Theoretical study of LiK and LiK+ in adiabatic representation

ISSN 0036�0244, Russian Journal of Physical Chemistry A, 2014, Vol. 88, No. 1, pp. 73–84. © Pleiades Publishing, Ltd., 2014.

73

1 INTRODUCTION

Theoretical studies done one ultracold molecules arecurrently a great challenge to spectroscopic studies ofalkali dimers, because of their importance in the coolingand trapping of atoms [1, 2] and molecules [3–6], theirrole in high precision spectroscopy [7, 8]. Ultracold mol�ecules should prove to be useful in spectroscopy and thestudy of molecular structure, especially in ultrahigh reso�lution spectroscopy, which requires cold and trappedsamples. Neutral–atom collisions at ultralow tempera�tures may be characterized by s�wave scattering lengths.Collisions of ions and atoms involve higher–order partialwaves because of the long–range attractive polarizationforces and differ also because of the possibility of chargetransfer. The cooling and manipulation of cold moleculesis likely to open up new branches of research. There areproposed experiments to study polar molecular systemsin order to measure the electron’s permanent electricdipole moment (EDM), the lifetime of long–livedenergy levels, and the effects of the dipole–dipole inter�actions on the molecular samples properties [9].

1 The article is published in the original.

Investigations on the LiK molecule have beenimpeded for a long time by difficulties in production ofmolecular vapor of concentration sufficient for spec�troscopic experiments. The two lowest singlet andtriplet states, X1Σ+ and a3Σ+ were characterized withhigh accuracy in a series of studies [10–13]. Severalexperiments were performed on the excited statesaccessible in one–photon transitions from the groundstate, including the B(1)1Π, C1Σ+, D(2)1Π, 41Σ+, 41Π,61Π, and 71Π, states [14–19].

Although LiK is not particularly well suited forphoto–association from cold atoms [20], its isotopicforms offer all possible boson/fermion pair combina�tions. The use of pseudo potentials for Li and K coresreduce the number of active electrons to only one valenceelectron, where the SCF calculation produces the exactenergy in the basis and the main source of errors corre�sponds to the basis–set limitations. Furthermore we cor�rect the energy by taking into account the core–core andcore–electron correlation following the formalism ofFoucrault et al. [21]. This formalism was used for severalsystems (KH, RbH, CsH, NaH, LiNa, LiNa+, LiCs, and

Theoretical Study of LiK and LiK+ in Adiabatic Representation1

Omar M. Al�dossarya,b and Neji Khelific

aDepartment of Physics, College of Science King Saud University, Saudi ArabiabNational Center for Mathematics and Physics, KACST, Saudi Arabi

cPhysics Department, College of Science, Shaqra University

e�mail: [email protected] January 31, 2013

Abstract—The potential energy curves have been calculated for the electronic states of the molecule LiKwithin the range 3 to 300 a.u., of the internuclear distance R. Using an ab initio method, through a semi–empirical spin–orbit pseudo–potential for the Li (1s2) and K (1s22s22p63s23p6) cores and core valence corre�lation correction added to the electrostatic Hamiltonian with Gaussian basis sets for both atoms. The corevalence effects including core–polarization and core�valence correlation are taken into account by using anl�dependent core–polarization potential. The molecular orbitals have been derived from self–consistent field(SCF) calculation. The spectroscopic constants, dipole moments and vibrational levels of the lowest elec�tronic states of the LiK molecule dissociating into K (4s, 4p, 5s, 3d, and 5p) + Li (2s, 2p, 3s, and 3p) in 1, 3Σ,1, 3Π, and 1, 3Δ symmetries. Adiabatic results are also reported for 2Σ, 2Π, and 2Δ electronic states of themolecular ion LiK+ dissociating into Li (2s, 2p, 3s, and 3p) + K+ and Li+ + K (4s, 4p, 5s, 3d, and 5p). Thecomparison of the present results with those available in the literature shows a very good agreement in spec�troscopic constants of some lowest states of the LiK and LiK+ molecules, especially with the available theo�retical works. The existence of numerous avoided crossing between electronic states of 2Σ and 2Π symmetriesis related to the charge transfer process between the two ionic systems Li+K and LiK+.

Keywords: adiabatic representation, pseudo–potential, operatorial core valence, correlations, full valence CIapproach, spectroscopic constants, potential energy curves, crossing, neutral, ionic limit, vibration levels,spacings, dipole moments.

