Symbolic Calculation of Two-Center Overlap Integrals Over Slater-Type Orbitals

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1 Journal of the Chinese Chemical Society, 2004, 51, Symbolic Calculation of Two-Center Overlap Integrals Over Slater-Type Orbitals Sedat Gümü and Telhat Özdoan Department of Physics, Amasya Education Faculty, Ondouz Mayis University, Amasya, Turey Two-center overlap integrals over Slater type orbitals (STOs) have been expressed in terms of the well-nown Mullien s integrals Bnptusing Rodrigues s formula for normalized associated Legendre functions. A computer program is written in Mathematica 4.0 for the evaluation of two-center overlap integrals over STOs. Using this computer program, symbolic tables are presented for two-center overlap integrals up to quantum numbers 1 nn, 3, 0 ll, 2, 2 mm, 2. Numerical results of this wor, for some quantum sets, have also been compared with prior literature and best agreement achieved with recent wors of Barnett while some discrepancies were obtained with wors of Öztein et al. and Guseinov et al. Keywords: Overlap integrals; Slater type orbitals; Symbolic calculation. INTRODUCTION The calculation of multicenter integrals have prime importance when LCAO-MO approximation is employed in the molecular properties calculations. Two-center overlap integrals constitute basic building blocs of other multicenter integrals since the expectation values of one-electron operators can be expressed as linear combinations of two-center overlap integrals, and also since Coulomb integrals can be calculated by taing quadrature of two-center overlap integrals. 1 Therefore, an accurate and efficient method for the numeric calculation of two-center overlap integrals is essential. It is well nown that the accuracy of molecular properties calculations depends deeply on the nature of the type of the basis functions used in the calculations. Among the basis functions, Slater-type orbitals (STOs) are able to represent density of electrons more accurately than other basis functions. 2,3 A variety of approaches on the evaluation of two-center overlap integrals over STOs exists in the literature, for example, expansion of STOs, 4-8 elliptical coordinates method, 9 integral transformation methods, 10,11 and recurrence schemes and other methods It is well nown from the literature that in the calculation of two-center overlap integrals some discrepancies occur for smaller quantum numbers, higher quantum numbers, higher or lower internuclear distances and, equal or nearly equal orbital exponents. Therefore, the symbolic calculation of two-center overlap integrals and also other multicenter integrals are needed to compare with the values in the literature. Recently, we have presented recurrence relations 13 and a series expansion formula 15 for the evaluation of two-center overlap integrals over STOs. In order to test the efficiency of our recent wor and formula in the literature, the symbolic calculation of two-center overlap integrals over STOs constitutes the aim of this wor. Symbolic calculation of two-center overlap integrals for researchers in this field can be highlighted to the relative benefits of all the different methods. In addition, the symbolic results from this paper can be used for the energy calculations and the crystal field properties of some small molecules, such as molecules H 2, H 2, CO 2, HCN and LiH. DEFINITIONS Two-center overlap integrals in line-up coordinate systems are as follows * S (1) nl, n l, ; R nl, ra n l, rb dv where R Rab ra rb is the radius vector of the sphericalpolar coordinates and nl, r a and nl, r bare normalized complex or real STOs centered on the nuclei a and b,respectively, defined by * Corresponding author. sedatg@omu.edu.tr

