Ab initio calculations of the electronic structure of the low-lying states for the ultracold LiYb molecule Samir N. Tohme, Mahmoud Korek, and Ramadan Awad Citation: The Journal of Chemical Physics 142, 114312 (2015); doi: 10.1063/1.4914472 View online: http://dx.doi.org/10.1063/1.4914472 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in An ab initio study of the ground and low-lying excited states of KBe with the effect of inner-shell electrons J. Chem. Phys. 139, 074305 (2013); 10.1063/1.4818452 Ab initio many-electron study for the low-lying states of the alkali hydride cations in the adiabatic representation J. Chem. Phys. 136, 124304 (2012); 10.1063/1.3695997 Relativistic calculations of ground and excited states of LiYb molecule for ultracold photoassociation spectroscopy studies J. Chem. Phys. 133, 124317 (2010); 10.1063/1.3475568 Ab initio potential energy surfaces, total absorption cross sections, and product quantum state distributions for the low-lying electronic states of N 2 O J. Chem. Phys. 122, 054305 (2005); 10.1063/1.1830436 Ab initio spin-orbit CI calculations of the potential curves and radiative lifetimes of low-lying states of lead monofluoride J. Chem. Phys. 116, 608 (2002); 10.1063/1.1423944

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THE JOURNAL OF CHEMICAL PHYSICS 142, 114312 (2015)

Ab initio calculations of the electronic structure of the low-lying states for the ultracold LiYb molecule Samir N. Tohme, Mahmoud Korek,a) and Ramadan Awad Faculty of Science, Beirut Arab University, P.O. Box 11-5020 Riad El Solh, Beirut 1107 2809, Lebanon

(Received 20 October 2014; accepted 27 February 2015; published online 19 March 2015) Ab initio techniques have been applied to investigate the electronic structure of the LiYb molecule. The potential energy curves have been computed in the Born–Oppenheimer approximation for the ground and 29 low-lying doublet and quartet excited electronic states. Complete active space self-consistent field, multi-reference configuration interaction, and Rayleigh Schrödinger perturbation theory to second order calculations have been utilized to investigate these states. The spectroscopic constants, ωe, Re, Be, . . ., and the static dipole moment, µ, have been investigated by using the two different techniques of calculation with five different types of basis. The eigenvalues, Ev, the rotational constant, Bv, the centrifugal distortion constant, Dv, and the abscissas of the turning points, Rmin and Rmax, have been calculated by using the canonical functions approach. The comparison between the values of the present work, calculated by different techniques, and those available in the literature for several electronic states shows a very good agreement. Twenty-one new electronic states have been studied here for the first time. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914472]

I. INTRODUCTION

Cold molecules are promising candidates in many applications, ranging from ultracold chemistry and precision measurements to quantum computing. Particularly, polar molecules are interesting objects to study because of their permanent electric dipole moment responsible for dipolar interaction among the molecules. This interaction is long-range and anisotropic, and many applications using polar molecules have been experimentally and theoretically proposed.1,2 In addition, polar molecules can interact with each other via both electric dipoledipole and magnetic spin-spin forces, when placed in an optical lattice. So they are promising candidates for quantum information,3 controlled preparation of many-body entangled states,4 high precision measurements of fundamental constants,5 and lead to fundamentally new condensed-matter phases and complex quantum dynamics.6 A large fraction of investigated systems in the ultracold molecular sciences is composed of alkali metal diatomic systems.7–9 However, closed shell systems having singlet ground states such as alkali-metal dimers are more challenging to cool and trap than open-shell systems, since they do not have a magnetic moment in their ground state that enables trapping. Therefore, there is a great interest in producing ultracold molecules with doublet or triplet ground states. Alkali-metals with 2 S atom are among the most promising candidates for these molecules. Rb + Yb10–12 and Li + Yb 13–17 are among quantum mixtures that are being studied by a number of experimental groups, in which such molecules can be formed in the ultracold regime.

a)Author to whom correspondence should be addressed. Electronic addresses:

[email protected] and [email protected]. Fax: +961 1 818 402.

0021-9606/2015/142(11)/114312/10/$30.00

The LiYb molecule is one of the 2Σ molecules, since the single unpaired electron is predominantly comprised of an selectron from the Li-atom. Because of the difference in the electronegativity, the Yb atom has the tendency to attract more electrons than the Li atom. Therefore, the spatial distribution of the electron cloud becomes asymmetrical, giving rise to a permanent dipole moment (PDM) in the LiYb molecule. The theoretical calculations obtained by Zhang et al.,18 Gopakumar et al.,19 and Kotochigova et al.20 show that the permanent dipole moment at the internuclear distance of the ground state is very small. Our calculated values of this permanent dipole moment, using five different basis sets (Table II), are greater than those calculated in the literature18–20 (Sec. III C). This significant deviation would provide a new handle for the control of the reaction dynamics, studying few- and manybody systems, and may be advantageous for the investigation of long-range dipole-dipole forces. This molecule has seven stable isotopes: five bosons and two fermions, and an open-shell doublet-spin. Both bosonic21–24 and fermionic25,26 isotopes have been cooled to quantum degeneracy. Moreover, 6Li174Yb was chosen due to the largest mass ratio MYb/MLi ∼ 29 between its constituents species Yb and alkali metal elements. The large mass ratio minimizes the impact of the micromotion which facilitates cooling. The candidate bond is expected to be weak and stable upon collisions and promising for the study of massimbalanced pairs such as Efimov trimer states.27 An accurate theoretical calculation of the molecule LiYb is very important to understand the mechanisms of production, collisional cooling, and relaxation of molecules, which may help and support experimental investigations which are already underway.28,29 In particular, providing an accurate calculation of the properties for the X2Σ+ ground state (such as the bond length, Re, dissociation energy, De,

142, 114312-1

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J. Chem. Phys. 142, 114312 (2015)

TABLE I. Basis sets used in the present ab initio calculations of the potential energy curves of the LiYb molecule. Basis Basis 1 Basis 2 Basis 3 Basis 4 Basis 5

