Ab initio multi-reference perturbation theory calculations of the ground and some excited electronic states of the RbYb molecule

The potential energy curves of the ground and twelve low-lying excited electronic states of the RbYb molecule have been calculated using the multi-reference perturbation theory method at the CASSCF/ XMCQDPT2 level of theory including the spin-orbit coupling. The electronic term energies, equilibrium internuclear distances, dissociation energies, transition dipole moments, the sequences of vibrational energies, the harmonic vibrational frequencies, and the Franck–Condon factors have been predicted. The potential energy curves at the asymptotic limits are in a good agreement with the experimental data for rubidium and ytterbium atoms. The obtained data would be useful for spectral experiments with RbYb molecules.


Introduction
The RbYb molecule belongs to heteronuclear diatomics, which due to the availability of an unpaired electron have both electric and magnetic permanent dipole moments.This makes possible trapping and manipulating of the weakly bound RbYb molecules in the ground X 2 R + 1/2 state using electric as well as magnetic fields [1] enabling their usage in many applications under cold and ultracold conditions [2].
The electronic states of the RbYb molecule were studied experimentally in [1,3] and theoretically in [4][5][6][7][8].Some experimental difficulties preventing the formation of RbYb molecules are discussed by Tassy et al. [9] and Baumer et al. [10].Nemitz et al. [1] obtained the energies of 15 vibrational states near the dissociation limit of the excited 1 2 G 1/2 state of the 87 Rb 176 Yb and 87 Rb 174 Yb molecules.The energies of 7 vibrational states near the dissociation limit of the ground X 2 R + 1/2 state of the 87 Rb 176 Yb molecule are determined by Münchow et al. [3].
Recently [11,12] we have shown that the ab initio calculations at the CASSCF/XMCQDPT2 (Extended Multi-Configuration Quasi-Degenerate 2nd Order Perturbation Theory) [13] level of theory predict the spectroscopic parameters of the ground state for the KRb molecule with a high accuracy.For the dissociation energy D e of the ground X 1 R + state, we have 4213.1 cm À1 that is in a better agreement with the experimental value (4217.815cm À1 [14]) than those in other ab initio calculations (for details see [12]).The calculated equilibrium internuclear distance of the KRb ground X 1 R + state (R e = 4.07581 Å) also agrees considerably well with the experimental data by Pashov et al. [14] (R e = 4.06770 Å).We assume that the CASSCF/XMCQDPT2 level of theory is well applicable to the electronic structure calculations of heavy diatomic molecules.Especially high accuracy is expected for the ground electronic state.
The main goals of this work are: (i) to calculate the potential energy curves (PECs) of the ground and some low-lying excited states of the RbYb molecule at the multi-reference perturbation level of theory including the spin-orbit coupling (SOC) and (ii) to determine other spectroscopic characteristics of the electronic states (transition dipoles, vibrational energies, Franck-Condon factors, etc.).The main attention is paid to the ground X 2 R + 1/2 and lowlying excited 1 2 G 1/2 , 1 2 G 3/2 , and 2 2 R + 1/2 states of the RbYb molecule.

