Ground State Properties of the Polar Alkali-Metal–Ytterbium and

Article pubs.acs.org/JPCA

Ground State Properties of the Polar Alkali-Metal−Ytterbium and Alkaline-Earth-Metal−Ytterbium Molecules: A Comparative Study Qinqin Shao,† Lijuan Deng,† Xiaodong Xing,† Dezhi Gou,† Xiaoyu Kuang,*,† and Hui Li*,‡ †

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, Université Paris-Saclay, 91405 Orsay, France

‡

S Supporting Information *

ABSTRACT: The accurate knowledge of electronic properties is important for creating and manufacturing ultracold molecules. We report here the ab initio quantum chemistry calculations on the properties of alkali-metal−ytterbium AM−Yb (AM = Li, Na, K, Rb, Cs) and alkaline-earth-metal−ytterbium AEM−Yb (AEM = Be, Mg, Ca, Sr, Ba) molecules for their electronic ground state. The potential energy curves (PECs) and permanent dipole moments (PDMs) are calculated on the basis of the multireference configuration interaction (MRCI) level of theory, where the core−valence correlations and scalar relativistic effects are included. The related spectroscopic constants are also determined. The results demonstrate that the dissociation energies and PDMs of AEM−Yb are smaller than those of AM−Yb molecules, and an interesting trend of the dissociation energy has been observed. This work provides favorable information for the experimental study of forming ultracold molecules via photoassociation technique. (Alk = Li, Na, K),14 have been investigated theoretically. The results indicated that these molecules in their electric ground state X2Σ possessed the van der Waals character with large equilibrium bond distance and weak binding energy. Along with their large PDMs, these molecules have become promising candidates for the study of dipole−dipole interaction. Besides, the ultracold heteronuclear diatomic molecules have attracted considerable attention for their rich internal structure, which can offer new opportunities for conducting fundamental studies in quantum chemistry and precision tests of fundamental physics.15 Recently, the high-atomic-number (high-Z) lanthanides have become of interest in ultracold atomic physics. The ultracold gases of Er,16−18 Dy,19−21 Tm,22 and Yb23 have been produced and the strong magnetic molecule Er224 has also been reported. Hence, diatomic molecules composed of AM/AEM and lanthanides can be the potential candidates in the study of polar molecules with large electronic and magnetic dipole moment. Europium-alkali-metal diatomic molecules25 are examples of such interesting species that have been predicted from ab initio calculations. Yb atom has a closed-shell structure [Xe]4f146s2 and its absence of electronic spin simplifies the operation and theoretical understanding of the magneto-optical trapping process. The Feshbach resonances in the mixtures of the Yb atom in the metastable 3P2 state with the ground state Yb and

1. INTRODUCTION Due to the successes of cooling atomic gases to ultracold temperature, the manufacture and control ultracold molecules have attracted increasing attentions in recent years. In particular, the photoassociation1 and Feshbach resonance2 techniques have been successfully applied to create ultracold molecules from ultracold atomic gases. Besides, ultracold atoms have also generated new avenues of researches related to quantum physics, like Bose−Einstein condensation3 and atom optics.4 With the permanent electric and magnetic dipole moment, ultracold polar molecules can extend many new applications, such as the quantum computation and information,5 the investigation of many-body quantum phenomena6 and high-precision measurement of the variation of the fine structure constant.7 For a long time, alkali-metal dimers and bialkali-metal molecules have been the active systems in the study of ultracold gases. Especially, the bialkali-metal molecules have been focused due to their large PDMs and mutual strong anisotropic interactions. In 2008, two groups reported the formation of ultrocold gases of alkali-metal dimer Rb2 molecule8 and bialkalimetal KRb molecule9 in ultracold temperature by using stimulated Raman adiabatic passage (STIPAP). These experiments offered the possibilities to profoundly understand and flexibly manipulate the heteronuclear diatomic molecules. In addition, they could inspire the further theoretical and experimental study on the heteronuclear diatomic molecules. Later on, a series of heteronuclear diatomic molecules, such as XY (X = Li, Na, K, Rb, Cs; Y = Mg, Ca, Sr, Ba)10−13 and AlkBe © 2017 American Chemical Society

