Analytical Potential Energy Function, Spectroscopic Constants and Vibrational Levels for A 1 Σ + u

Transcription

1 Commun. Theor. Phys. (Beijing, China) 48 (2007) pp c International Academic Publishers Vol. 48, No. 6, December 15, 2007 Analytical Potential Energy Function, Spectroscopic Constants and Vibrational Levels for A 1 Σ + u State of Dimer 7 Li 2 SHI De-Heng, 1, MA Heng, 1 SUN Jin-Feng, 1,2 and ZHU Zun-Lue 1 1 College of Physics & Information Engineering, Henan Normal University, Xinxiang , China 2 Department of Physics & Electron Science, Luoyang Normal University, Luoyang , China (Received November 8, 2006; Revised January 5, 2007) Abstract The symmetry-adapted-cluster configuration-interaction method is used to investigate the spectroscopic properties of 7 Li 2(A 1 Σ + u ) over the internuclear distance ranging from 2.4a 0 to 37a 0. The complete potential energy curves are calculated at numbers of basis sets. All the ab initio calculated points are fitted to the analytic Murrell Sorbie function and then employed to compute the spectroscopic constants. By comparison, the spectroscopic constants reproduced by the potential attained at D95(3df,3pd) are found to be very close to the experiments, and the values (T e, D e, R e, ω e, ω eχ e, α e and B e) are of ev, ev, nm, cm 1, cm 1, cm 1, and cm 1, respectively. With the potential obtained at D95(3df,3pd), the totally 75 vibrational states are found when J = 0. The vibrational levels, the classical turning points and the inertial rotation constants of the first 68 vibrational states are calculated for the first time and compared with the available measurements. Good agreement is obtained. The centrifugal distortion constants of the first 32 vibrational states are also reported for the first time. The reasonable dissociation limit for 7 Li 2(A 1 Σ + u ) is deduced using the calculated results at present. PACS numbers: Df, Ar Key words: analytic potential energy function, harmonic frequency, dissociation energy, vibrational level, classical turning point 1 Introduction Much attention has been given, experimentally as well as theoretically, to the diatomic analytic potential energy function (APEF), because detailed knowledge of APEF is very useful in understanding collision processes between atoms and molecules since the relatively slow nuclear motion is very sensitive to the details of the APEF. In order to determine the reliable collision dynamics at ultralow energies, [1] the accurate APEF, particularly the highly accurate APEF over a large internuclear separation range, is especially required. On the other hand, the accurate APEF plays a crucial role in investigating various molecular properties, such as determining the accurate spectroscopic constants, acquiring the trustworthy vibrational levels, the classical turning points, the inertial rotation and the centrifugal distortion constants, etc. As to lithium dimer 7 Li 2, the second smallest homonuclear molecule next to H 2, its molecular constants and potential energy curves (PECs) have been studied extensively by both theories and experiments [2 9] in the past several decades. But to our surprise, previous studies [10 15] on 7 Li 2 (A 1 Σ + u ) mainly focus on the dissociation energy, the equilibrium internuclear separation and the harmonic frequency. Very few APEFs can be found in the literatures to date. Furthermore, no theoretical investigations about the vibrational levels, the classical turning points, the inertial rotation and the centrifugal distortion constants have been found to the best of our knowledge, too. Comparison