DOI: 10.1134/S0036024414010336

STRUCTURE OF MATTERAND QUANTUM CHEMISTRY

74

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 1 2014

OMAR M. AL�DOSSARY, NEJI KHELIFI

LiCs+) and its use has demonstrated efficiency. Thenon–empirical pseudopotentials permit the use of verylarge basis sets for the valence and Rydberg states andallow accurate descriptions for the highest excited states.Despite the relative simplicity of the LiK+ system, fewtheoretical works have been done and for our best knowl�edge there are no experimental study done so far. Theneutral molecule LiK was explored by many theoretical[12, 14, 17, 22, 23, 34] and experimental [24–26] works.In this study LiK+, having only one active electron, willbe a one of the simplest hetero–molecules and the com�puting time is reduced. The present work succeeds ourstudy on many diatomic systems, such as KH, RbH,CsH, NaH, LiNa, LiNa+, LiCs, LiCs+, [24, 25, 27–39],where we used the same techniques. For all of them wegot a remarkable accuracy showing the validity of thisapproach. The present results for LiK and LiK+ systemscan be expected to reach a similar accuracy since themain restriction in the accuracy of the calculation is thebasis set limitation only. We present a complete set ofresults including all Rydberg states.

METHOD

The investigation of the electronic structure of theLiK molecule has been done by treating the atoms Liand K with the non�empirical relativistic effectiveone�electron core potential of the Durand and Bar�thelat type [34]. The interaction between the outer andthe atomic core is modulated with a non�empiricalrelativistic pseudo�potential:

where l is the orbital angular momentum and Pl corre�sponds to the projection operator on the subspace

defined by the spherical harmonics with a given l.The pseudo�potentials U(r) are written as:

with c, n, and α adjusted to fit the energy and wavefunctions of the valence Hartree–Fock orbitals. Thecore valence effects including core–polarization andcore valence correlation are taken into account byusing an l�dependent core–polarization potential ofthe Foucrault et al. type [35]:

.

The cutoff radii, reported in Table 1, were optimizedin order to reproduce the ionization potentials and thelowest valence s, p, and d, one�electron states asdeduced from the atomic data tables. The Gaussian�type orbitals basis sets on Li and K atoms were, respec�tively 6s/5p/3d/1f and 8s/6p/4d/2f. The core dipolepolarizability of Li+ and K+ are respectively 0.1997and 5.472 (a.u.)3.

In Table 2, a comparison between our ab initio andthe experimental [36], energy levels for K (4s, 4p, 5s,3d, and 5p) + Li (2s, 2p, 3s, and 3p) atomic states arepresented. Our atomic energies are in good agree�ment with the experimental ones. The differencesbetween our computed and the experimental valuesdo not exceed 25 cm–1. Such accuracies in the atomiccalculations would be expected in the molecular cal�culations. As can be seen in Table 3, where all disso�ciation limits are reported, reasonable agreementsbetween computed and experimental are obtainedeven for high energy dissociation limits. The largesterrors are for the Li (3s) + K (4s) state limit. Ourcomputed values agree generally better with availableexperimental values [37] than those from the mostrecent theoretical work of Dardouri et al. [40]. Thedifference is due to the Gaussian type orbital basissets, polarizabilities of Li+ and K+ and cutoff radiiused in the two computational works.

V r( ) Ul r( )Pl,

l 0=

2

∑=

Yml

U r( ) Cirni

eαr

2–

i 1=

2

∑=

VCPP12�� αk fk

k 0=

2

∑–=

Table 1. l�Dependent cutoff radii (in a.u.) for Li and K atoms

l Li K

s 1.315 2.114

p 0.999 1.973

d 0.600 1.988

f 0.900 1.988

Table 2. Atomic transition energies of the Li and K atoms(cm–1)

Atomic levels This work Exp. [36] ΔE, cm–1

Lithium

2s 0.00 0.00 0.002P (2p) 14901.79 14903.66 1.872S (3s) 27229.42 27206.12 23.302P (3p) 30922.71 30925.38 3.672D (3d) 31283.60 31283.08 0.52

Potassium

4s 0.00 0.00 0.002P(4p) 13028.6 13030.1 1.52S(5s) 21023.1 21026.8 3.72D(3d) 21532.3 21535.3 3.32P(5p) 24714.9 24716.0 1.12D(4d) 27401.9 27397.4 3.52S(6s) 27448.4 27451.5 3.12F(4f) 28132.3 28127.9 4.6

