Multi-reference perturbation theory study on the CsYb molecule including the spin-orbit coupling

Abstract We present CASSCF/XMCQDPT2 level of theory calculations of the ground and sixteen low-lying excited electronic states of the CsYb molecule taking into account the spin-orbit coupling. Spectroscopic constants (electronic term energies, equilibrium internuclear distances, dissociation energies, harmonic vibrational frequencies), transition dipole moments, Franck–Condon factors and vibrational energies of the CsYb molecule have been obtained. The energies of the ground and first exited states at the asymptotic limits definitely satisfy the experimental data for cesium and ytterbium atoms. All the data obtained allow to predict and realize two-photon schemes for producing ultracold CsYb molecules and carry out spectral experiments with them.


Introduction
Nowadays in the field of cold and ultracold molecules theoretical and practical studying of a new class of heteronuclear diatomic molecules, which include an atom of alkali metals and one of transition metals, is of great interest. In a diatomic molecule consisting of an alkali metal atom and, for example, of an atom of the lanthanide series with even atomic number one of the electrons is found out to be unpaired. This means that such a molecule besides a permanent electric dipole moment also has a permanent magnetic dipole moment. Thus, it is possible to effectively manipulate of the molecular quantum matter by external magnetic field. Moreover these molecules offer the possibilities for high precision measurements of fundamental constants, for quantum many-body physics and quantum information, for testing fundamental models, etc. [1,2]. The CsYb molecule is proposed to be an attractive candidate for the possibilities mentioned. For efficient production of ultracold CsYb molecules by two-photon schemes the knowledge of exact potential energy curves (PECs), vibrational energies, molecular spectroscopic and dynamic parameters are crucial.
Up to our knowledge, no experimental data are evaluable for a system of electronic states of the CsYb molecule, and there are only three ab initio calculations for the ground X 2 R + state in CCSD(T) (Coupled Cluster) [3] and CASSCF/MRCI (Complete Active Space Self-Consistent Field/Multi-Reference Configuration Interaction) approximation [4,5] respectively, and for the first excited states (namely spin-free 1 2 P and 2 2 R + states), performed by Meyer and Bohn [4] at the CASSCF/MRCI level of theory. The results of these calculations for the ground state are quite different: R e = 5.657 Å, D e = 182 cm À1 [4]; R e = 5.161 Å, D e = 542 cm À1 [5] and R e = 5.144 Å, D e = 621 cm À1 [3] (where R e is equilibrium internuclear distance, D e is dissociation energy). In this connection it is important to compare the calculated characteristics of other alkali-metal-ytterbium diatomic molecules: LiYb, NaYb, KYb, and RbYb.
Turning back to the CsYb molecule it is worth mentioning that the calculations by Shao et al. [5] (D e = 542 cm À1 ) and Brue and Hutson [3] (D e = 621 cm À1 ) were performed only for the ground state, i.e. without state averaged (SA) procedure on ground and some excited states. It means that the influence of the excited states on form and depth of the ground state well through density matrix averaging was absent. Note that for the weakly-bounded systems where the depth well is small (CsYb case), such influence might play an important role in forming of the ground state potential energy function.
Lately MRPT calculations at the CASSCF/XMCQDPT2 (Extended Multi-Configuration Quasi-Degenerate 2nd Order Perturbation Theory) [16] level of theory were performed for the KRb [17] and RbYb [12] molecules. In study [17] we obtained the spectroscopic parameters of the ground state for the KRb molecule with a high accuracy. As a result of calculations in [12] the energies obtained for the first two asymptotic limits exactly respond to the experimental data for rubidium and ytterbium atoms. An explanation of such sufficient results of the calculations is also discussed in [12]. Thus we use the same technique expecting to achieve the best results for CsYb molecule.
The aim of this work is to carry out ab initio calculations of PECs of the low-lying electronic states taking into account the spin-orbit coupling (SOC), to determine molecular spectroscopic constants, vibrational energies, Franck-Condon factors (FCFs) for the vibronic transitions of diatomic polar CsYb molecule at the high level of theory. Particular attention is paid to the ground X 2 R + 1/2 state and lowlying excited 1 2 G 1/2 , 1 2 G 3/2 , and 2 2 R + 1/2 states.
The next excited term (7/2, 3/2) 2,3,4,5 originates from the 4f 13 5d 1 6s 2 excited configuration with energies of the states lying in the range of 23 288-28 184 cm À1 [18]. The triplet 3 D 1,2,3 term arising from the 4f 14 5d6s configuration with the energies ranging 24 489-25 271 cm À1 [18] and the 1 P term (see above) are located between the components of the (7/2, 3/2) 2,3,4,5 term. Therefore calculations of the molecular electronic states corresponding to given or higher excited configurations of the Yb atom require including felectrons into the active space. The states just mentioned are not considered here.
We carry out the PECs calculations for spin-free (doublet and quartet) and spin-mixed states corresponding to the five lowest dissociation limits: Cs(6s) + Yb(6s 2 ) (limit I), Cs(6p) + Yb(6s 2 ) (limit II), Cs(5d) + Yb(6s 2 ) (limit III), Cs(7s) + Yb(6s 2 ) (limit IV), Cs(6s) + Yb(6s6p) (limit V). It is worth mentioning that 2 S 1/2 term of the Cs atom occurs between the 3 P 1 and 3 P 2 components of the 3 P term of the Yb atom. The molecular electronic terms relating to the Cs (5d) + Yb(6s 2 ) and Cs(7s) + Yb(6s 2 ) dissociation limits are supposed to be highly perturbed by the terms relating to the Cs(6s) + Yb (6s6p) dissociation limit. As we do not take into account during the calculations the terms corresponding to the Cs(6s) + Yb (4f 13 5d 1 6s 2 ) limit, situated much more above just mentioned terms, the terms corresponding to the Cs(5d) + Yb(6s 2 ) dissociation limit lift up and the term corresponding to the Cs(7s) + Yb(6s 2 ) dissociation limit is pushed out. Consequently, if excitations of the felectrons of the Yb atom are not taken into account in the active space, PECs of these terms [6,10,11,14] will be of the artifact character.
It is typical for heavy diatomic molecules containing an alkali metal to use (n)X notation, as spin-mixed states arise from interaction between the states with the different spin. But it is worth using (2S+1) K X notation to lay stress on the relationship between the spin-mixed and the spin-free states. Both notations ( (2S+1) K X and (n)X) are given in Table 1.
To obtain more reliable energies XMCQDPT2 method [16] was used. All the eight lowest double occupied orbitals were involved in the perturbation calculations, the ISA shift [21] of 0.190 [22] being used. The dimensions of the effective Hamiltonian were 11 Â 11 and 5 Â 5 for doublet and quartet states, respectively.
For the spin-orbit coupling (SOC) calculations we used the oneelectron Pauli-Breit operator. The effective nuclear charges Z eff , which always turn out to be of high value due to errors coursed by nodeless ECPs wave functions (for details see [23][24][25]), for Yb and Cs were 7137.0 and 7965.0 respectively.
We obtained potential curves for CsYb using the Firefly quantum chemical package [26]. For determining the electronic energy terms T e and the equilibrium internuclear distances R e we used the fourth degree polynomial approximation of the ab initio PECs near minima.
The calculations of transition dipole moments (TDMs) are performed at the CASSCF/XMCQDPT2 + SOC level of theory using GAMESS suit of codes [27].
The calculations of the vibrational states energies and the Franck-Condon factors are performed using the LEVEL program package [28]. The perturbations of the vibrational states, which belong to different electronic terms with the equal X values (for example, 1 2 G 1/2 and 2 2 R + 1/2 ) and overlap in some region of energies, were not calculated.

