Interatomic potentials, electric properties, and spectroscopy of the ground and excited states of the Rb{}_{2} molecule: Ab initio calculations and effect of a non-resonant field1footnote 11footnote 1 Dedicated to Professor Bretislav Friedrich on the occasion of his 60th birthday

Interatomic potentials, electric properties, and spectroscopy
of the ground and excited states of the Rb molecule: Ab initio calculations and effect of a non-resonant field111 Dedicated to Professor Bretislav Friedrich on the occasion of his 60th birthday

Michał Tomza    Wojciech Skomorowski    Monika Musiał    Rosario González-Férez    Christiane P. Koch    and Robert Moszynski
Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland
   Theoretische Physik, Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany Institute of Chemistry, University of Silesia, Szkolna 9, 40-006 Katowice, Poland Instituto ‘Carlos I’ de Física Teórica y Computacional and Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, 18071 Granada, Spain
Abstract

We formulate the theory for a diatomic molecule in a spatially degenerate electronic state interacting with a non-resonant laser field and investigate its rovibrational structure in the presence of the field. We report on ab initio calculations employing the double electron attachment intermediate Hamiltonian Fock space coupled cluster method restricted to single and double excitations for all electronic states of the Rb molecule up to dissociation limit of about 26.000cm. In order to correctly predict the spectroscopic behavior of Rb, we have also calculated the electric transition dipole moments, non-adiabatic coupling and spin-orbit coupling matrix elements, and static dipole polarizabilities, using the multireference configuration interaction method. When a molecule is exposed to strong non-resonant light, its rovibrational levels get hybridized. We study the spectroscopic signatures of this effect for transitions between the X electronic ground state and the A and b excited state manifold. The latter is characterized by strong perturbations due to the spin-orbit interaction. We find that for non-resonant field strengths of the order W/cm, the spin-orbit interaction and coupling to the non-resonant field become comparable. The non-resonant field can then be used to control the singlet-triplet character of a rovibrational level.

p
\doi

10.1080/0026897YYxxxxxxxx \issn \issnp \jvol00 \jnum00 \jyear2013


otential-energy curves, coupled-cluster theory, induced-dipole interaction, AC Stark effect, far-off-resonant laser field

1 Introduction

Rubidium was one of the first species to be Bose-condensed [1], and nowadays it can routinely be cooled and trapped. It has therefore become the drosophila of ultracold physics. Its long-range interatomic interactions have extensively been studied, and this has allowed to very accurately determine the scattering length and coefficient [2, 3, 4]. Rb molecules have been formed out of ultracold rubidium atoms using both photo- and magneto-association [5, 6]. Photoassociation and Feshbach spectroscopy have also served to measure the low-lying shape resonances of the rubidium dimer [7, 8, 9]. Trapping rubidium in an optical lattice has facilitated studies of atom-molecule dark states [10] and transferring the molecules into their vibrational ground state [11]. The Rb molecule continues to draw attention in the context of the coherent control of ultracold collisions [12, 13, 14, 15, 16] and femtosecond photoassociation [17, 18, 19, 20]. These experiments as well as those employing photoassociation with continuous wave lasers [21, 22, 23, 24, 25] require precise spectroscopic knowledge not only of the ground but also the excited states for both interpretation and detection.

The electronic ground and excited states have extensively been studied. According to Huber and Herzberg [26], the Rb molecule was first observed in a spectroscopic experiment by Lawrence and Edlefsen as early as 1929 [27]. Cold molecules studies have lead to a renewed interest in the Rb molecule. The ground X state has been investigated in Ref. [28], while the most accurate experimental results for the a state have been reported by Lozeille et al. [29], Beser et al. [30] and Tiemann and collaborators [31]. The most important excited states corresponding to the S+P dissociation limit, the A and b states, have extensively been analysed in Ref. [21]. Less experimental information is available for other excited states. Notably, the (1) state has been studied in Ref. [32], and Ref. [33] reports the experimental observation of the (2) state. The pure long-range state of symmetry, that is important for the photoassociation of ultracold Rb atoms, has been analysed in Ref. [34]. Several of these experimental data were successfully employed to derive empirical potentials that reproduce the spectroscopic data with the experimental accuracy, cf. Refs. [28, 31] for the ground state X and Refs. [30, 31] for the a potential. The coupled manifold of the A and b states was deperturbed by Bergeman and collaborators [21, 35] with the corresponding potential energy curves and spin-orbit coupling matrix elements reported in Ref. [35]. Potential energy curves for other electronic states fitted to the experimental data are older, cf. Ref. [36] for the empirical potential energy curve of the (1) state, and Refs. [37] and [38] for those of the (2) state and (2) states, respectively.

