Theoretical Study of the CsNa Molecule: Adiabatic and Diabatic

Article pubs.acs.org/JPCA

Theoretical Study of the CsNa Molecule: Adiabatic and Diabatic Potential Energy and Dipole Moment N. Mabrouk† and H. Berriche*,†,‡ †

Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences de Monastir, Avenue de l’Environnement 5019 Monastir, Tunisia ‡ Mathematics and Natural Sciences Department, School of Arts and Sciences, American University of Ras Al Khaimah, Ras Al Khaimah, UAE ABSTRACT: The adiabatic and diabatic potential energy curves of the low-lying electronic states of the NaCs molecule dissociating into Na (3s, 3p) + Cs (6s, 6p, 5d, 7s, 7p, 6d, 8s, 4f) have been investigated. The molecular calculations are performed using an ab initio approach based on nonempirical pseudopotential, parametrized l-dependent polarization potentials and full configuration interaction calculations through the CIPCI quantum chemistry package. The derived spectroscopic constants (Re, De, Te, ωe, ωexe, and Be) of the ground state and lower excited states are compared with the available theoretical and experimental works. Moreover, accurate permanent and transition dipole moment have been determined as a function of the internuclear distance. The adiabatic permanent dipole moment for the first nine 1Σ+ electronic states have shown both ionic characters associated with electron transfer related to Cs+Na− and Cs−Na+ arrangements. By a simple rotation, the diabatic permanent dipole moment is determined and has revealed a linear behavior, particularly at intermediate and large distances. Many peaks around the avoided crossing locations have been observed for the transition dipole moment between neighbor electronic states. study of CsNa molecule is made by Igel-Mann et al.32 on the basis of pseudopotential and CI calculations. They determined, in their work, the spectroscopic constants for the ground state. Korek et al.24 have performed an ab initio calculation for CsNa. They have studied the potential energy curves for 8, 6, and 2 states of, respectively, 1,3Σ+, 1,3Π, and 1,3Δ symmetries. In a recent study30 this same group has realized a calculation using nonempirical pseudopotentials and taking into consideration the spin−orbit effect. Aymar and Dulieu31 have extensively investigated several alkali metal dimers. They have determined accurate potential energy curves and dipole moments of the lowest 1,3Σ+ states. Several experimental studies of CsNa have been performed. Haimberger et al.39 have reported the translationally cold CsNa molecules starting from a laser cooled atomic vapor of Na and Cs atoms. In another study40 the same team has investigated the formation of the CsNa molecule in its ground state from ultra cold atoms by photoassociation. Docenko et al.35 have investigated the X1Σ+ and a1Σ+ states using high resolution Fourier-transform spectroscopy. The 1Π and B1Π states have been explored by Zaharova et al.41 using a dye laser for inducing fluorescence that was resolved by a high resolution Fouriertransform spectrometer. In another new study34 Zaharova by using the Fourier transform spectrometer with resolution of 0.03−0.05 cm−1 has measured the laser-induced fluorescence

1. INTRODUCTION An important theoretical and experimental effort has been devoted in the development of ultrashort and intense laser pulses during the past decade. This has opened a new research domain in molecular dynamics and molecular spectroscopy. Such an effort is motivated by the possible applications such as manipulation and controlling of ultracold chemical reactions,1−3 ultracold molecular collision dynamics,4−7 quantum computing,8,9 and experimental preparation of few-body quantum effects10 (such as Efimov states) where the aim is to prepare molecules in definite quantum states with respect to the center of mass, electronic, rotational, and vibrational motions.11 In addition, the long-range interaction will introduce new important physical phenomena. The theoretical investigation of the heteronuclear alkali diatomic molecule is encouraged by the pressing need for experimentalists as well as theorists for the simulation of the effect of the application of strong electric field on their rovibrational dynamics.12 In our group several theoretical studies of the electronic structure of these systems have been performed for CsLi,13,14 NaLi,15,16 RbLi,17 KLi and FrLi,18 and CsNa.19 Numerous other authors were interested in studying the heteronuclear alkali dimers using different quantum chemistry methods. They determined the potential energy of the ground and exited states for numerous alkali metal mixed molecules such as RbCs,20−22 NaRb and LiRb,23 CsNa, LiCs, and KCs,24 KRb,25 KLi,26 NaK,27,28 and NaLi.29 Several theoretical24,30−32 and experimental33−48 studies have been achieved for the CsNa molecule. The first theoretical © 2014 American Chemical Society

Received: May 3, 2014 Revised: July 21, 2014 Published: July 24, 2014 8828

dx.doi.org/10.1021/jp5043427 | J. Phys. Chem. A 2014, 118, 8828−8841

The Journal of Physical Chemistry A

Article

(LIF) spectra (4)Σ+1 → AΣ+1−bΠ3 and collisionally enhanced AΣ+1−bΠ3 → XΣ+1 LIF spectra. He has found about 1160 term values of the e-symmetry rovibronic levels of the fully mixed AΣ+1 and bΠ3 states of a NaCs molecule. In fact, the Zabawa team has made several studies on the formation of Ultra cold NaCs molecule. In 200842 they studied the collision between ultracold NaCs molecules and composite atoms. In 201043 they labeled several deeply bound vibrational quanta in the ground state of NaCs using pulsed laser depletion spectroscopy method. In the same year Zabawa et al.44 demonstrated that the photoassociation resonance detuned less than a wavenumber below the Cs 6 2P3/2 atomic line can be used to create a deeply bound molecular sample of ultracold polar NaCs. In 201145 the formation of ultracold X 1Σ+(v″=0) NaCs molecules via coupled photoassociation channels has been performed. In 201246 the same team presented the rovibrational cooling of translationally ultracold NaCs molecules using the broadband light for the vibrational cooling requires and narrow-line lasers for the rotational cooling. Haruza et al.47 presented the photoassociation channels associated with the A1Σ and b3Π electronic states of NaCs. They calculated dispersion coefficients and determined new b3ΠΩ=2 potential energy curve. Recently, Ashman48 have described a series of experiments designed to map out several electronic states of the NaCs molecule like 53Π and 11Σ +. This study completes our published work19 on the NaCs alkali dimer in the AIP Conference Proceedings as an extended abstract.19 In the latter we have reported only the spectroscopic data of the first nine 1Σ+ states. The used methodology and a generalization to the 3Σ+, 1,3Π and 1−31,3Δ symmetries are presented here in detail. In this context, we have investigated the adiabatic and diabatic potential energy curves of 1−91Σ+, 1−83Σ+, 1−61,3Π, and 1−31,3Δ states, which dissociate into Cs (6s, 6p, 5d, 7s, 7p, 6d, 8s, 4f) + Na (3s, 3p) atomic limits. In addition, the adiabatic and diabatic permanent dipole moment and the transition dipoles between neighbor states have been determined. This paper is organized as follows. In section 2, we summarize the ab initio and dibatisation methods used in our calculations. Section 3 is divided into three subsections. In the first subsection we present the potential energy curves and their spectroscopic constants for the ground and numerous excited states of 1,3Σ+, 1,3Π, and 1,3Δ symmetries. In the second and third subsections we report the adiabatic and diabatic permanent dipole moment, and the transition dipole moment between neighbor states. Finally, we conclude in section 4.

