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

Size: px

Start display at page:

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

Christopher Cameron
5 years ago
Views:

1 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, Çanaale, Turey B.A. Mamedov Department of Physics, Faculty of Arts and Sciences, Gaziosmanpaşa University,Toat, Turey Abstract In this study, using the analytical and recurrence relations suggested by the authors in previous wors, the new efficient and reliable program procedure for the overlap integrals over Slater type orbitals (STOs) is presented. The proposed procedure guarantees a highly accurate evaluation of the overlap integrals with arbitrary values of quantum numbers, screening constants and internuclear distances. It is demonstrated that the computational accuracy of the proposed procedure is not only dependent on the efficiency of formulas, as has been discussed previously, but also on a number of other factors including the used program language pacage and solvent properties. The numerical results obtained using the algorithm described in the present wor are in a complete agreement with those obtained using the alternative evaluation procedure. We notice that the program wors without any restrictions and in all range of integral parameters. Keywords: Slater type orbitals, Overlap integrals, Recurrence relations, Auxiliary functions I. Introduction In the study of the electronic structure of molecules, one has to evaluate overlap integrals over STOs accurately and efficiently. These integrals arise not only in the Hartree-Foc- Roothaan equations for molecules, but are also central to the calculation of arbitrary multicenter integrals based on the series expansion formulas about a new center and onerange addition theorems for STOs [] which necessitate to accurately calculate the overlap integrals especially for the large quantum numbers. It should be noted that the overlap integrals over STOs are also used in all of the semiempirical methods []. The aim of this report is to calculate the overlap integrals over STOs using the analytical approach containing well-nown auxiliary functions A and B and the recurrence relations for the basic overlap integrals presented in our previous wors [3] and [4, 5], respectively. These expressions are especially useful for computation of overlap integrals on the computer for high quantum numbers, internuclear distances and orbital exponents or vice versa.

2 In this wor, the differences and similarities in organization of existing overlap integral programs are discussed, and a new strategy is developed. This method is computationally simple and numerically well behaved. On the basis of formulas obtained in papers [3-5] we constructed a program for computation of the overlap integrals over STOs using Mathematica 5.0 international mathematical software and Turbo Pascal language pacages. The numerical results demonstrated that the computational accuracy of the established formulas is not only dependent on the efficiency of formulas, but also strongly dependent on the used program language pacages. Excellent agreement with benchmar results and stability of the technique are demonstrated. Since the overlap integrals over STOs are of considerable importance in the evaluation of arbitrary multicenter integrals, it is hoped that the present wor will prove useful in tacling more complicated molecular integrals appearing in the determination of various properties for molecules when the Hartree-Foc-Roothaan approximation is employed.. Definition The two-center overlap integrals over STOs with respect to lined-up coordinate systems are defined as * Snlλ, n l λ ( p, t) = χnlm ( ζ, ra ) χn l m ( ζ, rb ) dv, () R where 0 λ lm, =± λ, p= ( ζ + ζ ), t= ( ζ ζ )/( ζ + ζ ), R Rab = ra rb and (, ) ( ) n + ( )! n χ r nlm ζ r = ζ n r e ζ Slm ( θ, ϕ). () ) Here, is the complex ( S = Y or real spherical harmonic. It should be noted that our Slm lm lm definition of phases for complex spherical harmonics Y Shortley phases [6] by the sign factor. 3. Analytical relations in terms of auxiliary functions * lm = differs from the Condon- In Ref.[3], using the auxiliary function method for the overlap integrals have been established the following formula: Yl m () () 0 S p, t = N ( t) g ( lλ, l λ) F ( α + λ, β λ) nlλ, n l λ nn αβ q α= λ β= λ q= 0 n+ n α β m= 0 l l α+ β F n n A p B pt where N ( t), (, ) and N nn nn m n+ n + m( α, β) n+ n α β m+ q m+ q, F N N A [( + t)] [( t)] () t = ( n)!( n )! n+ / n + / n p are determined by (3) (4)