of the present theoretical determinations with those obtained by measurements [16] about the dissociation energy, the equilibrium internuclear separation and the adiabatic excitation energy, the present results deviate only by 0.795%, 0.142% and 0.63%, respectively. Thus it is very excellent. At the same time, the present work is the first theoretical effort to determine the vibrational levels, the classical turning points, the inertial rotation and the centrifugal distortion constants by solving the radial Schrödinger equation for this state. In this paper, the APEF, the spectroscopic constants, the vibrational levels, the classical turning points, the inertial rotation and the centrifugal distortion constants for this state are calculated using the symmetry-adaptedcluster configuration-interaction (SAC-CI) method [17] in complete active space presented in Gaussian 03 program package. [18] In the next section, we describe the detailed methodology. In Sec. 3, we present the calculated results of the APEF, the spectroscopic constants, the vibrational levels, the classical turning points, the inertial rotation and the centrifugal distortion constants, and make some useful discussion about them. Concluding remarks are made in Sec Methodological Details The SAC-CI method, which was originally published in 1978 and mainly used to investigate the excited, ionized and electron-attached states of molecules, has now The project supported by National Natural Science Foundation of China under Grant Nos and Corresponding author, scattering@sina.com.cn

2 1082 SHI De-Heng, MA Heng, SUN Jin-Feng, and ZHU Zun-Lue Vol. 48 been successfully employed to calculate the PECs of various states. [19 21] Here, we utilize this method to perform the PEC calculations of this A 1 Σ + u state in complete active space. In the calculations, a number of basis sets, such as G(3df,3pd), 6-311G(3df,3pd), G(2df,2pd), 6-311G(2df,2pd), 6-311G(df,pd), G(df,pd), G(d,p), G, AUG-cc- PVTZ, cc-pvtz, cc-pvqz, D95(3df,3pd) and D95V++, are used. The ab initio calculated points of PECs at the above-mentioned basis sets are all fitted to the analytic Murrell Sorbie (M-S) type function with the least squares fitting method, thus the dissociation energies D e and the APEFs are attained. Based on the APEFs obtained here, the force parameters, quadratic f 2, cubic f 3 and quartic f 4, are determined, and then the spectroscopic constants, ω e, ω e, χ e, α e, and B e, are calculated. By comparing between the computed spectroscopic constants and the measurements, [16] a most suitable basis set, D95(3df, 3pd), [22] is selected. Using the APEF obtained at the present SAC-CI/D95(3df,3pd) level of theory, the radial Schrödinger equation of nuclear motion has been solved to get all the vibrational levels and the corresponding classical turning points, R min and R max, then the inertial rotation and the centrifugal distortion constants. 3 Results and Discussions M-S function is a widely used APEF. It describes interaction potential energies of neutral diatomic well. The common M-S function [23] is n V (ρ) = D e (1 + a i ρ i) exp( a 1 ρ), (1) i=1 where ρ = R R e, R is the internuclear distance of diatoms, R e is regarded as a fixed parameter in the fitting process, which is attained by the fine single-point energy scanning calculations in this paper, and the parameters a i are determined by the fitting method using the ab initio calculated data. By iterating a system of normal equations based on a least-squares fit, the parameters a i and D e in Eq. (1) are fitted using the ab initio