IP 34099.6 35009.8 10.2


THEORETICAL STUDY OF LiK AND LiK+ 75

RESULTS AND DISCUSSION

Potential Energy Curves and SpectroscopicConstants for LiK

The potential energy curves are shown for the 1Σ+

and 3Σ+ molecular states, respectively, in Figs. 1 and 2of the LiK molecule, dissociating into K (4s, 4p, 5s,3d, and 5p) + (Li (2s, 2p, 3s, and 3p). They have beencalculated for a large and dense grid of interatomic

distances from 3 to 300 a.u. The spectroscopic constantsfor the 1Σ+ and 3Σ+ states are presented in Tables 4 and 5.The only possible comparisons, with the experimentalresults, relate to the fundamental state X1Σ. As can beseen in Table 4, our results for the equilibrium distanceare in good agreement with the experimental data[24], being better than the older theoretical results andcomparable to the most recent ones. As can be seen inTable 4, for the other states also, our results are in good

Table 3. Molecular states of LiK (in cm–1)

Asymptotes This work Exp. [36] ΔE, cm–1 ΔE, cm–1 [40] State

Li (2s) + K (4s) 0.00 0.00 0.00 0.00 1,3Σ

Li (2s) + K (4p) 13028.6 13030.1 1.5 0.00 1,3Σ, 1,3Π

Li (2p) + K (4s) 14901.79 14903.66 1.87 0.00 1,3Σ, 1,3Π

Li (2s) + K (5s) 21023.1 21026.8 3.7 6.00 1,3Σ

Li (2s) + K (3d) 21532.3 21535.3 3.3 0.00 1,3Σ, 1,3Π, 1,3Δ

Li (2s) + K (5p) 24714.9 24716.0 1.1 9.00 1,3Σ, 1,3Π

Li (3s) + K (4s) 27229.42 27206.12 23.30 63.30 1,3Σ

Li (2s) + K (4d) 27401.9 27397.4 3.5 13.00 1,3Σ, 1,3Π, 1,3Δ

Li (2s) + K (6s) 27448.4 27451.5 3.1 2.00 1,3Σ

Li (2p) + K (4p) 27930.39 27933.76 3.37 0.00 1,3Σ, 1,3Π, 1,3Δ

Li (2s) + K (4f) 28132.3 28127.9 4.6 21.00

Table 4. Spectroscopic constants (Re, a.u.; De, cm–1, and Te, cm–1) for the 3Σ+ states of LiK compared with the availableresults in the literature

Re De Te Reference Re De Te Reference

X 1Σ+ 5 1Σ+

6.18 6250 00.00 This work 7.43 3890.0 25780.00 This work

6.24 6216 00.00 Exp. [24] 7.54 3680 25600.00 [40]

6.26 6138 00.00 [25] 6 1Σ+

6.16 6220 00.00 [26] 7.38 4870.0 27094.00 This work

6.22 6197 00.00 [40] 7.34 4801 26794.00 [40]

2 1Σ+ 7 1Σ+

7.35 7234 12387.00 This work 7.91 3678.0 30136.00 This work

7.362 7186 12056.00 [26] 7.86 3578.0 29061.00 [40]

7.43 7167 12798.00 [40] 8 1Σ+

3 1Σ+ 7.51 4270.0 29753 This work

7.85 3344.6 17698.95 This work 7.48 4565 29631 [40]

7.876 3619.0 17501.178 Exp. [14] 9 1Σ+

7.922 3618.8 17501.239 Exp. [32] 7.18 4179.3 30124 This work

7.76 3477.0 17647.00 [26] 7.21 4208 30058 [40]

7.91 3554 17767 [40] 10 1Σ+

4 1Σ+ 7.29 3020.0 32140 This work

14.02 4930.56 23977.85 This work 7.26 2964 31197 [40]

13.97 4860 23898.00 [40]

13.98 Exp. [17]

76



0.36

10 20 30 40 50R, a.u.

0.30

0.24

−E, a.u.

Li(2s)+K(3d)

Li(2s)+K(5s)

Li(2p)+K(4s)

Li(2s)+K(4p)

Li(2s)+K(4s)

Fig. 1. Potential energy curves for the 1Σ+ states of LiK.

0.36

10 20 30 40 50R, a.u.

0.28

0.20−E, a.u.

Li(2s)+K(3d)

Li(2s)+K(5s)

Li(2p)+K(4s)

Li(2s)+K(4p)

Li(2s)+K(4s)

Li(2s)+K(5p)

Fig. 2. Potential energy curves for the 3Σ+ states of LiK.

agreement with the experimental data, being generallybetter than the previous theoretical results. This is par�ticularly true for the second excited state, where notice�able improvement can be observed for Re and De. Thestate 41Σ+ has an unusual shape and has attracted inter�esting studies, mainly experimental [17].

Table 5 presents the spectroscopic constants of the1–10 3Σ+ states. The 13Σ+ state has been experimentallyexplored recently by Salami et al. [12] it exhibits a verysmall potential well of 287 cm–1 located at an equilib�rium distance of 9.43 a.u. Here also, our results are inrather good agreement with the experimental and themost recent theoretical work of Dardouri et al. [40].