Received: November 22, 2016 Revised: January 20, 2017 Published: February 23, 2017 2187

DOI: 10.1021/acs.jpca.6b11741 J. Phys. Chem. A 2017, 121, 2187−2193

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The Journal of Physical Chemistry A Li atoms have been explored.26,27 Furthermore, properties of the YbHe28 and YbCr29 molecules in the ground state have also been investigated theoretically. The YbCr was considered as a candidate for such molecules possessing both large magnetic and electric dipole moments. Because the unpaired s-shell electron can be manipulated and trapped in the external magnetic fields,30 the AM−Yb molecules have been studied extensively31−39 in the past years. Very recently, the ultracold YbLi* molecule40 has been identified from photoassociation production. Until now, most works focused on the AM−Yb systems, whereas the structures of AEM−Yb molecules are still unclear either in theory or experiment. In 2011, Tomza et al.41 presented theoretical prospects of SrYb molecule in an optical lattice by photoassociation spectrum. The AEM−Yb molecules have their own advantages in the following species: (i) The closed-shell structures lead to simple molecular potentials with low radiative losses, giving novel possible applications in the manipulation of the scattering properties with low-loss optical Feshbach resonances.42 (ii) The total spin orbital angular momentum of these molecules is zero, leading to their simple hyperfine level structures. For generation, trapping, and manipulation of the quantum states of diatomic molecules, the knowledge of PECs is necessary. So, the purpose of this work is to carry out a systematic study of the ground state properties for both the AM−Yb and AEM−Yb molecules. The PECs and PDMs are calculated and analyzed on the basis of the ab initio techniques. The corresponding spectroscopic constants have been derived. The rest of the paper is organized as follows. The employed state-of-the-art ab initio calculation methods are described in section 2. Section 3 mainly discusses the characteristics of the PECs and PDMs for the ground state AM−Yb and AEM−Yb diatomic molecules. The brief comparison with previous results is also reported. We summarize this paper in section 4.

included nine electrons and ten molecular orbitals and was referred to as CAS (9, 10). For AEM−Yb molecules, only 12 electrons were explicitly treated in the MRCI calculations. The active space contained 5σ (Be 2s; Mg 3s; Ca 4s; Sr 5s; Ba 6s; Yb 5s, 5p0, 6s, 5d0), 2π (Yb 5p±1, 5d±1), and 1δ (Yb 5d±2) orbitals corresponding to 6a1, 2b1, 2b2, 1a2 [6, 2, 2, 1]. The more detailed description for basis set and active space orbital can be found in Table 1. Table 1. Basis Sets and Active Space Orbitals Used in the Present ab Initio Calculations for the AM−Yb and AEM−Yb Molecules elements Li Na K Rb Cs Be Mg Ca Sr Ba Yb

correlated basis sets 49

active space orbitals

aug-cc-pvqz (13s, 7p, 4d, 3f, 2g) aug-cc-pvqz50 (20s, 13p, 4d, 3f, 2g) ECP10MDF52 (11s, 11p, 5d, 3f) ECP28MDF52 (13s, 10p, 5d, 3f, 1g) ECP46MDF52 (12s, 11p, 5d, 3f, 2g) aug-cc-pvqz49 (13s, 7p, 4d, 3f, 2g) aug-cc-pvqz50 (17s, 13p, 4d, 3f, 2g) ECP10MDF53 (12s, 12p, 7d) ECP28MDF53 (14s, 11p, 5d, 4f, 1g) ECP46MDF53 (13s, 12p, 6d, 4f, 2g) ECP70MDF54 (7s, 6p, 5d)

