Calculation of adiabatic potentials of Li2

Chemical Physics 323 (2006) 563–573 www.elsevier.com/locate/chemphys

Calculation of adiabatic potentials of Li2 P. Jasik *, J.E. Sienkiewicz Department of Theoretical and Mathematical Physics, Faculty of Applied Physics and Mathematics, Gdan´sk University of Technology, ul. Narutowicza 11/12, 80-952 Gdan´sk, Poland Received 9 June 2005; accepted 19 October 2005 Available online 16 November 2005

Abstract We report adiabatic potential energy curves of the lithium dimer. Our curves are tabulated according to internuclear distance from 3.2a0 to 88a0. We compare our theoretical results with the ones calculated by other authors and potential energy curves derived from experiments. In our approach we use the configuration interaction method where only the valence electrons of Li atom are treated explicitly. The core electrons are represented by pseudopotential. All calculations are performed by means of MOLPRO program package. Ó 2005 Elsevier B.V. All rights reserved. PACS: 31.15.Ar; 31.15.Ne; 31.25.Nj; 32.80.Pj; 33.80.Ps; 34.20.Cf Keywords: Adiabatic potential energy curves; Li2 dimer; Cold molecules

1. Introduction The lithium dimer has attracted attention of experimentalists and theoreticians for many years, mainly because it is the second, just after the molecular hydrogen, smallest stable homonuclear molecule. It can serve as a convenient prototype for testing theoretical methods, which can be further applied to heavier alkali dimers. The first ab initio calculations on Li2 were performed by Konowalow and coworkers [1–5], including the work of Konowalow and Fish [5] where they computed and analyzed the potential energy curves in the long-range region of the 26 lowest lying states. They accounted for atomic polarizabilities in order to describe the atomic multiple interaction and the molecular valence-shell polarization. By using the atomic effectivecore potential (ECP) and high quality basis set they were able to perform self-consistent-field (SCF) and configuration-interaction (CI) calculations. Obtained results allowed for valuable comparisons with experimental spectroscopic

*

Corresponding author. Tel.: +48 58 347 29 49; fax: +48 58 347 28 21. E-mail addresses: [email protected] (P. Jasik), [email protected] (J.E. Sienkiewicz). 0301-0104/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.10.025

data of He et al. [6]. Nevertheless, as discussed by Mu¨ller and Mayer [7] the usage by Konowalow and co-workers of the originally atomic ECP leads to defect of an unbalanced treatment of core polarization effects in molecule since it lacks electron–other core polarization contributions. Next, Schmidt-Mink et al. [8] performed SCF/ valence CI calculation where they implemented the effective potential to reduce the above defect. In this approach, they employed the technique of self-consistent electron pairs (SCEP) to simultaneously optimize several electronic states, combined with a core polarization potential (CPP) for an efficient treatment of inter-shell correlation effect. Their results also achieved very good agreement with experimental data of He et al. [6]. In turn, Poteau and Spiegelmann [9] reported results for 49 states obtained by calculations, where effective core pseudopotentials and l-dependent core polarization potentials were employed. Recently, Song et al. [10] studied both theoretically and 1 þ experimentally the 51 Rþ g and 6 Rg Rydberg states. Several spectroscopic studies of lithium dimer were repeated. These 1 þ include analysis of the 11 Rþ u –1 Rg band system [11], near infrared two-photon laser spectroscopy of the 21 Rþ g state [6], ultrafast spectroscopy of wavelength-dependent coherent photoionization in the 51 Rþ g state [12], state to state

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P. Jasik, J.E. Sienkiewicz / Chemical Physics 323 (2006) 563–573

collision energy transfer in singlet state [13] and observation of predissociation in the 21Pu state in the 6Li7Li isotopomer [14]. Quite recently need for theoretical description of long range potential curves has been caused by molecular photoassociative spectroscopy of cold alkali atoms [15]. Two-photon photoassociation of colliding ultracold lithium atoms was used by Abraham et al. [16] to probe the lowest triplet state 13 Rþ u of Li2 in order to determine the scattering lengths necessary to predict the Bose–Einstein condensate (BEC). Also photoassociative spectroscopy was used to obtain high-lying vibrational levels of 11 Rþ u and 13 Rþ g states [17] and the intensity dependent rate and shift of a transition in a quantum degenerate gas [18]. Very recently our motivation to calculate the potential curves of highly excited states of the lithium dimer was strengthened by Johim et al. [19] who reported on the BEC of more than 105 Li2 molecules in an optical dipole trap and Chin et al. [20] who studied fermionic pairing in an ultracold twocomponent degenerate gas of lithium atoms by observing a pairing gap in radio-frequency excitation spectra. The experimentally observed states can be characterized by their potential curves which may serve for direct comparison with theoretical curves. For instance, the application of Doppler-free UV–visible optical–optical double resonance polarization spectroscopy allowed for determination of 1 21 Rþ ßbski u and C Pu states [21]. Quite recently, Jastrze et al. [22] and Pashov et al. [23] reported potential curves 1 þ for the 31 Rþ g and 4 Rg states. They used the inverted perturbation approach (IPA) developed by Pashov et al. [24]. In turn, Bouloufa et al. [25] constructed the potential energy curve for the 11Pu state by combining a short-range potential obtained by direct fit of accurate spectroscopic measurements with a long-range potential generated by asymptotic calculation of Coulombic and exchange energies. Our present calculations are based on the self-consistent-field configuration interaction (SCF CI) scheme. We use a substantially richer basis set than in our first report [26]. The present calculation also use the ECP which enables us to provide high lying adiabatic potential curves not calculated in previous theoretical approaches. 2. Theoretical method We consider the interaction between two alkali atoms. Let R be the separation between the nuclei of these atoms. In the Born–Oppenheimer approximation we seek for the solutions of the Schro¨dinger equation H Wi ðr; RÞ ¼ Ei ðRÞWi ðr; RÞ;