riλ and Rλ′λ are, respectively, the core−electron and core−core vectors. According to the formulation of Foucrault et al.59 the cut-off function F(riλ,ρλ) is taken as a function of the orbital angular momentum l. In this context, the interaction of valence electrons of different spatial symmetry with the core electrons is considered differently for each l. ∞

F(riλ ,ρλ ) =

i

riλ⃗ riλ

3

F(riλ ,ρλ ) −

∑ λ ′≠ λ

Rλ⃗ ′ λ Rλ ′ λ 3

Fl(riλ ,ρλl )|lmλ⟩⟨lmλ|

|lmλ⟩ is the spherical harmonic centered on λ. F(riλ,ρ1λ) is the cutoff operator expressed following the Foucrault et al.59 formalism by a step function defined by ⎧ 0, riλ < ρλ F(riλ ,ρλl ) = ⎨ ⎪ ⎩1, riλ > ρλ ⎪

This will exclude the valence electrons from the core region for calculating the electric field. In the formalism of Müller et al.,58 the cutoff function is unique for each atom, which is adjusted to reproduce the lowest atomic energy levels. Additionally, the SCF (self consistent field) calculation is followed by a full CI (configuration interaction) using the CIPCI algorithm of the Laboratoire de Physique et Chimie Quantique of Toulouse. We have used a 8s/5p/5d/2f Gaussian basis set for the cesium atom, which is taken from ref 61, whereas for the Na we have used the same basis set of Gaussian-type orbital (GTO’s) 6s/5p/4d/2f as in our previous works.51−57 The core dipole polarizabilities of Cs+ and Na+ are respectively, 15.81 and 0.9930 au. In Table 1 we present the cutoff radii for the s, p, d, Table 1. l-Dependent Cutoff Radii (in Bohr) for Cs and Na Atoms 1

Cs

Na

s p d f

2.7923 2.667 2.8130 2.8130

1.4423 1.625 1.5 1.5

and f lowest one electron valence orbitals. Our calculated values for the ionization potential (IP) and atomic energy levels are compared with the experimental60 ones. They are reported in Table 2. The difference between the theoretical and experimental energies does not exceed 321 × 10−6 au.

3. RESULTS AND DISCUSSION 3.1. Adiabatic Potential Energy and Spectroscopic Constants. The adiabatic potential energy curves of 35 electronic states of 1,3Σ+, 1,3Π, and 1,3Δ symmetries have been determined for the CsNa molecular system dissociating into Na (3s, 3p) + Cs (6s, 6p, 5d, 7s, 7p, 6d, 8s, 4f) limits. The potential energy has were calculated for a dense and large grid of internuclear distances, which varies from 5 to 100 au. The potential energy curves of the 1Σ+, 3Σ+ 1,3Π, and 1,3Δ states are displayed, respectively, in Figures 1−4. A general undulations behavior is observed for the higher excited states. It leads, for several states, to double and triple wells and sometimes to barriers. We noticed also many avoided crossings. They can be explained by the ionic interaction between Cs+ and Na− and Cs− and Na+. To confirm the precision of our potential energy curves, we

αλ is the dipole polarizability of the core λ and fλ is the electric field created by all other cores and the valence electrons on the core λ.

∑

∑∑ l = 0 m =−l

2. METHOD OF CALCULATION The Cs and Na alkali atoms are considered in this study as oneelectron pseudopotentials49,50 used in earlier works.51−57 The valence electrons and the polarizable cores (Na+ and Cs+) interactions, VCPP are considered using the formulation of Müller et al.58 1 VCPP = − ∑ αλfλ ·fλ 2 λ

fλ =

+l

Zλ 8829



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Table 2. Asymptotic Energy of CsNa Electronic States (au) asymptotic molecule state (1

Σ)

1,3

Na(3s) + Cs(6s) (2 1,3Σ, 1 1,3Π) Na(3s) + Cs(6p) (3 1,3Σ, 2 1,3Π, 1 Na(3s) + 1,3 Cs(5d) Δ) Σ, 3

1,3

1,3

Π)

Na(3p) + Cs(6s) (5 1,3Σ) Na(3s) + Cs(7s) (6 1,3Σ, 4 1,3Π) Na(3s) + Cs(7p) (7 1,3Σ, 5 1,3Π, 2 Na(3s) + 1,3 Cs(6d) Δ)

(4

(8

Σ)

1,3

Na(3s) + Cs(8s) (9 1,3Σ, 6 1,3Π, 3 Na(3s) + 1,3 Cs(4f) Δ)

ΔE

this work

expt60

−0.331957

−0.331958

10−6

−0.279342

−0.279342

0

−0.265626

−0.265626

0

−0.254648

−0.254649

10−6

−0.247490

−0.247502

12 × 10−6

−0.232264

−0.232238

25 × 10−6

−0.228596

−0.228917

321 × 10−6

−0.221137

−0.221162

25 × 10−6

−0.220401

−0.220401

0

Figure 2. Potential energy curves for the eight lowest 3Σ+ states of CsNa.

Figure 1. Potential energy curves for the nine lowest 1Σ+ states of CsNa.

have derived their spectroscopic constants Re, De, Te, ωe, ωexe and Be. They are compared in Tables 3−5 with the existing theoretical24,30−32 and experimental studies.34,36,38 We observe a very good agreement between our molecular constants and the theoretical and experimental ones. Our equilibrium distance for the ground state (Re = 3.804 Å) is in good accord with that of Docenko et al.36 and Korek et al.24 They found, respectively, 3.8506 and 3.813 Å. A similar good agreement is observed between our well depth De = 4994 cm−1, ωe = 98.04 cm−1 and those found by Korek et al.24 (De = 4923 cm−1, ωe = 99.97 cm−1) and Docenko et al.36 (De = 4954.24 cm−1). Korek et al.30 in a new study of this molecule introducing the spin−orbit effect found new values for Re = 3.769 Å and ωe = 101.9 cm−1.