Li basis set

Yb basis set

CRENBL ECP (4s, 4p) CRENBL ECP (4s, 4p) CRENBL ECP (4s, 4p) CRENBL ECP (4s, 4p) CRENBL ECP (4s, 4p)

Stuttgart RSC ANO/ECP (14s, 13p, 10d, 8 f , 6g)/[6s, 6p, 5d, 4 f , 3g] CRENBL ECP (6s, 6p, 6d, 6 f ) ECP60MHF (7s6p5d)/[5s4p3d] ECP60MWB (7s6p5d)/[5s4p3d] ECP60MDF (7s6p5d)/[5s4p3d]

and the permanent dipole moment, µ) reveals the strength of the dipolar interactions for the LiYb system. Further investigations of the potential energy curves with vibrationrotation study for the low-lying excited states, allow us to predict strong transitions from highly electronic states to the absolute vibronic ground state. These transitions allow the production of stable ultracold polar molecules. In a process known as photoassociation (PA), two colliding atoms in their ground state absorb a photon, and the molecule is promoted to a weakly bound ro-vibrational excited state. Another photon transfers this ro-vibrational level of the excited state to the electronic ground state with v = 0, through stimulated emission. This transition in molecules typically occurs at large interatomic distances. Providing an accurate description of the excited electronic states and the shape of the potential curves would be helpful in identifying the possible routes in which vibrational levels most highly populated by photoassociation. Furthermore, for laser cooling to be feasible, finding optimal pathways for transition via intermediate high-lying electronic states through two-photon absorption results in the formation of a given vibrational Rydberg molecular state. In addition, studying long-range vibrational states at the dissociation limit forms the basis of the Feshbach resonances required to access the continuum scattering states corresponding to the free atoms. Finally, investigation of adiabatic passages for such experiments requires presenting high-lying adiabatic energy curves, which is also important for the enhancement of nonlinear processes and for many laser cooling applications. The focus of this study is devoted to an accurate description of the ground and lowest electronically excited states. In order to emphasize the accuracy of our work, two methods have been chosen, the multi-reference configuration interaction (MRCI) model expansion and the second-order perturbation theory (RSPT2-RS2), to determine the ground and the lowest excited electronic states of the LiYb molecule. Then, the spectroscopic constants Re, Te, Be, ωe, . . . have been extracted for each of the corresponding electronic states. By taking advantage of the electronic structure of the investigated electronic states and by using the canonical functions approach,30–32 the eigenvalue, Ev, the rotational constant, Bv, the centrifugal distortion constant, Dv, and the abscissas of the turning points, Rmin and Rmax, have been calculated for several vibrational levels of the considered electronic states. The permanent dipole moment µ has been also calculated by using the MRCI and RSPT2-RS2 methods for the ground and excited electronic states of the LiYb molecule.

II. AB INITIO CALCULATION

In the present work, we study the low-lying doublet and quartet electronic states of the molecule LiYb using the two theoretical techniques: (a) the state averaged complete active space self-consistent field (CASSCF) procedure, followed by a multireference configuration interaction (MRDSCI with Davidson correction) and (b) RSPT2-RS2 treatment for the electron correlation. This calculation has been done via the computational chemistry program MOLPRO33 taking advantage of the graphical user interface GABEDIT.34 Table I shows the different basis sets used in our ab initio calculation. The basis of the lithium atom has been fixed for the uncontracted small effective core potential (ECP) with large orbital basis due to Christiansen et al.35–37 This basis has been tested systematically with each of the consecutive basis sets tabulated in Table I for the ytterbium atom. We will be pointing throughout this paper to the basis sets in numbers in ascending order (basis 1 → 5). For Li atom, 2 electrons are considered as inner electrons in the ECP, and 1 electron is treated as free. The Yb species with 70 electrons is treated as a system with ECP by using five different basis sets: (i) the QZ, Quadruple Zeta Stuttgart relativistic small core atomic natural orbital (RSC ANO)/ECP36–38 basis (basis 1), is the largest basis set employed in the present work. In this basis, 28 electrons are considered in the ECP, and the remaining 42 electrons are considered as valence electrons. The total free electrons in the LiYb molecule left are 43, where 40 electrons were frozen in the subsequent CI calculations, so that 3 electrons were explicitly treated. (ii) Basis 2 for the Yb atom is the uncontracted CRENBL ECP36,37 basis set, where 54 electrons are considered in the ECP, and the remaining 16 electrons are considered as valence electrons. The total free electrons in the LiYb molecule left are 17. Among the 17 electrons, 14 electrons were frozen in the subsequent CI calculations so that 3 electrons were explicitly treated. (iii) Bases 3, 4, and 5 consist of ECP60MHF,39 ECP60MWB,40 and ECP60MDF,41 respectively. In these basis sets, 60 electrons are considered in the core. Hartree-Fock (HF), Wood-Boring (WB), and DiracFock (DF) are for non-, quasi-, and fully relativistic basis, respectively. Among the 11 electrons, 8 electrons were frozen in the subsequent CI calculations, so that 3 electrons were explicitly treated for LiYb molecule. The active space for the LiYb molecule for the above basis sets used contains 7σ (Li : 2s, 2p0, 3s; Yb : 6s, 5d 0, 6p0, 7s), 3π (Li : 2p±1; Yb : 5d ±1, 6p±1), and 1δ (Yb : 5d ±2) orbitals distributed into irreducible representation a1, b1, b2, and a2 of C2v point group

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FIG. 1. Potential energy curves for the lowest doublet Σ(+/−) and ∆-states of the LiYb molecule.

FIG. 3. Potential energy curves for the lowest quartet Σ(+/−) and ∆-states of the LiYb molecule.

symmetry. This distribution has been done in the following way 8a1, 3b1, 3b2, and 1a2 noted by [8,3,3,1].