H ¼T þV;

ð3Þ

where T stands for the kinetic energy operator of the valence electrons and V represents the interaction operator. The latter is put into the form V ¼

X

ðV k þ V kpol Þ þ

k

N X 1 þ V cc . r j>i¼1 ij

ð4Þ

Here Vk describes Coulomb and exchange interaction as well as the Pauli repulsion between the valence electrons and core k and is defined as ! N X Qk X k k 2 k k V ¼ þ Bl;k expðbl;k rki ÞP l ; ð5Þ rki lk i¼1 where Qk = 1 denotes the net charge of lithium core k, P kl is the projection operator onto the Hilbert subspace of angular symmetry l with respect to the Li+-core and N is the number of the valence electrons. The parameters Bklk and bklk define the semi-local energy-consistent pseudopotentials. The second interaction term in Eq. (4) is the polarization term which describes, among others, core–valence correlation effects [20] and, in the case of atom A, is taken as 1 2 VA ð6Þ pol ¼ aA FA ; 2 where aA = 0.1915a0 [27] is the dipole polarizability of the A+ core and FA is the electric field generated at its site by the valence electrons and the other core. The latter can be written as X rAi Q R FA ¼ 1 exp dA r2Ai B3 1 exp dA R2 ; 3 r R Ai i ð7Þ 0:831a2 0

[27] is the cutoff parameter. The third where dA ¼ term in Eq. (4) represents the Coulomb repulsion between the valence electrons, whereas the last term describes the core–core interaction. Since the alkali atoms cores are well separated, we choose a simple point–charge Coulomb interaction in the latter case. Detailed formulas are given in the papers of Czuchaj and co-workers [28–33].

ð1Þ

where H is the Hamiltonian of a diatomic system, Ei (R) is the i-th adiabatic energy and Wi (r,R) describes the state of the system related to this energy, r represents the electronic coordinates and R is the relative position of the first atom to the second one. The Hamiltonian of the system can be written as H ¼ H A þ H B þ V AB ;

where HA and HB are the Hamiltonians of the isolated atoms and VAB is the interaction between them. In the present approach only the valence electrons are treated explicitly, but the atomic cores are represented by l-dependent pseudopotentials. Consequently the total Hamiltonian in Eqs. (1), (2) can be expressed as

ð2Þ

Table 1 Comparison of asymptotic energies with other theoretical and experimental results

2s + 2p 2s + 3s 2p + 2p 2s + 3p

Present work

Schmidt-Mink [8]

Poteau [9]

Bashkin [39]

14 911 27 201 29 822 30 931

14 924 27 209 29 876 30 934

14 904 27 212 – 30 927

14 904 27 206 29 808 30 925

Energies are shown in cm1 units.


565

Table 2 1 þ 1 þ 1 þ 1 þ 1 þ 1 þ The Li2 molecule adiabatic potential energy curves for 11 Rþ g , 2 Rg , 3 Rg , 4 Rg , 1 Ru , 2 Ru and 3 Ru states R(a0)

11 Rþ g

21 Rþ g

31 Rþ g

41 Rþ g

11 Rþ u

21 Rþ u

3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00 6.20 6.40 6.60 6.80 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 16.00 17.00 18.00 19.00 20.00 22.00 24.00 26.00 28.00 30.00 32.00 34.00 36.00 38.00 40.00 42.00 44.00 46.00 48.00 50.00 52.00 54.00 56.00 58.00

3152.98 113.42 2360.41 4319.57 5825.09 6940.40 7 725.27 8234.16 8515.69 8612.28 8560.29 8390.55 8129.00 7797.39 7413.95 6994.00 6550.41 6094.00 5633.86 5177.55 4621.95 4091.69 3593.92 3133.66 2713.92 2336.04 1999.86 1704.02 1223.52 869.96 615.93 436.27 310.36 222.46 161.08 118.08 87.76 66.21 50.70 39.41 24.77 16.31 11.14 7.84 5.65 3.11 1.82 1.42 0.72 0.48 0.33 0.24 0.17 0.13 0.10 0.08 0.07 0.06 0.05 0.05 0.04 0.04 0.03 0.03

30025.64 26708.38 23815.65 21350.95 19290.59 17594.63 16217.03 15111.26 14233.49 13544.32 13009.42 12599.50 12289.97 12060.44 11894.19 11777.77 11700.50 11654.19 11632.74 11631.90 11655.68 11704.33 11776.54 11871.87 11990.01 12130.31 12291.43 12471.31 12875.55 13314.59 13756.34 14170.58 14533.24 14829.31 15054.13 15212.27 15314.24 15372.76 15399.83 15405.41 15380.14 15334.30 15284.80 15238.37 15197.18 15130.94 15082.47 15063.32 15020.30 15000.18 14984.68 14972.55 14962.94 14955.23 14948.97 14943.84 14939.60 14936.06 14933.08 14930.56 14928.41 14926.57 14924.98 14923.60

36910.79 33099.65 29908.77 27280.51 25147.36 23440.98 22097.39 21059.21 20276.19 19704.92 19308.30 19054.76 18917.43 18873.29 18902.28 18986.63 19110.35 19259.04 19420.33 19584.69 19786.12 19980.75 20170.75 20358.27 20542.62 20720.04 20884.64 21029.85 21240.84 21337.98 21355.05 21350.53 21375.63 21461.25 21618.56 21844.20 22126.37 22450.24 22801.54 23168.32 23913.39 24636.13 25308.41 25908.46 26407.30 26970.36 27123.67 27150.98 27182.53 27189.78 27193.64 27195.91 27197.35 27198.30 27198.95 27199.41 27199.74 27199.98 27200.15 27200.28 27200.39 27200.46 27200.52 27200.57