Given this extensive amount of experimental data, it is not surprising that many theoretical calculations have tackled the ground and excited states of the rubidium dimer. The first ab initio calculation on the Rb molecule dates back to 1980 and was reported by Konowalow and Rosenkrantz [39]. Three recent studies have reported ab initio data of varying accuracy for the potential energy curves and in some cases further properties such as couplings and transition moments of Rb. The non-relativistic potentials for all molecular states by Park et al. [40] show a root mean square deviation (RMSD) between the theoretical well depths and the available experimental data of 235cm, i.e., 9.9% on the average. The 2003 calculations by Edvardsson et al. [41] were devoted to the ground state potential and six excited state potentials of ungerade symmetry. The spin-orbit coupling matrix elements were also reported. The overall accuracy of these results was about the same as in Ref. [40] with a RMSD of 180cm representing an average error of 25%. Note that since the number of states considered in Refs. [40] and [41] differs, the absolute RMSD may be smaller and the percentage error larger. Finally, in 2012 Allouche and Aubert-Frécon [42] reported calculations of all molecular states and spin-orbit coupling matrix elements corresponding to the dissociation limits , , and . These calculations are much more accurate than any other previously reported in the literature with a RMSD of 129 cm, i.e., an error of 5.5% only. However, they do not cover highly excited molecular states that are of interest for conventional spectroscopy experiments [43], for the detection of ultracold molecules [44] as well as photoassociation into states with ion-pair character [45, 46, 47].

Photoassociation into highly excited electronic states is at the core of a recent proposal for the production of ultracold Rb molecules [47], aimed at improving earlier femtosecond experiments [17, 18, 19, 20]. It is based on multi-photon transitions that can easily be driven by femtosecond laser pulses and allow to fully take advantage of the broad bandwidth of femtosecond laser pulses while driving the narrow photoassociation transition [48]. Moreover, multi-photon photoassociation populates highly excited electronic states with ion-pair character and strong spin-orbit interaction. These features are advantageous for an efficient stabilization of the photoassociated molecules into deeply bound molecules in the electronic ground state [47]. The theoretical modeling of the proposed photoassociation scheme required the knowledge of precise ab initio potential energy curves including those for highly excited states, spin-orbit and nonadiabatic coupling matrix elements, electric transition dipole moments and dynamical Stark shifts. These data were not available in the literature for the highly excited states, and the non-adiabatic couplings and dynamical Stark shifts have been missing even for the lowest states. Moreover, the newly developed tools of electronic structure theory based on the Fock space coupled cluster method [49, 50, 51] could possibly allow for reaching a better accuracy of the potentials than reported in Refs. [40, 41, 42]. Last but not least, calculations of the electric properties for diatomic molecules in spatially degenerate electronic states are scarce. To the best of our knowledge, only two studies considered this problem, in the context of the dispersion interactions between molecules [52, 53] rather than non-resonant interactions with an external field, and a systematic theoretical approach has not yet been proposed. Moreover, the presence of spin-orbit coupling between the electronic states has been neglected in a recent treatment of nuclear dynamics in a non-resonant field [54, 55]. Such an approximation does not allow to study the competition between the spin-orbit coupling and the interaction with a non-resonant field which may both significantly perturb the spectrum.

Here, we fill this gap and report the theoretical framework for a state molecule interacting with a non-resonant field and study its rovibrational dynamics in the presence of the field. We also report ab initio calculations of all potential energy curves, spin-orbit and nonadiabatic coupling matrix elements corresponding to the dissociation limits up to and including . We test our ab initio results by comparing the main spectroscopic characteristics of the potentials to the available experimental data. We devote special emphasis to the important manifold of the A and b states, comparing our results to Refs. [21, 35]. Since the electric properties of spatially degenerate electronic states were not extensively studied in the literature thus far, we report here, to the best of our knowledge, the first ab initio calculation of the irreducible components of the polarizability tensor, including their dependence on the interatomic distance , for the A and b states. Finally, we study the effect of a non-resonant field on the spectroscopy in the A and b manifold. This is motivated by our recent proposal for enhancing photoassociation by controlling shape resonances with non-resonant light [54, 55]. In order to significantly modify the scattering continuum of the atom pairs to be photoassociated, rather large non-resonant intensities are required. Since the bound rovibrational levels are much more affected by a strong non-resonant field than continuum states, it is important to investigate how the corresponding spectroscopic features change.