2s3s 3s4s 4s5s 5s6s 6s 2s 3s 4s 5s 6s 5p6s5d\5s5p6s5d

3. RESULTS AND DISCUSSION Owing to the lack of experimental data on the AM−Yb and AEM−Yb species, the first step is to confirm the reliability of our theoretical methods. To this goal, we perform the ab initio calculations of the dissociation energies (De), equilibrium bond distance (Re), and harmonic vibrational frequency (ωe) for homonuclear dimers AM2, AEM2, and Yb2 using the methods combined with basis sets described in section 2. The results together with the comparative experimental and theoretical data54−65 are listed in Table 2. Excellent agreement of our results with the available experimental and theoretical values confirms that the following calculations on AM−Yb and AEM− Yb are appropriate and reliable. 3.1. Potential Energy Curves and Spectroscopic Constants. The calculations of the PECs have been performed for both AM−Yb and AEM−Yb molecules in the range 2.0 Å ≤ R ≤ 12 Å at the MRCI+Q level. The results are exhibited in Figures 1 and 2, and other available values31,32 for AM−Yb are also displayed in Figure 1 to do comparison. From Figure 1, the PECs of NaYb, KYb, RbYb, and CsYb are in quite good agreement with the results from the coupled cluster method with single, double and triple excitations (CCSD(T)) calculations. For the LiYb molecule, however, there is a significant discrepancy between our curves and those of Zhang et al. from ref 32. The large deviations may be attributed to the different active space and correlated electrons used in calculations. In ref 32, Zhang et al. used the active space including three electrons in eight molecular orbitals (MOs) and 20 seven correlated electrons including 1s22s1 of Li and 5s25p64f146s2 of Yb, which may have overestimated the dissociation energy of LiYb. From Figures 1 and 2, we notice that both AM−Yb and AEM−Yb molecules have shallow potentials and large equilibrium bond distances. Due to the equal electrons in bonding and antibonding orbitals, we can clearly see that the potentials of AEM−Yb are shallower than those of AM−Yb.

2. COMPUTATIONAL METHOD In this paper, the ab initio calculations for the electronic ground state of AM−Yb (AM = Li, Na, K, Rb, Cs) and AEM−Yb (AEM = Be, Mg, Ca, Sr, Ba) molecules were performed by combining complete active space self-consistent field (CASSCF)43,44 and MRCI with Davidson corrections (MRCI +Q)45−47 methods. We applied the MOLPRO packages48 to execute all the calculations in the C2v symmetry point-group. The large augmented correlation consistent polarized valence quadruple-ζ quality (aug-cc-pVQZ) basis set was chosen to describe Li, Na, Be, and Mg atoms. The primitive Gaussian functions including (13s, 7p, 4d, 3f, 2g), (20s, 13p, 4d, 3f, 2g), (13s, 7p, 4d, 3f, 2g), and (17s, 13p, 4d, 3f, 2g) for Li,49 Na,50 Be,49 and Mg50 were included, respectively. For heavier atoms, the relativistic effects were extremely noticeable. We used the small-core relativistic energy-consistent pseudopotential51 (ECP) to consider their scalar relativistic effects. So, K or Ca, Rb or Sr, and Cs or Ba atoms were described with the ECP10MDF, ECP28MDF, and ECP46MDF pseudopotentials,52,53 respectively. The Yb atom with 70 electrons was described with the ECP60MDF including s, p, d functions (7s6p5d)/[5s4p3d].54 For AM−Yb molecules, only 9 electrons were explicitly treated in the MRCI calculations. The active space contained 5σ (Li 2s, 3s; Na 3s, 4s; K 4s, 5s; Rb 5s, 6s; Yb 5p0, 6s, 5d0), 2π (Yb 5p±1, 5d±1), and 1δ (Yb 5d±2) orbitals, which can correspond to irreducible representation 6a1, 2b1, 2b2, 1a2 [6, 2, 2, 1] in C2v symmetry for LiYb, NaYb, KYb, and RbYb molecules, respectively. The active space of CsYb 2188