38600.08 35351.71 33081.52 31283.14 29559.44 27802.53 26395.05 25275.40 24388.85 23687.39 23129.82 22682.32 22319.40 22024.42 21788.77 21609.86 21488.64 21427.35 21427.67 21489.02 21644.86 21873.31 22153.53 22465.18 22791.55 23120.19 23442.59 23753.80 24338.50 24886.61 25417.22 25930.29 26407.47 26822.23 27148.37 27373.47 27510.51 27587.01 27626.90 27645.62 27652.98 27649.11 27654.44 27687.70 27777.87 28265.30 29004.87 29370.28 29811.97 29815.96 29818.17 29819.52 29820.37 29820.94 29821.33 29821.59 29821.78 29821.92 29822.02 29822.10 29822.15 29822.19 29822.22 29822.25

23805.44 19921.24 16686.71 14031.58 11878.69 10154.24 8792.34 7734.62 6930.38 6336.43 5916.51 5640.35 5482.83 5423.09 5443.79 5530.44 5670.92 5855.04 6074.28 6321.42 6660.47 7023.90 7403.70 7793.30 8187.28 8581.13 8971.14 9354.22 10089.61 10771.22 11388.01 11933.40 12405.03 12804.56 13137.09 13410.12 13632.32 13812.42 13958.49 14077.49 14256.23 14381.82 14473.99 14544.17 14599.15 14679.00 14733.24 14771.65 14799.70 14820.67 14836.68 14849.12 14858.93 14866.77 14873.12 14878.31 14882.60 14886.17 14889.17 14891.71 14893.87 14895.72 14897.32 14898.70

40291.92 36302.24 32977.51 30250.43 28043.92 26282.04 24895.49 23823.01 23011.62 22416.30 21999.21 21728.76 21578.71 21527.25 21556.19 21650.18 21796.22 21983.12 22201.13 22441.53 22760.95 23085.33 23391.57 23638.05 23748.75 23668.09 23456.53 23198.46 22693.47 22284.85 21998.86 21836.14 21788.15 21842.02 21982.53 22193.45 22458.78 22763.65 23095.07 23442.27 24151.54 24843.40 25487.80 26059.95 26527.49 27026.23 27150.08 27181.42 27191.52 27195.55 27197.49 27198.56 27199.21 27199.64 27199.93 27200.13 27200.28 27200.39 27200.47 27200.54 27200.59 27200.62 27200.65 27200.67 (continued on

31 Rþ u 45645.37 41591.55 38214.89 35445.64 33204.93 31415.18 30005.55 28913.55 28085.09 27473.47 27035.66 26710.54 26423.57 26203.19 26053.22 25958.83 25903.36 25869.93 25841.35 25800.41 25708.88 25558.10 25360.50 25168.37 25077.64 25161.95 25373.24 25634.13 26164.63 26626.13 26988.21 27251.50 27432.13 27551.08 27627.17 27674.38 27702.26 27717.05 27722.89 27722.72 27712.76 27701.26 27701.50 27731.69 27823.15 28320.66 29050.11 29747.75 30346.21 30766.47 30896.14 30919.88 30926.13 30928.35 30929.36 30929.92 30930.26 30930.49 30930.66 30930.78 30930.87 30930.94 30931.00 30931.04 next page)

566


Table 2 (continued) R(a0)

11 Rþ g

21 Rþ g

31 Rþ g

41 Rþ g

11 Rþ u

21 Rþ u

31 Rþ u

60.00 62.00 64.00 66.00 68.00 70.00 72.00 74.00 76.00 78.00 80.00 82.00 84.00 86.00 88.00

0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03

14922.40 14921.35 14920.42 14919.61 14918.89 14918.24 14917.67 14917.16 14916.70 14916.28 14915.91 14915.57 14915.26 14914.99 14914.73

27200.61 27200.64 27200.66 27200.68 27200.69 27200.70 27200.71 27200.72 27200.72 27200.73 27200.73 27200.74 27200.74 27200.74 27200.75

29822.27 29822.29 29822.30 29822.31 29822.32 29822.33 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.36 29822.36 29822.36

14899.91 14900.97 14901.89 14902.71 14903.44 14904.08 14904.65 14905.17 14905.63 14906.05 14906.42 14906.76 14907.07 14907.35 14907.60

27200.69 27200.70 27200.71 27200.72 27200.73 27200.73 27200.74 27200.74 27200.75 27200.75 27200.75 27200.75 27200.75 27200.75 27200.75

30931.07 30931.10 30931.11 30931.13 30931.14 30931.15 30931.16 30931.17 30931.17 30931.18 30931.18 30931.18 30931.19 30931.19 30931.19


Table 3 3 þ 3 þ 3 þ 3 þ 3 þ 3 þ 3 þ The Li2 molecule adiabatic potential energy curves for 13 Rþ u , 2 Ru , 3 Ru , 4 Ru , 5 Ru , 1 Rg , 2 Rg and 3 Rg states R(a0)

1 3 Rþ u

23 Rþ u

33 Rþ u

43 Rþ u

53 Rþ u

13 Rþ g

23 Rþ g

33 Rþ g

3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00 6.20 6.40 6.60 6.80 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 16.00

18614.28 15220.69 12325.19 9888.40 7860.91 6189.69 4823.09 3713.33 2817.77 2099.34 1526.41 1072.32 714.85 435.64 219.51 54.03 71.00 163.90 231.40 278.94 316.85 336.93 344.17 342.32 334.14 321.69 306.48 289.58 253.70 218.09 184.84 154.92 128.71 106.24 87.29 71.55 58.60 48.04 39.44 32.46 22.16