data obtained at the abovementioned basis sets. In order to attain the satisfactory result, we try it from n = 3 to 8, and find the best result n = 6. When the APEFs are obtained, the quadratic, cubic and quartic force constants derived from Eq. (1) can be computed, f 2 = D e (a 2 1 2a 2 ), (2) f 3 = 6D e (a 3 a 1 a ) 3 a3 1, (3) f 4 = D e (3a a 2 1a a 1 a 3 ). (4) Then the spectroscopic parameters, f 2 ω e = 8πcµRe 2, (5) h B e = 8πcµRe 2, (6) ( α e = 6B2 e f3 R ) e + 1, (7) ω e 3f 2 ω e χ e = B e 8 [ f 4R 2 e f ( 1 + ω eα e 6B 2 e ) 2 ]. (8) The calculated spectroscopic parameters at these basis sets are tabulated in Table 1. Table 1 Equilibrium constants and spectroscopic parameters for 7 Li 2(A 1 Σ + u ). Basis sets T e (ev) D e (ev) R e (nm) ω e (cm 1 ) ω eχ e (cm 1 ) B e (cm 1 ) α e (cm 1 ) G(3df,3pd) G(3df,3pd) G(2df,2pd) G(2df,2pd) G(df,pd) G(df,pd) G(d,p) G AUG-cc-PVTZ cc-pvtz cc-pvqz D95V D95(3df,3pd) Experiments [16] From Table 1, one can easily find that the basis set D95(3df,3pd) is an excellent one, since the present T e, D e, and R e values at the basis set are in agreement with the measurements [16] within ev or 0.63%, ev or 0.795% and nm or 0.21%, though the present ω e value is somewhat smaller than that obtained in experiments [16]

3 No. 6 Analytical Potential Energy Function, Spectroscopic Constants and Vibrational Levels for 1083 by 3.52 cm 1 or 1.378%. Therefore, it sustains us to use this basis set for further calculations. For the convenient comparison, we tabulate the T e, R e, D e, ω e, ω e χ e, B e, and α e values obtained at the present SAC-CI/D95(3df,3pd) level of theory in Table 2 together with the measurements [16] and other theories. [10 15] Table 2 Spectroscopic constants for 7 Li 2(A 1 Σ + u ) at the SAC-CI/D95(3df,3pd) level of theory and the corresponding comparisons with the measurements and other theories. State Source T e (ev) R e (nm) D e (ev) ω e (cm 1 ) ω eχ e (cm 1 ) B e (cm 1 ) α e (cm 1 ) A 1 Σ + u This work Exp. [16] Theory [10] Theory [11] Theory [12] Theory [13] Theory [14] Theory [15] From Table 2, we find that the best R e value of the previous theories is presented by Poteau et al., [12] who used the effective core pseudopotential (ECP) and the 7s5p3d1f basis set centered on the two Li atoms (88 contracted Gaussians). And their R e discrepancy deviating from the measurements [16] is nm or 0.455%. This deviation is somewhat smaller than the result obtained by the present theories. However, one must notice that the present D e and T e values are much closer to the measurements [16] than all other theories. [10 15] Compared with those theoretical results, [12,14] the difference between the present ω e value and the measurements [16] is somewhat large, but other spectroscopic constants are in good accord with the measurements. [16] Thus, the present calculations are encouraging. Table 3 Parameters of M-S APEF for 7 Li 2(A 1 Σ + u ) at the SAC-CI/D95(3df,3pd) level of theory. D e (ev) R e (nm) a 1 (nm 1 ) a 2 (nm 2 ) a 3 (nm 3 ) a 4 (nm 4 ) a 5 (nm 5 ) a 6 (nm 6 ) RMSE (ev) In order to be convenient for further calculations, the APEF parameters at the SAC-CI/D95 (3df, 3pd) level of theory are all tabulated in Table 3. At the same time, the fitting results and the curve of the ab initio calculated points over the internuclear separation range from about 2.4a 0 to 37a 0 are intuitively illustrated in Fig. 1, too. In order to evaluate the fitting quality of the APEF obtained at the present SAC-CI/D95 (3df, 3pd) level of theory, we calculate the root-mean square error (RMSE), RMSE = 1 N (V