

+u

diag 0 ϕ1 + CLS ζ1

−CJL 0 0 0 Hb

3

+u

diag ϕ2 + CLS ζ2

CJS 0 0 ϕ1 + CLS ϕ2 + CLS Hc3u

diag ϕ3

0√

2CJL

√2CJL ζ1 ζ2 ϕ3 H

B′1u

diag

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (20)

where the diagonal term is given by

H(n)(2S+1)||udiag ≡ 1

2μp2

R + V(2S+1)||un (R) + J (J + 1) + S(S + 1) + L(L + 1) − 2 − 2 − 2

2μR2(21)

with pR denoting the radial momentum operator, μ the re-duced mass of the dimer, and is the projection of the to-tal electronic angular momentum on the molecular axis, so = + . The first three rows of the H matrix correspondto the states with || = 0 while the last four rows to the stateswith || = 1. The angular (Coriolis-type) couplings are de-fined as

CJL(R) = − [J (J + 1)]1/2 L(R)

2μR2, (22)

TABLE II. Parameters of the analytical fits of the spin-orbit coupling matrixelements (in atomic units) for Sr2. Numbers in parentheses denote the powerof 10.

Parameter A(R) ξ1(R) ξ2(R)

X∞ 8.824756(−4) − 9.860287(−4)BX 0.1281809 − 8.7008164(−3)αX

1 0.255215595 0.40876725

AX0 0.254210634 − 8.18616724(−2) 2.96378955(−6)

AX1 − 3.332647733(−2) 9.17710750(−3) − 3.39493117(−7)

AX2 − 4.77060945(−4) 1.0586325(−8)

αX2 0.7905806772 − 0.92238515

γ X 1.40267678(−2) 2.8575636(−2)βX 1.3932739215 6.8008294 2.22429216X3 2.64888569(−2)X4 − 1.684296847X6 − 35.590169218

CJS(R) = − [2J (J + 1)]1/2

2μR2, (23)

CLS(R) = 21/2 L(R)

2μR2, (24)

where L(R) stands for L(n ↔ n′) defined by Eq. (3) with nand n′ properly chosen, the quantum number L appearing inEq. (21) is the electronic angular quantum number of the ex-cited state atom, and all other symbols appearing in Eq. (20)are defined in Eqs. (4)–(6) and (7)–(11). We refer the readerto Ref. 37 for a rigorous justification of the above expressions.Note that the term L(L+1)

2μR2 is not rigorously correct, since it re-sults from the so-called adiabatic (diagonal) correction for thenuclear motion and the above mentioned value is true only inthe separated atoms limit. At present, there is no ab initio elec-tronic structure code that could provide us with (even approx-imate) values of the angular part of the adiabatic correction,so we keep it at its asymptotic atomic value. This approxima-tion should work very well for the rovibrational levels nearthe dissociation threshold. If the nonadiabatic angular cou-pling matrix elements are small, we can set the Coriolis cou-pling constants CJL, CJS, and CLS equal to zero, and the matrixH becomes block diagonal with two blocks corresponding



separately to the 0+u and 1u levels:

H0u =

⎛⎜⎜⎜⎝

Hc3u

diag − A(R) ξ1 ξ2

ξ1 HA

1

+u

diag 0

ξ2 0 HB

1

+u

diag

⎞⎟⎟⎟⎠ , (25)

H1u =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

Ha

3

+u

diag 0 ϕ1 ζ1

0 Hb

3

+u

diag ϕ2 ζ2

ϕ1 ϕ2 Hc3u

diag ϕ3

ζ1 ζ2 ϕ3 HB′1u

diag

⎞⎟⎟⎟⎟⎟⎟⎟⎠

. (26)

The line strength in the spectra, S(v′J ′ ← v′′J ′′), from therovibrational level |v′′, J ′′〉 of the ground electronic state tothe rovibrational level |v′, J ′〉 of the A 1

+u , c3u, and a 3

+u

manifold is given by

S(v′J ′ ← v′′J ′′) = (2J ′ + 1)HJ ′

×∣∣∣ ∑′=0,±1

1∑σ=−1

∑n′

〈χ1J ′′0(v′′)|rσ (n′ ← X)|χn′J ′′(v′)〉∣∣∣2

,

(27)

where HJ ′ is the so-called Hönl-London factor,

HJ ′ =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

J ′ + 1

2J ′ + 1for J ′ = J ′′ − 1,

1

2J ′ + 1for J ′ = J ′′,

J ′

2J ′ + 1for J ′ = J ′′ + 1,

(28)

and χ1J ′′0(v′′) is the rovibrational wave function of the groundstate, while χn′J ′′(v′) is an eigenfunction of the Hamilto-nian, either H given by Eq. (20), or H0+

u and H1u given byEqs. (25) and (26), if the helicity decoupling approximation(neglecting of the angular Coriolis couplings) is employed.

We also study the positions and lifetimes of Feshbachresonances appearing just above the 1S0 + 3P1 dissociationthreshold. The Feshbach resonances of interest are quasi-bound rovibrational levels of the (2)0+

u electronic state lyingabove the dissociation limit of the (1)0+

u state, i.e., in the con-tinuum of this state. Formally, the Feshbach resonances canbe characterized by complex energies of the form Er − (i/2),where Er denotes the position of the resonance state and itswidth which is directly related to the lifetime τ by τ = ¯/.We have determined these complex energies by diagonalizingthe Hamiltonian for the 0+

u states, Eq. (25), with an imaginaryabsorbing potential VCAP added to the diagonal Hamiltonianterms (21),64–66

VCAP(R)=

⎧⎪⎨⎪⎩

0 , R≤Ra,

4Emin

C2

[1

(1−x)2+ 1

(1+x)2−2

], Ra ≤R<Rmax,

(29)where C = 2.62206 and x = (R − Ra)/(Rmax − Ra). Theparameters Emin, Ra, and Rmax were adjusted to obtain sta-ble results with respect to small variations of these parame-

ters. Approximate positions of the resonances were first de-termined by using the stabilization method67 with respect tothe size of the grid. With these positions at hand, a set of pa-rameters leading to stable complex eigenvalues correspond-ing to the positions and widths of the resonances was easilyfound, yielding Ra = 30 bohrs, Rmax = 200 bohrs, and Emin

= 30 cm−1. The knowledge of the approximate positions wasparticularly useful to determine the value of Emin.


A. Ab initio electronic structure data

Before discussing the potential energy curves, we firstdiscuss the atomic excitation energies obtained from theLRCC3 calculations and the atomic lifetimes. In Table IIIwe present calculated excitation energies in comparisonwith fully relativistic atomic calculations of Porsev andcollaborators68 and experimental data. Our predicted positionof the nonrelativistic 3P state is 14 570.8 cm−1, to be com-pared with the experimental value of 14 704.9 cm−1 (Ref. 69)deduced from the positions of the states in the 3P multipletand the Landé rule. For the 1P1 state we obtain 21 764.3 cm−1,again in very good agreement with the experimental value of21 698.5 cm−1.69 For the nonrelativistic 3D state we obtain18 668.8 cm−1, to be compared with 18 255.2 cm−1 (Ref. 69)deduced from the positions of the states in the 3D multipletand the Landé rule. Finally, our term energy for the 1D2 stateis 20 650.3 cm−1, again in a satisfactory agreement with ex-periment, 20 149.7 cm−1.69 The accuracy of the atomic spin-orbit couplings can be judged by comparing the computedand observed splittings of the energy levels in the 3P and 3Dmultiplets. For the 3P multiplet, theoretical splittings betweenthe 3P2 and 3P1, 3P2 and 3P0, and 3P1 and 3P0 levels amountto 389.9 cm−1, 575.3 cm−1, and 185.4 cm−1, respectively,to be compared with the experimental numbers, 394.2 cm−1,581.0 cm−1, and 186.8 cm−1, respectively. A somewhat lessgood agreement is observed for the 3D multiplet. The the-oretical fine splittings for the 3D3–3D2, 3D3–3D1, and 3D2–3D1 states read 88.9 cm−1, 143.1 cm−1, and 54.2 cm−1, re-spectively, while the experimental numbers are 100.5 cm−1,160.2 cm−1, and 59.7 cm−1, respectively. Finally, we alsonote that the lifetimes of the 3P1, multiplet 3D, 1D2, and 1P1

states of Sr are accurately reproduced. Our calculated life-times together with the most recent experimental and other

TABLE III. Excitation energies (in cm−1) for the low-lying energy levelsof strontium atom.

Excited state Present Ref. 68 Experiment, Ref. 69

3P0 14 187.3 14 241 14 317.53P1 14 372.7 14 448 14 504.43P2 14 762.6 14 825 14 898.63D1 18 582.9 18 076 18 159.13D2 18 637.2 18 141 18 218.83D3 18 726.0 18 254 18 319.31D2 20 650.3 19 968 20 149.71P1 21 764.3 21 469 21 698.5



TABLE IV. Comparison of the present and most recent theoretical and ex-perimental values of the lifetimes of low-lying excited states of strontiumatom.

Excited state Lifetime Ref.

1P1 5.09 ns Present5.38 ns Theory, Ref. 685.35 ns Theory, Ref. 135.22(3) ns Experiment, Ref. 705.263(4) ns Experiment, Ref. 79

3P1 21.40 μs Present24.4 μs Theory, Ref. 8019.0 μs Theory, Ref. 8121.5(2) μs Experiment, Ref. 26

1D2 0.23 ms Present0.412(10) ms Experiment, Ref. 820.30 ms Experiment, Ref. 71

3D 2.72 μs Present2.4 μs Theory, Ref. 682.5(2) μs Experiment, Ref. 72

theoretical results are listed in Table IV. For the 1P1 statewe obtained 5.09 ns to be compared with the experimentalvalue of 5.22(3) ns.70 For 3P1 the theoretical and experimen-tal numbers are 21.4 μs and 21.5(2) μs,26 respectively. For theD states we observe a slightly worse agreement. The theoret-ical lifetime of the 1D2 state is 0.23 ms, to be compared withthe experimental value of 0.30 ms.71 The same numbers forthe average multiplet 3D are 2.72 μs and 2.5(2) μs.72 Such agood agreement between theory and experiment for the atomsgives us confidence that the molecular results will be of sim-ilar accuracy, i.e., at worst a few percent off from the exactresults.

One of the important issues in ab initio electronic struc-ture calculations is the quality of the basis set and of the wavefunctions. To further judge the quality of the basis set used inour calculations we have computed the leading C6 van derWaals coefficient for the ground X 1

+g state. This coeffi-

cient was obtained by using the explicitly connected repre-sentation of the expectation value and polarization propagatorwithin the coupled cluster method,73, 74 and the best approx-imation XCCSD4 proposed by Korona and collaborators.75

Our ab initio result is 3142 a.u. which compares very favor-ably with the value fitted to high-resolution Fourier trans-form spectra, 3168(10) a.u.1 The agreement between theoryand experiment is better than for most of the other ab initiocalculations.9–13 Comparison between theory and experimentfor the well depth of the ground state X 1

+g potential is some-

what less satisfactory. Our theoretical value is 1124.0 cm−1,to be compared with the experimental result of 1081.64(2)cm−1,1 i.e., 3.8% too large. However, in the case of the groundstate interaction, the FCI correction for the valence-valencecorrelation turned out to be important. Due to computationallimitations, we could obtain it only in the Sadlej pVTZ basis76

which is comparatively small and does not allow for a betteraccuracy.

The nonrelativistic potential energy curves relevant forthe spectroscopy in the A+c+a manifold are plotted in Fig. 1,while the spectroscopic characteristics of these states are re-

9000

12000

15000

18000

21000

6 8 10 12 14 16 18 20

Pot

entia

l ene

rgy

/ cm

−1

R / bohr

1S+3P

1S+3D

1S+1D

a3Σu+

c3Πu

b3Σu+

A1Σu+

FIG. 1. Ab initio potential energy curves for the A 1

+u , c3u, a 3

+u , and

b 3

+u states of the strontium dimer.

ported in Table V. The separated atoms energy for each statewas set equal to the experimental value. Due to the compu-tational limitations in the present work we did not considerthe B 1

+u , A′1u, and B′1u states. Fortunately enough, the

calculations of these potentials were not crucial for our study.Indeed, the potential energy curve for the B 1

+u state, with

the correct Cres3 R−3 asymptotics, could be fitted to the exper-

imental data, and is available in the literature.6 According toRef. 6, the A′1u state is isolated, and its rovibrational levelsare only weakly perturbed. Thus, it can safely be omitted fromour model for the rovibrational levels of the 1u states near the1S0 + 3P1 threshold. However, would this state become in-teresting from an experimental point of view, an accurate po-tential energy curve fitted to the observed spectroscopic tran-sitions is available in Ref. 6, while the important spin-orbitcoupling matrix elements can be obtained from the presentauthors upon request. The second state dissociating into the1S + 1P atoms, the B′1u state, was not observed exper-imentally. In our work the potential energy curve for thisstate was approximated with its long-range form Cres

3 R−3,with the proper long-range coefficient adapted from Ref. 13.While this is an approximation, it is not a crucial one, sincethe spin-orbit coupling of this state with the c3u state is

TABLE V. Spectroscopic characteristics of the non-relativistic electronicstates of Sr2 dimer.