39539.21 35578.15 32280.65 29573.41 27377.86 25618.25 24225.63 23139.25 22306.78 21683.63 21232.21 20920.97 20723.50 20617.53 20583.86 20604.97 20661.94 20723.64 20704.47 20495.35 20108.86 19713.34 19337.42 18986.23 18660.34 18359.20 18081.92 17827.46 17382.36 17013.92 16711.40 16464.07 16261.99 16096.38 15959.83 15846.31 15750.95 15669.95 15600.34 15539.84 15439.78

44903.74 40941.79 37637.68 34914.35 32693.66 30899.16 29456.02 28288.01 27310.75 26440.01 25631.62 24882.01 24193.44 23564.08 22990.12 22468.36 21999.57 21599.26 21339.88 21319.56 21474.81 21688.21 21920.94 22160.57 22401.42 22640.56 22876.46 23108.40 23559.11 23990.84 24401.11 24786.13 25141.32 25461.95 25744.10 25985.65 26187.27 26352.51 26486.84 26596.19 26760.40

46180.63 42286.27 39032.05 36317.32 34032.97 32093.77 30468.32 29138.88 28084.03 27287.72 26713.55 26306.22 26021.55 25831.08 25715.50 25660.12 25652.86 25683.60 25743.95 25827.09 25955.01 26102.36 26263.37 26433.52 26608.73 26784.90 26957.81 27123.21 27416.28 27642.77 27801.53 27910.07 27992.84 28071.00 28158.45 28262.02 28382.69 28517.27 28660.25 28805.45 29081.20

47733.56 43471.91 39863.70 36890.23 34479.41 32535.04 30963.96 29692.76 28679.93 27899.72 27327.29 26934.63 26692.34 26572.39 26550.21 26605.55 26722.06 26886.60 27088.44 27318.76 27635.58 27973.41 28321.98 28671.86 29013.28 29334.58 29619.93 29848.21 29630.18 29605.01 29598.00 29603.89 29618.10 29636.94 29657.70 29678.50 29698.18 29716.12 29732.03 29745.87 29767.88

26768.74 22450.56 18944.63 16133.01 13903.37 12155.91 10805.34 9779.72 9018.96 8473.24 8101.53 7870.21 7751.86 7724.11 7768.58 7870.17 8016.36 8196.80 8402.92 8627.69 8926.12 9235.62 9549.13 9861.22 10167.75 10465.62 10752.59 11027.08 11534.96 11985.51 12379.25 12719.34 12874.97 13030.58 13186.18 13341.78 13497.39 13652.99 13808.60 13964.20 14275.52

37336.86 33454.56 30193.81 27500.96 25309.34 23550.17 22158.81 21077.39 20255.56 19650.27 19225.13 18949.60 18798.25 18749.88 18786.78 18894.07 19059.16 19271.33 19521.49 19801.88 20184.88 20593.98 21020.19 21456.07 21895.37 22332.82 22763.95 23184.88 23983.00 24704.06 25328.47 25839.36 25953.40 26067.43 26181.45 26295.48 26409.50 26523.52 26637.55 26751.57 26979.70

41348.13 37235.96 33830.75 31054.29 28821.74 27051.27 25668.83 24609.73 23818.56 23248.54 22860.53 22621.99 22506.00 22490.31 22556.45 22688.99 22875.00 23103.59 23365.57 23653.24 24039.12 24444.51 24860.83 25281.10 25699.56 26111.46 26512.85 26900.42 27623.46 28263.03 28802.14 29137.83 29223.78 29309.71 29395.64 29481.57 29567.50 29653.43 29739.36 29825.30 29997.22


567


13 Rþ u

23 Rþ u

33 Rþ u

43 Rþ u

53 Rþ u

13 Rþ g

23 Rþ g

3 3 Rþ g

17.00 18.00 19.00 20.00 22.00 24.00 26.00 28.00 30.00 32.00 34.00 36.00 38.00 40.00 42.00 44.00 46.00 48.00 50.00 52.00 54.00 56.00 58.00 60.00 62.00 64.00 66.00 68.00 70.00 72.00 74.00 76.00 78.00 80.00 82.00 84.00 86.00 88.00

15.34 10.79 7.72 5.61 3.11 1.82 1.12 0.72 0.48 0.33 0.24 0.17 0.13 0.10 0.08 0.07 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03

15360.42 15296.12 15243.23 15199.24 15131.31 15082.53 15046.86 15020.30 15000.18 14984.68 14972.55 14962.94 14955.23 14948.97 14943.84 14939.60 14936.06 14933.08 14930.56 14928.41 14926.57 14924.98 14923.60 14922.40 14921.35 14920.42 14919.61 14918.89 14918.24 14917.67 14917.16 14916.70 14916.28 14915.91 14915.57 14915.26 14914.99 14914.73

26875.76 26959.43 27021.08 27066.71 27125.37 27157.28 27174.76 27184.59 27190.31 27193.78 27195.95 27197.36 27198.30 27198.95 27199.41 27199.74 27199.98 27200.15 27200.28 27200.39 27200.47 27200.53 27200.57 27200.61 27200.64 27200.66 27200.68 27200.69 27200.70 27200.71 27200.72 27200.72 27200.73 27200.73 27200.74 27200.74 27200.74 27200.75

29313.18 29488.05 29607.77 29684.30 29761.18 29791.73 29805.34 29812.21 29815.98 29818.17 29819.52 29820.37 29820.94 29821.33 29821.59 29821.78 29821.92 29822.02 29822.10 29822.15 29822.19 29822.22 29822.25 29822.27 29822.29 29822.30 29822.31 29822.32 29822.33 29822.33 29822.33 29822.34 29822.34 29822.34 29822.35 29822.35 29822.36 29822.36

29783.62 29794.70 29802.47 29807.93 29814.55 29817.99 29819.83 29820.87 29821.46 29821.81 29822.02 29822.15 29822.23 29822.28 29822.31 29822.33 29822.35 29822.36 29822.36 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.36 29822.37 29822.37