Computations
The electronic configuration and the ground state term for the rubidium atom are [Kr]5 s and 2 S 1/2 , respectively.The first three excited configurations, atomic states, and their energies of the Rb atom are: 5p ? 2 P 1/2 (12,578.950cm À1 ), 2  The electronic configuration and the ground state term for the ytterbium atom are [Xe]4f 14 6s 2 and 1 S 0 , respectively.The first excited configuration, appropriate triplet and singlet terms, and energies of the states for the Yb atom are: 4f 14 6s6p ? 3 P 0 (17,288.499cm À1 ), 3 P 1 (17,992.007cm À1 ), 3  The next excited term (7/2, 3/2) 2,3,4,5 corresponds to the 4f 13 5d 1 6s 2 excited configuration with energies of the states lying within the range 23,288-28,184 cm À1 [15].The triplet 3 D 1,2,3 term originating from the 4f 14 5d6 s configuration with the energies ranging 24,489-25,271 cm À1 [15] and the 1 P term (see above) are located between the components of the (7/2, 3/2) 2,3,4,5 term.Consequently, calculations of the molecular electronic states which correlate to the dissociation limits with given or higher excited configurations of the Yb atom require consideration of the felectrons in the active space.Such states are not considered here.
The PECs calculations have been performed for the doublet and quartet states corresponding to the three lowest dissociation limits: Rb(5 s) + Yb(6s 2 ) (limit I), Rb(5p) + Yb(6s 2 ) (limit II), and Rb (5s) + Yb(6s6p, only triplet term) (limit III).Note that the 2 D term of the Rb atom lies between the 3 P 1 and 3 P 2 components of the 3 P term of the Yb atom.Consequently, it is assumed that the molecular electronic terms corresponding to the Rb(4d) + Yb(6s 2 ) and Rb(6s) + Yb(6s 2 ) dissociation limits are strongly perturbed by the terms corresponding to the Rb(5s) + Yb(6s6p) dissociation limit.Since the terms corresponding to the Rb(5s) + Yb (4f 13 5d 1 6s 2 ) dissociation limit, which must be directly above the above-mentioned terms, are not included into the calculations, the terms corresponding to the Rb(4d) + Yb(6s 2 ) and Rb(6s) + Yb (6s 2 ) dissociation limits are pushed up.Consequently, if excitations of the f-electrons of the Yb atom are not taken into account in the active space, PECs of these terms [6] will be of the artifact character.x e , cm À1 D e , cm À1 Asymptotic limit, cm À1 The ground X 2 R + state corresponds to the asymptotic limit I; the doublet 2 2 R + and 1 2 G states correspond to the asymptotic limit II; the doublet 3 2 R + , 2 2 G states, and the quartet 1 4 R + and 1 4 G states correspond to the asymptotic limit III.When SOC is taken into account, the X 2 R + , 2 2 R + and 3 2 R + states correspond to the X 2 R + 1/2 , 2 2 R + 1/2 and 3 2 R + 1/2 states, respectively; the 1 2 G and 2 2 G states split into the 1 2 G 3/2 , 1 2 G 1/2 and 2 2 G 3/2 , 2 2 G 1/2 states, respectively; the 1 4 R + state splits into the 1 4 R + 3/2 and 1 4 R + 1/2 states; the 1 4 G state splits into the 1 4 G 5/2 , 1 4 G 3/2 , 1 4 G 1/2 and 1 4 G -1/2 states.Note that, since the spin-orbit coupling mixes states of different multiplicity, the (2S + 1) K X designation is rather conditional in character, especially for the states belonging to the asymptotic limit III.Then (n) X designation, where n is a number of the state with the given X, is more appropriate.The (2S + 1) K X designation is used here to emphasize the relationship between the spin-mixed states and the spin-free ones, and hence such designation sometimes is useful.Both designations ( (2S + 1) K X and (n)X) are given in Table 1.
Then the calculations for doublet and quartet states were performed in the XMCQDPT2 [13] approximation.All the eight lowest double occupied orbitals were included in the perturbation based calculations.The ISA shift of 0.048 [18] was used for the present approach.The dimensions of the effective Hamiltonian are: 9 Â 9 and 5 Â 5 for doublet and quartet states, respectively.
Finally, the spin-orbit coupling (SOC) calculations were performed with the one-electron Pauli-Breit operator.The effective nuclear charges (see, for example, [19][20][21]) are: 2035.0 and 8925.0 for the rubidium and ytterbium atoms, respectively.Large values of Z eff should compensate errors of the expectation value ‹1/r 3 › in the small r region arising with the use of the ECP basis sets [21].
So, the following parameters of the ab initio calculations at the MRPT + SOC level of theory influence the calculation accuracy: a number of the states considered in the perturbation theory method [12] (i); qualitative composition (R, P, D) of the above-mentioned states [12] (ii); the energy denominator shift (EDS) for the intruder state avoidance (ISA) [12,18] (iii), and Z eff values [21] (iv).These parameters allow one to vary the energies of the low-lying atomic states, and the optimal choice of these parameters provides the required accuracy for the characteristics of the low-lying electronic molecular states.
All these calculations were performed using the Firefly quantum chemical package [22].The electronic energy terms T e and the equilibrium internuclear distances R e are obtained using the fourth degree polynomial approximation of the ab initio PECs near their minima.
The transition dipole moments (TDMs) are computed based on the CASSCF functions at the CASSCF/XMCQDPT2 + SOC level of theory.The calculations of the TDMs are performed using the GAMESS quantum chemical package [23].
The calculations of the vibrational states energies and the Franck-Condon factors are performed using the LEVEL program package [24].