Figure 3. CsNa adiabatic potential energy curves for the 1Π (solid lines) and the 3Π (dashed lines).

An excellent agreement is noticed for the 13Σ+ state with the experimental results of Docenko et al.36 We obtained Re = 5.735 Å, De = 215 cm−1 and they obtained Re = 5.7448 Å, De = 217.17 cm−1. The 5−9 3Σ+ and 3 3Π states are studied here for 8830



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The diabatization method has been tested, first, for the CsH molecule61 and applied later for LiH62,65,66 and RbH67 and more recently for CsNa19 molecules. This diabatization technique can be considered, at present, to be the most effective for molecular ab initio calculations. The origin of the molecule is taken on the heavier atom, the Cs atom. The reference states are fixed as the adiabatic ones at large internuclear distance, here R = 100 au. Figure 5 presents the diabatic potential energy curves for the 1−9 1Σ+ adiabatic states and they are named D1−9. We notice that the first diabatic curve called D1, which dissociates into Cs+Na−, crosses all the neutral states at different internuclear distances. The crossing happens with the Na(3s) + Cs(6s) state around 11.02 au and with Na(3s) + Cs(6p) state around 17.85 au. The remaining crossings arise at much longer distances. The D1 state exhibits an ionic character, Cs+Na−, and the real crossings in the diabatic representation are transformed into avoided crossings in the adiabatic. The ionic curve kindly behaves as 1/R at large distances, with some corrections proportional to 1/R4 corresponding to Na− dipolar polarization and also, but to a minor extent, to core−valence corrections. We observe the existence of another ionic associated with the Cs−Na+ ionic character. The potential energy curve of this state crosses higher neutral excited states at shorter distances. These crossings turn into avoided crossings in the adiabatic representation. The diabatic potential energy curves for the 1−8 3Σ+, 1−6 1Π, 1−6 3Π, 1−3 1Δ and 1−3 3Δ states are presented in Figure 6−10, respectively. The same feature is observed for the diabatic potential energy curves of 3Σ+ (Figure 6) symmetry. There is, essentially, one potential curve D8, which cross the others curves at intermediate and large internuclear distance. For example, the curve called D8 crosses D4 at 12.51 and at 15.81 au and D5 at 17.78 au and D6 at 21.77 au and D7 at 22.78 au. It seems that these curves, as observed for the symmetry, correspond to the ionic characters related to Cs+Na−. The ionic curve D8 encloses all other curves, except the D1 as it is repulsive. The same thing is observed for the diabatic potential curves of the other symmetries. Such real crossings are transformed into avoided ones in the adiabatic representation. The repulsive feature of the diabatic potential energy curves at short distances can be explained by the penetration of the Rydberg orbitals by the cesium atom. It is polarized by the electric field created by the core Na+ and the other electron. In the Rydberg series, whereas these effects are not evidently apparent for the 3s orbital, because the 3s is a valence orbital; the attractive and repulsive interactions take place for the 3p state and beyond. For the higher states, the attractive effect is dominant because the Rydberg state is very diffuse and the potential energy curves become much more similar to the Na−Cs+ ones. The attractive character increases with l the orbital quantum number of the Rydberg state. This can be due to the increase of directionality of the corresponding orbital and the resulting field observed by the Cs atom. 3.3. Permanent and Transition Dipole Moments. Accurate electronic properties are necessary to predict the formation of cold and ultracold dipolar heteronuclear metal alkali dimers. In addition, their dipole function can be considered as a sensitive assessment for the accuracy of the determined energies and electronic wave functions. In this context, we have investigated the adiabatic and the diabatic permanent dipole moments for the same grid of internuclear distances (5−100 au). To understand the ionic character of the excited states, we have presented the adiabatic permanent dipole moment of the

Figure 4. CsNa adiabatic potential energy curves for the 1Δ (solid lines) and the 3Δ (dashed lines).

the first time. Once more we remark the very good agreement with the theoretical work of Korek et al.24 This is not surprising as we use similar methods, which are pseudopotential and configuration interaction methods. For the 2 1Π state we obtained Re = 4.206 Å, Te = 18470 cm−1, ωe = 66.69 cm−1 to be compared with the results of Korek et al.,24 who obtained Re = 4.205 Å, Te = 18470 cm−1, ωe = 69.34 cm−1. In a recent experimental study of the 13Π state, Zaharova et al.34 obtained Re = 3.78 Å, Te = 10236.048 cm−1, De = 6081.142 cm−1, which are in good accord with our results: Re = 3.735 Å, Te = 10266 cm−1, De = 6276 cm−1. We report also in this paper the 1Δ and 3 Δ states. We remark that all these sates are attractive and exhibit well depths of some thousands of cm−1 in accord with the theoretical results of Korek et al.24 Avoided crossings have been located at large as well as at short distances for all these states. 3.2. Diabatic Potential Energy Curves. The diabatization technique has been presented in details in several works.62−64 Its application has shown its efficiency on several dimers such as LiH.61 Nevertheless, this study can be considered as the first application for the CsNa alkali diatomic molecule. In this subsection, we briefly present the outlines of the diabatization method, which is based on the effective Hamiltonian theory. The procedure is to cancel the numerical nonadiabatic coupling between the adiabatic states by an appropriate unitary transformation. Thus, it will lead to a unitary transformation matrix U, which when applied for each distance will provide us with the diabatic energy and wave functions. In this context, the diabatic wave functions are written as a linear combination of adiabatic states. In fact, the nonadiabatic coupling is strictly related to an overlap matrix between the R-dependent adiabatic multiconfigurational states and the reference states. The states correspond to a large distance fixed wave functions. 8831



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Table 3. Spectroscopic Constants of the 1Σ+ State of the CsNa Molecule state 1 Σ

1 +

2 1Σ+

3 1Σ+ 4 1Σ+ 1st min

2nd min 5 1Σ+ 1st min



2nd min 8 1Σ+ 1st min 2nd min 9 1Σ+ 1st min 2nd min a

Re (Å)

De (cm−1)