14388.20 14475.15 14543.43 14597.89 14677.92 14732.56 14771.23 14799.43 14820.50 14836.57 14849.04 14858.87 14866.73 14873.09 14878.29 14882.58 14886.16 14889.16 14891.70 14893.86 14895.72 14897.32 14898.70 14899.91 14900.97 14901.89 14902.71 14903.44 14904.08 14904.65 14905.17 14905.63 14906.05 14906.42 14906.76 14907.07 14907.35 14907.60

27028.52 27069.25 27101.78 27126.93 27160.06 27178.01 27187.58 27192.78 27195.71 27197.42 27198.48 27199.15 27199.59 27199.90 27200.11 27200.27 27200.38 27200.47 27200.53 27200.58 27200.62 27200.65 27200.67 27200.69 27200.70 27200.71 27200.72 27200.73 27200.73 27200.74 27200.74 27200.74 27200.75 27200.75 27200.75 27200.75 27200.75 27200.75

30193.17 30360.25 30498.53 30608.26 30754.00 30833.31 30876.67 30900.55 30913.63 30920.80 30924.77 30927.03 30928.36 30929.20 30929.75 30930.12 30930.38 30930.57 30930.71 30930.82 30930.90 30930.96 30931.01 30931.05 30931.08 30931.10 30931.12 30931.13 30931.14 30931.15 30931.16 30931.17 30931.17 30931.18 30931.18 30931.19 30931.19 30931.19


Table 4 The Li2 molecule adiabatic potential energy curves for 11Pg, 21Pg, 11Pu, 21Pu, 13Pg, 23Pg, 13Pu, 23Pu, 11Dg and 13Du states R(a0)

11Pg

21Pg

11Pu

21Pu

13Pg

23Pg

13Pu

23Pu

3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00 6.20 6.40 6.60

31693.55 28814.78 26251.45 24003.14 22066.05 20424.10 19050.60 17913.54 16979.96 16218.71 15601.78 15104.77 14706.91 14390.72 14141.63 13947.53 13798.44 13686.10

41342.87 37609.54 34540.10 32031.99 29990.21 28337.39 27011.38 25960.86 25142.30 24518.31 24056.89 23730.85 23517.30 23397.05 23354.01 23374.66 23447.60 23563.20

26364.14 23321.78 20690.58 18488.62 16697.36 15277.02 14179.94 13357.98 12766.18 12364.45 12118.01 11997.20 11976.98 12036.27 12157.32 12325.03 12526.52 12750.76

37458.06 34298.10 31669.37 29469.27 27642.17 26150.95 24959.86 24031.10 23326.83 22811.48 22452.97 22223.27 22098.27 22057.44 22083.37 22161.33 22278.93 22425.95

35180.66 32500.84 29625.86 26966.24 24712.23 22855.57 21348.36 20138.32 19177.12 18422.33 17837.53 17391.81 17059.00 16816.90 16646.56 16531.69 16458.22 16414.06

37947.11 34449.77 32111.26 30400.75 29009.62 27837.25 26834.33 25967.30 25210.28 24544.03 23956.14 23438.94 22986.89 22595.37 22260.56 21979.45 21749.78 21569.86

11248.38 8892.35 6998.49 5519.10 4403.12 3599.88 3062.04 2746.67 2615.66 2635.93 2779.10 3021.14 3341.75 3723.91 4153.32 4617.93 5107.56 5613.59

38267.28 34464.34 31306.49 28731.86 26667.12 25038.60 23778.07 22825.12 22127.52 21640.87 21327.80 21157.04 21102.58 21142.72 21259.20 21436.49 21661.12 21921.11

11Dg 29252.64 27501.14 26026.99 24787.17 23754.14 22910.11 22241.37 21734.45 21374.56 21145.73 21031.60 21016.23 21084.77 21223.71 21420.99 21665.96 21949.31 22262.87 (continued on

13Du 50257.52 47684.43 45451.03 43508.25 41815.81 40339.73 39050.84 37923.87 36936.94 36071.19 35310.50 34641.14 34051.44 33531.48 33072.77 32668.03 32310.97 31996.11 next page)

568



11Pg

21Pg

11Pu

21Pu

13Pg

23Pg

13Pu

23Pu

11Dg

13Du

6.80 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 16.00 17.00 18.00 19.00 20.00 22.00 24.00 26.00 28.00 30.00 32.00 34.00 36.00 38.00 40.00 42.00 44.00 46.00 48.00 50.00 52.00 54.00 56.00 58.00 60.00 62.00 64.00 66.00 68.00 70.00 72.00 74.00 76.00 78.00 80.00 82.00 84.00 86.00 88.00

13603.76 13545.89 13501.17 13480.64 13478.84 13491.47 13515.17 13547.27 13585.64 13628.60 13723.01 13822.39 13921.40 14016.57 14105.77 14187.84 14262.28 14329.12 14388.69 14441.52 14488.25 14529.51 14598.10 14651.72 14693.94 14727.51 14754.47 14794.35 14821.68 14841.06 14855.20 14865.76 14873.81 14880.06 14884.99 14888.93 14892.11 14894.72 14896.87 14898.65 14900.16 14901.43 14902.51 14903.44 14904.24 14904.93 14905.54 14906.06 14906.53 14906.94 14907.30 14907.62 14907.91 14908.17 14908.40 14908.61 14908.79 14908.96 14909.12 14909.26 14909.38