The comparison of our calculated values of atomic energy using the five basis sets with those given in the literature (NIST atomic spectra) is not enough informative to choose the compatible basis for the study of the LiYb molecule. For this reason and in order to test the accuracy of these basis sets, we calculated the relative difference between the available data in the literature and our calculated values of the spectroscopic constants ωe, Be, and Re, and the energy with respect to the ground states Te for the ground and 8 excited electronic states. These are tabulated in Tables S1 and S2 in the supplementary material.53 The calculations have been done by using 3 valence electrons, and 2 different techniques MRCI and RSPT2-RS2, with different types of basis sets. The average values of these relative errors are given in Table S3 in the supplementary material.53 The study of this table shows that for the values of Te, the average relative error is nearly the same by using either MRCI or RSPT2-RS2 techniques. Its value decreases from one basis to another where the least average relative error is for basis 5. Neither the change of basis nor the techniques of calculations have influence on the average relative errors for

the values of ωe and Re. From these results, we will adopt basis 5 for further theoretical calculation by increasing the number of valence electrons to 5 and then to 7, in order to calculate the potential energy curves and the spectroscopic constants. The best results by comparing with the data in the literature are with 7 valence electrons (column 14 in Table S3 in the supplementary material53). In this case, 4 electrons were frozen in the subsequent CI calculation, so that 7 electrons were explicitly treated. The active space contains 6σ (Li: 2s, 2p0; Yb: 5p0, 5d 0, 6p0, 7s), 4π (Li: 2p±1; Yb: 5p±1, 5d ±1, 6p±1), and 1δ (Yb: 5d ±2) orbitals distributed as 7a1, 4b1, 4b2, and 1a2 noted by [7,4,4,1]. The potential energy curves for the 30 low-lying doublet and quartet electronic states of the molecule LiYb were generated, where we assumed that the LiYb molecule is mainly ionic around the equilibrium position. These curves have been calculated using the multi-reference singly and doubly excited configuration interaction (MRSDCI) method with basis 5 (ECP60MDF41) and 7 valence electrons in the representation 2s+1Λ(±). These curves are drawn versus the internuclear distance 2.0 Å ≤ R ≤ 8.0 Å in Figures 1–4. The equilibrium bond distances, Re, the harmonic vibrational frequencies, ωe, and the relative energy separations, Te, for these electronic states have been calculated and given in Tables II and III. The comparison of our calculated internuclear distance Re of the ground state X2Σ+ with those obtained by Zhang et al.,18 Gopakumar et al.,19 and Kotochigova et al.,20 shows a very good agreement with relative difference of 0.2% (Ref. 19) ≤ ∆Re/Re ≤ 7.5% (Ref. 19). Exception

FIG. 2. Potential energy curves for the lowest doublet Π-states of the LiYb molecule.

FIG. 4. Potential energy curves for the lowest quartet Π and Φ-states of the LiYb molecule.

III. RESULTS AND DISCUSSIONS A. Potential energy curves of the ground and excited electronic states of the LiYb molecule

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TABLE II. Spectroscopic constants of the X2Σ+ ground state of the LiYb molecule using CASSCF/MRCI method with 7 valence electrons of basis 5 and calculated permanent dipole moment µ using the five basis sets. State

Te (cm−1)

ωe (cm−1)

X2 Σ +

0a(1)

127.85a(1)

0b 0c 0d(1) 0d(2) 0d(3) 0d(4) 0d(5) 0d(6) 0d(7) 0d(8) 0d(9) 0d(10) 0d(11) 0d(12) 0d(13) 0d(14) 0d(15) 0d(16) 0d(17) 0d(18)

181.5b 134.81c

δωe/ωe%

Re (Å)

δRe/Re%

Te (cm−1)

Re (Å)

δRe/Re%

3.724a(1) 29.5 5.1

3.550b 3.535c 3.480d(1) 3.636d(2) 3.714d(3) 3.701d(4) 3.755d(5) 3.846d(6) 3.812d(7) 3.809d(8) 3.785d(9) 3.780d(10) 3.361d(11) 3.472d(12) 3.517d(13) 3.518d(14) 3.541d(15) 3.564d(16) 3.564d(17) 3.563d(18)

|µ e | (a.u.) 0.2264a(1)

4.9 5.3 7.0 2.4 0.2 0.6 0.8 3.1 2.3 2.2 1.6 1.4 10.8 7.2 5.8 5.8 5.1 4.4 4.4 4.5

0d(19) 0d(20) 0d(21) 0d(22) 0d(23) 0d(24) 0d(25) 0d(26) 0d(27) 0d(28) 0d(29) 0d(30) 0e(1) 0e(2) 0e(3) 0e(4) 0e(5) 0e(6) 0e(7) 0e(8)

3.553d(19) 3.553d(20) 3.352d(21) 3.461d(22) 3.508d(23) 3.511d(24) 3.535d(25) 3.557d(26) 3.558d(27) 3.558d(28) 3.511d(29) 3.547d(30) 3.519e(1) 3.519e(2) 3.550e(3) 3.492e(4) 3.545e(5) 3.497e(6) 3.598e(7) 3.545e(8)

4.8 4.8 11.0 7.5 6.1 6.0 5.3 4.6 4.6 4.6 6.0 4.9 5.8 5.8 4.9 6.6 5.0 6.4 3.5 5.0

0.5302a(2) 0.3680a(3) 0.0316a(4) 0.1575a(5) 0.2006a(6) 0.2312a(7) 0.011b 0.0586f(1) 0.1758f(2) 0.0432f(3) 0.0228g(1) 0.0110g(2) 0.0531g(3) 0.0373g(4) 0.0448g(5)