State De/cm−1 Re/bohr Ref. Dissociation

a 3

+u 6298 7.95 Present 1S + 3P

6895 7.74 156683 7.79 14

b 3

+u 5293 7.33 Present 1S + 3D

4698 7.25 155763 7.08 14

c3u 1422 8.41 Present 1S + 3P1892 8.02 151785 8.16 14

A 1

+u 8433 7.54 Present 1S + 1D

5440 7.12 159066 7.28 14



−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

10 15 20 25 30

Non

adia

batic

cou

plin

g

R / bohr

⟨a3Σu+|L−|c3Πu⟩

⟨b3Σu+|L−|c3Πu⟩

−200

−100

0

100

200

10 15 20 25 30 35

Ene

rgy

/ cm

−1

R / bohr

⟨c3Πu|HSO|A1Σu+⟩

⟨c3Πu|HSO|B1Σu+⟩

⟨c3Πu|HSO|c3Πu⟩

⟨c3Πu|HSO|b3Σu+⟩

⟨c3Πu|HSO|a3Σu+⟩

FIG. 2. Nonadiabatic matrix elements (left-hand panel) and spin-orbit cou-pling matrix elements (right-hand panel) as functions of the interatomic dis-tance R.

important only at large interatomic distances, where it be-comes constant, and affects the rovibrational dynamics of theSr2 molecule only near the intercombination line 1S0 + 3P1.

Let us compare our results with other available ab initiodata from nonrelativistic calculations.15 The spectroscopicconstants are listed in Table V and compared to the resultsof Ref. 15. Inspection of Table V shows that the agreementwith the data of Czuchaj et al.15 is not satisfactory. For mostof the states the computed electronic binding energies agreewithin a few hundred cm−1 at best, while the differences inthe positions of the minima are 0.4 bohr at worst. The moststriking difference between the present results and the data ofRef. 15 is the spectroscopically active A 1

+u state. Here, the

difference in the position of the minimum is 0.42 bohr, andthe difference in De is as much as 4000 cm−1. Surprisingly,the agreement of the present results with the older 1996 cal-culations by Aubert-Frécon and collaborators14 is compara-tively good. Except for the c3u state, the well depths agreewithin 6%–8% and the well positions within 0.3 bohr at worst.For the c3u state we note a serious disagreement to all pre-vious results. Our potential is considerably shallower and theminimum is shifted to larger distances. However, as will beshown in Sec. III B the present picture of the interatomic in-teractions in the A+c+a manifold reproduces all features ofthe available experimental data.

The nonadiabatic and spin-orbit coupling matrix ele-ments as functions of the interatomic distance R are reportedin Fig. 2. Note the maximum in the angular coupling betweenthe c3u and the b 3

+u states. The position of this maximum

corresponds to the crossing of the potential energy curves ofthese states. Also worth noting is the broad maximum of thespin-orbit coupling between the A 1

+u and c3u states. This

maximum extends to the region where the potential energycurves cross, and is responsible for the strong mixing of thesinglet and triplet rovibrational energy levels. At large dis-tances this particular coupling tends to zero, but the couplingbetween the B 1

+u and the c3u states becomes important

and is responsible for the nonvanishing relativistic dipole mo-ment between the ground state and the triplet levels in theA+a+c manifold.

The relativistic 0+u and 1u potentials as functions of the

interatomic distance R are depicted in Fig. 3. Their spectro-

9000

12000

15000

18000

21000

6 8 10 12 14 16 18 20

Pot

entia

l ene

rgy

/ cm

−1

R / bohr

1S0+3P2

1S0+3P1

1S0+3D1

1S0+1D2

(1)1u(1)0u+

(2)1u

(3)1u

(2)0+u

FIG. 3. Relativistic potential energy curves for the lowest 0+u and 1u states

of the strontium dimer. The dots represent the effective empirical potentialfitted to the high-resolution Fourier transform spectroscopic data of Ref. 6.

scopic parameters are reported in Table VI. First, we notean excellent agreement between the present ab initio poten-tial for the (1)0+

u state and the effective empirical potentialfitted to the high-resolution Fourier transform spectroscopicdata.6 The well depths, 2782 cm−1 on the theory side and2790 cm−1 from the fit to the experimental data, agree within8 cm−1, while the well positions agree to within 0.05 bohr.It is gratifying to observe that the present ab initio calcu-lations also reproduce satisfactorily the position and energyof the avoided crossing. Theory predicts the avoided cross-ing between the (1) and (2)0+

u potentials at R = 9.1 bohrsand V = –1357 cm−1, while the experimental numbers areR = 9.1 bohrs and V = –1470 cm−1. We note a substantialdisagreement between the present ab initio results and thosereported by Kotochigova.16 The difference in the well depthis as large as 946 cm−1. This means that Ref. 16 does not pre-dict any interaction between the (1) and (2)0+

u states. Thus,our result fully confirms the disagreement already noticedby Tiemann and collaborators.6 For other states the agree-ment between the two calculations is quite erratic. For in-stance, the well depths for the second state of 0+

u symmetryand the first state of 1u symmetry differ substantially, while

TABLE VI. Spectroscopic characteristics of the relativistic electronic statesof Sr2 dimer.

State De/cm−1 Re/bohr Ref. Dissociation

(1)0+u 2782 7.51 Present 1S0 + 3P1

2790 7.46 61837 8.2 16

(2)0+u 7039 8.39 Present 1S0 + 1D2

5292 7.2 16(1)1u 6097 7.95 Present 1S0 + 3P1

6921 7.8 16(2)1u 1942 7.34 Present 1S0+ 3P2

1907 8.2 16(3)1u 4863 7.97 Present 1S0+ 3D1

4849 7.4 16



0

1

2

3

4

5

10 15 20 25 30 35

Tra

nsiti

on d

ipol

e m

omen

t / a

t. un

its

R / bohr

⟨X1Σg+|z|B1Σu

+⟩

⟨X1Σg+|z|A1Σu

+⟩

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

10 15 20 25 30 35

R / bohr

⟨(1)0g+|z|(2)0u

+⟩

⟨(1)0g+|z|(1)0u

+⟩

FIG. 4. Transition dipole moments between the ground electronic stateand the excited states of Sr2 in the non-relativistic basis (left-hand panel)and the relativistic basis (right-hand panel) as functions of the interatomicdistance R.

the results for the (2)1u and (3)1u states are very close in en-ergy, but the positions of the wells are shifted by as much as0.5–0.9 bohrs.

Figure 4 shows the nonrelativistic electric transitiondipole moments from the ground state to the 1

+u states and

the relativistic transition dipole moments to the 0+u states. In-

spection of Fig. 4 reveals a broad maximum in the transitionmoment to the A 1

+u state at distances around the minimum

of the ground X 1

+g state, and decay to zero at large dis-

tances. The transition moment to the B 1

+u state exhibits

a broad minimum in the very same region and tends to theatomic value at large distances. The two relativistic curves re-ported in the right-hand panel of Fig. 4, when superimposed,reproduce the transition moment to the A 1

+u state. This is

not surprising, since at small distances the relativistic transi-tion moment to the (1)0+

u is dominated by the singlet compo-nent. Around the crossing between the A 1

+u and the c3u

it drops off drastically due to the predominantly triplet char-acter of the electronic wave function. Note that at large dis-tances it does not decay to zero, but rather to a small but con-stant atomic value reflecting the finite lifetime of the atomic3P1 state. The opposite picture holds for the transition mo-ment to the (2)0+

u state. At small distances this state is domi-nated by the triplet component, and the transition moment isvery small. Starting from the curve crossing, the (2)0+

u state ispredominantly singlet, and the transition dipole moment be-comes very large.

B. Energy levels and rovibrational spectra of Sr2in the A 1

+u , c3u, and a 3

+u manifold

Before comparing our results with the existing experi-mental data, let us discuss the validity of the decoupling be-tween the 0+

u and 1u states. Similarly, as for the Ca2 (Ref. 37)and SrYb (Ref. 42) molecules, the nonadiabatic angular cou-pling is completely negligible also for the strontium dimer,except for a few most weakly bound levels. We have calcu-lated the bound states for all possible values of the rotationalquantum number J by diagonalization of the full Hamiltonianmatrix (20) with the variable step Fourier grid representa-tion, and submatrices corresponding to the 0+

u and 1u blocks,

Eqs. (25) and (26). We found that the eigenvalues of the de-coupled matrices differ from the eigenvalues of the full matrixby less than 10−3 cm−1, justifying the decoupled representa-tion. Actually, experimental data for the rovibrational energylevels of the (1)1u state are scarce and limited to three levelsclosest to the 1S0 + 3P1 dissociation limit, so we will focusour discussion on the 0+

u levels.We have generated all energy levels for the 0+

u and 1u

states up to and including J′ = 219. Comparison of the com-puted energy levels for the 0+

u state with the data derived fromthe high-resolution Fourier transform spectroscopy show aroot-mean-square deviation (RMSD) between theory and ex-periment of 10.5 cm−1. While such an agreement betweentheory and experiment is very good for a system with 76 elec-trons, we decided to adjust the ab initio data to the existingexperimental data to lower the RMSD. This allows us to makereliable predictions for the ongoing experiments on ultracoldstrontium molecules.33 It turned out that by slightly changingthe A0 and C12 parameters by 0.03% and 0.19%, respectively,in the analytical expression for the A 1

+u potential, Eq. (14),

we could reduce the RMSD for J′ = 1 to 0.64 cm−1.With the new values of the A0 and C12 parameters, the root-mean-square deviation of our results for J′ ≤ 50, as comparedto the raw data of Tiemann and collaborators (see the sup-plementary material of Ref. 6) is 4.5 cm−1. We must admit,however, that the present fit is not perfect for very high valuesof J′. Fortunately, these values of the rotational quantum num-ber J′ are not of interest for the ultracold experiments such asphotoassociation spectroscopy26 or precision measurementsleading to the determination of the time variation of the elec-tron to proton mass ratio.27, 28 To better reproduce the levelsnear the 1S0 + 3P1 threshold,26 we have adjusted the spin-orbit coupling between the c3u and B 1

+u states, Eq. (6).

Specifically, we varied the parameters X∞, BX, and X3 inthe fit of ξ 2(R). The adjustments introduced to our ab initiodata are illustrated in Fig. 5. Quantitatively, the adjusted andab initio potentials for the A 1

+u state differ by 290 cm−1

in the repulsive region, at R = 6.5 bohrs, by −2.6 cm−1 atthe minimum, R = 7.54 bohrs, and by 0.07 cm−1 in the longrange, at R = 20 bohrs. All these differences represent at most2.5% of the original ab initio potential. Such an accuracy ofthe ab initio calculations for a system of this size would bepurely accidental. For the spin-orbit term ξ 2(R), the adjust-ment results in a change by nearly 50% in the repulsive re-gion and less than 0.3% in the asymptotic value. Here, theshifts were relatively more important than in the case of theA 1

+u potential, and this can be attributed to the lower ac-

curacy of the results from the MRCI method, which was em-ployed for the calculation of the spin-orbit coupling. It shouldbe stressed, however, that only the long-range value of theξ 2(R) coupling has some significance for the described dy-namics of the Sr2 molecule, as it affects positions of the mostweakly bound levels below the 1S0 + 3P1 threshold. Note thatthe parameters reported in Tables I and II are those adjustedto the experimental data for J′ = 1.