23713.32 23891.11 24143.54 24420.86 24715.44 25021.04 25332.49 25645.51 25956.51 26262.42 26848.89 27387.79 27866.27 28275.57 28611.94 28877.65 29080.68 29232.64 29345.79 29430.82 29495.98 29547.12 29621.94 29673.69 29710.84 29737.92 29757.78 29783.36 29797.77 29806.24 29811.44 29814.74 29816.91 29818.38 29819.39 29820.11 29820.63 29821.01 29821.30 29821.51 29821.68 29821.80 29821.90 29821.99 29822.05 29822.10 29822.14 29822.17 29822.20 29822.23 29822.24 29822.26 29822.27 29822.28 29822.29 29822.30 29822.31 29822.31 29822.32 29822.32 29822.33

12988.40 13231.53 13533.23 13823.05 14093.36 14338.99 14556.87 14745.81 14906.11 15039.22 15233.22 15349.37 15409.31 15431.09 15428.32 15410.63 15384.54 15354.34 15322.79 15291.58 15261.71 15233.73 15184.16 15142.89 15108.85 15080.81 15057.63 15022.28 14997.30 14979.21 14965.82 14955.71 14947.93 14941.86 14937.05 14933.20 14930.07 14927.50 14925.38 14923.61 14922.12 14920.86 14919.79 14918.87 14918.07 14917.38 14916.78 14916.26 14915.80 14915.39 14915.03 14914.70 14914.42 14914.16 14913.93 14913.72 14913.54 14913.37 14913.21 14913.08 14912.95

22594.23 22777.54 23021.24 23276.41 23540.74 23813.69 24095.46 24386.08 24685.04 24991.02 25615.56 26238.51 26836.07 27385.96 27869.99 28276.32 28602.17 28854.65 29047.51 29195.56 29310.98 29402.49 29535.75 29624.42 29684.10 29724.44 29751.85 29783.86 29799.97 29808.66 29813.64 29816.62 29818.48 29819.67 29820.45 29820.98 29821.35 29821.60 29821.79 29821.92 29822.02 29822.09 29822.15 29822.19 29822.22 29822.25 29822.27 29822.28 29822.30 29822.31 29822.32 29822.32 29822.33 29822.33 29822.33 29822.35 29822.35 29822.35 29822.35 29822.35 29822.35

16388.98 16374.56 16361.63 16346.18 16323.17 16290.79 16249.45 16200.76 16146.84 16089.77 15973.01 15860.54 15757.44 15665.51 15584.71 15514.14 15452.59 15398.83 15351.73 15310.32 15273.80 15241.50 15187.33 15144.16 15109.35 15081.00 15057.70 15022.29 14997.30 14979.21 14965.82 14955.71 14947.93 14941.86 14937.05 14933.20 14930.07 14927.50 14925.38 14923.61 14922.12 14920.86 14919.79 14918.87 14918.07 14917.38 14916.78 14916.26 14915.80 14915.39 14915.03 14914.70 14914.42 14914.16 14913.93 14913.72 14913.54 14913.37 14913.21 14913.08 14912.95

21438.30 21353.71 21311.39 21335.28 21419.32 21556.19 21737.83 21956.23 22203.88 22474.11 23060.36 23679.57 24307.58 24928.10 25530.08 26106.06 26650.80 27160.26 27630.72 28058.11 28437.65 28763.99 29242.05 29507.73 29639.94 29707.63 29745.34 29782.82 29799.79 29808.63 29813.63 29816.62 29818.48 29819.67 29820.45 29820.98 29821.35 29821.60 29821.79 29821.92 29822.02 29822.09 29822.15 29822.19 29822.22 29822.25 29822.27 29822.28 29822.30 29822.31 29822.32 29822.32 29822.33 29822.33 29822.33 29822.34 29822.35 29822.35 29822.35 29822.35 29822.35

6128.70 6646.66 7290.12 7921.12 8532.57 9118.67 9674.77 10197.24 10683.45 11131.69 11911.89 12541.53 13035.34 13414.74 13702.90 13921.19 14087.35 14215.15 14314.86 14393.92 14457.69 14509.96 14590.18 14648.55 14692.68 14727.01 14754.28 14794.32 14821.68 14841.06 14855.20 14865.76 14873.81 14880.06 14884.99 14888.93 14892.11 14894.72 14896.87 14898.65 14900.16 14901.43 14902.51 14903.44 14904.24 14904.93 14905.54 14906.06 14906.53 14906.94 14907.30 14907.62 14907.91 14908.17 14908.40 14908.61 14908.79 14908.96 14909.12 14909.26 14909.38

22205.34 22502.85 22875.14 23221.07 23505.04 23697.08 23800.94 23852.81 23889.46 23933.98 24088.17 24348.75 24708.06 25143.59 25628.97 26139.54 26654.71 27158.28 27637.46 28081.67 28480.99 28825.24 29315.15 29559.54 29668.12 29721.27 29751.09 29782.22 29797.57 29806.20 29811.43 29814.74 29816.91 29818.38 29819.39 29820.11 29820.63 29821.01 29821.30 29821.51 29821.68 29821.81 29821.90 29821.99 29822.05 29822.10 29822.14 29822.17 29822.20 29822.23 29822.24 29822.26 29822.27 29822.28 29822.29 29822.30 29822.31 29822.31 29822.32 29822.32 29822.33

22599.58 22953.29 23411.34 23878.88 24348.47 24813.81 25269.54 25711.07 26134.42 26536.23 27264.44 27880.00 28377.57 28762.79 29050.33 29259.43 29409.28 29516.16 29592.59 29647.65 29687.74 29717.28 29756.01 29778.71 29792.67 29801.64 29807.59 29814.49 29817.98 29819.83 29820.87 29821.46 29821.81 29822.02 29822.15 29822.23 29822.28 29822.31 29822.33 29822.35 29822.36 29822.36 29822.36 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.36 29822.37 29822.37 29822.37 29822.37 29822.36 29822.37 29822.37 29822.36