APEF V ab initio ) N 2, (9) i=1 where V APEF and V ab initio are energies attained by the fitting and the ab initio calculations, respectively. N is number of fitted points (here N = 173). The present RMSE for the A 1 Σ + u state is of only ev ( Kcal/mol). Very obviously, our fitting accuracy about the APEF is greatly superior to the chemical accuracy (1.0 Kcal/mol). [24] It shows that the fitting process is of high quality and the APEF tabulated in Table 3 is in excellent agreement with the ab initio calculated points on the whole. The following calculated results about the vibrational levels and their corresponding classical turning points of the first 26 vibrational states further affirm that the APEF tabulated in Table 3 is credible. Fig. 1 PEC of the first singlet excited state of 7 Li 2. The more stringent way to compare the ab initio results with the measurements is to attempt to reproduce the RKR data: the vibrational levels and their corresponding classical turning points. Since only in that way, it can be clearly demonstrated whether the present APEF accurately describes the PEC of this state. In order to attain the vibrational levels and their corresponding classical turning points, we must solve the following radial

4 1084 SHI De-Heng, MA Heng, SUN Jin-Feng, and ZHU Zun-Lue Vol. 48 Schrödinger equation of nuclear motion in the adiabatic approximation, [ h2 2µ d 2 dr 2 + ] h2 2µr 2 J(J + 1) + V (r) Ψ ν,j (r) = E ν,j Ψ ν,j (r). (10) Here V (r) is the adiabatic rotationless APEF tabulated in Table 3. ν and J are the vibrational and rotational quantum numbers, respectively. µ is the reduced mass of molecule 7 Li 2. The rotational sublevel of a given vibrational level is represented by the following power series, [25] E ν,j = G(v) + B v [J(J + 1)] D v [J(J + 1)] 2 + H v [J(J + 1)] 3 + L v [J(J + 1)] 4 + M ν [J(J + 1)] 5 + O ν [J(J + 1)] 6. (11) Table 4 The vibrational levels, classical turning points and inertial rotation constants of the first 26 vibrational states and the corresponding comparisons with the available measurements for 7 Li 2(A 1 Σ + u ) (J = 0) at the SAC-CI/D95(3df,3pd) level of theory. ν SAC-CI/D95(3df,3pd) measurements [16] G(ν) (cm 1 ) R min (nm) R max (nm) B ν (cm 1 ) G(ν) (cm 1 ) R min (nm) R max (nm) By solving Eq. (10), we have obtained the totally 75 vibrational states for dimer 7 Li 2 (A 1 Σ + u ) when J = 0. For each vibrational level G(ν), one inertial rotation constant B v and six centrifugal distortion constants D ν, H ν, L ν, M ν, N ν, and O ν are attained. Here the vibrational levels, their corresponding classical turning points, R max and R min, and the inertial rotation constants of the first 26 vibrational states together with the available measurements [16] are tabulated in Table 4. The vibrational levels, their corresponding classical turning points and the inertial rotation constants of the vibrational states from ν = 26 to 67 when J = 0 are tabulated in Table 5, and the six centrifugal distortion constants of the first 32 vibrational states when J=0 are tabulated in Table 6 as the predicting calculations. Table 5 The vibrational levels, classical turning points and inertial rotation constants of the vibrational states from ν = 26 to 67 for 7 Li 2(A 1 Σ + u ) (J = 0) at the SAC-CI/D95(3df,3pd) level of theory. ν G(ν) (cm 1 ) R min (nm) R max (nm) B ν (cm 1 ) ν G(ν) (cm 1 ) R min (nm) R max (nm) B ν (cm 1 )