From Figs. 1 and 3 we expect that some of the rovibra-tional levels can be assigned to a single state, A 1

+u or c3u,

while some of them will show a strongly mixed singlet/tripletcharacter. This is indeed the case, as is nicely illustrated



−20

−10

0

10

20

30

40

8 12 16 20−2

−1.5

−1

−0.5

0

0.5

ΔV /

cm−

1

(ΔV

/V)

/ %

R / bohr

−90

−70

−50

−30

−10

10

10 15 20 25 30 35 0

10

20

30

40

50

ΔVS

O /

cm−

1

(ΔV

SO

/VS

O)

/ %

R / bohr

FIG. 5. Left-hand panel: Comparison of the original A 1

+u potential with the fitted one to the experimental data6 for J′ = 1. The solid blue line shows the

absolute difference, V = Vfitted-Vab initio, while the green dashed line shows the relative difference, VV

= Vfitted−Vab initioVab initio

· 100%. Right-hand panel: the same

for the spin-orbit coupling term ξ2 = 〈c3u|HSO|B 1

+u 〉.

by Fig. 6. The contour plots of the A 1

+u and c3u state

populations clearly show that for lower values of J′ all levelswith v′ ≤ 18 are predominantly singlet levels of the A 1

+u

state. By contrast, the v′ = 19 level is the first level that canfully be assigned to the c3u state. This is also clear bycomparing the energies of this particular level in the cou-pled model and in the Born-Oppenheimer approximation:

1 41 81 121 161 201 241 281

J’

0

20

40

60

80

100

v’

0

0.2

0.4

0.6

0.8

1

Pop

ulat

ion

of th

e c3 Π

u st

ate

1 41 81 121 161 201 241 281

J’

0

20

40

60

80

100

v’

0

0.2

0.4

0.6

0.8

1

Pop

ulat

ion

of th

e A

1 Σ u+ s

tate

FIG. 6. Population of the c3u and A 1

+u components of the 0+

u rovibra-tional levels lying below the 1S0 + 3P1 asymptote.

1363 cm−1 vs. 1398 cm−1. At higher values of v′ the situationbecomes quite erratic and most of the levels are of stronglymixed singlet/triplet character. Only at high values of v′ thelevels can again be assigned, this time to the c3u state, al-though some regions of strong mixing are observed, e.g., nearv′ = 84 or v′ = 95. Such a mixing in the rovibrational lev-els close to the dissociation limit is very important for theongoing photoassociation experiments of ultracold strontiumatoms and is discussed in detail in Ref. 77.

More detailed information on the strongly mixed levelsis obtained by examining the rotational constants Bv′ as func-tion of the binding energy of the rovibrational level |v′, J ′〉.This is illustrated in the left-hand panel of Fig. 7, showing thefirst 19 levels from v′ = 0 to 18 not to be perturbed, with auniform nearly linear dependence of the rotational constanton the binding energy. Starting from v′ = 18 the linear de-pendence is clearly broken, and the curve is characterized byirregular behaviour. However, it is still possible to distinguishlevels which are almost purely triplet. The rotational spacingsas functions of J′(J′ + 1) reported in the right-hand panel ofFig. 7 give information about the perturbed levels. Indeed, therotational spacing for v′ = 0 shows a purely linear character

0

0.005

0.01

0.015

0.02

0.025

−3000 −2000 −1000 0

Rot

atio

nal c

onst

ant /

cm

−1

Binding energy / cm−1

0.017

0.019

0.021

0.023

0.025

0 1 2 3 4

Rot

atio

nal s

paci

ng /

cm−

1

J(J+1) / 104

v’=0

v’=16

v’=18

v’=19

FIG. 7. Rotational constant of the 0+u rovibrational levels for J = 1′ (left-

hand panel) and rotational spacing, defined by EvJ /(4J − 2) (right-handpanel).



0

1

2

3

4

5

13400 13450 13500 13550

Line

str

engt

h S

/ at

omic

uni

ts

Transition energy / cm−1

v’=18 ← v’’=27

0

0.2

0.4

0.6

0.8

1

13520 13530 13540

J’’=120

122

124126

128130132

0

0.5

1

1.5

2

2.5

3

3.5

4

13400 13450 13500 13550 13600

Line

str

engt

h S

/ at

omic

uni

ts

Transition energy / cm−1

v’=19 ← v’’=27

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

13480 13490 13500

J’’=116

114

112

110108

10610410210098

FIG. 8. Line strengths corresponding to the P (red solid) and R (blue dashed)branches in the band ν′ = 18 ← ν′ ′ = 27 (upper panel) and ν′ = 19 ← ν′ ′= 27 (lower panel). The spectral lines in the figures are labeled by the rota-tional quantum number J′ ′ of the initial state.

as a function of J′(J′ + 1). For v′ = 16 this linear dependenceis broken for high values of J′, but the nonlinear characteris due to the centrifugal distortion, and not to the triplet per-turbations only. Starting from v′ = 18, the rovibrational lev-els can have very different character (purely triplet or singlet,strongly mixed), depending on the value of J′. This is clearlyillustrated by the sudden changes in the slope of the lineardependence of the rotational spacing on J′(J′ + 1).

Figure 8 reports the line strengths for the transitions v′

= 18 ← v′′ = 27 and v′ = 19 ← v′′ = 27 for different val-ues of the rotational quantum number J′′ of the initial state.The level v′′ = 27 was chosen since it plays a central rolein proposed experiments to determine a time variation of theelectron to proton mass ratio.27, 28 Inspection of Fig. 8 revealsthat the spectral line strengths for the v′ = 18 ← v′′ = 27transitions show a typical dependence on the transition fre-quency, with a slow increase as a function of J′′, and a rela-tively steep and fast decrease for large transition frequencies.By contrast, the line strengths for the v′ = 19 ← v′′ = 27transitions show a completely different dependence on thetransition frequency. First, we observe a very slow increaseof the line strength with the increasing J′′, see the insert andnote the difference in the intensity scale, and a very slow de-crease at large values of the transition frequency. Such a be-

TABLE VII. Positions (relative to the 1S0 + 3P1 threshold), widths, andlifetimes of the Feshbach resonances of 88Sr2 above the 1S0 + 3P1 dissocia-tion limit.

v′ Er/cm−1 /cm−1 τ /ns

110 49.8 0.8761 0.0037111 112.2 0.4658 0.0067112 174.4 0.1924 0.0162113 236.3 0.0410 0.0759114 297.9 0.0008 3.8821115 359.2 0.0638 0.0488

haviour is again a signature of strong perturbations from thec3u state that should be observable in experiment.

C. Feshbach resonances abovethe 1S0 + 3P1 asymptote

The rovibrational states that could be assigned to the(2)0+

u state lying above the 1S0 + 3P1 dissociation limit areembedded in the continuum of the (1)0+

u state, and thus areFeshbach resonances, i.e., quasi-bound states with finite life-times. Two decay mechanisms are possible depending onthe strength of the coupling with the continuum. The reso-nances can decay to the 1S0 + 3P1 continuum, i.e., disso-ciate into atoms (predissociation), or, if the lifetime is longenough, decay to bound or continuum states of the electronicground state (spontaneous emission). Our computed posi-tions, widths, and lifetimes of a few low-lying Feshbach res-onances for J′ = 1 are reported in Table VII. The width of theresonances varies quite strongly, between almost 1 cm−1 forthe very broad one to ≈0.001 cm−1 for the very narrow one.The corresponding lifetimes also vary considerably, from thepicosecond to the nanosecond scale. This implies that someof the resonances should be observable in high-resolutionFourier transform spectroscopic experiments.

In Table VII we have numbered the resonant states bythe vibrational quantum number v′ corresponding to boundvibrational states above the 1S0 + 3P1 dissociation limit thatresult from the diagonalization of the H0+

u Hamiltonian. ForJ′ = 1 we have found 110 bound levels of the 0+

u symmetrywhich lie below the 1S0 + 3P1 asymptote, assigning them vi-brational quantum numbers ranging v′ = 0 to v′ = 109. Thus,the first quasi-bound level reported in Table VII located abovethe 1S0 + 3P1 threshold has the vibrational quantum num-ber v′ = 110. Only eigenvalues that were stable with respectto the grid size were selected, and the exact positions andlifetimes were determined with the complex absorbing po-tential of Eq. (29). It turns out that the width/lifetime is avery sensitive function of the strength of the spin-orbit cou-pling. To illustrate this point, we report the widths of theresonant states as a function of the triplet state populationof the rovibrational wave function in the left-hand panel ofFig. 9. The narrow resonances are found to be dominated bythe singlet component of the wave function, for which thecoupling through the spin-orbit interaction is weak. By con-trast, those resonances that are almost purely triplet states arevery broad with short, picosecond lifetimes. This is easily



−0.2

0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8

Γ / c

m−

1

Population of the c3Πu state

v’=110

v’=111

v’=112

v’=115

v’=113v’=114

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.9 0.95 1 1.05 1.1

λ

Γ(v’=110)

Γ(v’=111)

Γ(v’=112)

Γ(v’=113)

FIG. 9. Widths of the Feshbach resonances as a function of the popu-lation of the c3u state and scaling parameter λ of the spin-orbit term〈c3u|HSO|A 1

+u 〉.

rationalized in terms of a state with a predominantly singletcharacter decaying more slowly to the triplet continuum thanan (almost) purely triplet state. Visibly, the spin-orbit cou-pling is responsible for the predissociation decay of the Fesh-bach resonances, i.e., a smaller spin-orbit interaction favourslonger lasting quasi-bound states, cf. v′ = 114 reported inTable VII.

It was shown in Ref. 78 that the widths of the resonancesare very sensitive to the quality of the ab initio data. Since theexistence of low-lying Feshbach resonances is mostly due tothe spin-orbit coupling ξ 1(R), between the c3u and A 1

+u

states, we have checked how scaling this particular term bya parameter λ ranging from 0.9 to 1.1, λ · ξ 1(R), affects theresults. This is illustrated in the right-hand panel of Fig. 9. In-spection of this figure shows the width to be indeed a rathersensitive function of the coupling, and a 10% change in thecoupling may result in a change of the width by a factor oftwo. However, since scaling within ±10% does not changethe order of magnitude of the lifetimes, our conclusions con-cerning a possible experimental observation of the Feshbachresonances in the A 1

+u , c3u, and a 3

+u manifold remain

valid.


We have reported a theoretical study of interatomic inter-actions and spectroscopy of the strontium dimer in the A 1

+u ,

c3u, and a 3

+u manifold of electronic states. The spectro-

scopic characteristics of the potential energy curves for thenonrelativistic A 1

+u , c3u, and a 3

+u states of Sr2 and of

the relativistic states of 0+u and 1u symmetry deviates signif-

icantly from most of the previous ab initio results.15, 16 Thisis particularly true for the optically active A 1

+u state and its

Hund case (c) analog, (1)0+u . By contrast, the present spectro-

scopic characteristics of the (1)0+u state are in a very good

agreement with the experimental values deduced from thehigh-resolution Fourier transform spectroscopic data.6 Therovibrational energy levels corresponding to the spin-orbitcoupled (1)0+

u state dissociating into 1S0 + 3P1 atoms lyingbelow the avoided crossing with the (2)0+

u state dissociatinginto 1S0 + 1D2 atoms are almost unperturbed, and the corre-sponding energies are very close to the energies obtained in

the Born-Oppenheimer approximation. The rovibrational lev-els of the (1)0+

u state lying above the avoided crossing withthe (2)0+

u state are all heavily perturbed by the rovibrationalstates of the c state. These perturbations are exclusively dueto the spin-orbit interaction. In all cases, the nonadiabatic ef-fects due to the Coriolis couplings were shown to be negligi-ble, with the exception of a few least bound levels. We havelocated several quasi-bound Feshbach resonances lying abovethe 1S0 + 3P1 dissociation limit. Their lifetimes suggest thatthey should be observable in high-resolution spectroscopicexperiments.

Overall, our results reproduce (semi)quantitatively theexperimental data observed thus far.6 Our spectroscopic pre-dictions for on-going and future experiments concerningthe photoassociation of ultracold strontium atoms26, 33 andprecision measurements of the time variation of the electronto proton mass ratio27, 28 are reported elsewhere.77

ACKNOWLEDGMENTS

We would like to thank Eberhard Tiemann, PaulJulienne, Svetlana Kotochigova, and Tanya Zelevinsky formany useful discussions. Financial support from the PolishMinistry of Science and Higher Education through the projectN N204 215539 and from the Deutsche Forschungsgemein-schaft through the Emmy Noether programme are gratefullyacknowledged.