2 244 J. Chin. Chem. Soc., Vol. 51, No. 2, 2004 Gümü and Özdoan n12 2 n1 r nl, r r e Sl,, 2 n! Here is orbital exponent and the functions, are complex or real spherical harmonics defined by: 16 where S l are orthonormal functions given by (2) 1 i e (4a) 2 for complex spherical harmonics and 1 cos for 0, (4b) 1 sin for 0, 0 for real spherical harmonics. In Eq. (3), P l is a normalized associated Legendre function and can be defined as in the following 17 by using Rodrigues s formula: with S l l, P cos l l l 2 P (cos ) W 1cos i0 j! is the usual bi- i! j i! In Eqs.(6), the quantity Fi nomial coefficients. j (3) (5) (6a) (6b) THE EVALUATION OF TWO-CENTER OVERLAP INTEGRALS USING ELLIPTICAL COORDINATES 2 i li i A cos 1 cos 1, l 2l1F l 2l1 2 F l. Wl A F l F l i l i i 12 With substitution of the relationships into Eq. (5) and the expansions below (8) (9) (10) (11) we have the expression for the two-center overlap integrals over STOs as follows: Qii, jj nl, nl, ; p Bs pt (12) Here, the variables p a, p b, p and t are defined as follows 1 pa R, pb R, p pa pb, t, (13) 2 and the symbol F N N product cos 1 1 1, a cos 1 1 1, b i i Fi 1 1, i0 i Fi 1 2 1, i0 ii, j, j s, m, is the expansion coefficient in N N NNm m F, (14) m N N m0 and defined by 18,19 min mn, F N, N 1 F N F N. (15) i n12 n12 l pa pb S, nl n l, ; R1 e nn1 p m p F lii j, lii j s NN m 1 mnmn 2 In Eq. (12), the quantity Q ii, jj is defined as It is well nown that two-center overlap integrals over STOs are dealt with at their best in the elliptical coordinates. Therefore, for the integration in Eq. (1), we use the definitions for elliptical coordinates ; ; Q, nl, nl, ; p ii jj q nl p q nl p ij i j!2 F ii nn j j p (16) r r R, 1, a b r r R, 1 1, a b, 0 2, a 3 R 2 2 dv d d d 2 b. (7a) (7b) (7c) (7d) with li Wl i j qij nl; p1 Al Fj nlp (17) 2 n! and the summation indices ii,, j, j, and s run as follows:

3 Symbolic Calculation of Two-Center Overlap Integrals J. Chin. Chem. Soc., Vol. 51, No. 2, Table 1. Symbolic Results of Two-Center Overlap Integrals Over STOs for t 0

4 246 J. Chin. Chem. Soc., Vol. 51, No. 2, 2004 Gümü and Özdoan

5 Symbolic Calculation of Two-Center Overlap Integrals J. Chin. Chem. Soc., Vol. 51, No. 2,

6 248 J. Chin. Chem. Soc., Vol. 51, No. 2, 2004 Gümü and Özdoan

7 Symbolic Calculation of Two-Center Overlap Integrals J. Chin. Chem. Soc., Vol. 51, No. 2, Table 2. Symbolic Results of Two-Center Overlap Integrals Over STOs for t =0 In Eq. (12), B pt (18) is the well-nown Mullien s integral 9 and for readers interest in the calculating of this integral we refer to Refs. [15a, 20, 21]. RESULTS AND DISCUSSIONS An algorithm has been presented for the evaluation of two-center overlap integrals over STOs using ellipsoidal coordinates methods and Rodrigues formula for normalized associated Legendre functions. The obtained formula contains the finite sum of binomial coefficients and Mullien integral Bn pt. On the basis of Eq. (12), a computer program, in Mathematica 4.0 symbolic programming language, 22 has been constructed and presented in the Appendix for twocenter overlap integrals over STOs. Symbolic computer results for two-center overlap integrals over STOs between some quantum sets have been

8 250 J. Chin. Chem. Soc., Vol. 51, No. 2, 2004 Gümü and Özdoan n l n l p t Literatu a Ref. 27. b Ref. 26. listed in Table 1 for unequal orbital exponents and in Table 2 for equal orbital exponents. Numerical values obtained from the results in Tables 1 and 2 agree best with the available literature. Also, comparative values of some pilot calculations have been presented in Table 3 for higher quantum numbers and for a wide range of molecular parameters. As can be seen from Table 3 our computer results agree best with the recent wor of Barnett. 23 We mention also that our results are in best agreement with the values in Refs. [8-12, 15, 24, 25]. On the other hand, despite the fact that Öztein et al. 26 and Guseinov 27 claim that their algorithm can be used in the evaluation of other multicenter integrals when translation of STOs is employed in the calculations, we have found some discrepancies with our values even for small quantum numbers. We thin that these discrepancies may originate from their recurrence relations, that is, it is well nown that the subtraction of two smaller numbers is a big problem for coding in computers. This fact is investigated in a detailed way in the excellent wor of Barnett 23 and called "Digital Erosion". The agreement with the prior literature shows that the algorithm presented here does not suffer from possible instability problems (higher or lower quantum numbers, higher or lower internuclear distances and, equal and nearly equal orbital exponents), and we stress that the algorithm we presented can be used in molecular orbital calculations when Hartree-Foc-Roothaan approximation is employed. Wor is being continued in our laboratory and some preliminary symbolic results on two-center two-electron integrals over STOs will be reported shortly.