Our paper is organized as follows. In Sec. 2 we formulate the theory of the interaction of a homonuclear molecule with an external non-resonant field. In Sec. 3 we provide the theoretical description of the perturbation of spectra by a non-resonant field, using as an example the spin-orbit coupled manifold of the A and b electronic states of Rb. We briefly summarize the ab initio methods employed in our calculations in Sec. 4 and discuss the results of these calculations in Sec. 5. In particular, we compare our data with results available in the literature and discuss the ability of the ab initio results to reproduce the high-resolution spectroscopic data for the A and b manifold [21, 35]. We then describe the interaction with a non-resonant field and study its spectroscopy signatures on the transitions between the electronic ground state and the A and b manifold. Finally, Sec. 6 concludes our paper.

2 Diatomic molecule in a non-resonant electric field

We consider the interaction of a diatomic molecule with an electric field with the direction taken along the axis of the space-fixed coordinate system, . To the second order, the Hamiltonian for the interaction of the molecule with the electric field in the space-fixed frame can be written as,

(1)

where and denote the appropriate components of the electric dipole moment and electric dipole polarizability in the space-fixed frame. Since we deal with a homonuclear molecule, only the second term of the above Hamiltionian will be relevant in the present analysis. To evaluate the matrix elements of the Hamiltonian in the electronic and rovibrational basis, we rewrite in terms of the polarizability components in the body-fixed frame. The dipole polarizability component can be expressed in terms of space-fixed irreducible tensor components [56],

(2)

For the irreducible tensor components, the transformation from the space-fixed to the body-fixed coordinate system is given by the rotation matrices ,

(3)

Hence, we have

(4)

For simplicity, we omit the superscripts SF/BF in the rest of the paper as from now we will use only the body-fixed quantities. We assume in this paper that the molecular axis defines the body-fixed axis. For a diatomic molecule the set of the Euler angles can be chosen as , where is the angle between the molecular axis and the space-fixed axis. This particular choice of the Euler angles is consistent with the requirement that the space-fixed and body-fixed axes coincide. The other possible set would be which correspond to the coincidence of the space-fixed and body-fixed axes. Note that for our specific choice of the Euler angles, the Wigner functions appearing in Eqs. (4) reduce to:

(5)

where are the associated Legendre polynomials. For any diatomic molecule, the non-zero irreducible components of the dipole polarizability are and . In addition, for a diatomic molecule in a electronic state, the and terms do not vanish. They should be viewed as off-diagonal polarizability tensor components connecting two degenerate electronic states, and , with opposite projection of the total electronic orbital angular momentum on the molecular axis. See, for instance, Eq. (16) of Ref. [53].

The non-vanishing body-fixed polarizability components are most conveniently expressed in terms of the Cartesian tensor elements , . Then is related to the trace of the polarizability,

(6)

to the anisotropy of the polarizability,

(7)

and, for a molecule in a electronic state, and reflect the difference between two perpendicular components,

(8)

For a diatomic molecule in a state, the definitions of the Cartesian components of the polarizability tensor in Eqs. (6) to (8) are unambigous. The and components are simply the parallel and perpendicular components, and , respectively. Thus, the irreducible tensor components appearing in Eqs. (6) to (8) are simply related to the trace and the anisotropy of the polarizability tensor,

(9)

Obviously, for a state molecule the and components are equal, and .

In the case of a molecule in a degenerate electronic state (, etc.) some caution is needed when employing the Cartesian components , since one has to specify the basis of the electronic states, in which these quantities are expressed. Equation (8) assumes the Cartesian components, and , to be calculated for the state. However, the Cartesian basis for the electronic state is not convenient for the dynamical calculations, since the spin-orbit coupling matrix elements are complex in this basis. Therefore, we prefer to use the spherical basis for the state over the Cartesian basis since it avoids complex quantities in the calculations and allows for a simple adaptation of the Hund’s case wave function to a given symmetry of the rovibrational level. Therefore, we will use the irreducible polarizability components rather than the Cartesian .

Combining Eqs. (1) to (8) and making use of properties of the rotation matrices , one arrives at the following Hamiltonian for the interaction of the homonuclear diatomic molecule with the static electric field,

(10)

The above Hamiltonian is valid for any isolated electronic state of a diatomic homonuclear molecule. Albeit, the last two terms in this equation are relevant only for molecules in a electronic state. Let us stress here that although this form of the Hamiltonian seems a bit elaborate at first glance, it simplifies the evaluation of the matrix elements in the symmetry-adapted basis set, and it also avoids any ambiguities when employing the Cartesian polarizability components for degenerate electronic states. Equation (10) also assumes the frequency of the non-resonant field to be far from any resonance which allows for using the static polarizability and the two-photon rotating-wave approximation. Such a field can be produced for example by a carbon dioxide laser with a wavelength of about 10m,