3.804 3.8506 3.813 3.769 3.809 3.69 3.83 3.85 4.62 4.65 4.48 4.466 4.417

4994 4954.24 4923

5.286 4.536 5.155 9.101 9.067

3184

4.397 4.379 4.347 7.757 7.765 4.471 4.474 4.423 12.693

Te (cm−1)

ωe (cm−1) 98.04

ωexe (cm−1)

Be (cm−1)

ref

0.38

0.0594

0.16

0.04040

0.25

0.04276 0.04312 0.04406

this Work expta CIb CIc CId expte CIf exptg this work expte CIb this work CIb

0.02

0.03078 0.04181 0.0323 0.01038 0.104

this work CIb CIc this work CIc

74.33 88.82 76.9 39.23 36.7

0.67

0.04449 0.04487 0.0455 0.01429 0.0142

this work CIb CIc this work CIc

23626 23582 23693 23738

64.78 64.96 65.3 −34.24

0.35

0.04303 0.04297 0.0439 0.00533

this work CIb CIc this work

24700 24659 24757 26888

47.41 50.52 50.9 9.66

0.11

0.03827 0.03924 0.0392 0.00198

this work CIb CIc this work

65.36 65.4 58.35

0.40

0.04164 0.0422 0.02406

This Work CIc this work

65.55 28.38

1.02

0.04429 0.02890

this work this work

99.97 101.9

0.05917 0.0605 0.0631

4839.29 4919.94 6014 5992.060 2816

1387

2591

2038

3248

3136

10527

98 99 57.79

10511 16736 16665

105.08 62.24 91.60

18777 19068 18973 20575 20847

41.52 139.98 40.8 23.23 23.6

20941 20887 21005 21495 21809

4.741 4.682 4.681 20.862

2979

4.545 4.515 5.979

2505 1925

25689 25925 26268

4.407 5.455

3147 2818

26168 26498

790

Reference 36. bReference 24. cReference 30. dReference 31. eReference 34. fReference 32. gReference 38.

Σ , 1,3Π, 1,3Δ symmetries states. Figure 11 presents the adiabatic permanent dipole moment of the 1−3 1Σ+ states. We remark that their dipole moment is insignificant and does not behave as a linear function of R. Figure 12 displays the permanent dipole moment for the 4−9 1Σ+ electronic states. We remark that the dipole moment of these states, one after another, behaves as a −R function and then drops to zero at specific internuclear distances, which correspond to the avoided crossings between the two neighbor molecular states. When these curves are combined, they produce piecewise the whole −R function, which is related to the ionic character of these states. The discontinuities between the consecutive parts are due to the avoided crossings and such nodes will disappear in the diabatic representation and will lead to a full linear curve corresponding to the dipole moment of the pure ionic state as it was observed previously for the LiH, RbH, and CsH

molecules.66,67,61 This observation is well demonstrated by Figure 13, which presents the diabatic permanent dipole moment of the 1−9 electronic 1Σ+ states. They are obtained by a simple rotation of the adiabatic dipole moment. In the diabatic scheme such nodes will disappear, and a full linear curve will appear. It corresponds to the pure ionic state dipole moment as it was demonstrated earlier in the LiH,66 RbH,67 and CsH61 molecules. The internuclear location when the two dipole moments of neighbor states are the same is correlated to the avoided crossing between them. For a better explanation, the dipole moment is calculated for a large number of internuclear distances particularly around the avoided crossing locations. The difference between two successive distances in this region is 0.01 au. This demonstrates the ionic arrangement and the long-range polar character of the excited states. The same ionic character was detected previously for the LiH66 molecule. But,

1,3 +

8832



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Table 4. Spectroscopic Constants of the 3Σ+ State of the CsNa Molecule state 1 Σ

3 +

2 3Σ+ 3 3Σ+ 4 3Σ+ 5 3Σ+ 1st min 2nd min 6 3Σ+ 1st min 2nd min 7 3Σ+ 1st min 2nd min 8 3Σ+ a

Re (Å)

De (cm−1)

Te (cm−1)

ωe (cm−1)

ωexe (cm−1)

Be (cm−1)

ref

5.735 5.7448 4.640 4.61 4.275 4.297 4.513 4.5

215 217.17 19.38

4779 −217.168 14603 14538 16994 16919 21747 21698

22.30

2.20

0.026184

53.85 55.50 67.68 67.43 60.97 61.33

0.49

0.039912

0.50

0.047029

0.46

0.042211

this work expta this work CIb this work CIb this work CIb

2558 214

4.455 5.995

294 707

23238 22825

63.30 82.65

0.39

0.043342

this work this work

4.434 5.265

1878 3055

24996 23819

68.03 143.95

1.74

0.043728

this work this work

4.725 8.815 4.624

2582 791 3548

25097 26887 25767

168.43 56.83 116.77

15.62

0.038507

4.16

0.040203

this work this work this work

Reference 36. bReference 24.

Table 5. Spectroscopic constants of the 1Π, 3Π, 1Δ and 3Δ of CsNa molecule state

Re (Å)

De (cm−1)

Te (cm−1)

ωe (cm−1)

1Π

4.360 4.381 4.206 4.205 4.894 5.895 4.365 4.369 4.418 4.424 4.450 4.456 3.735 3.78 3.737 4.365 4.398 6.444 4.497 4.540

1149

15393 15341 18470 18470 20052 20047 24268 24251 25101 25056 26301 26526 10266 10236.048 10277 16810 16737 21618 23800 23785 24258 24640 24701 25372 25489 18208 18252 24564 24730 25652 26084 18465 18490 24656 24802 26081 26526

54.16 53.75 66.69 69.34 64.33 65.68 65.40 66.12 59.93 60.37 64.63 65.43 96.21

1

21Π 31Π 41Π 5 1Π 6 1Π 13Π

23Π 33Π 43Π

53Π 63Π 1 1Δ 21Δ 31Δ 13Δ 23Δ 33Δ b

4.804 4.768 4.402 4.439 4.185 4.215 4.429 4.415 4.296 4.294 4.302 4.342 4.392 4.383 4.370 4.418

1082 1909 2607 2578 3177 6276 6081.142 2742 344 3075 2616 3039 4106 1344 3116 3825 1087 3023 3397

99.28 55.57 56.51 25.45 65.85 64.94 7.96 61.29 57.78 67.53 68.86 58.85 58.98 71.30 67.45 70.44 72.52 54.91 53.42 67.13 68.95 68.62 81.16