Figures 3 and 4 illustrate the singlet and triplet adi�abatic 1, 3Π and 1, 3Δ states. Since these symmetriesinvolve orbitals perpendicular to the axis of the mole�cule, which overlap only loosely, the adiabatic curvesare rather flat. The singlet and triplet adiabatic Δcurves are almost degenerate.

The spectroscopic constants (Re, De, and Te) of the1, 3Π and 1, 3Δ states are presented in Tables 6 and 7. Forall these states of different symmetries, many avoidedcrossings have been located at large internuclear dis�tance as well as at short distance. Their existences willgenerate large non�adiabatic coupling and lead to anundulating behavior of the higher excited states atlarge inter�nuclear distances. Very recently, Pashov

0.30

10 20 30 40 50R, a.u.

0.26

0.22

−E, a.u.

Li(2s)+K(3d)

Li(2p)+K(5s)

Li(2p)+K(4s)

Li(2s)+K(5p)

Li(2s)+K(4p)

Fig. 3 Potential energy curves for the 1Π (solid lines) and3Π (dashed lines) states of LiK.

0.27

10 20 30 40R, a.u.

0.25

0.23

−E, a.u.

Li(2s)+K(3d)

Li(2p)+K(4d)

Li(2s)+K(4p)

Fig. 4. Potential energy curves for the 1Δ (solid lines) and3Δ (dashed lines) states of LiK.



et al. [14] have studied the 11Π and Grochola et al.[39] the 21Π excited states. They found well depth of1686 and 1664.3 cm–1 respectively for the 11Π and 21Πstates located at an equilibrium distance of 6.979 and

7.60 a.u. Our equilibrium distances are in good agree�ment with these data, while De values are somewhatunderestimated as in the previous theoretical study[42]. Our term energies values Te for the 11Π and 21Π

Table 5. Spectroscopic constants (Re, a.u.; De, cm–1, andTe, cm–1) for the 3Σ+ states of LiK molecule compared withthe available results in the literature

Re De Te Reference

1 3Σ+

9.45 278 5874 This work

9.4 273 5947 [26]

9.43 ± 0.09 287.0 ± 4 Exp. [12]

9.42 276.33 5760 [40]

2 3Σ+

7.39 4268 15120 This work

7.36 4309 14931 [26]

7.43 4099 15996 [40]

3 3Σ+

7.25 836.26 23865 This work

7.13 917.60 22616 [40]

4 3Σ+

7.49 2673 24648 This work

7.56 2530 24233 [40]

5 3Σ+

8.74 2654 25796 This work

8.45 2739 25313 [40]

6 3Σ+

7.65 3876 27986 This work

7.80 3646 28140 [40]

7 3Σ+

7.45 4987 28975 This work

7.53 5094 29160 [40]

8 3Σ+

7.14 4678 30593 This work

7.23 4540 30484 [40]

9 3Σ+

7.28 3784 31354 This work

7.32 3604 31127 [40]

10 3Σ+

7.19 2989 32453 This work

7.21 3141 32332 [40]

Table 6. Spectroscopic constants (Re, a.u.; De, cm–1; andTe, cm–1) for the 1,3

Π states of LiK compared with theavailable results in the literature

Re De Te Reference

1 1Π

7.12 1573.0 17365.98 This work

6.98 1686.0 17572.76 Exp. [14]

7.06 1517.0 17725.12 [26]

7.19 1454 17672.00 [40]

2 1Π

7.54 1438.0 20154.54 This work

7.6 1664.3 19455.73 [41]

7.56 1584.0 19541.56 [26]

7.69 1300 20410.00 [40]

3 1Π

7.85 1598.0 25986.00 This work

8.05 1624.0 26131.00 [26]

7.29 1584 26029.00 [40]

4 1Π

7.18 3987.0 26475.00 This work

7.28 4144 26595.00 [40]

5 1Π

7.38 4997.0 28265.00 This work

7.45 5011 28427.00 [40]

1 3Π

6.12 8674.0 10395.00 This work

6.06 8767.0 10476.84 [26]

6.09 8504 10534.00 [40]

2 3Π

7.62 688.0 20457.69 This work

7.56 736.0 20388.00 [26]

7.58 612.49 21090.00 [40]

3 3Π

9.78 3478.0 24564.00 This work

9.84 3384.0 24373.00 [26]

9.85 3283.0 26652.00 [40]

4 3Π

7.54 4789.0 28064.32 This work

7.63 4627.0 27870.00 [40]

5 3Π

7.62 5347.0 29785.54 This work

7.69 5265.0 29591.00 [40]

78



states are overestimated compared to the experimentalones [14, 39, 41] (see Table 6). However, it is not clearif zero point energy corrections are needed or not. The

present values are taken from the bottom of the groundstate well. If we made the correction, then the agree�ment improves considerably.