3 min( mn, ) σ Fm( N, N ) = ( ) Fm σ( N) Fσ( N ), (5) n σ = [( m n ) + m n ) A p = p A p. (6) n Here, Fm( n) = n!/[ m!( n m)!] are the binomial coefficients and n+. It should be noted that, Eq.(5) for the generalized binomial coefficients with different notation been presented by N. Rosen in Ref. []. The quantities An ( p ) and Bn (3) and (6) are well nown auxiliary functions [8] (see also Ref. [9]). NN D m firstly has pt occurring in Eqs. The quantities g 0 αβ ( lλ, l λ) in Eq.(3) are the expansion coefficients for a product of two normalized Legendre functions in elliptic coordinates. The relationship for these coefficients in terms of factorials was given in [0]. In Ref.[], these coefficients were expressed in terms of binomial coefficients. 4. Use of recurrence relations for basic overlap integrals In Ref.[5], using the expansion formula for product of two spherical harmonics both with the same center [0], the overlap integrals, Eq.(), were expressed through the basic overlap integrals: S nlλ [ ( + )] ( + )( )! ( + ) ( + ) ( ),, = l l p t l l F n n l Fl + λ l λ Fl λ l λ n l λ p t l l = λ [ p( t) ] ( l + )( l )! F ( n n l ) ( l, l λ ) S ( p, t ), L L L + C λ n l 00, n + l L 0 () L where C ( l λ, l λ) are the Gaunt coefficients. With the aid of recurrence relations given in Ref [5], the basic overlap integrals S ( p ) the functions n00, nl 0, t (, ) (, ) and S ( p,0 ) S ( p,0) S p t S p t ,000 which we can use the following analytical formulas: S S p( t ) p(+ t) ( p, t) ( p t){ e e 00 00, 00 appearing in () can be expressed in terms of ,000 for the calculation of = η } (8) t p ( p,0) e. = (9) 5. Numerical results and discussion On the basis of Eqs.(3) and (), obtained in our papers [3-5], we constructed the programs which were performed in the Mathematica 5.0 international mathematical software and Turbo Pascal.0 language pacages. The computational results of overlap integrals by the use of Turbo Pascal.0 language pacage program have been examined in our published papers [3-

4 5]. The Barnett s data [] and results of our calculation using Mathematica 5.0 international mathematical software and Turbo Pascal.0 language pacages for various values of parameters are represented in Table. Barnett s data are reproduced by using our scheme with Mathematica while we get different results using the same scheme with Turbo Pascal. Thus, in this paper we show that the discrepancies can be arisen in the case of different programming environments. We note that, the difference between the numerical results of Eqs.(3) and () arise only after forty fifth digits. It should be noted that for the comparison of the accuracy of computer results obtained from the formulas of overlap integrals, one should use the same program language pacages. It is well nown from the expert of this field that the problems occur in the evaluation of overlap integrals are as follow: small internuclear distances and small orbital exponents, and high internuclear distances and high orbital exponents. The results of calculation in these cases are given in Table. As is clear from our tests that the recurrence and analytical formulas presented in this study are useful tool for exact evaluation of the overlap integrals with arbitrary values of quantum numbers, internuclear distances and orbital parameters. Thus, our program calculates the overlap integrals over STOs with arbitrary quantum numbers ( nln,,, l, λ) and variables (p,t). References. I. I. Guseinov, J.Chem. Phys., 69 (98) 4990; Phys.Rev. A, (980) 369; 3 (985) 85; 3 (985) 864; 3 (988) 34; Int. J. Quant. Chem., 90 (00) 4 ; J.Mol.Model., 9 (003) 90.. M. J. S. Dewar and Y. Yamaguchi, Comput. Chem., (98) I.I. Guseinov, B.A. Mamedov, J.Mol.Model., 8 (00). 4. I.I. Guseinov, B.A. Mamedov, MATCH, 5 (004) I.I. Guseinov, B.A. Mamedov, J. Mol. Struct.(Theochem), 465 (999). 6. E.U.Condon, G.H.Shortley, The theory of a atomic spectra, Cambridge University Press, Cambridge, 90.. N. Rosen, Phys. Rev., 38 (93) R.S. Mullien, C.A. Riee, D. Orloff, and H. Orloff, J. Chem. Phys., (949) I.I. Guseinov, B.A. Mamedov, J. Math.Chem., 38 (005). 0. I. I. Guseinov, J. Phys. B., 3 (90) I.I. Guseinov, J. Mol. Struct. (Theochem), 4 (99).. M.P.Barnett, Theor.Chem.Acc., 0 (00) 4.

5 n l n l λ p t Eqs.(3) and () in Turbo Table. Comparison with results of Barnett [] Eqs.(3) and () in Ref.[] in Mathematica Pascal procedure Mathematica procedure procedure E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-

6 Table. The comparative values of the two-center overlap integrals over STOs in lined-up coordinate systems for small and high values of integral parameters n l n l λ p t Eqs.(3) and () in Eqs.(3) and () in Turbo Mathematica procedure Pascal procedure E-4 E E-6 E E E E E E-8 E E E E-5 E E-6 E E E E E E E E E E E E E E E E E E-8

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

Journal of the Chinese Chemical Society, 2004, 51, 243-252 243 Symbolic Calculation of Two-Center Overlap Integrals Over Slater-Type Orbitals Sedat Gümü and Telhat Özdoan Department of Physics, Amasya