1A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 57, 171 (2010).2T. Bergeman and P. F. Liao, J. Chem. Phys. 72, 886 (1980).3G. Gerber, R. Moller, and H. Schneider, J. Chem. Phys. 81, 1538 (1984).4J. C. Miller, B. S. Ault, and L. Andrews, J. Chem. Phys. 67, 2478 (1977).5J. C. Miller and L. Andrews, J. Chem. Phys. 69, 936 (1978).6A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 64, 227 (2011).7R. O. Jones, J. Chem. Phys. 71, 1300 (1979).8G. Ortiz and P. Ballone, Phys. Rev. B 43, 6376 (1991).9J. F. Stanton, Phys. Rev. A 49, 1698 (1994).

10N. A. Lima and M. J. Caldas, Phys. Rev. B 72, 033109 (2005).11J. Mitroy and M. W. J. Bromley, Phys. Rev. A 68, 052714 (2003).12S. Porsev and A. Derevianko, J. Exp. Theor. Phys. 102, 195 (2006).13J. Mitroy and J. Zhang, Mol. Phys. 108, 1999 (2010).14N. Boutassetta, A. R. Allouche, and M. Aubert-Frécon, Phys. Rev. A 53,

3845 (1996).15E. Czuchaj, M. Krosnicki, and H. Stoll, Chem. Phys. Lett. 371, 401 (2003).16S. Kotochigova, J. Chem. Phys. 128, 024303 (2008).17S. Kraft, F. Vogt, O. Appel, F. Riehle, and U. Sterr, Phys. Rev. Lett. 103,

130401 (2009).18S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck, Phys. Rev.

Lett. 103, 200401 (2009).19Y. N. M. de Escobar, P. G. Mickelson, M. Yan, B. J. DeSalvo, S. B. Nagel,

and T. C. Killian, Phys. Rev. Lett. 103, 200402 (2009).20S. Stellmer, M. K. Tey, R. Grimm, and F. Schreck, Phys. Rev. A 82, 041602

(2010).21P. G. Mickelson, Y. N. Martinez de Escobar, M. Yan, B. J. DeSalvo, and

T. C. Killian, Phys. Rev. A 81, 051601 (2010).22T. Fukuhara, S. Sugawa, and Y. Takahashi, Phys. Rev. A 76, 051604 (2007).23Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura,

T. Yabuzaki, and Y. Takahashi, Phys. Rev. Lett. 91, 040404 (2003).24R. Ciuryło, E. Tiesinga, and P. S. Julienne, Phys. Rev. A 71, 030701 (2005).25S. Tojo, M. Kitagawa, K. Enomoto, Y. Kato, Y. Takasu, M. Kumakura, and

Y. Takahashi, Phys. Rev. Lett. 96, 153201 (2006).26T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye, R. Ciuryło,

P. Naidon, and P. S. Julienne, Phys. Rev. Lett. 96, 203201 (2006).27T. Zelevinsky, S. Kotochigova, and J. Ye, Phys. Rev. Lett. 100, 043201

(2008).28S. Kotochigova, T. Zelevinsky, and J. Ye, Phys. Rev. A 79, 012504 (2009).



29K. Beloy, A. W. Hauser, A. Borschevsky, V. V. Flambaum, and P. Schw-erdtfeger, Phys. Rev. A 84, 062114 (2011).

30D. DeMille, Phys. Rev. Lett. 88, 067901 (2002).31S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis,

G. Quéméner, P. S. Julienne, J. L. Bohn, D. S. Jin, and J. Ye, Science 327,853 (2010).

32Y. N. Martinez de Escobar, P. G. Mickelson, P. Pellegrini, S. B. Nagel,A. Traverso, M. Yan, R. Côté, and T. C. Killian, Phys. Rev. A 78, 062708(2008).

33T. Zelevinsky, private communication (2011).34B. Bussery-Honvault, J.-M. Launay, and R. Moszynski, Phys. Rev. A 68,

032718 (2003).35B. Bussery-Honvault, J.-M. Launay, and R. Moszynski, Phys. Rev. A 72,

012702 (2005).36B. Bussery-Honvault and R. Moszynski, Mol. Phys. 104, 2387 (2006).37B. Bussery-Honvault, J.-M. Launay, T. Korona, and R. Moszynski,

J. Chem. Phys. 125, 114315 (2006).38C. P. Koch and R. Moszynski, Phys. Rev. A 78, 043417 (2008).39L. Rybak, Z. Amitay, S. Amaran, R. Kosloff, M. Tomza, R. Moszynski,

and C. P. Koch, Faraday Discuss. 153, 383 (2011).40L. Rybak, S. Amaran, L. Levin, M. Tomza, R. Moszynski, R. Kosloff, C. P.

Koch, and Z. Amitay, Phys. Rev. Lett. 107, 273001 (2011).41M. Krych, W. Skomorowski, F. Pawłowski, R. Moszynski, and

Z. Idziaszek, Phys. Rev. A 83, 032723 (2011).42M. Tomza, F. Pawłowski, M. Jeziorska, C. P. Koch, and R. Moszynski,

Phys. Chem. Chem. Phys. 13, 18893 (2011).43H. Sekino and R. J. Bartlett, Int. J. Quantum Chem. 26, 255 (1984).44H. Koch and P. Jorgensen, J. Chem. Phys. 93, 3333 (1990).45H. Koch, O. Christiansen, P. Jorgensen, A. M. S. de Meras, and T. Helgaker,

J. Chem. Phys. 106, 1808 (1997).46P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy (NRC,

Ottawa, 1998).47J. M. Brown and A. Carrington, Rotational Spectroscopy of Diatomic

Molecules (Cambridge University Press, Cambridge, 2003).48H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-

electron Atoms (Academic, New York, 1957), p. 170.49K. M. Jones, P. S. Julienne, P. D. Lett, W. D. Phillips, E. Tiesinga, and C. J.

Williams, Europhys. Lett. 35, 85 (1996).50W. Skomorowski and R. Moszynski, “Long-range spin-orbit interac-

tions between like atoms and their QED retardation: application to theSr(3P)+Sr(1S) interaction” (in preparation).

51I. S. Lim, H. Stoll, and P. Schwerdtfeger, J. Chem. Phys. 124, 034107(2006).

52H. Partridge and C. W. Bauschlicher, Mol. Phys. 96, 705 (1999).53S. Boys and F. Bernardi, Mol. Phys. 19, 553 (1970).54T. Helgaker, H. J. A. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. Ågren, A.

A. Auer, K. L. Bak, V. Bakken, O. Christiansen, S. Coriani, P. Dahle, E.K. Dalskov, T. Enevoldsen, B. Fernandez, C. Hättig, K. Hald, A. Halkier,H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch,K. V. Mikkelsen, P. Norman, M. J. Packer, T. B. Pedersen, T. A. Ruden, A.Sanchez, T. Saue, S. P. A. Sauer, B. Schimmelpfennig, K. O. Sylvester-

Hvid, P. R. Taylor, and O. Vahtras, DALTON, release 2.0, an ab initioelectronic structure program, 2005, see http://www.kjemi.uio.no/software/dalton/dalton.html.

55H.-J. Werner, P. J. Knowles, F. R. M. R. Lindh, M. Schütz et al.,MOLPRO, version 2006.1, a package of ab initio programs, 2006, seehttp://www.molpro.net.

56K. T. Tang and J. P. Toennies, J. Chem. Phys. 80, 3726 (1984).57W. J. Meath, J. Chem. Phys. 48, 227 (1968).58P. E. S. Wormer, F. Mulder, and A. Van der Avoird, Int. J. Quantum Chem.

11, 959 (1977).59T. Cwiok, B. Jeziorski, W. Kołos, R. Moszynski, J. Rychlewski, and

K. Szalewicz, Chem. Phys. Lett. 195, 67 (1992).60T. G. A. Heijmen, R. Moszynski, P. E. S. Wormer, and A. Van der Avoird,

Mol. Phys. 89, 81 (1996).61V. Kokoouline, O. Dulieu, R. Kosloff, and F. Masnou-Seeuws, J. Chem.

Phys. 110, 9865 (1999).62K. Willner, O. Dulieu, and F. Masnou-Seeuws, J. Chem. Phys. 120, 548

(2004).63S. Kallush and R. Kosloff, Chem. Phys. Lett. 433, 221 (2006).64U. V. Riss and H. D. Meyer, J. Phys. B 26, 4503 (1993).65D. E. Manolopoulos, J. Chem. Phys. 117, 9552 (2002).66T. Gonzalez-Lezana, E. J. Rackham, and D. E. Manolopoulos, J. Chem.

Phys. 120, 2247 (2004).67H. S. Taylor, G. V. Nazaroff, and A. Golebiewski, J. Chem. Phys. 45, 2872

(1966).68S. G. Porsev, A. D. Ludlow, M. M. Boyd, and J. Ye, Phys. Rev. A 78,

032508 (2008).69See http://physics.nist.gov/PhysRefData for information about energy

levels of strontium atom.70S. B. Nagel, P. G. Mickelson, A. D. Saenz, Y. N. Martinez, Y. C. Chen,

T. C. Killian, P. Pellegrini, and R. Côté, Phys. Rev. Lett. 94, 083004 (2005).71I. Courtillot, A. Quessada-Vial, A. Brusch, D. Kolker, G. D. Rovera, and

P. Lemonde, Eur. Phys. J. D 33, 161 (2005).72C. Redondo, M. S. Rayo, P. Ecija, D. Husain, and F. Castao, Chem. Phys.

Lett. 392, 116 (2004).73B. Jeziorski and R. Moszynski, Int. J. Quantum Chem. 48, 161 (1993).74R. Moszynski, P. S. Zuchowski, and B. Jeziorski, Collect Czech. Chem.

Commun. 70, 1109 (2005).75T. Korona, M. Przybytek, and B. Jeziorski, Mol. Phys. 104, 2303 (2006).76A. Sadlej, Collect Czech. Chem. Commun. 53, 1995 (1988).77W. Skomorowski, R. Moszynski, and C. P. Koch, Phys. Rev. A 85, 043414

(2012).78R. Moszynski, B. Jeziorski, A. van der Avoird, and P. E. S. Wormer,

J. Chem. Phys. 101, 2825 (1994).79M. Yasuda, T. Kishimoto, M. Takamoto, and H. Katori, Phys. Rev. A 73,

011403 (2006).80R. Santra, K. V. Christ, and C. H. Greene, Phys. Rev. A 69, 042510

(2004).81S. G. Porsev, M. G. Kozlov, Y. G. Rakhlina, and A. Derevianko, Phys. Rev.

A 64, 012508 (2001).82D. Husain and G. Roberts, Chem. Phys. 127, 203 (1988).


APPENDIX G

PAPER VII

“Formation of deeply bound ultracold Sr2 molecules by photoassociation

near the 1S → 3P1 intercombination line ”

W. Skomorowski, R. Moszynski and C.P. Koch

Physical Review A 85, 043414 (2012)

157

PHYSICAL REVIEW A 85, 043414 (2012)

Formation of deeply bound ultracold Sr2 molecules by photoassociationnear the 1S + 3P1 intercombination line

Wojciech Skomorowski and Robert MoszynskiQuantum Chemistry Laboratory, Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland

Christiane P. Koch*

Theoretische Physik, Universitat Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany(Received 20 March 2012; published 17 April 2012)

We predict feasibility of the photoassociative formation of Sr2 molecules in arbitrary vibrational levels of theelectronic ground state based on state-of-the-art ab initio calculations. Key is the strong spin-orbit interactionbetween the c 3u, A 1+

u , and B 1+u states. It creates not only an effective dipole moment allowing free-to-

bound transitions near the 1S + 3P1 intercombination line but also facilitates bound-to-bound transitions viaresonantly coupled excited-state levels to deeply bound levels of the ground X 1+

g potential, with v′′ as lowas v′′ = 6. The spin-orbit interaction is responsible for both optical pathways. Therefore, those excited-statelevels that have the largest bound-to-bound transition moments to deeply bound ground-state levels also exhibita sufficient photoassociation probability, comparable to that of the lowest weakly bound excited-state levelpreviously observed by Zelevinsky et al. [Phys. Rev. Lett. 96, 203201 (2006)]. Our study paves the way for anefficient photoassociative production of Sr2 molecules in ground-state levels suitable for experiments testing theelectron-to-proton mass ratio.

DOI: 10.1103/PhysRevA.85.043414 PACS number(s): 34.80.Qb, 33.80.−b, 33.70.−w

I. INTRODUCTION

The cooling and trapping of alkaline-earth metals andsystems with a similar electronic structure have attractedsignificant attention over the last decade. The interest inultracold gases of alkaline-earth-metal atoms was triggeredby the quest for new optical frequency standards [1]. Theextremely narrow linewidth of the intercombination 1S + 3P1

transition, together with the magic wavelength of an opticallattice [2], is at the heart of the clock proposals. Strontium isthe atomic species of choice in many current clock experiments[3–6]. The narrow width of the intercombination line impliesDoppler temperatures as low as 0.5 μK for laser cooling [7].It also allows for easy optical control of the atom-atominteractions via optical Feshbach resonances that involve onlysmall losses [8,9].