Vibrational level spacings values (Ev + 1 – E

v) of the

1Σ+, 1Π, and 1Δ electronic states of the LiK are given inTables 8 and 9.

Adiabatic Permanent and Transition Dipole Moments

The permanent dipole moments are illustrated inFig. 5 for the adiabatic 1Σ+ states of LiK. We consid�ered a large range of internuclear distances because weexpected interesting features to arise from a global pic�ture also involving the highly excited states, whichbecome ionic at large internuclear distances. Weobserve that one after other, each adiabatic state has itsdipole that reaches the curve and then drops to zero.When combined, these curves reproduce piecewise thewhole R curve characteristic of the ionic dipole andcross forming nodes between consecutive pieces. Thepermanent dipole gives actually a direct illustration ofthe ionic character of the adiabatic electronic wavefunction. We thus access directly a visualization of theR�dependence of the charge distribution of the wavefunction. The distance for which two consecutive adi�abatic states have the same dipole locates the crossingof the ionic state with the corresponding neutral one.The sharpness of the slopes around the node for thedipole is closely related to the weakness of the avoidedcrossing for energy. The permanent dipoles for the X,A, and C, states are illustrated in Fig. 6, the results areconsistent with the previous analysis, the crossingsbeing shifted to larger internuclear distances.

The permanent dipole moment has been alsodetermined for the electronic states of the 3Σ+ symme�tries. Their permanent dipole moments are displayed

Table 7. Spectroscopic constants (Re, a.u.; De, cm–1; andTe, cm–1) for the 1,3Δ states of LiK compared with the avail�able results in the literature

Re De Te Reference

11Δ

6.83 3976 23467 This work6.66 4230 23525 [26]6.78 4026 23515 [40]

21Δ

7.22 4526 28964 This work6.86 4800 28837 [26]7.30 4310 29097 [40]

31Δ

6.95 5679 27854 This work7.03 5354 27604 [40]

13Δ

7.12 2997 24238 This work7.06 3278 24478 [26]7.03 3124 24418 [40]

23Δ

7.37 4125 28975 This work7.26 4521 29116 [26]7.31 4340 29066 [40]

33Δ

7.22 4739 30864 This work7.16 4477 29670 [26]7.17 4561 30691 [40]

3010 20 30 40 50

R, a.u.

20

10

−μ, a.u.

0

61Σ

+

51Σ

+

41Σ

+

31Σ

+21Σ

+

11Σ

+

Fig. 5. Permanent dipole moments (μ) for the low�lying1Σ+ states of the LiK, as a function of the internuclear dis�tance.

6 12 15 18 21R, a.u.

4

2

−μ, a.u.

0

21Σ

+

9 24

11Σ

+

31Σ

+

Fig. 6. Permanent dipole moments (μ) for X, A, and Cstates of LiK molecule.



in Fig. 7. They are not negligible and become particu�larly significant for higher excited states. At large dis�tance they decrease and vanish. They present strongvariations but only some of them can be related toavoid crossings. Adiabatic transition dipole momentsare shown in Fig. 8 for the coupled pairs 1–2, 1–3, 2–3,and 3–4. These adiabatic transition dipoles areimprinted by the ionic curve, as can be seen by theirchanges when the ionic curve is modified. Consistentwith the 2–3 potential curves avoided crossing around10.0 a.u. and also with the related crossing of the per�manent dipoles of the A and C states, the 1–2 and 1–3transition dipoles present an avoided crossing at thesedistances while the 2–3 transition dipole shows a peak.This behavior can be understood recalling that,around 10.0 a.u., the A and C states exchange theirionic character while the ground state X keeps its neu�tral character.

This change in the electronic wave function gener�ates a peak in the 2–3 transition dipole around

10.0 a.u. Similarly, the large peak in the 3–4 transitiondipole curve is related to the 3–4 potential curvesavoided crossing around 8.5 a.u. Being less avoided,the peak is more wide and higher when compared tothe case. Asymptotically, the X, A, C, and D statesreach the potassium atomic states, respectively, andthe transition dipole reaches the corresponding atomictransitions. Strong variations are clearly present.Again, the avoided crossings between the potentialcurves illustrate changes in the electronic wave func�tions and found some reflect here.