ωexe (cm−1)

Be (cm−1)

ref

0.85

0.045229

0.37

0.048573

0.01

0.035929

0.40

0.045146

0.47

0.044014

0.41

0.043434

0.28

0.061703

0.28

0.045085

−0.03 0.85

0.020721 0.042492

this work CIb this work CIb this work CIb this work CIb this work CIb this work CIb this work expte CIb This Work CIb this work this work CIb

0.04

0.037258

0.35

0.044416

0.22

0.049135

3.74

0.043838

0.43

0.046574

0.80

0.46493

0.13

0.044628

0.33

0.045076

this work CIb this work CIb this work CIb this work CIb this work CIb this work CIb this work CIb this work CIb

Reference 24. eReference 34. 8833



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Table 6. Avoided Crossing Positions states

position (au)

11Σ+/21Σ+ 2 1Σ+/3 1Σ+ 3 1Σ+/4 1Σ+

11.02 17.85 11.02 19.01 9.81 27.20 10.49 43.02 12.28 50.75 10.49 16.48 6.39 11.02 8.86 16.90 8.55 7.97 26.99 5.48 9.56 21.39

5 1Σ+/6 1Σ+ 6 1Σ+/7 1Σ+ 7 1Σ+/8 1Σ+ 8 1Σ+/9 1Σ+ 2 3Σ+/3 3Σ+ 4 3Σ+/5 3Σ+ 5 3Σ+/6 3Σ+ 7 3Σ+/83Σ+ 8 3Σ+/93Σ+ 33Π/43Π 43Π/53Π

Figure 6. Diabatic potential energy curves for the eight lowest 3Σ+ states of CsNa.

Figure 5. Diabatic potential energy curves for the nine lowest 1Σ+ states of CsNa. Figure 7. Diabatic potential energy curves for the six lowest 1Π states of CsNa.

the linear comportment has taken place with the ground state X 1 + Σ in the LiH molecule. In fact, the LiH system is polar in its ground state in contrast to the CsNa. The LiH molecule has an important permanent dipole moment. In addition, the LiH permanent dipole has shown a pure linear feature in the first ionic diabatic state and quasi-zero for the higher neutral diabatic states. We remark that no second ionic curve appears clearly

either in the adiabatic and or in the diabatic representations. However, we notice that some adiabatic states exhibit also positive permanent dipole moment at short and intermediate distance. This is in addition to the negative one, which is the case for the 7, 8, and 9 1Σ+ states. This proves their inverse 8834



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Figure 10. Diabatic potential energy curves for the three lowest 3Δ states of CsNa.

Figure 8. Diabatic potential energy curves for the six lowest 3Π states of CsNa.

Figure 9. Diabatic potential energy curves for the three lowest 1Δ states of CsNa. −

Figure 11. Adiabatic permanent dipole moment of the first three 1Σ+ states of CsNa.

+

ionic behavior connected to Cs Na arrangement in this region. The feature of the permanent dipole is comparable to that observed at larger distance as when it disappears for one state it increases for the next one. This is linked to the avoided

crossings between neighbor states in the short and intermediate region. Figure 12 has a positive part with a +R behavior of the 8835



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Figure 14. Adiabatic permanent dipole moment of the first eight 3Σ+ states of CsNa. Figure 12. Adiabatic permanent dipole moment of the first 1−9 1Σ+ states of CsNa.

Figure 15. Adiabatic permanent dipole moment of the first six 1Π states of CsNa. Figure 13. Diabatic permanent dipole moment of the first nine 1Σ+ states of CsNa.

form a perfect straight line. In fact, the positive and negative R conducts of the permanent dipole moment are associated with the two series of the avoided crossings between the 1Σ+ states. The first series is linked to the higher excited states avoided crossings, which happen at short and intermediate internuclear

dipole moment demonstrates such hypothesis. Nevertheless, the consecutive parts of the positive dipole moment do not 8836



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Figure 16. Adiabatic permanent dipole moment of the first six 3Π states of CsNa.

Figure 18. Adiabatic permanent dipole moment of the first three 3Δ states of CsNa.

Figure 19. Transition dipole moment between neighbor states of 1Σ+ symmetry.

this range is negative. Accordingly, the both ionic character is proved for the higher excited states associated with two ionic charged arrangements, Cs+Na− and Cs−Na+. The permanent dipole moments are also determined for the remaining 3Σ+, 1Π, 3 Π, 1Δ, and 3Δ electronic states. As expected, the dipole moment of these states is not insignificant. It becomes much more significant for the higher excited states. This demonstrates once

Figure 17. Adiabatic permanent dipole moment of the first three Δ states of CsNa. 1

locations. In this range of distance the dipole moment is positive. The second series of avoided crossings occurs at larger internuclear distances and the permanent dipole moment in 8837



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Figure 20. Transition dipole moment between neighbor states of 3Σ+ symmetry.

Figure 22. Transition dipole moment between neighbor states of 3Π symmetry.

Figure 21. Transition dipole moment between neighbor states of 1Π symmetry.

Figure 23. Transition dipole moment between neighbor states of 1Δ and 3Δ symmetries.

more their ionic character. The permanent dipole moments of these states are presented in Figure 14−18. For example, Figure 18 presents the permanent dipole moment for the 1Δ and 3Δ studied states. We observe a significant dipole moment at intermediate distance. This range of distance corresponds to repulsive interactions in their deep potential wells. To understand the ionic character of the electronic states around the avoided crossings, we have performed the transition dipole moment between neighbor 1Σ+, 3Σ+, 1Π, 3Π, 1Δ, and 3Δ states. They are displayed in Figures 19−23. We detect many peaks positioned at specific distances very close to the avoided

crossings locations in the adiabatic representation. These positions were reported in Table 6. These accurate transition dipole moments will be used in the near future to evaluate the radiative lifetimes and for formation prediction of the CsNa alkali dimer via photoassociation.