The diatomic strontium molecule represents a candidate forhigh-precision spectroscopy that aims at determining the timevariation of the electron-to-proton mass ratio [10]. The ideais to prepare tightly confined Sr2 molecules in their electronicground state by photoassociation in an optical lattice and carryout high-precision Raman spectroscopy on the ground-statevibrational level spacings [10,11]. Photoassociation refers tothe excitation of colliding atom pairs into bound levels ofan electronically excited state [12]. Molecules in their elec-tronic ground state are obtained by spontaneous decay [13].Whether the excited-state molecules redissociate or decay intobound ground-state levels is determined by the shape of theexcited-state potential curve and possibly its coupling to otherexcited states. Long-range potential wells and strong spin-orbitinteraction in the excited state of alkali-metal dimers were

*[email protected]

found to yield significant bound-to-bound transition matrixelements [14].

To date, Sr2 molecules in their excited state have beenformed by photoassociation, using both a dipole-allowedtransition [15,16] and a dipole-forbidden transition near the1S + 3P1 intercombination line [17,18]. The formation ofSr2 molecules in their electronic ground state has not yetbeen demonstrated except for the very last bound level [18].After photoassociation using the dipole-allowed transition,the majority of the excited-state molecules redissociates, andonly the last two bound levels of the electronic ground statecan be populated [19]. This is due to the long-range R−3

nature of the electronically excited state (with R denoting theinteratomic separation) that does not provide any mechanismfor efficient stabilization to bound ground-state levels [20].The situation changes for photoassociation near the intercom-bination line where the excited-state potential curve in theasymptotic region behaves predominantly as R−6 with a smallδCres

3 R−3 correction, where δCres3 is proportional to α4 (with α

the fine-structure constant). Large bound-to-bound transitionmatrix elements with the electronic ground state that behavesasymptotically as R−6 are then expected [17]. However,quantitative estimates on which ground-state levels can beaccessed were hampered to date due to lack of reliable abinitio information on the excited-state potential-energy curvesand, importantly, the spin-orbit interaction. The latter is crucialbecause it yields the effective dipole moment that is utilizedin the photoassociation transition and also governs possiblebound-to-bound transitions following the photoassociation.

Here, we consider the photoassociation process of twoultracold strontium atoms into the manifold of the coupledc 3u(1S + 3P ) + A 1+

u (1S + 1D) + B 1+u (1S + 1P ) states.

The excited-state potential-energy curves, spin-orbit coupling,and transition dipole matrix elements are obtained by state-of-the-art ab initio calculations [21]. This allows us to

043414-11050-2947/2012/85(4)/043414(10) ©2012 American Physical Society

SKOMOROWSKI, MOSZYNSKI, AND KOCH PHYSICAL REVIEW A 85, 043414 (2012)

make quantitative predictions on the photoassociation rates,bound-to-bound transition matrix elements, and spontaneousemission coefficients. We find that the spin-orbit interactionalters parts of the excited-state vibrational spectrum quali-tatively, opening the way for transitions into deeply boundground-state levels. This implies that the standard pictureof pure Franck-Condon-type transitions near the classicalturning points in the ground- and a single excited-statepotential-energy curve yields qualitatively wrong predictions.The crossing between the c 3u(1S + 3P ) and A 1+

u (1S + 1D)states is found to also significantly affect the transitionmoments for the Raman spectroscopy envisioned for the testof the electron-to-proton mass ratio. The paper is organizedas follows: Sec. II introduces our model and briefly reviewsthe theoretical methods employed. The numerical results arepresented in Sec. III, and Sec. IV concludes our paper.

II. THEORY

When a pair of colliding atoms absorbs a photon, itundergoes a transition from the scattering continuum of theX 1+

g ground electronic state into a bound rovibrationallevel of an electronically excited state. Here, we considerphotoassociation using a continuous-wave laser that is reddetuned with respect to the 3P 1 intercombination line ofstrontium. This transition is dipole forbidden in the non-relativistic approximation. The c 3 state, correlating to the

asymptote of the intercombination line transition, is, however,coupled by the spin-orbit interaction to two singlet states,A 1+

u and B 1+u . Both singlet states are connected by

a dipole-allowed transition to the ground electronic stateX 1+

g . Thus an effective transition matrix element is createdwhich for moderate and large interatomic separations is wellapproximated by

dSO = 〈X 1+g |dz|B 1+

u 〉〈B 1+u |HSO|c 3u〉

Ec 3u− EB 1+

u

+〈X 1+g |dz|A 1+

u 〉〈A 1+u |HSO|c 3u〉

Ec 3u− EA 1+

u

, (1)

where HSO is the spin-orbit Hamiltonian in the Breit-Pauliapproximation [22]. The long-range part of dSO, dominated bythe first term in the above expression, is due to the couplingwith the B 1+

u state, ideally suited for photoassociation.The short-range part is due to the coupling with the A 1+

u

state, paving the way toward efficient stabilization of thephotoassociated molecules to the electronic ground state, aswe will show below. The scheme for photoassociation into thelowest manifold of Hund’s case (c) 0+

u states is depicted inFig. 1.

We will make use of nonadiabatic effects caused by the spin-orbit interaction and therefore employ the diabatic [Hund’scase (a)] picture for our calculations. The correspondingHamiltonian in the rotating-wave approximation reads

H =

⎛⎜⎜⎜⎜⎜⎝

HX 1+

g

diag 0 12dz(A ← X)E0

12dz(B ← X)E0

0 Hc 3+

u

diag − A(R) − ω1 ξ1(R) ξ2(R)12dz(A ← X)E0 ξ1(R) H

A 1+u

diag − ω1 012dz(B ← X)E0 ξ2(R) 0 H

B 1+u

diag − ω1

⎞⎟⎟⎟⎟⎟⎠

(2)

where dz(n ← X) denotes the z component of the electronic transition dipole moment from the X electronic ground state to anelectronically excited state n. R is the interatomic separation. The peak amplitude and the detuning of the photoassociation laserwith respect to the intercombination line are represented by E0 and ω1 , respectively. The diagonal terms for the (n)(2S+1)||state are given by

H(n)(2S+1)||diag ≡ 1

2μ

∂2

∂R2+ V

(2S+1)||n (R) + J (J + 1) + S(S + 1) − 2 − 2 + L(L + 1) − 2

2μR2, (3)

with μ denoting the reduced mass, V(2S+1)||n (R) the radial

potential energy curve, J the rotational quantum number, andS the electronic spin quantum number. , , and denotethe projections of the electronic orbital angular momentum,electronic spin angular momentum, and the total angularmomentum on the molecular axis, respectively. The terminvolving the electronic orbital quantum number L in Eq. (3) isan approximation to the true diagonal adiabatic correction [23],with L corresponding to the orbital quantum number in theseparated-atom limit, cf. the discussion following Eq. (40) inRef. [23]. The spin-orbit matrix elements are defined by

A(R) = 〈c 3u( = ±1, = ∓1)|HSO|c 3u

× ( = ±1, = ∓1)〉 (4)

and

ξ1(R) = 〈c 3u( = ±1, = ∓1)|HSO|A 1+u 〉, (5)

ξ2(R) = 〈c 3u( = ±1, = ∓1)|HSO|B 1+u 〉. (6)

The potential energy curve for the X 1+g ground electronic

state was taken from Ref. [24]. All other potential energycurves, spin-orbit coupling matrix elements (shown in the leftpanel of Fig. 2), and electronic transition dipole momentsdz(n ← X) (shown in the right panel of Fig. 2), were obtainedfrom state-of-the-art ab initio electronic structure calculations.The details of these calculations as well as their agreement withthe most recent experimental data [25], in particular, for thecrucial A 1+

u state, are reported elsewhere [21].

043414-2

FORMATION OF DEEPLY BOUND ULTRACOLD Sr2 . . . PHYSICAL REVIEW A 85, 043414 (2012)

0

5000

10000

15000

20000

25000

6 8 10 12 14 16 18 20 22 24

Ene

rgy

(cm

−1 )

R (bohr)

1S+1S

3P+1S

1D+1S

1P+1Sx100

ω1ω2

Δω1

X1Σg+

c3Πu

A1Σu+

B1Σu+

0

5000

10000

15000

20000

25000

6 8 10 12 14 16 18 20 22 24

Ene

rgy

(cm

−1 )

R (bohr)

1S+1S

3P+1S

1D+1S

1P+1Sx100

ω1ω2

Δω1

X1Σg+

c3Πu

A1Σu+

B1Σu+

FIG. 1. (Color online) Proposed scheme for the production of ul-tracold Sr2 molecules by photoassociation near the intercombinationline. The green wave function represents a scattering state of twoSr atoms and the red, blue, and brown wave functions represent thediabatic components of the excited-state vibrational level with bind-ing energy Ev′=−15 = 12.9 cm−1. Spin-orbit interaction facilitates atransition from this level to X 1+

g v′′ = 6 (with the correspondingwave function depicted in purple) via spontaneous or stimulatedemission.

The most promising route to form Sr2 molecules in theirelectronic ground state via photoassociation and subsequentspontaneous emission is determined by diagonalization of thefull Hamiltonian (2) and analysis of its rovibrational structure.In order to connect our model to experimental observables,we calculate the photoassociation rate, K(ω1,T ), and thebranching ratios for spontaneous emission, P (v′′ ← v′). Theabsorption coefficient K(ω1,T ) at laser frequency ω1 is givenby [26,27]

K(ω1,T ) = 2πρ2

hQT

∑v′J ′

∑J ′′

gJ ′′ (2J ′′ + 1)

×∫ ∞

0e−E/kBT |Sv′J ′ (E,J ′′,ω1)|2 dE, (7)

0

50

100

150

200

250

10 15 20 25 30 35

Ene

rgy

(cm

−1 )

R (bohr)

⟨c3Πu|HSO|A1Σu+⟩

⟨c3Πu|HSO|B1Σu+⟩

⟨c3Πu|HSO|c3Πu⟩

0

0.5

1

1.5

2

2.5

3

3.5

10 15 20 25 30 35

Tra

nsiti

on d

ipol

e m

omen

t (at

. uni

ts)

R (bohr)

dz(B ← X)

dz(A ← X)

FIG. 2. (Color online) Spin-orbit couplings (left) and transitiondipole moments (right) between the relevant electronic states of theSr2 dimer that enter the Hamiltonian (2).

where ρ denotes the gas number density, T the temperature, kB

the Boltzmann constant, v′ and J ′ the vibrational and rotationalquantum numbers in the electronically excited state, J ′′ therotational quantum number of the initial scattering state, gJ ′′

the spin statistical weight depending on the nuclear spin, equalto one for 88Sr, and QT = (μkBT/2πh2)3/2. Sv′J ′ (E,J ′′,ω1)is the S-matrix element for the transition from a continuumstate with scattering energy E and rotational quantum numberJ ′′ into the bound level |v′,J ′〉. Throughout this paper,the quantum numbers J ′′ and v′′ denote the rovibrationallevels of the ground electronic state, while J ′,v′ refer to therovibrational levels of the excited electronic state. The squareof the S-matrix element in Eq. (7) can be approximated by theresonant scattering expression for an isolated resonance [26],

|Sv′J ′ (E,J ′′,ω1)|2

= γ sv′J ′ (E,J ′′)γ d

v′J ′

[E − v′J ′ (ω1)]2 + 14

[γ s

v′(E,J ′′) + γ dv′J ′

]2 , (8)

where γ sv′J ′ (E,J ′′) is the stimulated emission rate and

γ dv′ (E,J ′′) is the rate of the spontaneous decay, both in units

of h, and v′J ′ (ω1) is the detuning relative to the position ofthe bound rovibrational level |v′,J ′〉, i.e., v′J ′ = Ev′J ′ − hω1,where Ev′J ′ is the binding energy of the level |v′,J ′〉. In Eq. (8),we assume the decay rate due to any other undetected processesto be negligible.