Electronic states
The calculated PECs versus internuclear distances are shown in Fig. 1 and are also given in the supplementary material.The calcu- lated molecular spectroscopic constants (the electronic term energy T e , the equilibrium internuclear distance R e , the dissociation energy D e , and the harmonic vibrational frequency x e ) are presented and compared with the previous theoretical results in Table 1.
Our calculated T e values of the 1 2 G 1/2 , 2 2 R + 1/2 and 1 2 G 3/2 states are higher than those of Sørensen et al. [4].And, since the asymptotic behavior of these PECs in our and Sørensen's [4] calculations differs slightly, our calculated D e values are lower than those of Sørensen et al. [4] (see Table 1).The equilibrium internuclear distances of the PECs of 1 2 G 1/2 , 2 2 R + 1/2 and 1 2 G 3/2 states obtained in our calculations are similar to the values of Sørensen et al. [4].
PECs of the asymptotic limit III are underestimated approximately by 626 cm À1 relative to the experimental energies [15] and the calculated total SOC splitting (2422.8cm À1 ) of this limit corresponds to the experimental [15] value (2421.889cm À1 ).

Vibrational states
The 85 Rb 174 Yb molecule is the most abundant isopotomer (23%) among 14 natural isotopomers of the title molecule.Thus, the vibrational energies for the 85 Rb 174 Yb molecule (and some other isotopomers) are calculated for all spin-mixed PECs up to the dissociation limit of the respective state, and the harmonic vibrational frequencies are also obtained (see Table 1).For the ground and excited 1 2 G 1/2 , 1 2 G 3/2 , and 2 2 R + 1/2 states we have 46, 168, 143 and 126 vibrational levels, respectively.For all of these states, the obtained x e values are the largest among all the earlier calculated values (see Table 1).
Since the experimental data for the vibrational states of the RbYb molecule are few in number, we compare the calculated energies of the weakly-bound vibrational states for the ground and for the first excited electronic states near the dissociation limits with the experimental data of Münchow et al. [3] and Nemitz et al. [1] in the case of 87 Rb 176 Yb isotopomer.The results are presented in Fig. 3.Here Dv = v À v max is the relative vibrational quantum number and v max is the vibrational quantum number of the last bound state.
The ground state PEC has a shallow well and hence PEC has a sloping behavior near the dissociation limit.Since our calculations are limited to 20 Å, a faster convergence of the vibrational states to the dissociation limit than that observed in the experiment is predicted [3].So, the calculated energy of the v max vibrational state is artificially overestimated relative to the lower vibrational states.In Fig. 3a and b we compare the calculated energies with Dv and the corresponding experimental values with Dv À 2. For the first excited 1 2 G 1/2 state, a more exact agreement between the calculated and experimental [1] vibrational energies for the same Dv values is observed (Fig. 3c and d).

Franck-Condon factors
Basing on our ab initio PECs and calculated vibrational eigenfunctions, we predict the Franck-Condon factors (FCFs) between the relevant states of the 85 Rb 174 Yb molecule.FCFs have been calculated for the rovibronic transitions between the all spin-mixed electronic states with the DX = 0, ±1 selection rules.These calculations are performed for the transitions between the fixed vibrational level v 0 of the upper electronic state and a sequence of vibrational levels v'' of the lower electronic state (applying the selection rules DJ = 0 or DJ = ±1).
The selected FCFs between the vibrational levels of the first three exited states and the vibrational levels of the ground state with DJ = 0 selection rule are presented in Fig. 4. FCFs shown here are calculated for the transitions from the lowest vibrational levels (v 0 = 0. ..4) of the excited electronic states (Fig. 4a, d, g) (1), for the transitions with the largest FCFs to the ground rovibronic X 2 R + 1/2 (v'' = 0, J = 0) state (Fig. 4b, e, h) (2), and for the transitions with the largest values of all possible FCFs (Fig. 4c, f, i

Conclusions
We have calculated the potential energy curves for the ground and twelve low-lying excited states of the RbYb molecule at the CASSCF (3,18)/XMCQDPT2 + SOC level of theory.We have predicted the molecular spectroscopic constants and other characteristics for all of the above-mentioned states.Since at the Rb(5p) + Yb(6s 2 ) dissociation limit our calculated PECs are in a good agreement with the experimental [15] energies and since our earlier calculations for the KRb molecule [12] in a similar approximation predict the spectroscopic parameters of the ground state of the KRb molecule with a high accuracy, we assume that the results of our calculations for the ground and low-lying excited states of the RbYb molecule are also close to the realistic values.The obtained data seem useful for spectral experiments with RbYb molecules.