a Present theoretical CASSCF/MRCI calculations using: (1) basis 5 with 7 valence electrons, (2) basis 1 with 3 valence electrons, (3) basis 2 with 3 valence electrons, (4) basis 3 with 3 valence electrons, (5) basis 4 with 3 valence electrons, (6) basis 5 with 3 valence electrons, (7) basis 5 with 5 valence electrons. b Values calculated by S. Kotochigova et al.20 in Table VI using UCCSD(T) method with basis set for Li: (15s6p2d1f)/[6s4p2d1f] and for Yb: (14s13p10d8f6g)/[10s8p5d4f3g]. c Values calculated by G. Gopakumar et al.19 in Table III using CASPT2 method neglecting basis set superposition error (BSSE) effect with RCC-ANO basis (8s7p4d2f) for Li atom and RCC-ANO basis (12s11p8d7f4g2h) for Yb atom. d Values calculated by G. Gopakumar et al.19 in Table II: (1–5) theoretical values calculated using the open-shell MP2 model without CP correction with five incremental RCC-ANO basis set for both Li and Yb atoms named as bases 1, 2, 3, 4, and 5 relates to Li:5s4p2d1f–Yb:8s7p4d3f2g1h, Li:5s4p2d1f-Yb:9s8p5d4f3g2h, Li:5s4p2d1f-Yb:10s9p6d5f4g2h, Li:8s7p4d2f-Yb:11s10p7d6f4g2h, and Li:8s7p4d2f-Yb:12s11p8d7f4g2h, respectively, (6–10) theoretical values calculated using the open-shell MP2 model with CP correction with the 5 incremental RCC-ANO basis set for Li and Yb atoms, (11–15) theoretical values calculated using the CCSD(T) method without CP correction with the five incremental RCC-ANO basis set for Li and Yb atoms, (16–20) theoretical values calculated using the CCSD(T) method with CP correction with the five incremental RCC-ANO basis set for Li and Yb atoms, (21–25) theoretical values calculated using the CASPT2 method without CP correction with the five incremental RCC-ANO basis set for Li and Yb atoms, (26–30) theoretical values calculated using the CASPT2 method with CP correction with the five incremental RCC-ANO basis set for Li and Yb atoms. e Values calculated by Zhang et al.18 in Table III: (1 and 2) ab initio values calculated using CCSD(T)/ECP method with (1)/without (2) CP correction with aug-cc-pCV5Z basis set for Li atom and ECP28MDF basis set for Yb atom, (3 and 4) ab initio values calculated using CCSD(T)/ANO method with (3)/without (4) CP correction with aug-cc-pVQZ-DK basis set for Li atom and RCC-ANO basis set for Yb atom, (5 and 6) ab initio values calculated using SS-MRCI/ANO method with (5)/without (6) CP correction with aug-cc-pVQZ-DK basis set for Li atom and RCC-ANO basis set for Yb atom, (7 and 8) ab initio values calculated using SA-MRCI/ANO method with (7)/without (8) CP correction with aug-cc-pVQZ-DK basis set for Li atom and RCC-ANO basis set for Yb atom. f Values calculated by Gopakumar et al.:19 (1) computed PDM at the CASSCF level of correlation at the internuclear distance, R , = 3.535 (Å) using basis 5 without CP correction, (2) e computed PDM at the CASPT2 level of correlation invoking finite field perturbation theory (FFPT) with 0.0001 a.u. external field strength at the internuclear distance Re = 3.535 (Å) using basis 5 without CP correction, (3) Computed PDM at the CCSD(T) level of correlation invoking FFPT with 0.0001 a.u. external field strength at the internuclear distance Re = 3.541 (Å) using basis 5 without CP correction. g Values calculated by P. Zhang et al.18 in Table VI: (1) computed PDM at the RECP-UCCSD(T) level of theory at the internuclear distance, R , using the finite field (FF) calculation, e (2) computed PDM at the DKH3-UCCSD(T) level of theory at the internuclear distance Re using the FF calculation, (3) computed PDM at the SS-MRCISD(Q) level of theory at the internuclear distance Re using the FF calculation, (4) computed PDM at the SS-MRCISD level of theory at the internuclear distance Re using the FF calculation, (5) computed PDM at the SS-MRCISD level of theory at the internuclear distance Re using the expectation formalism (EV) calculation.

goes for the two values which are in acceptable agreement with our values with a relative difference of ∆Re/Re = 10.8% (Ref. 19) by using the CCSD(T) method without counterpoise (CP) correction,19 and ∆Re/Re = 11.0% (Ref. 19) using the CASPT2 method without CP correction.19 We notice that the values with CP correction in comparison with our values, show a better agreement (1.4% (Ref. 19) ≤ ∆Re/Re ≤ 6.0% (Ref. 19)) than the values without CP correction (0.2% (Ref. 19) ≤ ∆Re/Re ≤ 11.0% (Ref. 19)). The value of the harmonic frequency, ωe, of the ground state is in very good agreement with Gopakumar et al.19 with a relative error of 5.1% (Ref. 19). However, a large deviation in the value of ωe (∆ωe/ωe = 29.5% (Ref. 20)) calculated by Kotochigova et al.20 has been observed.

In Table III, the comparison of our calculated values of Te by using basis 5 with 7 valence electrons with those obtained theoretically shows a very good agreement with relative difference 0.3% (Ref. 19) ≤ ∆Te/Te ≤ 8.3% (Ref. 19). But the value of Te for the state (1)2Π calculated by Gopakumar et al.19 is in disagreement with our values, where the relative difference is large compared to our value with ∆Te/Te = 20.3% (Ref. 19). The calculated values of Te using bases 2 and 3 in the present work for the first excited state (1)2Π are closer to the values calculated by Gopakumar et al.19 However, our values calculated using bases 1, 4, and 5 show a deviation toward the value calculated by Zhang et al.18 Further experiments on the LiYb molecule based on several techniques could be useful to determine

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TABLE III. Calculated spectroscopic constants and permanent dipole moments for the doublet and quartet excited electronic states for the LiYb molecule using CASSCF/MRCI method with 7 valence electrons of basis 5. State (1)2Π

Te (cm−1) 7 114.58a 6 687b(1)

(2)2Σ+

(3)2Σ+

(2)2Π

(1)4Π

6 955b(2) 5 911.46c 10 739.97a

6.3 2.2 20.3

6.9

10 798b(2) 10 782.63c 14 153.39a 15 070b(1)

0.5 0.3

14 775.69c 15 122.01a 15 047b(1) 14 388.11c 15 815.18a

4.2

16 304.75c 18 223.73a 19 321b(3)

ωe (cm−1)

δωe/ωe%

268.01a

11 541b(1)

16 271b(3)

(1)4Σ+

δTe/Te%

6.0

0.4 5.1

2.8 3.0

5.6

7.2

... ... 176.77c 142.33a ...