The spontaneous emission rates γ dv′J ′ are obtained from the

Einstein coefficients Av′J ′,v′′J ′′ ,

γ dv′J ′ =

∑v′′J ′′

Av′J ′,v′′J ′′ , (9)

and related to the natural lifetimes τv′J ′ , γ dv′J ′ = h/τv′J ′ . The

Einstein coefficient Av′J ′,v′′J ′′ is given by

Av′J ′,v′′J ′′ = 4α3

3e4h2 HJ ′ (Ev′J ′ − Ev′′J ′′ )3

×∣∣∣∣∑n′

⟨χX

v′′J ′′∣∣dz(n

′ ← X)∣∣χn′

v′J ′⟩∣∣∣∣

2

, (10)

where HJ ′ is the so-called Honl-London factor equal to(J ′ + 1)/(2J ′ + 1) for J ′ = J ′′ − 1 and J ′/(2J ′ + 1) forJ ′ = J ′′ + 1, and e denotes the electron charge. The labeln′ represents all considered (singlet) dissociation limits ofthe excited diatomic molecule; in our case these are 1S + 1P

and 1S + 1D. The nonadiabatic rovibrational wave functionsχn

vJ (R) = 〈R|χnvJ 〉 are obtained as the eigenfunctions of the

coupled-channel Hamiltonian, Eq. (2), in the absence of thephotoassociation laser field, i.e., for E0 = 0. In principle, inHund’s case (a), the rovibrational wave functions χn

vJ (R)could also be labeled, in addition to n, v, and J , by thequantum numbers p, S, , , and , denoting the parity,total electronic spin, its projection on the molecular axis, theprojection of the orbital electronic angular momentum, andprojection of the total electronic angular momentum on themolecular axis [23]. Since here we consider bosonic 88Sratoms which are photoassociated to form molecules in therovibrational states of the 0+

u potential, the parity is equal toone, and the projection of the total electronic angular ′ iszero, which in turn implies ′ = 0 for singlet excited states n′.

043414-3


At low laser intensity I , the stimulated emission rate isgiven by Fermi’s golden rule expression:

γ sv′J ′ (E,J ′′) = 4π2 I

c

J ′′∑M ′′=−J ′′

J ′∑M ′=−J ′

|〈EJ ′′M ′′ |d · ε|v′J ′M ′ 〉|2,(11)

where ε denotes the vector of the laser polarization, c isthe speed of light, and EJ ′′M ′′ and v′J ′M ′ denote the totalnonadiabatic (electronic and rovibrational) wave functions ofthe initial and final states, respectively. M is the quantumnumber of the projection of the total angular momentum J

on the space-fixed Z axis, and d denotes the electric dipolemoment operator in the space-fixed coordinate system. Afterintroducing the Born-Huang expansion of the nonadiabaticwave functions, Eq. (11) can further be simplified to thefollowing form [28]:

γ sv′J ′ (E,J ′′) = 4π2 I

c(2J ′ + 1)HJ ′

×∣∣∣∣∑n′

⟨χX

EJ ′′∣∣dz(n

′ ← X)∣∣χn′

v′J ′⟩∣∣∣∣

2

, (12)

where χXEJ ′′ (R) are energy normalized continuum wave func-

tions of the ground electronic state with scattering energy E.Using this notation, the transition matrix elements betweencoupled-channel rovibrational eigenstates become

〈v′′,J ′′|dz|v′,J ′〉 ≡∑n′

⟨χX

v′′J ′′∣∣dz(n

′ ← X)∣∣χn′

v′J ′⟩. (13)

They are almost J independent as a result of the extremelysmall spacings between the rotational levels of Sr2. We maytherefore assume 〈v′′,J ′′|dz|v′,J ′〉 ≈ 〈v′′|dz|v′〉 (of course, theselection rule J ′′ = J ′ ± 1 holds).

Finally, the branching ratio

P (v′′ ← v′J ′) =∑

J ′′ Av′J ′,v′′J ′′∑v′′J ′′ Av′J ′,v′′J ′′

(14)

describes the probability for the spontaneous decay from thelevel |v′,J ′〉 of the electronically excited state to rovibrationallevels |v′′,J ′′ = J ′ ± 1〉 of the ground electronic state. Again,the branching ratio P (v′′ ← v′J ′) is nearly independent of theJ ′ quantum number.


We consider 88Sr atoms trapped at a temperature ofT ∼ 2 μK, typical for the two-color mangeto-optical trapsemployed for the alkaline-earth-metal species [29]. At such alow temperature, the collisions are purely s wave, i.e., J ′′ = 0.The Hamiltonian (2) is represented on a Fourier grid with anadaptive step size [30–32].

The photoassociation yield is determined by the ground-state scattering length and the rovibrational structure ofthe levels in the excited c 3u-, A 1+

u -, and B 1+u -state

manifolds which couple to 0+u symmetry. The correct ground-

state scattering properties including the scattering length areaccounted for by employing the empirical X 1+

g potentialreported in Ref. [24], reflecting the current spectroscopicaccuracy. The excited-state rovibrational levels are obtained

0

0.2

0.4

0.6

0.8

1

−250 −200 −150 −100 −50 0

Pop

ulat

ion

Binding energy (cm−1)

c3Πu

A1Σ+u

B1Σ+u

FIG. 3. (Color online) Population of the c 3u, A 1u and B 1u

components of the 0+u rovibrational levels for J ′ = 1. The binding

energies are taken with respect to the Sr(3P1) + Sr(1S) asymptote.

from diagonalization of the Hamiltonian (2) with E0 = 0.Their analysis reveals a significant singlet-triplet mixing, cf.Fig. 3 presenting the c 3u, A 1+

u , and B 1+u diabatic compo-

nents of the coupled wave functions. This mixing results fromthe crossing between the c 3u and A 1+

u states, which arecoupled by spin-orbit interaction. On average, the rovibrationallevels are predominantly of triplet character as expected for the1S + 3P asymptote. However, a sequence of peaks indicatesthe occurrence of rovibrational levels with very strong singlet-triplet mixing. These levels are particularly useful for bothphotoassociation and a subsequent bound-to-bound transition.This is illustrated in Fig. 4 showing the vibrational wavefunctions that correspond to the two rightmost peaks in theA 1+

u -state components of Fig. 3 (at binding energies of 12.9and 75.8 cm−1) and comparing them to the v′ = −6 wave func-

⎢ Ψ⎢

Σ

Σ

Π

Σ

Σ

Σ

Σ

Π

Π

FIG. 4. (Color online) Vibrational wave functions of the coupledc 3u, A 1u, and B 1+

u electronic states for v′ = −6, v′ = −15, andv′ = −26. The corresponding binding energies are Ev′=−6 = 0.27cm−1 = 8.09 GHz, Ev′=−15 = 12.9 cm−1, and Ev′=−26 = 75.8 cm−1.Note the different scale for the interatomic separation.

043414-4


tion, the lowest level previously observed experimentally [17].The v′ = −6 wave function is almost purely long-range andof predominantly triplet character, with the population of bothsinglet components being three orders of magnitude smallerthan the triplet one (note that the wave functions of both theA 1+

u and B 1+u components were scaled up by a factor of 100

to be visible in the figure). The picture changes completely forthe levels v′ = −15 and v′ = −26. Since the relative weightsof the c 3u and A 1+

u components are almost equal (cf. Fig.3), the v′ = −15 and v′ = −26 wave functions in Fig. 4 displayA 1+

u and c 3u components on the same scale. Remarkably,the triplet wave functions also show peaks at short internucleardistance. This is a clear signature of resonant, nonadiabaticcoupling between vibrational levels of the spin-orbit-coupledelectronic states [14,33,34]. It occurs when two potentialenergy curves that are coupled cross and the energies of thetwo corresponding vibrational ladders coincide [33]. Then thevibrational wave functions reflect the turning points of the twopotentials, as seen in Fig. 4. Resonant coupling was shown tolead to significantly enlarged bound-to-bound transition ratesto form deeply bound molecules in their electronic ground state[14,35,36]. According to Fig. 4, it is the coupling between thec 3u state and the A 1+

u state that becomes resonant, inducingstrong mixing between these components. The effect of thisresonant coupling will be further increased by the presence ofthe B 1+

u state in addition to the A 1+u state. The behavior of

the B 1+u component strictly follows the c 3u wave function,

but is two orders of magnitude smaller (cf. Fig. 4). This is easilyrationalized in terms of the B 1+

u component representingonly a small admixture, due to the spin-orbit coupling ξ2(R)in the Hamiltonian (2), to the principal part of the (1)0+

u statethat originates from the c 3u potential. The magnitude of theB 1+

u component is straightforwardly estimated by treatingthe spin-orbit coupling as a perturbation and calculating thefirst-order correction to the wave function, similarly to theexpression for the transition dipole moment, Eq. (1).

In the alkali-metal dimers, the spin-orbit coupling mixesin a triplet component that does not directly participate in theoptical transition between singlet states [14,35,36]. Therefore,the enhancement of the bound-to-bound transitions in thealkali-metal dimers is only due to the modification of thesinglet wave function. Here, for bound-to-bound transitions tothe electronic ground state, the effective dipole is mainly dueto the coupling between the c 3u and the A 1+

u states [cf. Eq.(1)]. Therefore, it is not only the modification of the c 3u wavefunction but also the presence of a large A 1+

u component thatis responsible for the enhancement of bound-to-bound transi-tions. Both effects together, the additional peaks in the c 3u

wave function at interatomic separations R < 10 bohr, and thelarge A 1+

u component at these interatomic separations leadto a significantly enhanced effective dipole moment accordingto Eq. (1). We thus find that for alkaline-earth-metal atomsnear the 1S + 3P1 intercombination line, the resonant couplingenlarges the singlet admixture to a predominantly tripletwave function and enhances both the bound-to-bound and thefree-to-bound transition matrix elements. The enhancementof the bound-to-bound transitions significantly reduces thelifetime of the excited-state bound levels. The lifetimes of thelevels v′ = −15 and v′ = −26 are found to be 30.9 and 27.2 ns,

0

2

4

6

8

10

-120 -100 -80 -60 -40 -20 0

K (

10-1

5 cm3 s-1

)

Binding energy (cm-1)

v’=-15

v’=-26

T = 2 μK

10-14

10-12

10-10

10-8

10-6

-8 -6 -4 -2 0

K (

cm3 s-1

)

Binding energy (GHz)

v’=-1

v’=-2

v’=-3v’=-4

v’=-5

v’=-6

0

2

4

6

8

10

-120 -100 -80 -60 -40 -20 0

K (

10-1

5 cm3 s-1

)

Binding energy (cm-1)

v’=-6v’=-15

v’=-26

T = 20 μK

10-15

10-13

10-11

10-9

10-7

-8 -6 -4 -2 0

K (

cm3 s-1

)

Binding energy (GHz)

v’=-1

v’=-2

v’=-3v’=-4v’=-5

v’=-6

FIG. 5. (Color online) Photoassociation into rovibrational levelsof the coupled c 3u, A 1+

u , and B 1+u states below the Sr(3P1)

+ Sr(1S) dissociation limit for a laser intensity I = 1 W/cm2 andtwo temperatures, 2 μK (upper panel) and 20 μK (lower panel). Thetransitions to the six least-bound levels that were reported in Ref. [17]are shown in a semilogarithmic plot in the insert (note the differentscales).

respectively, compared to 7.61 μs for v′ = −6, i.e., they aredecreased by two orders of magnitude. This is rationalizedby a larger spontaneous emission rate resulting from anenhancement in the bound-to-bound transitions according toEq. (10).