3 Hamiltonian for the Rb molecule in the manifold of the coupled A and b excited states interacting with a non-resonant field

We construct the Hamiltonian for the nuclear motion in Hund’s case coupling scheme with the primitive basis functions that are products of the electronic , electron spin and rotational functions. Here, is the total angular momentum quantum number, is the total electronic spin quantum number, and are the projections of the total electronic orbital and total electronic spin angular momenta onto the molecular axis, and is the projection of the total angular momentum onto the space-fixed axis. labels the nonrelativistic dissociation limit of the molecular state. We also define the projection of the total, electronic orbital plus spin, angular momentum onto the molecular axis, . For the coupled A and b manifold, we consider the rovibrational levels of the spectroscopic symmetry and odd parity. For simplicity, any hyperfine structure effects are neglected here. The properly symmetry-adapted Hund’s case wavefunctions read,

(11)

The first two states have a projection of the total angular momentum onto the molecular axis , while the third one has . In the field-free case, the state with is decoupled from the states with , and it is not accessible from the ground electronic state in the one-photon dipolar transitions considered here. Consequently, the field-free model Hamiltonian describing the nuclear motion in the manifol of the coupled A and b states can be represented by following matrix,

(12)

where is the sum of the vibrational and rotational kinetic energy operators with being the mechanical angular momentum of the molecule and , , , denotes the respective potential energy curves in the Born-Oppenheimer approximation. and are the spin-orbit coupling matrix elements, and only the electronic states with are included. Our model does not account for Coriolis-type angular couplings, i.e., the couplings of the states with states because their effect on the rovibrational dynamics is negligible compared to the spin-orbit couplings, the error of the electronic structure data and the influence of the weak non-resonant field. It is not surprising due to large reduced mass of Rb molecules whose inverse enters all coupling matrix elements.

When the electric field is switched on, the and components of the b state are coupled. The coupling results form the off-diagonal polarizability tensor components in the Hamiltonian of Eq. (10). Therefore, not only the interaction from Eq. (10) has to be added to the Hamiltonian for the A and b states with , but also the matrix (12) has to be extended so as to include the component originating from the b state since it has the projections exactly opposite to those found in the state with while all other quantum numbers are the same. Hence, in the presence of the electric field the rovibrational levels of the A and b manifold are obtained by diagonalizing the Hamiltonian represented by the following matrix,

(13)

The diagonal elements of the interaction potentials incorporating the interaction with non-resonant field are given by,

(14)

where A or b and is given by Eq. (10) for the electronic state labeled by . The off-diagonal term due to the non-resonant field, , couples the and components resulting from the b state. It is proportional to the off-diagonal polarizability of the molecule in the b state,

(15)

with defined by Eq. (8). Analogously to Eqs. (13) and (14), the Hamiltonian for the molecule in its electronic ground state interacting with a non-resonant field is simply given by .

4 Ab initio electronic structure and dynamical calculations

We adopt the computational scheme successfully applied to the ground and excited states of the calcium dimer [57, 58, 59, 60, 61], magnesium dimer [62, 63], strontium dimer [64, 65], (BaRb) molecular ion [66], and SrYb heteronuclear molecule [67]. The potential energy curves for the singlet and triplet gerade and ungerade states of the Rb molecule corresponding to the first seven lowest dissociation limits, , , , , , , and , have been obtained by a supermolecule method,

(16)

where denotes the energy of the dimer computed using the supermolecule method (SM), and , or , is the energy of the atom in the electronic state corresponding to the dissociation limit of the state . The full basis of the dimer was employed in the supermolecule calculations on the atoms and , and the molecule , and the Boys and Bernardi scheme was utilized to correct for the basis-set superposition error [68]. The calculations for the excited states employed the recently introduced Double Electron Attachment Intermediate Hamiltonian Fock Space Coupled Cluster method restricted to single and double excitations (DEA-IH-FS-CCSD) [49, 50, 51]. Starting with the closed-shell reference state for the doubly ionized molecule Rb that shows the correct dissociation at large interatomic separations, , into closed-shell subsystems, Rb+Rb, and using the double electron attachment operators in the Fock space coupled cluster ansatz makes our method size-consistent at any interatomic separation and guarantee the correct large- asymtptotics. Thus, the DEA-IH-FS-CCSD approach overcomes the problem of the standard coupled cluster method restricted to single and double excitations (CCSD) and the equation of motion CCSD method [50] with the proper dissociation into open-shell atoms. The potential energy curves obtained from the ab initio calculations were smoothly connected at intermediate interatomic separations with the asymptotic multipole expansion [56]. The coefficient of the electronic ground state and the coefficient of the first excited state were fixed at their empirical values derived from high-resolution spectroscopic experiments [4, 34], while the remaining coefficients were taken from Ref. [69].