DOI: 10.1021/acs.jpca.6b11741 J. Phys. Chem. A 2017, 121, 2187−2193

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The Journal of Physical Chemistry A Table 2. Dissociation Energy of AM2 (AM = Li, Na, K, Rb, Cs), AEM2 (AEM = Be, Mg, Ca, Sr, Ba), and Yb2 Molecules molecule

method

Re (Å)

De (cm−1)

ωe (cm−1)

Li2

MRCI exp55 MRCI exp56 MRCI exp57 MRCI exp58 MRCI exp59 MRCI exp60 MRCI exp61 MRCI exp62 MRCI exp63 MRCI CIPSI64 MRCI RCCSD(T)54 RCCSD(T)65

2.696 8517.03 3.165 3.080 4.163 3.924 4.670 4.21 5.022 4.651 2.465 2.45 3.851 3.890 4.473 4.277 4.758 4.672 5.086 4.879 4.970 4.861 4.809

8433.07

356.49

5848.56 6022.60 4299.12 4451.29 3724.44 3993.53 3329.28 3649.5 836.25 790.00 474.32 429.60 971.55 1095.40 1181.58 1061.58 1549.90 1629.24 542.25 476.80 538.30

155.57 159.2 86.17

Na2 K2 Rb2 Cs2 Be2 Mg2 Ca2 Sr2 Ba2 Yb2

53.41 58 37.63 42.02 246.87

Figure 2. Potential energy curves of the X1Σ+ electronic ground state of the AEM−Yb (AEM = Be, Mg, Ca, Sr, Ba) molecules.

55.75 51.12 57.98 65.07 41.52

(Be), and rovibrational coupling constant (αe), are of crucial importance in designing optical routes for the formation of ultracold molecules. In this work, they are derived by fitting the PEC with the Morse potential, which can be expressed as fiveparameters Murrell−Sorbie polynomial function,71 V (R ) = −De(1 + a1ρ + a 2ρ2 + a3ρ3 ) exp( −a1ρ)

35.13 35 19.41 18

ρ = R − Re

where V(R) and R are the dissociation energies and internuclear distances, respectively. All the results as well as previous theoretical data are given in Table 3 for AM−Yb and Table 4 for AEM−Yb. For each ab initio point, we have calculated the corresponding V(R) and R, and present the results in Tables S1and S2 (Supporting Information). As shown in Table 3, our present results are in excellent agreement with the CCSD(T) calculations,31,32 with the exception of LiYb where the difference of De is 14.5% due to the same reason explained for the PECs. Comparing the calculated spectroscopic constants of LiYb with two other available theoretical values in refs 33 and 34, shows a better agreement between them. For NaYb, KYb, and RbYb molecules, the obtained spectroscopic constants are generally in good agreement with the available values35−37 calculated at the MRCI level, particularly the difference of Re is less than 2.3%. As we know that the electronic correlation effect has a momentous impact in the multielectronic heavy molecular calculations. On account of the same correlated electrons used

The shallow potential and large bond length induce AM−Yb and AEM−Yb molecules appear to be of a van der Waals-type bonding nature. A similar situation has been reported in the study of CsXe66 and homonuclear alkaline-earth-metalatom67−69 and ytterbium molecules.70 By comparing the PECs of AM−Yb with those of AEM−Yb molecules, we observed a different variation trend. With the increase of the mass of the molecules, the dissociation energies of AM−Yb are monotonically decreasing whereas the dissociation energies of AEM−Yb are monotonically increasing, except for an abnormal phenomenon for MgYb. The different trends may be caused by the interaction of the electrons mainly from the outermost sshell of the Yb (6s2) atom, AM (ns1) and AEM (ns2) atoms. Basic spectroscopic constants, including R e, De, ω e, anharmonic vibrational frequency (ωeχe), rotational constant

Figure 1. Potential energy curves of the X2Σ+ electronic ground state of the AM−Yb (AM = Li, Na, K, Rb, Cs) molecules. 2189