97.67c 191.41a ... 241.50c 171.82a

2.8

45.7

20.7

... 193.83c 30.49a

11.3

...

0.0513a

2.963b(1) (3.016)b(1)

4.5 2.7

2.963b(2) 2.919c 3.589a 3.439b(1) (3.439)b(1)

4.5 6.1

3.439b(2) 3.473c 4.032a 3.862b(1) (3.915)b(1)

1.0755a 4.3 4.3 4.3 3.3 0.3485a 4.4 2.9

3.641c 3.380a 3.175b(1) 3.127c 3.550a 3.280b(3) (3.333)b(3)

10.7

3.318c 5.311a 4.561b(3) (4.550)b(3)

6.9

1.8813a 6.4 8.0 0.0050a 8.2 6.5

1.4135a 16.4 16.7

(4)2Π (2)4Σ+ (5)2Σ+ (5)2Π (1)4∆ (6)2Σ+ (1st min) (6)2Σ+ (2nd min) (6)2Π (3)4Π (2)2∆ (7)2Σ+ (1st min) (7)2Σ+ (2nd min) (7)2Σ+ (3rd min)

30 686.45a

69.58a

7.394a

5.4012a

(7)2Π (1st min) (7)2Π (2nd min) (1)2Σ− (3)2∆ (4)4Π (1st min) (4)4Π (2nd min) (2)4∆ (3)4Σ+ (2)4Σ−

32 291.78a 31 805.45a 31 097.88a 31 777.04a 32 295.22a 32 495.76a 32 408.06a 32 575.20a 32 707.26a

406.01a 92.37a

3.788a 4.891a 3.160a 3.371a 3.125a 4.947a 3.398a 3.323a 3.053a

0.5272a 1.2097a 1.0207a 0.0812a 0.7180a 1.1365a 0.6853a 0.7149a 0.3233a

(1)2∆ (1)4Σ− (2)4Π (3)2Π

8.3 0.7

3.6

146.55a 121.91a 233.28c 94.84a 115.61a 109.03a 145.27a 119.56a 236.34a 21.97a 135.50a 103.14a 100.22a 373.91a 234.53a

217.21a 267.17a 51.06a 225.40a

37.9 31.8

47.7

4.494c 3.354a 3.239c 3.177a 2.964a 4.014a 3.834a 3.418c 4.183a 3.927a 5.186a 3.874a 4.477a 3.923a 4.540a 4.217a 4.106a 4.135a 3.970a 6.018a

|µ e | (a.u.)

19 890.69c 18 850.08a 18 716.52c 20 709.87a 23 450.95a 23 762.32a 24 246.38a 25 173.37c 25 233.34a 25 452.50a 26 071.38a 27 769.10a 28 026.04a 28 191.47a 28 517.51a 28 769.78a 29 250.08a 29 548.33a 29 825.22a 30 285.12a

(4)2Σ+

49.10c 265.56a 389.81c 252.24a

δRe/Re%

3.099a

... ... 288.93c 181.85a

Re (Å)

18.1 3.5

0.3490a 0.9878a 1.1328a 0.9433a 1.3195a

12.7 0.5107a 1.5853a 2.0789a 0.7508a 0.7186a 0.3231a 0.3980a 0.1131a 1.6480a 1.3365a 0.6292a 2.8621a

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J. Chem. Phys. 142, 114312 (2015)

TABLE III. (Continued.)

(1)4Φ a Present

δTe/Te%

Te (cm−1)

State

34 941.49a

δωe/ωe%

ωe (cm−1)

Re (Å)

231.77a

δRe/Re%

3.145a

|µ e | (a.u.) 0.1533a

work using CASSCF/MRCI method with 7 valence electrons.

b Values calculated by Zhang et al.18 in Table IV: (1) theoretical values calculated using the SA-MRCISD(Q) method with and without CP corrections with the RCC-ANO basis with the

DKH3 Hamiltonian, (2) theoretical values calculated using the SS-MRCISD(Q) method without CP corrections with the RCC-ANO basis with the DKH3 Hamiltonian, (3) theoretical values calculated using the ROHF-UCCSD(T) method with and without CP corrections with the RCC-ANO basis with the DKH3 Hamiltonian. The tabulated values are without (with) CP corrections for (1) and (3). c Values calculated by G. Gopakumar et al.19 in Table III: theoretical values calculated using the CASPT2 method with the RCC-ANO basis set with Li:8s7p4d2f and Yb:12s11p8d7f4g2h, neglecting the BSSE effect.

the correct deviation of the (1)2Π state. The comparison of our calculated values of the harmonic frequency ωe with those given in the literature for the (2)2Σ+ and (3)2Σ+ excited electronic states shows a very good agreement with relative differences ∆ωe/ωe = 7.2% (Ref. 19) and ∆ωe/ωe = 2.8% (Ref. 19), respectively. However, the relative differences ∆ωe/ωe become larger for the higher excited states where 11.3% (Ref. 19) ≤ ∆ωe/ωe ≤ 47.7% (Ref. 19). This is due to the selection of a different basis (relativistic correlationconsistent atomic natural orbital) and to the choice in the number of valence electrons (three valence electrons are considered19 and seven valence electrons in the present work). This means, different description in the Complete Active Space (CAS) which allows in developing a different description of the wavefunction, which alters or affects the

TABLE IV. Positions of the crossings between the different electronic states of the molecule LiYb. State 1