4. CONCLUSION In this study, we have determined the potential energy, the spectroscopic properties and the dipole moment for 35 electronic states of the CsNa. These states dissociate into Na (3s, 3p) and Cs (6s, 6p, 5d, 7s, 7p, 6d, 8s, 4f) and lead 1−9 1Σ+, 8838



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1−8 3Σ+, 1−6 1,3Π, and 1−3 1,3Δ symmetries. The used method is based on nonempirical pseudopotentials, parametrized l-dependent polarization potentials, and full valence CI calculations. The adiabatic potential energy curves, permanent and transition dipole moment have been performed for a large and dense grid of internuclear distances, which varies from 5 to 100 au. The spectroscopic constants have been extracted and compared with the available theoretical24,30−32 and experimental works.34,36,38 A very good accord is noticed with the results of Korek et al.24 for the ground state X1Σ+ and excited states. A similar agreement is also observed with the experimental studies of Docenko et al.36 and Diemer et al.38 Our results, for the 11Σ+ and 1 3Π states were compared with the very recent experimental work of Zaharova et al.34 This comparison has shown a good agreement for the equilibrium distance and well depth. However, our well depths for both states are underestimated by more than 195 cm−1 compared to the experimental ones. For a better understanding of the ionic character of the electronic states of this molecule, we have evaluated their permanent and transition dipole moments. The permanent dipole moment has shown, as it is expected, a linear behavior at intermediate and large distances mainly for the higher excited states. The permanent dipole moment of the 1−3 1Σ+ states is found to be insignificant, in contrast to that of the 4−9 1Σ+ higher states. Moreover, it behaves as −R, giving an important dipole moment, which corresponds to a longrange ionic character. We observed also for the same excited states a quasi-similar behavior at smaller distances but with positive permanent dipole moment. These negative and positive dipole moments are related to Cs+Na− and Cs−Na+ ionic charge arrangements of the excited states. For each ionic character a series of avoided crossings is observed between the neighbor states. Although the −R behavior of the dipole moment was observed previously for LiH, RbH, and CsH, the +R behavior is observed for the first time for NaLi in our previous work and in this present study on CsNa. To determine the diabatic states, a diabatization is realized using the effective Hamiltonian theory. The behavior of these diabatic states has revealed also an ionic character connected to the two ionic arrangements Cs+Na− and Cs−Na+. We have observed two different diabatic curves that crossed the neutral ones. These lead to two series of avoided crossings located at all ranges of distances. Such crossings turn out to be avoided crossings in the adiabatic representation. The same observation is seen for the diabatic states of other symmetries. The accurate potential energy data will be used for prediction simulation of the CsNa molecule at low temperature via photoassociation process, for radiative lifetime evaluation, radial coupling, vibronic shifts, and adiabatic energy correction.

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(3) Rom, T.; Best, T.; Mandel, O.; Widera, A.; Greiner, M.; Hansch, T. W.; Bloch, I. Phys. State Selective Production of Molecules in Optical Lattices. Rev. Lett. 2004, 93, 073002−073006. (4) Kajita, M. Eur. Phys. Cold Collisions between Linear Polar Molecules. J. D 2003, 23, 337−342. (5) Krems, R. V. Molecules Near Absolute Zero and External Field Control of Atomic and Molecular Dynamics. Int. Rev. Phys. Chem. 2005, 24, 99−118. (6) Avdeenkov, A. V.; Kajita, M.; Bohn, J. L. Suppression of Inelastic Collisions of Polar 1Σ State Molecules in an Electrostatic Field. Phys. Rev. A 2006, 73, 022707−022714. (7) Krems, R. V. Controlling Collisions of Ultracold Atoms with dc Electric Fields. Phys. Rev. Lett. 2006, 96, 123202−123206. (8) Demille, D. Quantum Computation with Trapped Polar Molecules. Phys. Rev. Lett. 2002, 88, 067901−067905. (9) Yelin, S. F.; Kirby, K.; Côté, R. Schemes for Robust Quantum Computation with Polar Molecules. Phys. Rev. A 2006, 74, 050301− 050305. (10) Kraemer, T.; Mark, M.; Walburger, P.; Danzl, J. G.; Chin, C.; Engesser, B.; Lange, A. D.; Pilch, K.; Jaakkola, A.; Nagerl, H. C.; et al. Evidence for Efimov Quantum States in an Ultracold Gas of Caesium Atoms. Nature 2006, 440, 315−318. (11) Kraft, S. D.; Staanum, P.; Lang, J.; Vogel, L.; R. Wester, R.; Weidemuller, M. Formation of Ultracold LiCs Molecules. J. Phys. B: At. Mol. Opt. Phys. 2006, 39, S993−S1000. (12) Smirov, A. D. Calculation of Spectroscopic Constants for the Ground Electronic States of CsK, CsLi, and RbLi Molecules. J. Struct. Chem. 2007, 48, 21−27. (13) Mabrouk, N.; Berriche, H.; Gadea, F. X. Theoretical Study of the CsLi Molecule Beyond the Born-Oppenheimer Approximation. AIP Conf. Proc. 2007, 963, 23−27. (14) Mabrouk, N.; Berriche, H.; Ben Ouada, H.; Gadea, F. X. Theoretical Study of the LiCs Molecule: Adiabatic and Diabatic Potential Energy and Dipole Moment. J. Phys. Chem. A 2010, 114, 6657−6668. (15) Mabrouk, N.; Zrafi, W.; Berriche, H.; Ben Ouada, H. Theoretical Study of the NaLi Molecule. Lect. Ser. Comp. Comput. Sc. 2006, 7, 1451−1454. (16) Mabrouk, N.; Berriche, H. Theoretical Study of the NaLi Molecule: Potential Energy Curves, Spectroscopic Constants, Dipole Moments and Radiative Lifetimes. J. Phys. B: At. Mol. Opt. Phys. 2008, 41, 155101−155116. (17) Jendoubi, I.; Berriche, H.; Ben Ouada, H.; Gadea, F. X. Structural and Spectroscopic Study of the LiRb Molecule Beyond the Born-Oppenheimer Approximation. J. Phys. Chem. A 2012, 116, 2945−2960. (18) Belayouni, S.; Jendoubi, I.; Mabrouk, N.; Berriche, H. Systematic Study of the Electronic Properties and Trends in the LiX (X = Na, K, Rb, Cs and Fr) Molecules. Adv. Quantum Chem. 2014, 68, 203−238. (19) Mabrouk, N.; Berriche, H. Adiabatic and Diabatic Study of the Lowest 1Σ+ States of CsNa Molecule. AIP Conf. Proc. 2009, 1148, 326−329. (20) Pavolini, D.; Gustavsson, T.; Spigelmann, F.; Daudey, J. P. Theoretical Study of The Excited States of The Heavier Alkali Dimers: I. The RbCs Molecule. J. Phys. B: At. Mol. Opt. Phys. 1989, 22, 1721− 1731. (21) Allouche, A. R.; Korek, M.; Fakherddin, K.; Chalaan, A.; Dagher, M.; Taher, F.; Aubert-Fécon, M. Theoretical Electronic Structure of RbCs Revisited. J. Phys. B: At. Mol. Opt. Phys. 2000, 33, 2307−2316. (22) Fahs, H.; Allouche, A. R.; Korek, M.; Aubert-Frécon, M. The Theoretical Spin−Orbit Structure of the RbCs Molecule. J. Phys. B 2002, 35, 1501−1508. (23) Korek, M.; Allouche, A. R.; Kobeissi, M.; Chaalan, A.; Dagher, M.; Fakherddin, F.; Aubert-Frecon, M. Theoretical Study of The Electronic Structure of the LiRb and NaRb Molecules. Chem. Phys. 2000, 256, 1−6.