The two effects, i.e., an increase in the bound-to-bound andfree-to-bound transition matrix elements, have an oppositeimpact on the photoassociation probability, with the formerhindering and the latter facilitating the photoassociation pro-cess. The photoassociation absorption coefficient [cf. Eq. (7)]is shown in Fig. 5 for all bound levels below the 1S + 3P1

dissociation limit for two temperatures, T = 2 μK [29] andT = 20 μK [17]. The absorption coefficient for the levelsthat were experimentally observed [17] are shown in theinset of Fig. 5 using a logarithmic scale. At T = 2 μK, thepeak rate coefficients for the strongly mixed levels v′ = −15and v′ = −26 amount to K = 1.6 × 10−15 cm3 s−1 andK = 1.8 × 10−15 cm3 s−1, respectively, compared to K =1.9 × 10−14 cm3 s−1 for the lowest previously observed level,v′ = −6, i.e., about one order of magnitude smaller. However,at T = 20 μK and also at higher temperatures, the levels with

043414-5


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−6 −4 −2 0 2 4 6

K (

10−

15cm

3 s−1 )

Detuning (MHz)

v’=−15

FIG. 6. (Color online) Shape of the photoassociation line fortransition into the rovibrational level |v′ = −15,J ′ = 1〉 as a functionof the detuning, v′J ′ , from this level (I = 1 W/cm2) at T = 2 μK(solid red line) and T = 20 μK (dashed blue line).

strong resonant coupling have absorption coefficients that arevery similar to that of v′ = −6, K = 1.3 × 10−15 cm3 s−1,and K = 1.6 × 10−15 cm3 s−1 for v′ = −15 and v′ = −26,respectively, compared to K = 2.2 × 10−15 cm3 s−1 for v′ =−6 [see also bottom panel of Fig. 5]. The peak rate coefficientsfor the strongly mixed levels are less affected by temperaturebroadening. This is rationalized in terms of their large naturalwidth, of the order of a few MHz. In constrast, for the level v′ =−6 the natural width amounts to merely 20 kHz. The naturalwidths need to be compared to thermal widths of 42 kHz and0.42 MHz for T = 2 μK and T = 20 μK, respectively. Forthe strongly mixed levels, the photoassociation line shapes,shown in Fig. 6 for |v′ = −15,J ′ = 1〉, are thus governedby the natural width, about 4.5 MHz in Fig. 6, and thermalbroadening is of secondary importance even at a temperatureof T = 20 μK. Due to the relatively short lifetime of the level,the profile manifests also only a very weak asymmetry.1 Forregular levels such as v′ = −6 the opposite holds, i.e., thethermal width is larger than the natural width. An increasein temperature from 2 to 20 μK therefore has a noticeableeffect on the photoassociation rate (cf. Ref. [37] for a detailedanalysis of the effect of thermal broadening on the peak ratecoefficients). We conclude that photoassociation of strontiumatoms into strongly perturbed levels, albeit challenging, iswithin reach for an experimental setup such as that of Ref. [17].

After observing that photoassociation into resonantly per-turbed levels such as v′ = −15 or v′ = −26 should be feasibleexperimentally, the transition moments from these levels intobound levels of the electronic ground state are examinedin Fig. 7. Furthermore, Fig. 8 shows the modulus squaredof the vibrationally averaged transition moments governingthe spontaneous emission coefficients [cf. Eq. (10)] and thebranching ratios [cf. Eq. (14)]. While the level v′ = −6

1Note that the detuning in Fig. 6 is taken with respect to the bindingenergy Ev′=−15,J ′=1 = 12.9 cm−1, not with respect to the atomictransition frequency.

0 10 20 30 40 50 60ground state level v″

0

0.1

0.2

0.3

| ⟨v″

| dz |v

′⟩ |2 (

ato

mic

uni

ts )

v′ = −6 ( τ ≈ 7.6 μs )

v′ = −15 ( τ ≈ 31 ns )

v′ = −26 ( τ ≈ 27 ns )

0

3×10-3

6×10-3

FIG. 7. (Color online) Modulus squared of the vibrationallyaveraged bound-to-bound electric transition dipole moments betweenexcited-state rovibrational levels v′ = −6, v′ = −15, and v′ = −26,all with J ′ = 1 (shown in Fig. 4) and all vibrational levels |v′′,J ′′ = 0〉of the ground electronic state, X 1+

g . τ denotes the lifetime forspontaneous decay to the X 1+

g state.

decays predominantly, with a branching ratio of more than80%, into v′′ = −3, a very weakly bound ground-state levelwith a binding energy of 0.17 cm−1, the strongly perturbedlevels v′ = −15 or v′ = −26 decay into a range of theground-state levels, including deeply bound ones. The largesttransition moment is observed for the ground-state level v′′ = 6with a binding energy of 836.4 cm−1. The correspondingbranching ratios amount to about 17% for both v′ = −15and v′ = −26, compared to less than 2% for v′ = −6. Notethat the branching ratios to v′′ = 6 in Fig. 8 are almost equalfor v′ = −15 and v′ = −26, while the transition momentsin Fig. 7 are not. This is due to the dependence of thespontaneous emission coefficients on the transition frequencyin addition to the transition moment [cf. Eq. (10)]. Based on thefavorable transition moments between the strongly perturbedexcited-state levels and v′′ = 6, stimulated emission using a

0 10 20 30 40 50 60ground state level v″

0

5

10

15

bran

chin

g ra

tio

( in

per

cen

t )

v’ = -15v’ = -26

020406080

v’ = -6

FIG. 8. (Color online) Branching ratio for the spontaneous decayfrom levels v′ = −6, v′ = −15, and v′ = −26 to bound rovibrationallevels of the ground electronic state.

043414-6


10 20 30 40 50

v’

0

10

20

30

40

50

60

v’’

0.0001

0.001

0.01

0.1

1

|⟨v’|d

z|v’

’⟩|2 (

atom

ic u

nits

)

60 70 80 90 100 110

v’

0

10

20

30

40

50

60

v’’

0.0001

0.001

0.01

0.1

|⟨v’|d

z|v’

’⟩|2 (

atom

ic u

nits

)

FIG. 9. (Color online) Modulus squared of the vibrationallyaveraged bound-to-bound electric transition dipole moments betweenall rovibrational levels |v′,J ′ = 1〉 of the 0+

u potential and allvibrational levels |v′′,J ′′ = 0〉 of the ground electronic state, X 1+

g

(for other possible combinations of J ′ and J ′′ the pattern is almostidentical).

nanosecond pulse could be employed in order to pump theexcited-state population selectively into the ground-state levelv′′ = 6. Alternatively, final state selectivity could be achievedby photoassociation via stimulated Raman adiabatic passage(STIRAP) [38]. It requires a sufficiently steep trap to ensure awell-defined phase of the initial state |E,J ′′〉, which is expectedto be feasible in a deep optical lattice [39]. Due to theirlarge transition moments for both pump and Stokes steps,the pathways E → v′ = −15(−26) → v′′ = 6 would be themost promising routes for STIRAP photoassociation from anoptical lattice into deeply bound levels.

A complete overview over transitions between the rovibra-tional levels v′ of the excited 0+

u states below the 1S + 3P1

asymptote and all ground-state levels v′′ is given by Fig. 9.For clarity, the figure has been separated into two parts,showing the highly excited state levels v′ in the top paneland the lower excited state levels v′ in the bottom panelof Fig. 9. Note that we find 110 excited state 0+

u levelsv′ below the 1S + 3P1 asymptote with J ′ = 1, i.e., v′ = −6corresponds to v′ = 104, v′ = −15 to 95, and v′ = −26 to84. Considering first levels close to the 1S + 3P1 dissociationlimit, we notice that the last two excited-state levels haveextremely weak bound-to-bound transition moments. The nextten lower levels display a single peak in their transitionmoments, indicating pure Franck-Condon transitions close

to the outer turning point. This is typical for weakly bound,regular levels. Transferring the molecular population to shorterbond lengths is extremely difficult for such levels and requiresmany excitation-deexcitation cycles [38].

The first strongly perturbed level, v′ = −15 (or v′ = 95),leads to a prominent series of peaks in the squared transitionmoment matrix. Figure 9 indicates that also the neighboringlevels of v′ = −15 are significantly perturbed. This wouldbe important for pump-dump schemes using picosecond laserpulses [40,41]. An excited-state wave packet ideally suitedfor selective population transfer into v′′ = −6 is obtainedby superimposing levels v′ = 92, . . . ,98. This translates intoa spectral width of the photoassociation pulse of 15 cm−1,corresponding to a transform-limited pulse duration of 1 ps.Note that a previous study considering only the experimentallyobserved weakly bound levels concluded that short-pulsepump-dump photoassociation near the intercombination linetransition is not viable [19]. The main obstacle is the quasi-R−6

behavior of the excited-state potential that leads to a reduceddensity of vibrational levels for very small photoassociationdetunings. The number of vibrational levels present is thentoo small to obtain a truly nonstationary wave packet [19].However, the picture changes completely for more deeplybound excited-state levels such as those around v′ = 95.The spectral width of the pulse can easily be chosen suchthat several vibrational levels are within the photoassociationwindow, without exciting the atomic intercombination linetransition that would lead to loss of atoms [41]. The advantageof a time-dependent photoassociation scheme in the presenceof nonresonant coupling lies in the dynamical interplay thatarises between the interaction of the molecule with the laserlight and the spin-orbit interaction. In such a situation, adynamical enhancement of the final state population wasfound for strong dump pulses, indicating that the efficiency ofpopulation transfer is not determined by the transition matrixelements anymore [41].

A key question is how accurate our predictions are regardingthe position of the perturbed levels such as v′ = −15 orv′ = −26. There is no doubt about the presence of suchlevels since it results from the crossing between the c 3u

and A 1+u potential energy curves, and this crossing was

confirmed by a recent experimental study [25]. Our abinitio data reported in Ref. [21] are able to reproduce therovibrational energy levels for J ′ = 1 obtained from the fitof the experimental data to a Dunham-type expansion [25] towithin 0.64 cm−1. Considering all experimentally observedlevels with J ′ 50, the root-mean-square deviation betweentheoretically calculated levels and the raw experimental data is4.5 cm−1. Perhaps this value, ±4.5 cm−1, should be consideredas a very conservative estimate of the error bars in the bindingenergies reported in the present study. The main sources oferror in the binding energies are the inaccuracy of the c 3u

potential and its spin-orbit correction, A(R). Scaling of thec 3u potential or the A(R) coupling by ±5% leads to shiftsin the binding energies by 2–2.5 cm−1, in particular, for thelevels with strong singlet-triplet mixing. However, very goodresults for the A 1+

u state and for the atomic spin-orbit splittingof the 3P and 3D multiplets, obtained in Ref. [21], suggestthe accuracy of the c 3u potential and the A(R) couplingto be better than 5%. Note that scaling the other spin-orbit

043414-7


-2500 -2000 -1500 -1000 -500 0

binding energy Ev′bind

( cm-1

)

00.10.20.30.4

0

0.1

⎢⟨v″

| dz |

v′⟩⟨

v′| d

z |v⟩ ⎢

2 (

atom

ic u

nits

)

0

0.005

0.01

0.015 v″ = -3 → v′ → v″ = 0v″ = -3 → v′ → v″ = 27

v″ = 27 → v′ → v″ = 0

v″ = 6 → v′ → v″ = 0

FIG. 10. (Color online) Vibrationally averaged bound-boundRaman transition moments as a function of the binding energy ofthe intermediate 0+

u rovibrational levels for three different pathwaysdiscussed in proposals for the measurement of the time variation ofthe electron-to-proton mass ratio, me/mp [10,11]. Note the differentscales for the transition moments.

couplings, ξ1(R) and ξ2(R), by ±5% has a negligible effect onthe position of the bound levels. This confirms our assessmentof the estimated error bars of ±4.5 cm−1 as rather conservative.While such error bars might appear to be relatively large froman experimental perspective, they are not surprising for asystem with 78 electrons, strong relativistic effects, and theA 1+

u potential as deep as 8433 cm−1 that are found in thestrontium dimer.

The Franck-Condon parabola typical for transitions be-tween regular vibrational levels [42,43] is absent in Fig. 9.This reflects the strong perturbation of the vibrational spectrumof the excited-state levels due to the spin-orbit interaction. Areasoning on possible optical pathways solely based on theshape of the adiabatic potentials will therefore give a wrongpicture. To emphasize this point, Fig. 10 presents the transitionmatrix elements for Raman transitions that are relevant in theproposal for the measurement of the time variation of theelectron-to-proton mass ratio, me/mp [10,11]. The idea is totransfer molecules into the X 1+

g ground vibrational levelstarting from the weakly bound ground-state level v′′ = −3(corresponding to v′′ = 60) that is populated by spontaneousdecay from v′ = −6, the lowest excited-state level previouslyobserved in a photoassociation experiment [17]. One couldexpect the efficiency of a direct transfer v′′ = −3 → v′ →v′′ = 0 to be much smaller than the efficiency of a two-Raman-step transfer employing an intermediate state, v′′ = 27.Inspection of Fig. 10 reveals, however, that this expectationis not confirmed. There exist a few excited-state levels, withbinding energies between 2000 and 1400 cm−1, that have largetransition dipole moments with both v′′ = −3 (or v′′ = 60) andv′′ = 0, yielding a high efficiency for Raman transfer directlyfrom v′′ = −3 to v′′ = 0. The maximum Raman momentsare found for v′ = 14 and v′ = 16 [cf. Fig. 9]. These levelsare almost pure singlet rovibrational states belonging to theA 1+

u potential, and are only marginally perturbed by the

spin-orbit coupling. For all the pathways presented in Fig. 10the most favorable intermediate levels v′ are those which areenergetically the highest and yet almost unperturbed, i.e., thelevels located just below the crossing between the c 3u andA 1+

u potential-energy curves. This is easily rationalized interms of the strong transition dipole moment, dz(A ← X),of these levels and their relatively good overlap with therovibrational levels of the X 1+

g potential. The decrease ofthe Raman transition moments for the deeply bound levelsexcited-state levels, with v′ 10 and binding energies largerthan 2000 cm−1, is due to shift of equilibrium positions of theA 1+

u and X1+g potential wells [cf. Fig. 1].