5 No. 6 Analytical Potential Energy Function, Spectroscopic Constants and Vibrational Levels for 1085 continued ν G(ν) (cm 1 ) R min (nm) R max (nm) B ν (cm 1 ) ν G(ν) (cm 1 ) R min (nm) R max (nm) B ν (cm 1 ) Table 6 The centrifugal distortion constants of the first 32 vibrational states for 7 Li 2(b 3 Π u) when J = 0 at the SAC- CI/D95(3df,3pd) level of theory. ν D ν( 10 6 )(cm 1 ) H ν( )(cm 1 ) L ν( )(cm 1 ) M ν( )(cm 1 ) N ν( )(cm 1 ) O ν( )(cm 1 )

6 1086 SHI De-Heng, MA Heng, SUN Jin-Feng, and ZHU Zun-Lue Vol. 48 continued ν D ν( 10 6 )(cm 1 ) H ν( )(cm 1 ) L ν( )(cm 1 ) M ν( )(cm 1 ) N ν( )(cm 1 ) O ν( )(cm 1 ) Only the measurements [16] of the first 26 vibrational levels and the classical turning points can be found in the literatures to date to the best of our knowledge. From the comparisons between the first 26 vibrational levels and their classical turning points with the measurements, [16] we find that the largest deviations do not exceed 1.5% for the vibrational levels and 0.8% for the classical turning points, respectively. And with the level increasing, the deviations from the measurements gradually decrease. For example, the deviations are of 1.194% for the vibrational level and 0.721% for the classical turning point when ν = 5. Whereas the deviations are of only 0.614% for the vibrational level and 0.448% for the classical turning point when ν = 25. According to these, we conclude that the predicting calculations tabulated in Tables 5 and 6 should be accurate. Therefore, the APEF obtained at the present SAC-CI/D95(3df,3pd) level of theory well describes this A 1 Σ + u state. Finally, we determine the reasonable dissociation limit for the A 1 Σ + u state using the computed results tabulated in Table 2. As shown in Fig. 2, the atomic excitation energy E a for an excited state of a molecule equals E a = D e + T e D 0 e. (12) Here, De is the dissociation energy for a given excited state. De 0 is the dissociation energy for the ground state and equals ev for 7 Li 2, which has taken into consideration of the vibrational zero energy. [26] T e is the adiabatic excitation energy from the ground to a given state. And E a is the atomic excitation energy sum of the separated atoms for a given excited state in the dissociation limit when the atomic excitation energy sum of the separated atoms in the dissociation limit for the ground state is set to equal zero. The ground-state dissociation limit [10] of dimer 7 Li 2 is 7 Li 2 (X 1 Σ + g ) Li( 2 S g ) + Li( 2 S g ). (13) Obviously, the two separated atoms in the dissociation limit for the ground state are both in the ground state. Thus their atomic excitation energy sum in the dissociation limit equals zero. Fig. 2 PECs of the ground and excited states for a molecule. The calculated T e and De values for the A 1 Σ + u state in this paper are of ev and ev, respectively, therefore, we get a value of E a ev ( cm 1 ). Table 7 tabulates the energy levels [27] of two lithium atoms in several electronic configurations. According to the Table, we conclude that the two Li atoms in the dissociation limit for this state must be one in the configuration 1s 2 2p 1 and the other in the configuration 1s 2 2s 1. Thus the reasonable dissociation limit for the A 1 Σ + u state must be 7 Li 2 ( 1 Σ + u ) Li( 2 S g ) + Li( 2 P u ). (14) Table 7 Energy levels [27] of two lithium atoms in several electronic configurations. Configuration Excitation energy (cm 1 ) Configuration Excitation energy (cm 1 ) (1s 2 2s 1 )+(1s 2 2s 1 ) 0 (1s 2 2s 1 )+(1s 2 3s 1 ) (1s 2 2s 1 )+(1s 2 2p 1 ) (1s 2 2p 1 )+(1s 2 2p 1 ) (1s 2 2s 1 )+(1s 2 3p 1 ) (1s 2 2s 1 )+(1s 2 3d 1 ) (1s 2 2p 1 )+(1s 2 3p 1 ) Conclusions We have calculated the interaction potentials of the A 1 Σ + u state for dimer 7 Li 2 using the SAC-CI method at a number of basis sets and found that the best potential can be obtained at the basis set D95(3df,3pd). We have employed the calculated potentials to compute the spectroscopic constants. With the APEF obtained at the SAC-CI/D95(3df,3pd)