Potential Energy Curves and SpectroscopicConstants for the LiK+

Potential energy curves were obtained for all theelectronic states, for ionic system LiK+, dissociatinginto Li (2s, 2p, 3s, and 3p) + K+ and Li+ + K (4s, 4p,5s, 3d, and 4p), in the range of internuclear distancesvarying from 3 to 300 a.u. The potential energy curves

Table 8. Vibrational level spacings (Ev + 1 – E

v) of the 1Σ+ states of LiK (cm–1)

v X 1Σ+ 2 1Σ+ 3 1Σ+ 4 1Σ+ 5 1Σ+ 6 1Σ+

0

1 209.52 154.74 96.97 136.59 168.92 52.71

2 206.74 153.15 97.31 136.13 149.92 74.35

3 203.98 151.65 97.72 138.92 138.52 82.51

4 201.22 150.15 98.04 137.62 130.16 95.84

5 198.43 148.63 98.32 129.59 121.66 103.26

6 195.58 147.15 98.53 117.04 111.10 109.27

7 192.72 145.66 98.65 100.43 90.99 114.12

8 189.80 144.17 98.70 90.05 67.00 117.29

9 186.84 142.68 98.67 90.72 75.96 119.42

10 183.85 141.21 98.58 91.05 81.38 120.53

11 180.83 139.73 98.43 87.66 83.09 121.10

12 177.75 138.24 98.20 52.45 84.96 120.96

13 174.62 136.76 97.89 51.38 85.90 120.13

14 171.40 135.29 97.52 48.09 86.39 118.73

15 168.12 133.81 97.08 40.95 85.66 114.55

16 164.77 132.32 96.58 18.21 84.71 111.92

17 161.36 130.84 96.03 25.91 83.69 109.09

18 157.86 129.36 95.42 29.15 82.53 106.10

19 154.27 127.86 94.75 30.10 81.29 102.95

20 150.59 126.38 94.03 32.50 79.86 99.65

21 146.82 124.87 93.25 33.74 78.24 96.18

22 142.92 123.36 92.43 34.98 76.36 92.55

23 138.88 121.84 91.55 36.07 74.22 88.68

25 134.74 120.30 90.61 37.05 71.83 84.36

26 130.45 118.74 89.61 37.89 79.24

27 126.03 88.55 38.67 72.80

80



are displayed in Fig. 9 for 2Σ states, in Fig. 10 for 2Πand 2Δ states.

The ground state has the deepest well(De = 4857.9 cm–1) compared to the 2Σ excited statesor other symmetries showing the electron delocaliza�tion and the formation of a chemical bond. In addi�

tion, numerous avoided crossings exist between neigh�bor states for a given symmetry. Most of the avoidedcrossings are due to the charge transfer processbetween the two ionic systems K+Li and LiK+, andfinally between the two atoms Li and K of molecularsystem.

Table 9. Vibrational–level spacings (Ev + 1 – E

v) of the 1Π and 1Δ states of LiK (cm–1)

v 1 1Π 2 1Π 3 1Π 4 1Π 1 1Δ 2 1Δ 3 1Δ

0

1 234.25 105.39 94.14 113.83 86.34 110.42 89.85

2 137.18 103.95 91.89 112.26 86.82 107.14 95.53

3 50.22 102.43 89.64 110.63 94.86 106.00 104.09

4 117.41 100.87 87.59 108.94 83.02 104.32 93.03

5 134.43 99.19 85.63 107.13 81.90 103.49 94.50

6 93.33 97.42 83.88 105.22 78.64 101.36 94.91

7 27.47 95.58 82.29 103.23 75.61 100.63 92.65

8 91.64 91.47 80.84 10111.82 71.41 98.47 93.30

9 30.62 89.06 79.54 98.79 69.72 97.14 91.44

10 54.62 86.02 78.43 96.19 64.32 95.24 90.53

11 47.53 81.18 77.36 92.90 61.72 93.15 89.09

12 36.29 69.76 76.41 87.68 58.08 91.04 87.38

13 37.38 43.98 75.50 75.34 53.75 88.68 86.08

14 36.81 35.39 74.62 47.51 50.15 86.29 84.35

15 32.22 31.76 73.76 38.23 46.16 83.95 82.98

16 25.78 30.20 72.92 34.30 41.87 81.80 81.41

17 39.45 72.02 32.62 37.61 79.83 80.04

18 48.35

19 27.49

10R, a.u.

−4

0

μ, a.u.8

23Σ

+

13Σ

+

33Σ

+

20 30 40

4

−8

Fig. 7. Permanent dipole moments (μ) for the low�lying3Σ+ states of the LiK.

6R, a.u.