The transitions from the ground X state to the and states and from the a to the and states are electric dipole allowed. The transition dipole moments for the electric transitions were computed from the following expression [70],

(17)

where the , or , denotes the th component of the electric dipole moment operator. Note that in the first term of Eq. (17) or corresponds to transitions to states, while corresponds to transitions to states. The transitions from the a state connect this state with the and states, through the and and operators, respectively.

We expect the rovibrational energy levels of the excited electronic states of Rb to show perturbations due to the nonadiabatic couplings between the states. Analysing the pattern of the potential energy curves, we have found that many potential energy curves display avoided crossings, suggesting strong radial couplings between these electronic states. We have therefore computed the most important radial coupling matrix elements, defined by the expression,

(18)

where signifies that the electronic states and are coupled. Note that the radial derivative operator couples states with the same projection of the electronic orbital angular momentum on the molecular axis .

Electric transition dipole moments, radial non-adiabatic coupling and spin-orbit coupling matrix elements were obtained using the Multireference Configuration Interaction method (MRCI) restricted to single and double excitations with a large active space. Scalar relativistic effects were included by using the small-core fully relativistic energy-consistent pseudopotential ECP28MDF [71] from the Stuttgart library. Thus, in the present study the Rb molecule was treated as a system of effectively 18 electrons. The basis set was employed in all calculations. This basis was obtained by decontracting and augmenting the basis set of Ref. [71] by a set of additional functions improving the accuracy of the atomic excitation energies of the rubidium atom with respect to the NIST database [72]. The DEA-IH-FS-CCSD calculations were done with the code based on the ACES II program system [73], while the MRCI calculations were performed with the MOLPRO code [74]. All ab initio results reported in the present paper are available from the Authors on request.

The rovibrational levels of the A and b excited state manifold are computed by diagonalizing the Hamiltonian (12) represented on a mapped Fourier grid, employing about radial grid points. For the calculations in the field we complement our Fourier grid representation for the radial part by a basis set expansion in terms of Legendre polynomials for the angular part, taking advantage of the magnetic quantum number being conserved. We find that is sufficient to obtain converged results for W/cm. Presence of an intense non-resonant field leads to strong hybridization of the rovibrational levels, and an adiabatic separation of rotational and vibrational motion is not applicable [54, 55]. We account for this fact by diagonalizing the full two-dimensional Hamiltonian, Eq. (13), represented by a matrix. For , the non-resonant field mixes different partial waves, and and are not good quantum numbers anymore. For the sake of simplicity, we label the field-dressed rovibrational levels by the field-free quantum numbers. Note that the field-dressed levels are adiabatically connected to their field-free counterparts even for very large intensities.

5 Numerical results and discussion

5.1 Potential energy curves

\topruleasymptote energy energy molecular states
(present) (exp.)
\colrule+ 0 0 ,
+ 12731 12737 , , , , , ,
,
+ 19471 19355 , , , , , ,
, , , , ,
+ 20126 20133 , , ,
+ 23732 23767 , , , , , ,
,
+ 25462 25475 (2), , , , ,
(2), , , ,
+ 25736 25707 , , , , , ,
, , , , ,
Rb+Rb 29741 29771 ,
\botrule
Table 1: Asymptotic enrgies (in cm) and molecular states arising from different states of rubiudium atoms [26].

To test the ability of the ab initio approach adopted in the present work to reproduce the experimental data, we first check the accuracy of the atomic results. In Table 1 we report the excitation energies at the dissociation limit computed with the DEA-IH-FS-CCSD method and compare the results to non-relativistic excitation energies obtained with the Landé rule from the experimental excitation energies. Inspection of Table 1 shows that the agreement between the theoretical and experimental excitation energies is very good. For the and dissociation limits, the RMSD is only 21cm, which represents an error of 0.08%. When the D states are included this good agreement is somewhat degraded. The RMSD is now 49cm, i.e., 0.26%. This is due to the lack of symmetry functions in the basis set used in our calculations. Note parenthetically that we could not include functions in the basis, because the ACESS II program does not support orbitals in the calculations involving pseudopotentials. Our method reproduces very well the electron affinity of the Rb atom, 3893cm on the theory side vs. 3919cm measured in Ref. [75], as well as the ionization potential, 33630cm vs. 33690cm [72]. Finally, we note that the ground state static electric dipole polarizability of the atom obtained from our molecular calculations is 319.5a compared to 318.6a from the most sophisticated atomic calculations by Derevianko et al. [76].