7 No. 6 Analytical Potential Energy Function, Spectroscopic Constants and Vibrational Levels for 1087 level of theory, we have found the totally 75 vibrational states when J = 0 for 7 Li 2 (A 1 Σ + u ). The vibrational levels, the classical turning points and the inertial rotation constants of the first 68 vibrational states are reported for the first time. Favorable agreement has been obtained in comparing with the available measurements. And the six centrifugal distortion constants of the first 32 vibrational states are also reported for the first time. At the same time, the reasonable dissociation limit for 7 Li 2 (A 1 Σ + u ) is attained using the results obtained at the SAC-CI/D95(3df,3pd) level of theory in this paper. References [1] M.A.P. Lima, T.L. Gibson, and V. McKoy, Phys. Rev. A 38 (1998) [2] A. Pashov, W. Jastrzçbski, and P. Kowalczyk, J. Chem. Phys. 113 (2000) [3] D. Danovich, W. Wu, and S. Shaik, J. Am. Chem. Soc. 121 (1999) [4] A.M. Maniero and P.H. Acioli, Int. J. Quant. Chem. 103 (2005) 711. [5] C. Linton, F. Martin, A.J. Ross, I. Russier, P. Crozet, A. Yiannopoulou, L. Li, and A.M. Lyyra, J. Mol. Spectrosc. 196 (1999) 20. [6] N. Bouloufa, P. Cacciani, R. Vetter, and A. Yiannopoulou, J. Chem. Phys. 114 (2001) [7] I. Russier, F. Martin, C. Linton, P. Crozet, A.J. Ross, R. Bacis, and S. Churassy, J. Mol. Spectrosc. 168 (1994) 39. [8] A.A. Zavitsas, J. Mol. Spectrosc. 221 (2003) 67. [9] M. Aubert-Frécon, G. Hadinger, S. Magnier, and S. Rousseau, J. Mol. Spectrosc. 188 (1998) 182. [10] D.H. Shi, A.D. Xie, Z.L. Zhu, T. Gao, and H.Y. Wang, J. At. Mol. Phys. 21 (2004) 622 (in Chinese). [11] D.K. Watson, C.J. Cerjan, S. Guberman, and A. Dalgarno, Chem. Phys. Lett. 50 (1977) 181. [12] R. Poteau and F. Spiegelmann, J. Mol. Spectrosc. 171 (1995) 299. [13] D.D. Konowalow and J.F. Fish, Chem. Phys. 84 (1984) 463. [14] D.D. Konowalow and M.L. Olson, J. Chem. Phys. 71 (1979) 450. [15] D.W. Davies and G.J.R. Jones, Chem. Phys. 81 (1981) 279. [16] P. Kusch and M.M. Hesse, J. Chem. Phys. 67 (1977) 586. [17] H. Nakatsuji, M. Hada, M. Ehara, et al., SAC/SAC- CI Program Combined with Gaussian for Calculating Ground, Excited, Ionized, and Electron-Attached States and Singlet, Doublet, Triplet, Quartet, Quintet, Sextet, and Septet Spin States and Their Analytical Energy Gradients, Kyoto University Press, Kyoto (2002) pp. 19, 20, 38, 40. [18] M.J. Frisch, G.W. Trucks, H.B. Schlegel, et al., Gaussian 03 Revision A1, Gaussian Inc., Pittsburgh, PA (2003). [19] D.H. Shi, H. Ma, J.F. Sun, and Z.L. Zhu, Commun. Theor. Phys. (Beijing, China) 47 (2007) [20] T. Nakajima and H. Nakatsuji, Chem. Phys. 242 (1999) 177. [21] M. Ishida, K. Toyota, M. Ehara, and H. Nakatsuji, Chem. Phys. Lett. 347 (2001) 493. [22] M. Kaupp, P.V.R. Schleyer, H. Stoll, and H. Preuss, J. Chem. Phys. 94 (1991) [23] J.N. Murrell, S. Carter, S.C. Farantos, P. Huxley, and A.J.C. Varandas, Molecular Potential Energy Functions, Chinchester, Wiley (1984). [24] A. Aguado and M. Paniagua, J. Chem. Phys. 96 (1992) [25] G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 1, Van Nostrand Reinhold, New York (1951). [26] B. Barakat, R. Bacis, F. Carrot, S. Churassy, P. Crozet, F. Martin, and J. Verges, J. Chem. Phys. 102 (1986) 215. [27] C.E. Moore, Atomic Energy Levels, US Governments Printing Office, Washington (1971) p. 9.