AUTHOR INFORMATION

Corresponding Author

*H. Berriche. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Balakrishnan, N.; Dalgarno, A. Chemistry at Ultracold Temperatures. Chem. Phys. Lett. 2001, 341, 652−656. (2) Bodo, E.; Gianturco, F. A.; Dalgarno, A. F+D2 Reaction at Ultracold Temperatures. J. Chem. Phys. 2002, 116, 9222−9228. 8839



Article

(24) Korek, M.; Allouche, A. R.; Fakhreddine, K.; Chaalan, A. Theoretical Study of the Electronic Structure of LiCs, NaCs, and KCs Molecules. Can. J. Phys. 2000, 78, 977−988. (25) Rousseau, S.; Allouche, A. R.; Aubert-Frécon, M. Theoretical Study of the Electronic Structure of the KRb Molecule. J. Mol. Spectrosc. 2000, 203, 235−243. (26) Rousseau, S.; Allouche, A. R.; Aubert-Frécon, M.; Magnier, S.; Kowalczyk, P.; Jastrzebski, W. Theoretical Study of the Electronic Structure of KLi and Comparison With Experiments. Chem. Phys. 1999, 247, 193−199. (27) Magnier, S.; Aubert-Frécon, M.; Millié, Ph. Potential Energies, Permanent and Transition Dipole Moments for Numerous Electronic Excited States of NaK. J. Mol. Spectrosc. 2000, 200, 96−103. (28) Magnier, S.; Millié, Ph. Potential Curves for the Ground and Numerous Highly Excited Electronic States of K2 and NaK. Phys. Rev. A 1996, 54, 204−218. (29) Schmidt-Mink, I.; Müller, M.; Meyer, W. Potential Energy Curves for Ground and Excited States of NaLi from Ab Initio Calculations with Effective Core Polarization Potentials. Chem. Phys. Lett. 1984, 112, 120−128. (30) Korek, M.; Bleik, S.; Alloche, A. R. Theoretical Calculation of the Low Laying Electronic States of the Molecule NaCs with SpinOrbit Effect. J. Chem. Phys. 2007, 126, 124313−124322. (31) Aymar, M.; Dulieu, O. Calculation of Accurate Permanent Dipole Moments of the Lowest 1,3Σ+ States of Heteronuclear Alkali Dimers Using Extended Basis Sets. J. Chem. Phys. 2005, 122, 204302− 204311. (32) Igel-Mann, G.; Wedig, U.; Fuentealba, P.; Stoll, H. GroundState Properties of Alkali Dimers XY (X, Y=Li to Cs)a). J. Chem. Phys. 1986, 84, 5007−5012. (33) Docenko, O.; Tamanis, K.; Ferber, R.; Pashov, A.; Knockel, H.; Tiemann, E. Potential of the Ground State of NaRb. Phys. Rev. A 2004, 69, 042503−042508. (34) Zaharova, J.; Tamanis, M.; Ferber, R. Solution of the FullyMixed-State Problem: Direct Deperturbation Analysis of the A1Σ+-b3Π Complex in a NaCs Dimer. Phys. Rev. A 2009, 79, 012508−012521. (35) Docenko, O.; Tamanis, K.; Zaharova, J.; Ferber, R.; Pashov, A.; Knockel, H.; Tiemann, E. High Resolution Spectroscopy and Potential Determination of the (3)Π1 state of NaCs. J. Chem. Phys. 2006, 124, 174310−174311. (36) Docenko, O.; Tamanis, M.; Zaharova, J.; Ferber, R.; Pashov, A.; Konckel, H.; Tiemann, E. The Coupling of the X1Σ+ and a3Σ+ States of the Atom Pair Na+Cs and Modelling Cold Collisions. J. Phys. B:At. Mol. Opt. Phys. 2006, 39, S929−S943. (37) Dagdigian, P. J.; Wharton, L. Molecular Beam Electric Deflection and Resonance Spectroscopy of the Heteronuclear Alkali Dimers: 39K7Li, Rb7Li, 39K23Na, Rb23Na, and 133Cs23Na. J. Chem. Phys. 1972, 57, 1487−1497. (38) Diemer, U.; Weickenmeier, H.; Whal, M.; Demtroder, W. SubDoppler Spectroscopy of the NaCs Molecule. Chem. Phys. Lett. 1984, 104, 489−495. (39) Haimberger, C.; Kleinert, J.; Bhattacharya, M.; Bigelow, N. P. Formation and Detection of Ultracold Ground-State Polar Molecules. Phys. Rev. A 2004, 70, 021402−021406. (40) Haimberger, C.; Kleinert, J.; Dulieu, O.; Bigelow, N. P. Processes in the Formation of Ultracold NaCs. J. Phys. B: At. Mol. Opt. Phys. 2006, 39, S957−S963. (41) Zaharova, J.; Docenko, O.; Tamanis, M.; Ferbera, R. The B1Π State of NaCs: High Resolution Laser Induced Fluorescence Spectroscopy and Potential Construction. J. Chem. Phys. 2007, 127, 224302−224310. (42) Wakim, A. E.; Zabawa, P. J.; Haimberger, C.; Kleinert, J. Bigelow, N. P. Collision Studies between Ultracold NaCs Molecules and Composite Atoms. Frontiers in Optics 2008, 1−3 (Laser Science XXIV, Plasmonics and Metamaterials. Optical Fabrication and Testing, OSA Technical Digest (CD), optical Society of America, paper JSuA18). (43) Zabawa, P.; Wakim, A.; Neukirch, A.; Bigelow, N.; Pazyuk, E.; Stolyarov, A.; Tamanis, M.; Ferber, R. Pulsed Laser Depletion