Figure 1: Potential energy curves for the and states of the Rb molecule.
Figure 2: Potential energy curves for the and states of the Rb molecule.
Figure 3: Potential energy curves for the and states of the Rb molecule.
Figure 4: Potential energy curves for the and states of the Rb molecule.
Figure 5: Potential energy curves for the and states of the Rb molecule.
\toprulestate Ref. asymptote
(bohr) (cm) (cm) (cm)
\colruleX present 7.99 3912 56.1 0
[28] (exp.) 7.96 3994 57.8 0
[42] 7.96 3905 58.4 0
present 10.29 3102 32.0 13545
[37] (exp.) 10.28 2963 31.5 13602
[42] 10.17 3084 31.2 13559
present 10.32 4210 32.9 19180
[42] 10.20 4072 31.9 19189
present 9.34 4144 62.0 19898
present 9.21 3483 37.8 24166
2nd. min. present 22.22 2968 11.0 24681
present 8.93 3055 46.6 24594
2nd. min. present 12.02 2056 50.6 25593
3rd. min present 34.60 86 4.7 27734
present 11.26 1852 92.8 25797
present 9.47 183 41.3 27465
\colrule present 10.25 1230 21.7 15417
[36] (exp.) 10.24 1290 22.3 15510
[42] 10.24 1198 22.0 15545
present 9.92 1326 31.0 22063
[42] 9.88 1238 22.0 22023
present 9.25 2833 43.1 22149
present 9.48 2598 37.1 22099
present 9.13 1994 42.9 22187
\colrule present 8.18 5026 48.7 18449
[42] 8.14 4871 50.5 18390
present 8.76 5291 57.6 24165
present 9.22 2528 56.5 27212
\botrule
Table 2: Spectroscopic characteristics of the non-relativistic electronic states of Rb molecule.
\toprulestate Ref. asymptote
(bohr) (cm) (cm) (cm)
\colrule present 9.91 3367 37.8 13279
[42] 9.73 3345 36.6 13298
present 8.58 5372 51.1 18017
[42] 8.47 5347 51.5 17914
present 9.31 1657 38.2 22384
present 8.95 3335 46.7 24313
present 9.72 3488 19.4 26065
present 9.19 3292 43.8 26953
present 9.12 3268 38.5 27832
\colrule present 9.54 -267 30.3 16914
[42] 9.47 -268 30.3 16911
present 10.56 3104 34.2 20285
[42] 10.53 2927 33.6 20334
present 9.08 3416 45.4 24232
present 9.06 2646 27.4 26735
present 9.09 2170 45.8 27484
\colrule present 8.36 4181 48.3 19284
[42] 8.31 4017 48.9 19244
present 8.85 5152 46.2 24588
\botrule
Table 3: Spectroscopic characteristics of the non-relativistic electronic states of Rb molecule.
\toprulestate Ref. asymptote
(bohr) (cm) (cm) (cm)
\colrule present 9.24 5967 44.1 10680
[35] (exp.) 9.21 5981 44.6 10750
[42] 9.20 5896 44.4 10747
present 10.21 3128 20.5 20261
[42] 10.09 3003 22.1 20258
2nd. min present 14.11 3112 13.5 20277
[42] 13.81 2926 11.5 20335
present 9.37 1737 42.4 22305
present 9.46 2390 31.3 25258
2nd. min. present 12.64 2702 24.3 24946
3rd. min present 22.26 2973 10.7 24675
present 9.28 3565 39.1 26088
2nd. min present 34.69 1920 5.0 27733
present 10.38 3308 52.9 26937
\colrule present 8.57 1971 46.9 14676
[38] (exp.) - 1907 47.5 14666
[42] 8.48 1989 47.9 14654
present 8.92 2369 31.6 21021
[38] (exp.) - 2454 36.4 20895
[42] 8.77 2157 36.1 21104
present 9.23 4927 40.4 22721
present 9.03 4216 43.1 25166
present 10.06 3189 31.4 26465
\colrule present 9.80 639 28.0 22825
[42] 9.78 542 26.9 22718
present 9.31 3638 48.1 25818
present 9.40 2630 34.2 27110
\botrule
Table 4: Spectroscopic characteristics of the non-relativistic electronic states of Rb molecule.
\toprulestate Ref. asymptote
(bohr) (cm) (cm) (cm)
\colrulea present 11.46 250 13.5 3662
[31] (exp.) 11.51 242 13.5 -
[42] 11.45 237 13.3 3669
present repulsive - - -
present 11.02 2761 40.0 20628
[42] 10.96 2646 40.6 20614
present 10.06 1340 43.0 22701
present 9.18 2493 44.7 25155
present 9.29 3235 40.9 26147
present 9.09 938 47.2 28444
\colrule present 7.91 6969 57.2 9677
 [35] (exp.) 7.81 7039 60.1 9601
[42] 7.88 7015 59.7 9624
present 8.73 3527 43.5 19862
[42] 8.60 3497 43.3 19764
present 9.28 5117 40.0 22531
present 8.99 4189 43.3 25193
present 10.04 3711 56.5 25943
\colrule present 9.83 719 27.3 22746
[42] 9.86 619 25.8 22641
present 9.30 3695 40.7 25761
\botrule
Table 5: Spectroscopic characteristics of the non-relativistic electronic states of Rb molecule.