The Raman transition moments from v′′ = −3 → v′ =14/16 → v′′ = 0 are larger than any of the moments fortransfer from v′′ = −3 to v′′ = 27. Of course, even higherRaman transition moments are found for optical pathwaysto v′′ = 0 that start in v′′ = 6 (cf. bottom panel of Fig. 10),the level that is populated by photoassociation into a stronglyperturbed excited-state level followed by spontaneous or stim-ulated emission or pump-dump photoassociation, as explainedabove. We thus conclude that a single Raman transition afterphotoassociation is sufficient to obtain molecules in the X 1+

g

ground vibrational level. The least intensity of the Ramanlasers is required for optical pathways starting from v′′ = 6,i.e., after photoassociation into strongly perturbed levels suchas v′ = −15 or v′ = −26. The pathways starting from v′′ = 6come with the additional advantage that the transition frequen-cies of the Raman lasers differ only by Ev′′=0 − Ev′′=6 ≈ 225cm−1 compared to 792 cm−1 for v′′ = 27 → v′′ = 0 or 1061cm−1 for v′′ = −3 → v′′ = 0. We would like to stress here thatall these conclusions concerning the Raman transitions shouldstrictly be valid as the intermediate v′ levels between 2000 and1400 cm−1 are located in the bottom of the A 1+

u well wherethe potential is known precisely [21,25], and are almost notperturbed by the spin-orbit interaction. This also means thatdoing high-precision Raman spectroscopy with these statesshould be feasible and the spectra will not be obscured by thespin-orbit perturbation effects.


Based on state-of-the-art ab initio calculations, we havecalculated photoassociation rates and spontaneous emissioncoefficients for the photoassociation of Sr2 molecules nearthe 1S + 3P1 intercombination line transition. We have alsoanalyzed bound-to-bound transition moments as well asRaman transition moments connecting vibrational levels inthe electronic ground state, relevant to achieve transfer intothe X 1+

g ground vibrational level. The vibrational spectrumof the coupled c 3u, A 1+

u , B 1+u excited-state manifold

is found to be strongly perturbed. Therefore, optical pathwayscannot be predicted based on the turning points of the adiabaticpotentials. Consequently, the theoretical analysis needs to fullyaccount for the spin-orbit coupling of the electronically excitedstates.

For excited-state binding energies of about 13 cm−1 andlarger, up to 2000 cm−1, strongly perturbed vibrational levelsare identified. The strong perturbations result from the resonantinteraction of the coupled vibrational ladders of the c 3u andA 1+

u states. For Sr2, these levels are found to be particularly

043414-8


well suited for the stabilization of photoassociated moleculesto the electronic ground state, either via spontaneous orstimulated emission. The photoassociation rate of the stronglyperturbed levels is calculated to be comparable to that ofthe lowest level previously observed [17] at a temperatureof T = 20 μK and about one order of magnitude smaller atT = 2 μK. We therefore conclude that photoassociation intostrongly perturbed levels should be feasible with the currentlyavailable experimental techniques.

Strongly perturbed levels display large bound-to-boundtransition moments with deeply bound vibrational levels ofthe electronic ground state. If photoassociation is followed byspontaneous emission, this will show up as a dominant decayinto X 1+

g (v′′ = 6), although a large range of ground-statevibrational levels will be populated as well. State selectivity ofthe ground-state levels can be achieved by stimulated emission,either employing STIRAP photoassociation in a deep opticallattice [39] or pump-dump photoassociation with picosecondpulses [40,41].

Identifying in the experiment the strongly perturbed levelsof the c 3u,A 1+

u ,B 1+u manifold that are particularly suit-

able for efficient stabilization to deeply bound ground-statelevels requires a spectroscopic search since even state-of-the-art ab initio methods cannot predict the positions of therovibrational levels with precision better than a few wavenumbers for such a heavy system like Sr2. The theoreticalprecision is limited here mainly by uncertainty of the c 3u

state and its relativistic correction, and can be reduced onlyafter emergence of new experimental data concerning thec 3u,A 1+

u ,B 1+u manifold of Sr2.

Finally, the crossing between A 1+u and c 3u potentials

will be important not only for the initial formation of Sr2

molecules but also for any subsequent Raman-type transitionproceeding via the coupled c 3u,A 1+

u ,B1+u manifold of

states. The presence of unperturbed levels of the A 1+u state,

that are located just below the crossing with the c 3u curve,leads to the somewhat unexpected result that the weakly boundX 1+

g vibrational levels just below the dissociation limit showlarger Raman transition moments with the ground vibrationallevel than with levels half-way down the ground-state potentialwell. Direct Raman transitions to the ground vibrational levelthus become possible for both weakly and strongly boundlevels. When utilizing these transitions for population transferby STIRAP, deeply bound levels such as v′′ = 6 come withthe advantage of a smaller frequency gap between the pumpand Stokes pulse and significantly larger transition momentstranslating into lower pulse amplitudes.

There are thus at least two good reasons for futureexperiments on the strontium dimer to employ stronglyperturbed levels of the c 3u,A 1+

u ,B 1+u manifold: efficient

stabilization to deeply bound ground-state levels and largematrix elements for Raman transitions between ground-statelevels. Our calculations show these experiments to be feasiblewith currently available experimental technology.

ACKNOWLEDGMENTS

We would like to thank Tanya Zelevinsky, Paul Julienne,and Svetlana Kotochigova for many useful discussions. Thisstudy was supported by the Polish Ministry of Scienceand Higher Education through the project N N204 215539.Financial support from the Deutsche Forschungsgemeinschaft(Grant No. KO 2301/2) is also gratefully acknowledged.

[1] A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M.Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M.Foreman, J. Ye et al., Science 319, 1805 (2008).

[2] M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, Nature(London) 435, 321 (2005).

[3] M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin,A. M. Rey, and J. Ye, Science 331, 1043 (2011).

[4] S. Falke, H. Schnatz, J. S. R. V. Winfred, T. Middelmann,S. Vogt, S. Weyers, B. Lipphardt, G. Grosche, F. Riehle,U. Sterr et al., Metrologia 48, 399 (2011).

[5] A. Yamaguchi, N. Shiga, S. Nagano, Y. Li, H. Ishijima,H. Hachisu, M. Kumagai, and T. Ido, Appl. Phys. Express 5,022701 (2012).

[6] P. G. Westergaard, J. Lodewyck, L. Lorini, A. Lecallier, E. A.Burt, M. Zawada, J. Millo, and P. Lemonde, Phys. Rev. Lett.106, 210801 (2011).

[7] M. Chalony, A. Kastberg, B. Klappauf, and D. Wilkowski, Phys.Rev. Lett. 107, 243002 (2011).

[8] R. Ciuryło, E. Tiesinga, and P. S. Julienne, Phys. Rev. A 71,030701 (2005).

[9] S. Blatt, T. L. Nicholson, B. J. Bloom, J. R. Williams, J. W.Thomsen, P. S. Julienne, and J. Ye, Phys. Rev. Lett. 107, 073202(2011).

[10] T. Zelevinsky, S. Kotochigova, and J. Ye, Phys. Rev. Lett. 100,043201 (2008).

[11] S. Kotochigova, T. Zelevinsky, and J. Ye, Phys. Rev. A 79,012504 (2009).

[12] K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, Rev.Mod. Phys. 78, 483 (2006).

[13] A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, Phys. Rev. Lett. 80, 4402 (1998).

[14] C. M. Dion, C. Drag, O. Dulieu, B. Laburthe Tolra, F. Masnou-Seeuws, and P. Pillet, Phys. Rev. Lett. 86, 2253 (2001).

[15] S. B. Nagel, P. G. Mickelson, A. D. Saenz, Y. N. Martinez, Y. C.Chen, T. C. Killian, P. Pellegrini, and R. Cote, Phys. Rev. Lett.94, 083004 (2005).

[16] P. G. Mickelson, Y. N. Martinez, A. D. Saenz, S. B. Nagel, Y. C.Chen, T. C. Killian, P. Pellegrini, and R. Cote, Phys. Rev. Lett.95, 223002 (2005).

[17] T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye,R. Ciuryło, P. Naidon, and P. S. Julienne, Phys. Rev. Lett. 96,203201 (2006).

[18] Y. N. Martinez de Escobar, P. G. Mickelson, P. Pellegrini, S. B.Nagel, A. Traverso, M. Yan, R. Cote, and T. C. Killian, Phys.Rev. A 78, 062708 (2008).

[19] C. P. Koch, Phys. Rev. A 78, 063411 (2008).

043414-9


[20] C. P. Koch and R. Moszynski, Phys. Rev. A 78, 043417 (2008).[21] W. Skomorowski, F. Pawłowski, C. P. Koch, and R. Moszynski,

e-print arXiv:1203.4524.[22] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-

and Two-Electron Atoms (Academic Press, New York, 1957), p.170.

[23] B. Bussery-Honvault, J.-M. Launay, T. Korona, and R. Moszyn-ski, J. Chem. Phys. 125, 114315 (2006).

[24] A. Stein, H. Knockel, and E. Tiemann, Phys. Rev. A 78, 042508(2008).

[25] A. Stein, H. Knockel, and E. Tiemann, Eur. Phys. J. D 64, 227(2011).

[26] R. Napolitano, J. Weiner, C. J. Williams, and P. S. Julienne,Phys. Rev. Lett. 73, 1352 (1994).

[27] K. Sando and A. Dalgarno, Mol. Phys. 20, 103 (1971).[28] B. Bussery-Honvault, J.-M. Launay, and R. Moszynski, Phys.

Rev. A 72, 012702 (2005).[29] T. Zelevinsky, S. Blatt, M. M. Boyd, G. K. Campbell, A. D.

Ludlow, and J. Ye, Chem. Phys. Chem. 9, 375 (2008).[30] V. Kokoouline, O. Dulieu, R. Kosloff, and F. Masnou-Seeuws,

J. Chem. Phys. 110, 9865 (1999).[31] K. Willner, O. Dulieu, and F. Masnou-Seeuws, J. Chem. Phys.

120, 548 (2004).

[32] S. Kallush and R. Kosloff, Chem. Phys. Lett. 433, 221 (2006).[33] C. Amiot, O. Dulieu, and J. Verges, Phys. Rev. Lett. 83, 2316

(1999).[34] S. Ghosal, R. J. Doyle, C. P. Koch, and J. M. Hutson, New J.

Phys. 11, 055011 (2009).[35] H. K. Pechkis, D. Wang, Y. Huang, E. E. Eyler, P. L. Gould,

W. C. Stwalley, and C. P. Koch, Phys. Rev. A 76, 022504 (2007).[36] A. Fioretti, O. Dulieu, and C. Gabbanini, J. Phys. B 40, 3283

(2007).[37] R. Ciuryło, E. Tiesinga, S. Kotochigova, and P. S. Julienne, Phys.

Rev. A 70, 062710 (2004).[38] C. P. Koch and M. Shapiro, Chem. Rev. (2012).[39] M. Tomza, F. Pawłowski, M. Jeziorska, C. P. Koch, and

R. Moszynski, Phys. Chem. Chem. Phys. 13, 18893 (2011).[40] C. P. Koch, E. Luc-Koenig, and F. Masnou-Seeuws, Phys. Rev.

A 73, 033408 (2006).[41] C. P. Koch, R. Kosloff, and F. Masnou-Seeuws, Phys. Rev. A 73,

043409 (2006).[42] M. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu,

D. Comparat, and P. Pillet, Science 321, 232 (2008).[43] M. Viteau, A. Chotia, D. Sofikitis, M. Allegrini, N. Bouloufa, O.

Dulieu, D. Comparat, and P. Pillet, Faraday Discuss. 142, 257(2009).

043414-10

Date post:	04-Feb-2023
Category:	Documents
Upload:	khangminh22
View:	0 times
Download:	0 times

Production of ultracold molecules by photoassociation ...

Documents