State 2

2∆

2Σ + 2Σ + 2Σ +

2Σ −

2Σ + 2Σ + 2∆ 2Σ + 2Σ + 2∆

2∆

2Σ + 2Σ + 2Σ + 2Σ +

2∆

2Σ + 2Σ + 2Σ +

4Σ +

4Σ − 4∆

4Σ +

4Σ − 4∆ 4Σ −

4∆

4Σ −

4Σ −

4∆ 4Σ +

4Σ +

4∆ 4∆ 4∆





Crossing between (n1) state 1/(n2) state 2

Rc (Å)

1/3 1/4 1/5 1/4 1/5 1/2 1/6 1/7 1/3 2/5 2/6 2/7 2/7 3/5 3/6 3/7 1/1 1/1 2/1 2/1 2/2 1/1 2/2 2/3 3/2 3/2 3/2 1/4

2.140 2.886 4.862 2.149 2.813 3.063 3.109 3.316 4.051 2.657 3.127 5.830 6.650 2.172 2.932 3.110 3.050 2.316 3.686 2.546 2.433 4.233 3.161 3.128 2.106 3.214 4.827 2.367

shape of the potential energy curve, thus resulting a large relative difference in the ωe value. The internuclear distance Re shows also a very good agreement by comparing with the data in the literature for the considered excited electronic states with relative difference 2.7% (Ref. 18) ≤ ∆Re/Re ≤ 12.7% (Ref. 19) except for (1)4Σ+ state where the relative differences are ∆Re/Re = 16.4% (Ref. 18), 16.7% (Ref. 18), and 18.1% (Ref. 19). Since there is an overall good agreement between our calculated values for the lower electronic states with those given in the literature, and since we are using the same way of calculation for all these states, we can predict the accuracy of our calculated values of the higher excited electronic states and this prediction can be confirmed in future experiments. It is quite common for the molecular electronic states of the potential energy curves to make crossings or avoided crossings known as conical intersections. These points of the potential energy curves of a diatomic molecule are important in photochemistry. In fact, the avoided crossing regions are likely to be leakage channels along which the molecules flow from the higher down to the lower potential energy curves. Such crossings or avoided crossings can dramatically alter the stability of the molecules. If these crossings are overlooked, then low barrier transitions can be missed and an incorrect chemical picture will arise.42 In the range of R considered, several avoided crossings have been detected in the potential energy curves of the excited electronic states of the molecule LiYb. The positions of these crossings, Rc, and avoided crossings, Rav, together with their energy gap separation, ∆Eav, are reported in Tables IV and V. B. The vibration-rotation calculation

By using the canonical functions approach30–32 and the cubic spline interpolation between each two consecutive TABLE V. Avoided crossings between different electronic states of the molecule LiYb. State 2Σ + 2Σ + 2Π 2Π 4Σ + 4Π 4Π

(n1) state 1/(n2) state 2

RAC (Å)

∆EAC (cm−1)

5/6 6/7 3/4 5/7 2/3 2/3 4/5

3.49 2.87 4.91 6.35 2.23 2.41 3.79

492.42 208.61 46.27 267.58 1011.52 281.54 0.498

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J. Chem. Phys. 142, 114312 (2015)

TABLE VI. Values of the eigenvalues, Ev, the rotational constants, Bv, the centrifugal distortion constant, Dv, and the abscissas of the turning points for the different vibrational levels of X Σ+ and (2)2Σ+ states of the LiYb molecule using CASSCF/MRCI method with 7 valence electrons of basis 5. 2

X2 v

Ev (cm−1) 59.58a

0

Bv × 10 (cm−1)

+

(2)2

Dv × 106 (cm−1)

Rmin (Å)

Rmax (Å)

1.7941a

1.6874a

3.542a

3.956a

1.7422a

1.7711a

3.431a

1.6866a

1.9264a

1.6299a

Ev (cm−1)

+

Bv × 10 (cm−1)

Dv × 106 (cm−1)

Rmin (Å)

Rmax (Å)

91.96a

1.9524a

0.8833a

3.432a

3.764a

4.162a

274.69a

1.9368a

0.8760a

3.327a

3.904a

3.365a

4.331a

455.85a

1.9212a

0.8673a

3.259a

4.006a

2.0636a

3.316a

4.496a

635.51a

1.9058a

0.8627a

3.207a

4.093a

1.5635a

2.4003a

3.279a

4.661a

813.55a

1.8900a

0.8624a

3.163a

4.173a

1.5021a

2.3504a

3.248a

4.828a

989.83a

1.8742a

0.8480a

3.125a

4.246a

1.4335a

2.6986a

3.223a

5.003a

1164.59a

1.8596a

0.8320a

3.092a

4.315a

1.3691a

2.9096a

3.202a

5.185a

1338.01a

1.8446a

0.8458a

3.062a

4.380a

1.2992a

2.9749a

3.184a

5.379a

1509.70a

1.8285a

0.8539a

3.035a

4.444a

1.2288a

3.3618a

3.168a

5.586a

1679.41a

1.8117a

0.8666a

3.010a

4.507a

1.0870a

3.9333a

3.144a

6.052a

1846.93a

1.7943a

0.8558a

2.987a

4.568a

1.0116a

4.3010a

3.134a

6.323a

2012.41a

1.7780a

0.8533a

2.966a

4.629a

0.9363a

4.6291a

3.126a

6.620a

2175.83a

1.7598a

0.8926a

2.946a

4.689a

0.8599a

5.4316a

3.119a

6.957a

2336.75a

1.7417a

0.8674a

2.928a

4.749a

0.7808a

5.8458a

3.114a

7.345a

2495.40a

1.7229a

0.9140a

2.910a

4.809a

0.7032a

6.2306a

3.109a

7.792a

2651.35a

1.7029a

0.9100a

2.894a

4.869a

11.88b

16

174.24a 185.02b 282.54a 350.85b 383.78a 510.14b 477.77a 662.89b 563.65a 808.70b 642.96a 947.55b 715.24a 1079.37b 780.85a 1204.01b 840.27a 1321.36b 940.96a 1431.38b 982.73a 1534.10b 1019.10a 1629.51b 1050.47a 1717.64b 1076.91a 1798.56b 1098.87a 1872.43b 1939.37b