Electronic states
The calculated PECs are performed in Fig. 1 and are also given in the Supplementary Material. The calculated molecular spectroscopic constants (the electronic term energy T e , the equilibrium internuclear distance R e , the state binding energy D e and the harmonic vibrational frequency x e ) are presented in Table 1. Table 1 also shows the results of the ab initio calculations [3][4][5].
As there is no experimental values of spectroscopic parameters for ground and excited states to determine the accuracy of the calculations in Table 1 we compare the calculated energies of molecular states at the dissociation limits (at the internuclear distance of 17.0 Å) with the sum of the experimental energies [18] of separated atoms. The energies obtained for the 1 2 G 3/2 , 1 2 G 1/2 , and 2 2 R + 1/2 states corresponding to the limit II are 11 178.0, 11 728.7, 11 732.3 cm À1 and with the SOC splitting of 554.3 cm À1 . These data are in agreement with the experimental ones (see Table 1).
Our calculated equilibrium internuclear distance (R e = 5.763 Å) for the X 2 R +  [4] (182 cm À1 ) and significantly lower than Shao's [5] (542 cm À1 ) and Brue and Hutson's ones [3] (621 cm À1 ) (see Section 1). The shallow potential well is caused by weak polarizability of the 6s state of Cs atom and signals the van der Waals' character of the molecule. In contrast to the ground state PEC, the first excited states (1 2 G 1/2 , 1 2 G 3/2 , and 2 2 R + 1/2 ) have deeper wells due to stronger polarizability of the 6p state of cesium atom.
During the calculations the 1 2 D 1/2,3/2 states that should belong to limit III disappeared, the remaining ones being shifted and SOC splitting of this limit being overestimated by 1633.8 cm À1 . This may be caused by the not including f-electrons of Yb atom in active space through the calculations. The term, related to limit IV, must appear according to experimental data between fine-structure components of limit V. However, within the calculations such a situation did not arise.
''Intruded" term 5 2 R + 1/2 caused a perturbation and as a result dropped down by 2739.8 cm À1 and affected terms belonging to limit III.
The PECs obtained for limit V are shifted down approximately by 253 cm À1 in comparison with NIST energies [18]. Nevertheless, the total SOC splitting of this limit equals to 2 422.5 cm À1 , which almost coincides with experimental one (2 421.9 cm À1 ). Fig. 2 presents the calculated TDMs as the functions of internuclear distances for the transitions from the ground state to the all spin-mixed excited states. We concerned only the transitions allowed by DX = 0, ±1 selection rules. In the area of the dissociation limits only TDMs related to the limit II are nonzero due to fulfilling the selection rules for the atomic transitions.
Since in photoassociation processes diatomic molecules (atomic pairs) are formed in weakly-bounded vibrational states with large internuclear distances near the dissociation limit, the value of the TDM can be significant for the probabilities of the optical exciting at the first step of the two-photon optical scheme for the producing ultracold CsYb molecules in the ground rovibronic state. At the large internuclear distances for the transitions from the ground electronic state to the all excited electronic states under consideration (with the exception of 2 2 R + 1/2 , 1 2 G 1/2 , and 1 2 G 3/2 states, i.e. states of the limit II) TDMs are zero (see Fig. 2). Therefore, it is assumed that namely these exited states (2 2 R + 1/2 , 1 2 G 1/2 , and 1 2 G 3/2 ) are of more direct interest for the production of ultracold CsYb molecules by optical methods.