31718.66 31474.41 31210.13 30985.57 30795.21 30634.27 30498.59 30384.55 30288.97 30209.11 30087.26 30003.50 29946.43 29907.85 29881.93 29864.60 29853.02 29845.27 29840.04 29836.45 29833.96 29832.18 29829.89 29828.49 29827.51 29826.74 29826.11 29825.12 29824.39 29823.86 29823.48 29823.20 29823.00 29822.85 29822.75 29822.66 29822.60 29822.55 29822.52 29822.49 29822.46 29822.45 29822.43 29822.42 29822.41 29822.40 29822.39 29822.39 29822.39 29822.38 29822.38 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.37 29822.36



569

Table 5 ˚ ), De, xe and Te (cm1) for the ground and excited states of Li2 molecule Spectroscopic parameters Re (A State

De

Te

Dissociation

Method

Re

(2s + 2s)

Present work Exp. [36] Exp. [37] Exp. [40] Theory [9] Theory [8]

2.658 2.673 2.673

Present work Exp. [41] Exp. [38] Theory [9] Theory [8]

4.134 4.127 4.171 4.159 4.182

344 336 333 321 322

65.95 67.58 65.13 66.1 63.73

Present work Exp. [42] Theory [9] Theory [8]

3.655 3.651 3.667 3.655

3274 3319 3161 3276

129.94 128.67 126.3 129.04

20139 20101 20253 20128

13 Rþ g


3.053 3.068 3.055 3.067

7184 7091.5 7137 7071

253.19 251.5 253.3 252.2

16290 16328.8 16277 16333

11 Rþ u


3.092 3.108 3.094 3.108

9483 9352.5 9466 9356

257.54 257.47 257.4 256.06

13994 14068 13948 14048

23 Rþ u

Present work Theory [9] Theory [8]

3.181 3.185 3.166

11Pg


4.048 4.058

Present work

repulsive


2.934


2.577 2.590 2.581 2.595


3.076 3.086 3.074 3.085

23 Rþ g


21 Rþ u , inner well

1

1

Rþ g

13 Rþ u

21 Rþ g

(2s + 2p)

13Pg 1

1 Pu

13Pu

31 Rþ g

21 Rþ u , outer well

(2s + 3s)

2.660 2.675

xe

8613 8600 8516.78 8615 8510 8466

5669 5712 5709

352.41 351.39

0 0

353 351.01

0 0 8126 8180.9 8183.8 8189 8144

193.73 213.1

1426 1422.5 1421.9 1400 1406

93.37 93.35 92.77 92.5 91.85

2930 2984.42 2984.5 2716 2903

269.49

20553

270.69 269.7 270.28

20436 20698 20501

348.15 345.6 348.7 345.88

11225

8333 8312.8 8289 8279

246.62 245.9 250.4 245.7

27428 27410 27434 27396

3.073 3.075 3.079

8451 8428 8379

270.07 270.4 270.68

27322 27294 27296


3.081 3.096 3.094 3.083 3.089

5674

5660 5598

259.82 259.003 259.9 260.1 259.2

30094 30101.407 30100.3 30062 30077


6.072 6.037 6.080 6.052

5413

119.05

5453 5329

30285 30400.137 30269

117 121.96 (continued on next page)

4.063 4.073

2.936 2.939 2.942

12297 12145 12178 12148

21968 21988.5 21998.8 22013 21998

11236 11256

570


Table 5 (continued) Method

Re

De

xe

Te


3.658 3.667 3.695

5903 5866 5839

373.15 356.4 392.69

29922 29853 29836


3.545 3.548 3.543 3.551

8388 8349.3 8377 8319

229.19 227.3 231.1 229.55

29985 29975 29941 30023

41 Rþ g , outer well


9.107 8.750 8.940

2159 2158 2166

19.38 17

36110 36160

43 Rþ u


3.351 3.267 3.371

4157 4395 4075

187.36 206.4 186.17

34195 33923 34267

53 Rþ u

Present work Theory [9]

3.149 3.126

3260 3079

265.39 264.9

35132 35240

21Pg

Present work Exp. [49] Theory [9]

3.191 3.2014 3.201

6469 6455 6398

230.70 229.26 229.1

31905 31868.4 31920

23Pg


3.853 3.816 3.851

8511 8484 8435

189.83 189.1 191.3

29842 29840.5 29883

21Pu


3.077 3.074 3.089

7751 7648.4 7641

236.47 239.1 231.3

30627 30549 30677

23Pu


2.968 2.965

8720 8671

283.82 284.1

29681 29647

11Dg


2.822

276.30 271.44 273.1

29586

2.919

8811 9579 9368

13Du


repulsive 3.126

3430

251.6

34888


3.034 3.081 3.034 3.030

8446 8401 8346 8396

274.49 273.3 274.4 278.89

31060 31041.7 31091 31149


4.369 4.340 4.315

5848 5839 5875

295.30 258 256.16

33668 33594 33670

State

Dissociation

33 Rþ u

41 Rþ g , inner well

3

3

Rþ g

(2p + 2p)

(2s + 3p)

31 Rþ u

3. Computational method All calculations reported in this paper were performed by means of the MOLPRO program package [34]. The core electrons of Li atoms are represented by pseudopotential ECP2SDF [27], which was formed from the uncontracted (9s 9p 8d 3f) basis set. The basis for the s and p orbitals, which comes with this potential is enlarged by functions for d and f orbitals given by P. Feller [35] and assigned by CC-PV5Z. Additionally, our basis set was augmented by three s short range correlation functions (392.169555, 77.676373, 15.38523), three p functions (96.625417, 19.845562, 4.076012) and three d functions (3.751948, 1.9783, 1.043103). Also, we added to the basis a set of five diffused functions: two s functions (0.010159, 0.003894), two p functions (0.007058, 0.002598) and one d function (0.026579).