The computed potential energy curves are reported in Fig. 1 for the and symmetries, in Fig. 2 for the and symmetries, in Figs. 3 and 4 for the and , and and symmetries, respectively. Finally Fig. 5 shows the potential energy curves for the singlet and triplet gerade and ungerade states of symmetry. The spectroscopic characteristics of the singlet gerade states are reported in Table 2 while Table 3 collects these properties for the triplet gerade states. Tables 4 and 5 present the results for the singlet and triplet states of ungerade symmetry, respectively. Inspection of Figs. 1 to 5 reveals that almost all potential energy curves show a smooth behavior with well defined minima. Some higher states display perturbations, mostly in the form of avoided crossings, due to the interaction with other electronic states of the same symmetry that are located nearby. At high energies the density of states becomes so high that the avoided crossings produce some irregularities in the curves. This is especially true for the singlet and triplet gerade and ungerade states of symmetry. The states show less perturbations, except for the avoided crossings between the curves corresponding to the and , and and states. Interestingly, the states and the states do not show any irregularity due to nonadiabatic interactions between the states.

The agreement of the present potentials with those derived from the experimental data is very good. This is demonstrated in Tables 2 to 5, where we compare the potential characteristics with the available experimental data and with the most recent calculations [42]. For all the experimentally observed states, the RMSD of our calculation is only 75.9cm, i.e., the error is 3.2% on average, better than the most recent calculations by Allouche and Aubert-Frécon [42] with a RMSD of 129cm corresponding to an average error of 5.5%. It is gratifying to observe that we reproduce low lying and highly excited electronic states equally well. This is in a sharp contrast to Ref. [42] which reproduces the well depth of the state only with an error of 12% compared to 3.5% for our calculation. Such a good agreement between theory and experiment for the highest observed excited electronic state gives us confidence that our predictions for the photoassociative production of ultracold Rb molecules in even higher electronic states [47] are accurate. Tables 2 to 5 also report the fundamental vibrational frequencies for all electronic states considered in the present paper. Except for the ground state, the agreement between theory and experiment is within a few tenths of a wavenumber. Similar agreement was found in the calculations by Allouche and Aubert-Frécon [42].

5.2 Non-adiabatic coupling and spin-orbit coupling matrix elements

Figure 6: Non-adiabatic radial coupling matrix elements between states of and symmetry.

The importance of nonadiabatic interactions between electronic states, resulting in the avoided crossings of the corresponding potential energy curves observed in Figs. 1 to 5, can nicely be explained by analysing the nonadiabatic coupling matrix elements computed according to Eq. (18). The nonadiabatic coupling matrix elements are reported in Fig. 6 for singlet and triplet states of and symmetry (top) and the states (bottom). As expected, the nonadiabatic coupling matrix elements are smooth, Lorenzian-type functions, which, in the limit of an infinitely close avoided crossing, become a Dirac -function. The height and width of the curve depends on the strength of the interaction. The smaller the width and the larger the peak, the stronger is the interaction between the electronic states, and the corresponding potential energy curves are closer to each other at the avoided crossing. It is gratifying to observe that the maxima on the nonadiabatic coupling matrix elements agree well with the locations of the avoided crossing, and this despite the fact that two very different methods were used in ab initio calculations. Since the potential energy curves were shown to be accurate, cf. the discussion in Sec. 5.1, we are confident that also the nonadiabatic coupling matrix elements are essentially correct.

Figure 7: Spin-orbit coupling matrix elements between states of ungerade and gerade symmetries dissociating into . Black circles and red squares are analytical fit to high-resolution spectroscopic data from Ref. [35].