Vibrational states
For calculations of vibrational energies and frequencies reduced mass of the molecule is important. The Yb atom has seven stable isotopes: 168 Yb (0.13%), 170 Yb (3.04%), 171 Yb (14.28%), 172 Yb (21.83%), 173 Yb (16.13%), 174 Yb (31.83%) and 176 Yb (12.76%). For Cs atom there is only one ( 133 Cs) stable isotope. We calculate the vibrational energies for 133 Cs 174 Yb molecule, as the most abundant isotopomer, and some other isotopomers for all spin-mixed PECs up to the dissociation limit of the corresponding state, and then we obtained the harmonic vibrational frequencies x e for these electronic states (see Table 1).

Franck-Condon factors
On the basis of calculated PECs we obtain the Franck-Condon factors (FCFs) between the pointful states of the 133 Cs 174 Yb molecule. Using the DX = 0, ±1 selection rules we have calculated FCFs for vibronic transitions between the spin-mixed electronic states. We concern the transitions between the fixed vibrational level v 0 of the upper electronic state and a sequence of vibrational levels v 00 of the lower electronic state applying the selection rules DJ = 0 or DJ = ±1.
The FCFs obtained for transitions between the vibrational levels of the first three exited electronic states and the vibrational levels of the ground electronic 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) (3).
The FCFs distributions for the X 2 R + 1/2 1 2 G 1/2 and  Fig. 2), the 2 2 R + 1/2 (or (3)1/2) state is concerned to be a suitable candidate for two-photon optical scheme for the producing CsYb molecules in the ground rovibronic state. Fig. 3. The vibrational energies E v (a, c) and vibrational intervals DG(v 0 + 1/2) (b, d) for the ground X 2 R + 1/2 (a, b) and excited 3 2 R + 1/2 (c, d) states of the 133 Cs 176 Yb molecule. The insert in a shows sequences of vibrational energies near dissociation limit calculated without (1) and with asymptotic function (2).

Conclusions
The system of the low-lying electronic states of the CsYb molecule taking into account the spin-orbit coupling at the high level of theory was calculated for the first time. PECs of the ground and 16 excited states were constructed, for each of them the system of vibrational levels and vibrational frequencies were computed. The value of 158.7 cm À1 (or 160.8 cm À1 with approximation function) obtained for dissociation energy of the ground state indicates van der Waals' character of the molecule. The transition dipole moments were calculated, Franck-Condon factors for vibronic transitions were determined. Since at the Cs(6p) + Yb(6s 2 ) dissociation limit our obtained PECs are in a good agreement with the experimental [18] energies and since the calculations for the KRb [17] and RbYb [12] molecules in the same approximation show accurate values, we expect the results of our current calculations for the ground and low-lying excited states of the CsYb molecule to be close to the correct values. The effective schemes (optical cycles, for example) for transferring CsYb molecules to the ground rovibronic state by initial excitation into overlying states can be developed on the basis of the calculated spectral and dynamic characteristics. The results obtained can be used for spectroscopic studies of CsYb molecule, as well as for the synthesis of such molecules. The ytterbium-containing diatomics (such as RbYb and SrYb) now are considered as perspective molecular systems for the experiments under cold and ultracold conditions [13,29]. We assume that the molecule under consideration also can be used for the purposes mentioned.