−2

0

μ, a.u.4

10 14 18

2

−4

3→4

1→2

1→3

2→3

Fig. 8. Transition adiabatic dipole moment 1–2, 1–3, 2–3and 3–4 of LiK.



Some excited states present more than one or twopositions of avoided crossings leading to an undulatingfeature of their potentials. The spectroscopic con�stants (Re, De, Te) of the ground and low�lying statesare presented in Table 10 and compared with othertheoretical works. Compared with other theoreticalworks [26, 42–44], and our study. Müller et al. [25]report the spectroscopic constants for many neutraland ionic alkali dimers but only for the ground state.Our ground state equilibrium distance Re is in verygood agreement as well as the dissociation energy De

with their work. We find for Re and De, respectively,7.34 a.u. and 4809.36 cm–1 and they found 7.29 a.u.and 4864.90 cm–1. The difference between our andtheir values is 0.05 a.u. for Re and 55.54 cm–1 for De.This good agreement between our results and those ofMüller et al, for the spectroscopic constants for theground state is not surprising since we used similarmethods. The only difference between our and theirworks is the large basis set and the 1�dependent cut–off radius used in our study since they used a smallbasis set and an average cut off radius for all (s, p, d,and f ) orbitals. In the same time there is good agree�ment for the ground state between our spectroscopicconstants and that of Patil [43]. They found for Re andDe respectively, 7.25 a.u. and 4838.70 cm–1. For our bestknowledge, there are no experimental results for theground and excited states. Equilibrium distance of allthese states is found for intermediate and large values ofthe internuclear distance, 12.76 to 23.25 a.u. We havestudied 1–10 2Σ, 1–6 2Π, and 1–4 2Δ. In our work, wefound that l, 2, 3, 4, 5, and 6 2Σ were bond states.

−0.20

10 20 30 40R, a.u.

−0.10

0

E, a.u.

Li(2s)+K+

0.10

Li++K(4s)

Li(2p)+K+

Li++K(3p)

Li(3s)+K+

Li++K(5s)

42Σ

12Σ

22Σ

32Σ

52Σ

Fig. 9. Potential energy curves for the six lowest 2Σ electronicstates of LiK+ dissociating into Li (2s, 2p, 3s and 3p) + K+

and Li+ + K (4s, 4p, 5s, 3d and 5p).

10 20 30 40R, a.u.

−0.10

0

E, a.u.

0.10

Li(2p)+K+

Li++K(4p)

Li(3p)+K+

0.20

Li++K(5p)

Fig. 10. Potential energy curves for the 2Π (solid lines) andthe 2Δ (dashed lines) lowest electronic states of LiK+ disso�ciating into Li (2p and 3p) + K+ and Li+ + K (4p, 3d and 5p).

Table 10. Spectroscopic constants (Re, a.u.; De, cm–1; andTe, cm–1) for the 2Σ, 2Π, and 2Δ states of LiK+ system com�pared with the available results in the literature

State Re De Te Reference

1 2Σ 7.29 4864.90 00.00 This work

7.34 4663.00 00.00 [42]

7.32 4807.00 00.00 [26]

7.26 4847.00 00.00 [25]

7.40 3928.01 00.00 [44]

2 2Σ 12.65 1044.70 8477.18 This work

12.79 1038.00 [42]

12.76 1059.77 [44]

3 2Σ 16.07 749.06 14901.2 This work

15.80 751.49 [44]

4 2Σ 19.13 5297.48 21494.62 This work

18.98 5386.53 [44]

5 2Σ 23.25 1518.9 27207.75 This work

6 2Σ 32.0 930.04 29489.62 This work

1 2Π 9.13 246.00 00.00 This work

8.90 240.28 [44]

2 2Π 15.87 126.7 6597.17 This work

16.68 105.74 [44]

3 2Π 20.23 1647.65 15103.32 This work

19.98 1663.47 [44]

4 2Π 29.68 395.84 16003.38 This work

5 2Π Repulsive Repulsive 16405.46 This work

6 2Π Repulsive Repulsive 18328.05 This work

1 2Δ 8.57 1425.33 This work

2 2Δ 9.4 2543.76 1279.31 This work

3 2Δ 11.0 3678.95 5877.29 This work

82



For the 2 and 3 2Σ states, we remark a good agree�ment for Re between our values and those of the theo�retical work [44]. We find respectively 12.65 and 16.07a.u., and they found 12.75 and 15.80 a.u. The differ�ence between the two calculated Re is about 0.25 a.u.For De we find for 2, 3 2Σ states respectively, 1044.4 and749.06 cm–1 and they found 1059.77 and 751.49 cm–1.We remark that our De, for the two states, is lower thantheir values but the difference does not exceed 16 cm–1,showing a good agreement between the two calcula�tions.