DOI: 10.1021/acs.jpca.6b11741 J. Phys. Chem. A 2017, 121, 2187−2193

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The Journal of Physical Chemistry A Table 3. Spectroscopic Constants of the Ground State AM−Yb (AM = Li, Na, K, Rb, and Cs) Molecules molecule

methoda,b

Re (Å)

De (eV)

ωe (cm−1)

ωeχe (cm−1)

LiYb

MRCI CCSD(T)32 CASPT233 MRCI34 MRCI CCSD(T)31 MRCI35 MRCI35 MRCI CCSD(T)31 MRCI36 MRCI36 MRCI CCSD(T)31 MRCI37 CCSD38 CCSD(T)39 MRCI CCSD(T)31

3.676 3.519 3.529 3.724 4.113 4.027 4.041 4.022 4.672 4.699 4.683 4.646 4.909 4.911 4.999 4.895 4.683 5.161 5.144

0.1739 0.2035 0.1847

129.5

2.859

NaYb

KYb

RbYb

CsYb

0.1131 0.1254 0.1262 0.1296 0.0863 0.0914 0.1063 0.1204 0.0701 0.0814

135.5 127.9 55.87 62.41 63.94 35.99 39.28 42.26 23.60 27.82 24.07 28.81 19.56

0.0733 0.0974 0.0672 0.0770

Be (10−2cm−1)

αe (10−3 cm−1)

D0 (eV)

de (D)

4.571

0.1660

1.654

18.70 19.82

0.1763

0.8675

4.911

0.8092

0.1096

0.854

0.4872

5.080 5.130 2.422

0.3348

0.0841

1.316

0.2697

2.400 2.440 1.223

0.1387

0.0687

1.218

1.179 0.987 0.1882

0.8418

0.0796

0.0660

1.478

Reference 35. The first line: ECP60MDF (7s6p5d)/[5s4p3d] for Yb and 6-31G basis set for Na. The second line: ECP60MDF(7s6p5d)/ [5s4p3d4f] for Yb and 6-31G basis set for Na. bReference 36. The first line: ECP60MDF for Yb and Def2-QZVPP for K atom. The second line: ECP60MDF for Yb and wCQZ basis set for K atom. a

Table 4. Spectroscopic Constants of the Ground State AEM−Yb (AEM = Be, Mg, Ca, Sr, and Ba) Molecules BeYb method Re (Å) De (eV) ωe (cm−1) ωeχe (cm−1) Be (10−2 cm−1) αe (10−3 cm−1) D0 (eV) de (D)

MRCI 3.926 0.0475 60.27 3.506 12.77 7.103 0.0438 −0.127

MgYb MRCI 4.348 0.0587 40.70 1.041 4.183 1.033 0.0562 −0.245

CaYb MRCI 4.869 0.0557 29.42 0.5536 2.185 0.3902 0.0539 0.441

in the calculations, RbYb shows a good agreement in comparison to the results of ref 38 in Table 3, with a discrepancy of δRe/Re = 0.3%, δDe/De = 4.4%, and δωe/ωe = 1.9%, respectively. On the contrary, due to the different correlated electrons considered in the calculations, large deviations can be observed between our spectroscopic constants and the values in ref 39 for RbYb and in ref 41 for SrYb. Because of being short of comparable values, we wish the results in Table 4 could stimulate future experimental and theoretical investigation on the AEM−Yb molecules. 3.2. Permanent Dipole Moment. Originating from the asymmetric distribution of charge, the PDM is the most fundamental electrostatic property of a heteronuclear diatomic molecules, notably for polar molecules. It is the leading term for electrically neutral systems and is useful in assessing the strength of the long-range dipole−dipole interactions. The calculated PDM for the AM−Yb and AEM−Yb at the MRCI +Q level, as a function of the internuclear distance R, are depicted in Figures 3 and 4, respectively. The PDM values at the equilibrium bond distance are also gathered in Tables 3 and 4. As shown in Figure 3, the PDMs of KYb, RbYb, and CsYb demonstrate similar behavior that the PDMs rapidly increase to the maximum and then gradually drop with the increase of