28950

We checked the quality of our basis set performing the CI calculations for the ground and several excited states of isolated lithium atom. The calculated Li2 adiabatic potentials correlate to the (2s + 2s) ground atomic asymptote and (2s + 2p), (2s + 3s), (2p + 2p), (2s + 3p) excited atomic asymptotes. The comparison of experimental and theoretical asymptotic energies for different states is shown in Table 1. The spin–orbit coupling (SO) and core–core polarization effect contribute insignificant part to energy of our system, so we do not take them into consideration in our calculations. The potential energy curves for Li2 are calculated using the complete-active-space self-consistent-field (CASSCF) method to generate the orbitals for the subsequent CI calculations. The corresponding active space in the D2h point group involved the molecular counterparts of the 2s, 2p, 3s, 3p and 3d valence orbitals of the Li atom.


4. Results and discussion Calculations of the adiabatic energy curves are performed for the internuclear separation R in the range from 3.2a0 to 88a0 with the various steps adjusted to the internuclear distance. Numerical values of the calculated potential energies are shown in Tables 2–4. Equilibrium positions Re and depths of the potential wells De are obtained using cubic spline approximation to the calculated potentials around their equilibrium positions. Spectroscopic parameters xe and Te are calculated by solving the Schro¨dinger equation with calculated adiabatic potentials. These values are shown in Table 5. As it is seen, overall agreement of all our spectroscopic constants and other theoretical and

Fig. 1. Adiabatic potential energy curves for the ground and excited singlet states of the Li2 molecule correlating to the 2s + 2s, 2s + 2p, 2s + 3s, 2p + 2p and 2s + 3p asymptotes.

Fig. 2. Adiabatic potential energy curves for excited triplet states of the Li2 molecule correlating to the 2s + 2s, 2s + 2p, 2s + 3s, 2p + 2p and 2s + 3p asymptotes.

571

experimental data is very reasonable. Particularly, what is worth noting, there is a good agreement for the spectroscopic constants in the case of double wells for two states 1 þ 21 Rþ u and 4 Rg . Present results for all singlet and triplet states are plotted relative to the corresponding experimental values of the atomic terms in Figs. 1–3. The states 53 Rþ u, 21Pu, 21Pg, 23Pu, 23Pg, 11Dg and 13Du are calculated for the second time (besides Poteau and Spiegelmann [9]). All our states are calculated for the first time, for large internuclear separations R (up to 88a0). We obtain asymptotic energies for ground and excited states, which are in very good agreement with experimental and other theoretical values (Table 1.) We also present almost all states corresponding to the 2p + 2p asymptote (Fig. 3), except 11 R u and 13 R g states, which are not available by methods of the MOLPRO program package [34]. In Fig. 4, we

Fig. 3. Adiabatic potential energy curves for excited states of the Li2 molecule correlating to the 2p + 2p asymptote.

Fig. 4. Comparison of the ground and the first excited molecular state correlating to 2s + 2s asymptote with the theoretical results of Olson and Konowalow [3,4] and Schmidt-Mink et al. [8].

572


compare our ground and the first excited molecular state correlating to ground state of lithium atoms along with the theoretical results of Olson and Konowalow [3,4] and Schmidt-Mink et al. [8]. As one may see, the agreement is very good. In Fig. 5. we compare present results of the ground state Xð1Þ1 Rþ g with two curves derived from experiments by Hessel and Vidal [36] and Barakat et al. [37]. In the vicinity of the potential well we note better agreement with the curve of Hessel and Vidal [36], but for larger distances we note the perfect fit with the curve of Barakat et al. [37]. In Fig. 6, the comparison for að1Þ3 Rþ u with the curve derived from experiment by Linton et al. [38] shows some discrepancy for the repulsive part of potential curve. Also our theoretical curve displays a deeper well of about 11 cm1. In Fig. 7 we show our results along with the experimental curve given by Bouloufa et al. [25] and theoFig. 7. Comparison of the B(1)1Pu state correlating to 2s + 2p asymptote with results derived from experiment by Bouloufa et al. [25] and theoretical results of Schmidt-Mink et al. [8].

Fig. 5. Comparison of the ground state with the experimental results of Hessel and Vidal [36] and Barakat et al. [37].

1 þ Fig. 8. Comparison of the Eð3Þ1 Rþ g and Fð4Þ Rg states correlating, respectively, to 2s + 3s and 2p + 2p asymptotes with potential curves derived from experiment by Jastrzebski et al. [22] and Pashov et al. [23].

retical results of Schmidt-Mink et al. [8]. Here, we underline the good agreement in the distinctive hump shape which appears around 10–13a0. In turn, in the Fig. 8, we display the comparison with two 1 Rþ g states derived form experiment. Here, in the case of 41 Rþ g state once again the agreement is excellent, while in the case of the lower 31 Rþ g state correlating to 2s + 3s atomic energy [22,23] we find a disagreement for small internuclear distance. 5. Conclusion

Fig. 6. Comparison of the first excited molecular state correlating to 2s + 2s asymptote with the experimental result of Linton et al. [38].

We have calculated the adiabatic potential energy curves of the lithium dimer using CASSCF/MRCI method. Several curves correlated to the excited atomic states are pre-


sented for the second time (besides Poteau and Spiegel1 1 3 3 1 3 mann [9]): 53 Rþ u , 2 Pu, 2 Pg, 2 Pu, 2 Pg, 1 Dg and 1 Du. Comparisons with available lowly-lying theoretical and experimental curves provide almost perfect agreement. Only in the case of the repulsive part of the 31 Rþ g state, we note certain disagreement. In the near future, we plan to calculate adiabatic potential energy curves for heavier alkali dimers. Acknowledgements The authors gratefully acknowledge fruitful and helpful discussions with prof. E. Czuchaj. This research is partially supported by ESF Network – CATS (Collisions in Atom Traps). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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