Spectroscopy of Ultracold NaCs Molecules. 41st Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics American Physical Society: College Park, MD, 2010; Vol. 55, DAMOP.B2.7. (44) Zabawa, P.; Wakim, A.; Neukirch, A.; Haimberger, C.; Bigelow, N. P.; Stolyarov, A. V.; Pazyuk, E. A.; Tamanis, M.; Ferber, R. NearDissociation Photoassociative Production of Deeply Bound NaCs Molecules. Phys. Rev. A 2010, 82, 040501−040505. (45) Zabawa, P.; Wakim, A.; Haruza, M.; Bigelow, N. P. Formation of Ultracold X 1Σ+(v″=0) NaCs Molecules via Coupled Photoassociation Channels. Phys. Rev. A 2011, 84, 061401−061405. (46) Zabawa, P.; Wakim, A.; Haruza, M. Bigelow, N. P. Rovibrational Cooling of Translationally Ultracold, Polar NaCs Molecules. Frontiers in Optics 2012, 4, 1−3 (Laser Science XXVIII, OSA Technical Digest (online), optical Society of America, paper LTu4I). (47) Haruza, M.; Zabawa, P.; Wakim, A. Bigelow, N. P. Photoassociation Spectroscopy of the A1Σ-b3Π Complex in Ultracold Polar NaCs Molecules. Frontiers in Optics 2012, 2, 1−3 (Laser Science XXVIII, OSA Technical Digest (online), optical Society of America, paper LTu4I). (48) Ashman, Seth T. Experimental Studies of the NaCs 5(3)Pi(0) and 1(a)3Sigma+ Electronic States. Ph.D. Thesis, Lehigh University 2010, 3419329, pp 1−385. (49) Durand, P.; Barthelat, J. New Atomic Pseudopotentials for Electronic Structure Calculations of Molecules and Solids. Chem. Phys. Lett. 1974, 27, 191−194. (50) Durand, P.; Barthelat, J. A Theoretical Method to Determine Atomic Pseudopotentials for Electronic Structure Calculations of Molecules and Solids. Theor. Chim. Acta 1975, 38, 283−302. (51) Berriche, H.; Gadea, F. X. Ab initio Study of the LiH+ Molecule, Electronic Interaction Analysis and LiH UV Photoelectron Spectrum. Chem. Phys. 1995, 191, 119−131. (52) Berriche, H.; Tlili, C. Ab Initio Potential Energy Surface and Rotationally Inelastic Collisions of LiH (X1Σ+) With H. I. the Ab Initio Evaluation of the Potential Energy Surface. J. Mol. Struct. 2004, 682, 11−16. (53) Berriche, H. Rotationally Inelastic Collisions of LiH (X1Σ+) With H:State-to-State Inelastic Rotational Cross-Section. J. Mol. Struct. 2004, 682, 89−96. (54) Ghanmi, C.; Bouzouita, H.; Berriche, H.; Ben Ouada, H. Theoretical Investigation on CsLi+ and CsNa+ Ionic Molecules. J. Mol. Struct. 2006, 777, 81−86. (55) Berriche, H.; Ghanmi, C.; Ben Ouada, H. Theoretical Study of the Electronic States and Transition Dipole Moments of the LiK+ Molecule. J. Mol. Spectrosc. 2005, 230, 161−167. (56) Ghanmi, C.; Berriche, H.; Ben Ouada, H. A Vibrational Level Spacing Analysis of the LiK+ Lowest Electronic States: Long-Range Behavior and Evaluation of Li and K Polarizabilities. Comput. Math. Sci. 2007, 38, 494−501. (57) Berriche, H. Theoretical Study of the Lowest Electronic States of the LiNa+ Molecule. J. Mol. Struct. 2003, 663, 101−108. (58) Müller, W.; Flesh, J.; Meyer, W. Treatment of Intershell Correlation Effects in Ab Initio Calculations by use of Core Polarization Potentials. Method and Application to Alkali and Alkaline Earth Atoms. J. Chem. Phys. 1984, 80, 3297−3310. (59) Foucrault, M.; Millié, Ph.; Daudey, J. P. Nonperturbative Method for Core−Valence Correlation in Pseudopotential Calculations: Application to the Rb2 and Cs2 Molecules. J. Chem. Phys. 1992, 96, 1257−1264. (60) Moore, C. E. Atomic Energy Levels, Natl. Bur. Stand. Circ. No. 467; U.S GPO: Washington, DC, 1971. (61) Zrafi, W.; Oujia, B.; Gadea, F. X. Theoretical Study of the CsH Molecule: Adiabatic and Diabatic Potential Energy Curves and Dipole Moments. J. Phys. B: At. Mol. Opt. Phys. 2006, 39, 3815−3832. (62) Boutalib, A.; Gadea, F. X. Ab Initio Adiabatic and Diabatic Potential-Energy Curves of the LiH Molecule. J. Chem. Phys. 1992, 97, 1144−1156. (63) Gadea, F. X. Théorie des Hamiltoniens Effectifs: Application aux Probèmes de Diabatisation et de Collision Réactive. Thèse d’état, Université Paul Sabatier, Toulouse, 1987 8840



Article

(64) Gadea, F. X.; Pelissier, M. Approximately Diabatic States: A Relation Between Effective Hamiltonian Techniques and Explicit Cancellation of the Derivative Coupling. J. Chem. Phys. 1990, 93, 545− 551. (65) Gadea, F. X.; Boutalib, A. Computation and Assignment of Radial Couplings Using Accurate Diabatic Data for the LiH Molecule. J. Phys. B: At. Mol. Opt. Phys. 1993, 26, 61−74. (66) Berriche, H.; Gadea, F. X. Ab Initio Adiabatic and Diabatic Permanent Dipoles for the Low-lying States of the LiH Molecule. A Direct Illustration of the Ionic. Chem. Phys. Lett. 1995, 247, 85−88. (67) Khelifi, N.; Zrafi, W.; Oujia, B.; Gadea, F. X. Ab Initio Adiabatic and Diabatic Energies and Dipole Moments of the RbH Molecule. Phys. Rev. A 2002, 65, 042513−042523.

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Theoretical Study of the CsNa Molecule: Adiabatic and Diabatic

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