SrYb MRCI 4.955 0.0649 24.04 0.2977 1.181 0.1383 0.0635 −0.169

BaYb 41

CCSD 4.646 0.1027 32.8

MRCI 5.109 0.0725 21.31 0.2066 0.8435 0.0782 0.0712 −0.093

Figure 3. Permanent dipole moments of the X2Σ+ electronic ground state of the AM−Yb (AM = Li, Na, K, Rb, Cs) molecules at the MRCl + Q level.

internuclear distance R. The PDMs of KYb (1.316 D), RbYb (1.218 D), and CsYb (1.478 D) are large, making them potential candidates for ultracold collisional studies of dipolar molecules. In this paper, our PDM of RbYb (1.218 D) is larger than that of Sørensen et al.38 (0.987 D) using the CCSD(T) method. The differences may be attributed to the different basis sets used in calculations. From Figure 4, the PDMs of AEM− Yb at the equilibrium bond distance are smaller than 0.5 D. 2190

DOI: 10.1021/acs.jpca.6b11741 J. Phys. Chem. A 2017, 121, 2187−2193

The Journal of Physical Chemistry A

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AUTHOR INFORMATION

Corresponding Authors

*X.Y.K. E-mail: [email protected]. Telephone number: +86-28-85403803. *H.L. E-mail: [email protected]. Telephone number: +33-768062998. ORCID

Xiaoyu Kuang: 0000-0001-7489-9715 Notes

The authors declare no competing financial interest.

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Figure 4. Permanent dipole moments of the X1Σ+ electronic ground state of the AEM−Yb (AEM = Be, Mg, Ca, Sr, Ba) molecules at the MRCl + Q level.

ACKNOWLEDGMENTS We thank Olivier Dulieu and Xiaowei Sheng for fruitful discussions and comments. The work was supported by the National Natural Science Foundation of China (Nos.11574220 and 11274235).

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This can be explained from the similar electronegativity between AEM and Yb atoms. It is known that electronegativity reflects the ability of one atom to attract electron from other atoms. So, the small difference of electronegativity between two atoms of the diatomic molecules leads to small dipole moment. Even though their PDMs are small, the AEM−Yb molecules are still the interesting candidates for achieving ultracold polar molecules.

4. CONCLUSIONS In this work we have explored the ground state properties of AM−Yb and AEM−Yb molecules based on state-of-the-art ab initio techniques. The PECs and PDM functions have been achieved by utilizing the CASSCF+MRCI methods with large Dunning basis sets, which can accurately treat the core−valence correlation. The scalar relativistic effects are taken into account by means of the effective core potentials in the calculations. The corresponding spectroscopic constants are also determined and in line with the available data in literature. From the comparison, an opposite variation trend of the dissociation energy with the increasing mass of molecules is found between the AM−Yb and AEM−Yb molecules. And the dissociation energies of AM−Yb molecules are larger than those of AEM− Yb molecules. Due to the similar electronegativity between the AEM and Yb atoms, the PDMs of AEM−Yb molecules are smaller than those of AM−Yb molecules. Both species are interesting candidates for studies of ultracold polar molecules. Especially the AM−Yb molecules, with an electronic double-σ (2Σ) ground state, are a promising class of polar molecules possessing both electric and magnetic dipole moments. We hope the calculated PECs and PDMs provide primal information for further experimental and theoretical investigations.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b11741. Values of internuclear distance R and dissociation energy De for each ab initio point of AM−Yb (AM = Li, Na, K, Rb, Cs) molecules and AEM−Yb (AEM = Be, Mg, Ca, Sr, Ba) molecules (Tables S1 and S2) (PDF) 2191

DOI: 10.1021/acs.jpca.6b11741 J. Phys. Chem. A 2017, 121, 2187−2193

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