2804.61a

1.6825a

0.9464a

2.878a

4.930a

17

1999.59b

2954.89a

1.6605a

0.9717a

2.864a

4.992a

18

2053.27b

3101.99a

1.6375a

0.9994a

2.850a

5.056a

19

2100.62b

3245.69a

1.6127a

1.0539a

2.837a

5.121a

20

2141.90b

3385.68a

1.5867a

1.0862a

2.824a

5.189a

21

2177.33b

3521.72a

1.5581a

1.1502a

2.813a

5.259a

22

2207.18b

3653.47a

1.5284a

1.2169a

2.802a

5.333a

23

2231.74b

3780.62a

1.4959a

1.2531a

2.791a

5.410a

24

2251.30b

3902.99a

1.4616a

1.3595a

2.782a

5.492a

25

2266.16b

4020.23a

1.4259a

1.4062a

2.773a

5.579a

26

2276.64b

4132.20a

1.3876a

1.4529a

2.764a

5.671a

27

2283.34b

4238.82a

1.3482a

1.5550a

2.756a

5.768a

28

2287.06b

4339.95a

1.3080a

1.6063a

2.749a

5.871a

29

2288.64b

4435.61a

1.2663a

1.6372a

2.742a

5.981a

30

2288.99b

4525.84a

1.2235a

1.7172a

2.735a

6.097a

4610.65a

1.1802a

1.8112a

2.729a

4897.92a

1.0024a

2.0319a

2.710a

4957.28a 5011.77a 5061.55a 5106.66a 5147.27a

0.9567a 0.9094a 0.8617a 0.8134a 0.7641a

2.1879a 2.2343a 2.4059a 2.4731a 2.5687a

2.706a 2.702a 2.699a 2.696a 2.694a

6.222a 6.798a 6.968a 7.152a 7.352a 7.570a 7.809a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

31 32 33 34 35 36 37 a Present b Values

work using CASSCF/MRCI method with 7 valence electrons. calculated by Kotochigova et al.20 in Table V using UCCSD(T) method with basis set for Li: (15s6p2d1f)/[6s4p2d1f] and for Yb: (14s13p10d8f6g)/[10s8p5d4f3g].

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114312-8

Tohme, Korek, and Awad

points of the potential energy curves, the eigenvalue, Ev, the rotational constant, Bv, the centrifugal distortion constant, Dv, and the abscissas of the turning point, Rmin and Rmax, have been calculated up to v = 42. The values of these 2 2 constants for X Σ+ and (2) Σ+ (as illustration) are given in Table VI. The comparison of these values with those calculated by Kotochigova et al.20 shows a large deviation, except for the vibrational energy level v = 1 where the relative difference is 5.8%. We also note that the number of vibrational levels obtained by Kotochigova et al.20 for 2 X Σ+ is v = 30, whilst v = 15 obtained in the present work. This can be explained by the different values of dissociation energies obtained in the literature and in the 2 present work of the ground electronic state X Σ+. These values are De = 1577 cm−1 (Ref. 18), De = 1421.96 cm−1 (Ref. 19), De = 2289 cm−1 (Ref. 20), and De = 1151.08 cm−1 for the present work. The dissociation energy, De, obtained by Kotochigova et al.20 shows significantly deeper potential energy curve. Hence, a deeper potential leads to a higher number of vibrational levels. No comparison could be done for our ro-vibrational constants for the excited states with any results, since they are given here for the first time. 2 2 2 2 2 The electronic states, (4) Σ+, (5) Σ+, (2) ∆, (3) ∆, (1) Σ−, 4 − 4 4 − 4 + 4 2 + 2 2 (1) Σ , (2) ∆, (2) Σ , (3) Σ , (1) Φ, (6) Σ , (3) Π, (4) Π, 2 2 (5) Π, and (6) Π, of these constants are not calculated either because of the presence of crossing or avoided crossing near the minimum of the potential energy curves. The ro2 2 4 vibrational constants for the states (7) Σ+, (7) Π, and (4) Π have also not been obtained because of the existence of a double minimum or triple minimum. The reason that the ro-vibrational values of the corresponding states cannot be evaluated is the breakdown of the Born–Oppenheimer approximation in this approach. This breakdown occurs if either a crossing, or an avoided crossing, or a double minimum exist at any point around the minimum in 2 2 2 the potential energy curves. As for the (3) Σ+, (1) ∆, (1) Π, 2 4 + 4 + 4 4 4 4 (2) Π, (1) Σ , (2) Σ , (1) ∆, (1) Π, (2) Π, and (3) Π states, the values are given in the supplementary material.53

J. Chem. Phys. 142, 114312 (2015)

of charges (one end more positive than the other) between the two atoms. The electric dipole moment function Re(r) is given by

Re(r) = Ψe| µe |Ψe′ ,

The heteropolar molecule, LiYb, has a permanent dipole moment (µe ), due to the non-uniform and uneven distribution