9 Symbolic Calculation of Two-Center Overlap Integrals J. Chin. Chem. Soc., Vol. 51, No. 2, Appendix: Computer program written for two-center overlap integrals over STOs in Mathematica 4.0 Received June 3, REFERENCES 1. Ohata, K. O.; Ruedenberg, K. J. Math. Phys. 1966, 7, Kato, T. Commun. Pure Appl. Math. 1951, 10, Agmon, S. Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bound on Eigenfunctions of N-Body Schrödinger Operators; Princeton University Press: Princeton, NJ, Barnett, M. P.; Coulson, C. A. Phil. Trans. Roy. Soc. London. A. 1951, 243, (a) Harris, F. E.; Michels, H. H. J. Chem. Phys. 1965, 43, 165. (b) J. Chem. Phys. 1966, 45, Collidge, A. S. Phys. Rev. 1932, 42, 189.

10 252 J. Chin. Chem. Soc., Vol. 51, No. 2, 2004 Gümü and Özdoan 7. Sharma, R. R. Phys. Rev. A. 1976, 13, Jones, H. W. Int. J. Quantum Chem. 1997, 61, Mullien, R. S.; Orloff, C. A.; Orloff, H. J. Chem. Phys. 1949, 17, Weniger, E. J.; Grotendorst, J.; Steinborn, E. O. Phys. Rev. A. 1986, 33, Magnasco, V.; Rapallo, A.; Casanova, M. Int. J. Quantum Chem. 1999, 73, Rico, J. F.; Lopez, R.; Ramirez, G.; Ema, I. J. Mol. Struct. (Theochem). 1998, 433, Guseinov, I. I.; Mamedov, B. A.; Orbay, M.; Özdoan, T. Commun. Theor. Phys. 2000, 33, Chang, C.-H.; Hayashi, M.; Chang, R. J. Chin. Chem. Soc. 2000, 47, (a) Özdoan, T.; Orbay, M. Int. J. Quantum Chem. 2002, 87, 15. (b) Özdoan, T.; Orbay, M.; Gümü,S. Commun. Theor. Phys. 2002, 37, 711. (c) Özdoan, T.; Gümü, S.; Kara, M. J. Math. Chem. 2003, 33, 181. (d) Gümü, S.; Özdoan, T. Commun. Theor. Phys. 2003, 39, 701. (e) Özdoan, T. Collect. Czech. Chem. Commun. 2004, 69, 279. (f) Özdoan, T. Int. J. Quantum Chem. 2003, 52, Gradshteyn, I. S.; Ryzhi, I. M. Tables of Integrals, Series and Products; Academic Press: New Yor, Silver, D. M.; Ruedenberg, K. J. Chem. Phys. 1968, 49, Pople, J. A.; Beveridge, D. L. Approximate Molecular Orbital Theory; McGraw-Hill Boo Com.: New Yor, Guseinov, I. I. J. Phys. B 1970, 13, Corbato, F. J. J. Chem. Phys. 1956, 24, Carbo, R.; Besalu, E. Adv. Quantum Chem. 1992, 24, Wolfram, S. The Mathematica Boo; 4-th ed.; Cambridge Univ. Press, Reading: MA, (a) Barnett, M. P. Int. J. Quantum Chem. 2000, 76, 464. (b) Barnett, M. P. Theor. Chem. Acc. 2002, 107, Novosadov, B. K.; Niitin, O. Y. J. Struct. Chem. 2000, 41, Pendas, A. M.; Francisco, E. Phys. Rev. A. 1991, 43, Öztein, E.; Yavuz, M.; Atalay,. Theor. Chem. Acc. 2001, 106, Guseinov, I. I.; Mamedov, B. A. J. Mol. Struct. (Theochem). 1999, 465,1.

On the accurate evaluation of overlap integrals over Slater type orbitals using analytical and recurrence relations I.I. Guseinov. B.A.

On the accurate evaluation of overlap integrals over Slater type orbitals using analytical and recurrence relations I.I. Guseinov Department of Physics, Faculty of Arts and Sciences, Onseiz Mart University,