Rubidium is a heavy atom and the electronic states of the Rb molecule show strong couplings due to the relativistic spin-orbit interaction. Figure 7 reports the spin-orbit coupling matrix elements as a function of the interatomic separation. The matrix elements are all represented by smooth curves approaching the atomic limit at large . The fine splittings of the atomic states are very accurately reproduced by our calculations. For the first excited P state, the theoretical splitting between the 1/2 and 3/2 components is 236.2cm as compared to 237.6cm from the experiment. It is also gratifying to observe that our ab initio calculations reproduce very well the spin-orbit coupling functions obtained from fitting analytical functions to high-resolution spectroscopic data for the A and b manifold of states [35]. This gives us confidence that also perturbations in the molecular spectra due to the spin-orbit interaction will correctly be reproduced from the present ab initio data.

5.3 Electric transition dipole moments and electric dipole polarizabilities

Figure 8: Electric dipole transition moments: between the X ground state and excited states of and symmetry and between the a lowest triplet state and excited states of and symmetry.

A full characterization of the molecular spectra requires knowledge of the electric transition dipole moments. These were calculated according to Eq. (17) and are presented in Fig. 8 for transitions from the X ground state and in Fig. 9 for transitions from the a lowest triplet state. The strongest transitions from the ground singlet state are those to the A and states, i.e., to states corresponding to the first excited dissociation limit. All other transition moments are much smaller, suggesting that the corresponding line intensities in the spectra will be much weaker. The same is true for transitions departing from the a state. The transition moments do not show a strong dependence on , except at small interatomic separations, and smoothly tend to their asymptotic atomic value.

Figure 9: Electric dipole polarizabilities for the electronic ground state (left) and the first excited state (right).

The static electric dipole polarizabilities for the X electronic ground state, the a state, and the relevant excited A and b states are presented in Fig. 9. They show an overall smooth behavior and also tend smoothly to their asymptotic atomic values. The interaction-induced variation of the polarizability is clearly visible while changing the internuclear distance . It is significant for excited states, especially for the A state for which the isotropic part reaches 8000, and the anisotropic part reaches 6000. Such large values of both the interaction-induced variation of isotropic and anisotropic polarizabilities suggest that the influence of the non-resonant laser field on the rovibrational dynamics and transitions between the ground X state, and the A and b states should be significant even at relatively weak field intensities. Comparing the present polarizabilities of the X and a states with theoretical results by Deiglmayr et al. [77], we find good agreement. For example the isotropic polarizability given by trace of the polarizability tensor for the X and a states being 522a.u. and 675a.u. in the present study and 533a.u. and 678a.u. in Ref. [77], respectively.

Note parenthetically that the transition moments and matrix elements of the spin-orbit coupling also change when a DC or non-resonant AC field is applied, but the changes induced on the rovibrational spectrum are expected to be smaller compared to the effects introduced within Eq. (13). Therefore, the investigation of the field-induced variation of the transition moments and spin-orbit couplings is out of the scope of the present paper.

5.4 Rovibrational spectra in the manifold without a non-resonant field

Figure 10: Characteristics of the rovibrational levels for the component of the coupled A and b manifold of states in Rb: present and empirical potential energy curves [35], rotational spacings, and rotational constants for strongly bound levels and close to the dissociation limit .

We now compare in more detail the ability of our ab initio data to reproduce the fine details of high-resolution experiments of Ref. [35]. In Fig. 10, we report the ab initio and empirical potentials for the A and b states of Rb. Inspection of Fig. 10 shows a very good agreement. The ab initio calculations reproduce the well depth of the A state within 14 cm on the overall depth of 5981 cm, i.e., within 0.2%. The agreement for the b state is slightly less good. The difference in the well depths amounts to 70 cm for the well depth of 7039 cm. This represents an error of roughly 1%. Such an agreement between theory and experiment should be considered as very good. Also the crossing of the A and b potential energy curves is perfectly reproduced. Our dynamical calculations predict the level to be the first rovibrational level corresponding to the A state, see the rotational spacings in panel of Fig. 10. This is one quantum higher than predicted by the experiment [35], but the 70 cm disagreement in the well depths fully explains this difference.

Figure 10 also reports the rotational constants for the deeply bound rovibrational levels (panel ) and levels at the threshold (panel ). Inspection of Fig. 10 reveals that theory correctly locates all levels that are not perturbed by the spin-orbit interaction, and the first perturbed level. The agreement in the rotational constants for the rovibrational levels in the middle of the potential well is less good, but note the scale on the axis. Overall, we reproduce semi-quantitatively the pattern of the rovibrational levels in this region of the potentials. Also the oscillations of the rotational constants reflecting the perturbations due to the spin-orbit coupling between the A and the b states are correctly described. This is in accordance with the good agreement between the ab initio spin-orbit coupling and the data fitted to the experiment shown in Fig. 7. The agreement of the rotational constants