where Ψe and Ψe′ are the electronic wavefunctions of two different states, and µe (r) is the permanent dipole moment which can be considered as the response of the wavefunction (and energy) to the external field, in the limit where the field strength is vanishingly small. The dipole moments are analyzed for the 30 lowest electronic states of the LiYb molecule. All the calculations were performed with the MOLPRO33 program. The dipole moment operator is among the most reliably predicted physical properties, because the quantum mechanical operator is a simple sum of one-electron operators. The expectation value of this operator is sensitive to the nature of the least energetic and most chemically relevant valence electrons.43 The HF dipole moment is usually large, as the HF wavefunction overestimates the ionic contribution. The permanent dipole moment has been calculated by taking the lithium (Li) atom at the origin, and the ytterbium (Yb) atom moves along the positive z-axis. The positive sign of the dipole moment corresponds to a charge transfer from the Yb atom towards the Li atom. To obtain the best accuracy, MRCI and RSPT2-RS2 wavefunctions were constructed using MCSCF active space. Tables II and III show the present calculated values of the permanent dipole moments at the internuclear distance | µe | using CASSCF/MRCI method. In order to compare these values with those given in the literature,18–20 we calculated the values of | µe | for the ground state by using five different basis sets (Table II). The results obtained for | µe | by Zhang et al.18 using 5 different techniques, those of Gopakumar et al.,19 using 3 different techniques, and those of Kotochigova et al.20 are very small, except the value obtained from the CASPT2 method19 where | µe | = 0.1758 a.u. = 0.447 D. Similarly, we obtained small values of | µe | by using basis 3 with 3 valence electrons and a noticeable value using basis 4 with 3 valence electrons | µe | = 0.1575 a.u. = 0.400 D. However, bases 1, 2, and 5 with 3 valence electrons and basis 5 with 5 and 7 valence electrons show significant values (µ ∼ 0.5 D–2.0 D) of | µe | = 0.2006 a.u.–0.5302 a.u. (| µe | = 0.509 D–1.347 D). Moreover, the calculation

FIG. 5. Permanent dipole moment curves of the lowest doublet Σ(+/−) and ∆-states of the LiYb molecule.

FIG. 6. Permanent dipole moment curves of the lowest doublet Π-states of the LiYb molecule.

C. Permanent dipole moment

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114312-9

Tohme, Korek, and Awad

J. Chem. Phys. 142, 114312 (2015)

FIG. 7. Permanent dipole moment curves of the lowest quartet Σ(+/−) and ∆-states of the LiYb molecule.

FIG. 8. Permanent dipole moment curves of the lowest quartet Π and Φstates of the LiYb molecule.

has been repeated using the RSPT2-RS2 level of theory for bases 1-5 for the ground and excited electronic states. The values of the RSPT2-RS2 method of | µe | are in very good agreement with those calculated by the MRCI technique with relative difference 3.0%–4.3%. In the literature, the molecules LiCs,44 RbYb,45,46 RbCs,47 and KRb48 with the dipole moments 5.5 D, (0.98 D and 2.58 D), 1.25 D, 0.76 D, respectively, are considered polar. Accordingly, our molecule LiYb of | µe | = 0.509 D–1.347 D can be considered in the same category. The value of µ for neutral diatomic polar molecules depends on the difference in the electronegativity between the two considered atoms. In particular, the LiYb polar molecule is expected to have relatively small value of | µe | with respect to the other alkali-Yb diatomic systems, due to the small difference in the electronegativity between the Li (χ = 0.98 Pauling units) and Yb (χ = 1.1 Pauling units) atoms. These new predicted values of | µe | might be important and advantageous to the ongoing experiments28,29 of the ultracold polar LiYb molecule and can provide access to many laser cooling applications. In Table III, we present also the absolute values of | µe | for the electronic excited states using basis 5 with 7 valence electrons of the LiYb molecule. No comparison of these values with other results could be done, since they are given here for the first time. The values of the dipole moments for the 2,4Σ, 2,4Π, 2,4∆, 4 and Φ states are given in a.u. as a function of the internuclear distance R in Figures 5–8. The avoided crossing of the potential energy curves and the crossing at dipole moment 2 2 curves for the 2 states (6) Σ+ and (7) Σ+ occur at the same 2 value of internuclear distance R = 2.87 Å, states (3) Π and 2 4 + 4 + (4) Π at R = 4.91 Å, states (2) Σ and (3) Σ at R = 2.23 Å, 4 4 and states (2) Π and (3) Π at R = 2.41 Å, which is a good sign for the accuracy of the present results. The existence of this phenomenon is related to the interaction between the adiabatic potential curves of the corresponding states. The crossing between 2 potential energy curves of two electronic states of different symmetries is strictly allowed, and their wavefunctions are adiabatic solutions of the Schrödinger equation. But if these wavefunctions belong to the same symmetry, they will mix with each other to give two adiabatic solutions, which no longer cross between the corresponding potential energy curves (PECs) and the crossing becomes avoided. This avoided crossing causes an important change

in the dipole moment of the two states where the electronic character is interchanged in this region where the polarity of the atoms is reversed.

IV. CONCLUSION

In the present work, ab initio investigation for 30 lowlying electronic states in the representation 2s+1Λ(+/−) of the ultracold LiYb molecule was computed via CASSCF/ MRSDCI and CASSCF/RSPT2-RS2 methods for the electronic excited states. The potential energy curves, the electronic energy with respect to the ground state, Te, the harmonic frequency, ωe, and the internuclear distance, Re, have been calculated. Taking advantage of the electronic structure of the investigated molecular states of LiYb molecule, and by using the canonical function approach, the vibrational eigenvalues, Ev, the rotational constant, Bv, the centrifugal distortion constant, Dv, and the abscissas of the turning points, Rmin and Rmax, were calculated for several vibrational levels for the LiYb molecule. The comparisons of the present results with the available values in the literature show an overall a very good agreement. Finally, the permanent electric dipole moments have been calculated for the 30 low-lying electronic states. Using different basis sets with different techniques, and different number of valence electrons, our calculated values of the permanent electric dipole moments for the ground state are greater than those given in the literature. To the best of our knowledge, 21 new electronic states have been investigated for the first time through this work. With the recent interest on this molecule through various theoretical18–20,49–51 and experimental15–17,52 studies during the last 5 years, the present study of these new excited electronic states may assist in searching for bound states of the LiYb molecule. This may lead to further investigations and new experimental works on this molecule.

ACKNOWLEDGMENTS

This work was performed with the financial support of CNRS-Lebanon (N0 1-1-13). One of the authors wishes to express his thanks to Dr. Daniel Comparat and Dr. Hans Lignier for their endless support.

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114312-10 1K.-K.

Tohme, Korek, and Awad

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Ab initio calculations of the electronic structure of the low-lying states for the ultracold LiYb molecule.

Ab initio techniques have been applied to investigate the electronic structure of the LiYb molecule. The potential energy curves have been computed in...
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