In our work the 4 and 5 2Σ states have been foundattractive with potential wells of, respectively, 5297.4,1518.9 and 19.3 and 23.25 a.u. equilibrium distances.

For the 6 2Σ state we find respectively, 32.0 a.u. forequilibrium distance and 930.04 cm–1 for De. Further�more, only the 1 2Π and 2 2Π states were found to bebound states in their calculation. Our values Re and De

are, respectively, 9.13 a.u. and 246.0 cm–1 for 1 2Π stateand 16.68 a.u. and 105.74 cm–1 for 3 2Π state. Our val�ues are in good agreement with those of ref [44], inwhich, the values are respectively, 8.9 a.u. and240.28 cm–1 for 1 2Π state and 16.68 a.u. and105.74 cm–1 for 3 2Π state. For the 2Δ electronic stateswe present the three lowest states dissociating into Li(3d) + K+ and Li+ + K (3d and 4d). Their spectro�scopic constants are reported in Table 10.


v) of 2Σ of LiK+ (cm–1)

v 1 2Σ 2 2Σ 3 2Σ 4 2Σ 5 2Σ 6 2Σ

0

1 114.33 62.19 32.85 52.00 30.62 25.18

2 112.65 61.39 31.47 51.61 30.24 25.06

3 111.07 60.62 30.05 51.22 29.87 24.93

4 109.34 59.84 28.64 50.84 29.53 24.82

5 107.64 59.04 27.25 50.45 29.16 24.72

6 105.88 58.23 25.87 50.06 28.64 24.63

7 104.14 5.59 24.48 49.66 27.96 24.54

8 102.35 56.57 23.09 49.257 27.12 24.46

9 100.55 55.70 21.74 48.85 26.21 24.39

10 98.71 54.81 20.44 48.45 25.29 24.31

11 96.84 53.91 19.16 48.03 24.38 24.23

12 94.93 52.99 17.90 47.62 23.46 24.15

13 93.00 52.05 16.69 47.19 22.53 24.07

14 91.00 51.09 15.55 46.75 21.61 23.99

15 88.99 50.11 14.45 46.32 20.67 23.92

16 86.93 49.09 13.43 45.86 19.73 23.84

17 84.80 48.05 12.47 45.41 18.78 23.77

18 82.66 46.99 44.94 17.79 23.69

19 80.49 45.89 44.46 16.79 23.62

20 78.27 44.75 43.97 23.54

21 76.00 43.58 43.47 23.46

22 73.66 42.37 42.96 23.38

23 71.27 41.14 42.44 23.29

24 68.84 39.85 41.91

25 66.34 38.53 41.38

26 112.65

27 111.07

28 109.33



Vibrational level spacings (Ev + 1 – E

v) values of 2Σ,

2Π, and 2Δ electronic states of LiK+ are given inTables 11 and 12.

CONCLUSION

In this paper we have performed an accurate calcu�lation of the potential energy curves and their spectro�scopic constants for the ground and numerous excitedstates for LiK and LiK+.The computational methodused in this work is based on a non–empiricalpseudo–potentials approach for Li and K cores. Fur�thermore, computed energies have included the core–core and core–valence correlations. The adiabaticpotential energy curves of all electronic states and theirspectroscopic constants of LiK molecule dissociatinginto K (4s, 4p, 5s, 3d, and 5p) + Li (2s, 2p, 3s, and 3p)and of LiK+ ionic molecule dissociating into Li (2s,2p, 3s, and 3p) + K+ and Li+ + K (4s, 4p, 5s, 3d, and 5p)have been calculated. Using the pseudo–potentialapproach for Li and K atoms, the LiK+ becomes one–electron system and potential energy curves were com�puted with a good accuracy. The Agreement betweenthe spectroscopic obtained in our work and those ofavailable references for the ground and first excitedstates, is very good. The presented data can helpexperimental investigation on the LiK+ molecular ion.Despite the relative simplicity of LiK+ ionic system,few theoretical works have been done. To our bestknowledge, there is no experimental study on LiK+ incontrast to some neutral molecules, which wereexplored by theory and experience. Finally we areexpecting to reach a similar accuracy as many previous

works using the same techniques. Since the mainrestriction in the accuracy of the calculation is thebasis set limitation only.

ACKNOWLEDGMENTS

This work is supported by the National Plan forScience and Technology of King Abdulaziz City forScience and Technology (KACST) under grant num�ber 10�MAT1217�02.

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Documents

Theoretical study of LiK and LiK+ in adiabatic representation