Calculation of Multicenter Nuclear Attraction and Electron Repulsion Integrals over Slater Orbitals by Fourier Transform Method Using Gegenbauer Polynomials

In this study, using complete orthonormal sets of exponential type orbitals (ETOs), a single closed analytical relation is derived for a large number of different expansions of overlap integrals over Slater type orbitals (STOs) with the same screening parameters in terms of Gegenbauer coefficients. The general formula obtained for the overlap integrals is utilized for the evaluation of multicenter nuclear attraction and electron repulsion integrals appearing in the Hartree–Fock–Roothaan equations for molecules. The formulas given in this study for the evaluation of these multicenter integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, scaling parameters of STOs and internuclear distances.     

Evaluation of Multicenter Electronic Attraction, Electric Field and Electric Field Gradient Integrals with Screened and Nonscreened Coulomb Potentials over Integer and Noninteger n Slater Orbitals

By the use of complete orthonormal sets of -exponential-type orbitals, where ( = 1, 0, –1, –2,...) the multicenter electronic attraction (EA), electric field (EF) and electric field gradient (EFG) integrals of nonscreened and Yukawa-like screened Coulomb potentials are expressed through the two-center overlap integrals with the same screening constants and the auxiliary functions introduced in our previous paper (I.I. Guseinov, J. Phys. B, 3 (1970) 1399). The recurrence relations for auxiliary functions are useful for the calculation of multicenter EA, EF and EFG integrals for arbitrary integer and noninteger values of principal quantum numbers, screening constants, and location of slater-type orbitals. The convergence of the series is tested by calculating concrete cases.     

Use of Recursion and Analytical Relations in Evaluation of Hypergeometric Functions Arising in Multicenter Integrals with Noninteger <Emphasis Type="Italic">n</Emphasis> Slater Type Orbitals

In this work we present the new recursion and analytical relations for the calculation of hypergeometric functions F(1,b;c;z) occurring in multicenter integrals of noninteger n Slater type orbitals. The formulas obtained are numerically stable for 0 < z < 1 and all integer and noninteger values of parameters b and c The Author cordially congratulates Prof. I.I. Guseinov on his 70th birthday     

Calculation of Two-center Nuclear Attraction Integrals over Slater Type Orbitals in Molecular Coordinate System

A closed analytical relation is derived for the two-center nuclear attraction integrals over Slater type orbitals (STOs) in terms of binomial coefficients. This formula can be used in highly accurate calculations of the nuclear attraction integrals. The relationships obtained are valid for arbitrary values of quantum numbers and screening constants of STOs and location of nuclei.     

Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations

New downward recursion and analytical approaches are presented for the calculation of incomplete gamma functions (,x) and (,x). The efficiency of calculation with an adequate interval of these expressions is compared with those of other methods and great numerical stability and good rate of convergence is obtained for wide range of integral parameters. It is shown that the downward recursive relations are stable for all values of and x of (,x).     

Evaluation of Two-center One- and Two-electron Integrals over Slater Type Orbitals

A formulation previously presented by the authors for coulomb integrals was generalized to other two-center integrals, except exchange integral. Within this frame, molecular integrals were expressed in terms of some new functions closely related to the well-known incomplete gamma functions and these functions recursively evaluated. Special issues arising in the case of hybrid integrals were addressed, and the results were compared with the ones found in the literature.     

Use of Cartesian coordinates in evaluation of multicenter multielectron integrals over Slater type orbitals and their derivatives

Using addition theorems for interaction potentials and Slater type orbitals (STOs) obtained by the author, and the Cartesian expressions through the binomial coefficients for complex and real regular solid spherical harmonics (RSSH) and their derivatives presented in this study, the series expansion formulas for multicenter multielectron integrals of arbitrary Coulomb and Yukawa like central and noncentral interaction potentials and their first and second derivatives in Cartesian coordinates were established. These relations are useful for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules, and solids by Hartree–Fock–Roothaan and correlated theories. The formulas obtained are valid for arbitrary principal quantum numbers, screening constants and locations of STOs.     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. III. Auxiliary functions \documentclass{article}\pagestyle{empty}\begin{document}$Q_{nn'}^{q}$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$G_{-nn'}^{q}$\end{document}

The auxiliary functions $Q_{nn'}^{q}(p,pt)$ and $G_{-nn'}^{q}(p_{a},p,pt)$ which are used in our previous paper [Guseinov, I. I.; Mamedov, B. A. Int J Quantum Chem 2001, 81, 117] for the computation of multicenter electron‐repulsion integrals over Slater‐type orbitals (STOs) are discussed in detail, and the method is given for their numerical computation. The present method is suitable for all values of the parameters p_a, p, and pt. Three‐ and four‐center electron‐repulsion integrals are calculated for extremely large quantum numbers using relations for auxiliary functions obtained in this paper. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. II. Two‐center expansion method

Using expansion formulas for the charge‐density over Slater‐type orbitals (STOs) obtained by the one of authors [I. I. Guseinov, J Mol Struct (Theochem) 1997, 417, 117] the multicenter molecular integrals with an arbitrary multielectron operator are expressed in terms of the overlap integrals with the same screening parameters of STOs and the basic multielectron two‐center Coulomb or hybrid integrals with the same operator. In the special case of two‐electron electron‐repulsion operator appearing in the Hartree–Fock–Roothaan (HFR) equations for molecules the new auxiliary functions are introduced by means of which basic two‐center Coulomb and hybrid integrals are expressed. Using recurrence relations for auxiliary functions the multicenter electron‐repulsion integrals are calculated for extremely large quantum numbers. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 117–125, 2001     

Evaluation of Multicenter Electric Multipole Moment Integrals over Integer and Noninteger n‐STOs

Multicenter electric multipole moment integrals over Slater type orbitals with integer and noninteger principal quantum numbers are expressed in terms of overlap integrals. The computer results for the integer case agree best with the prior literature. The accuracy of the computer results for noninteger case is not compared with the literature due to the lack of relevant literature, but the limit of the noninteger case is compared with the integer case and good agreement is achieved for wide changes in the relevant molecular parameters.     

Numerical Evaluation of Three-Center Two-Electron Coulomb and Hybrid Integrals Over B Functions Using the HD and H\overline D Methods and Convergence Properties

The difficulties of the numerical evaluation of three-center two-electron Coulomb and hybrid integrals over B functions, arise mainly from the presence of the hypergeometric series and semi-infinite very oscillatory integrals in their analytical expressions, which are obtained using the Fourier transform method.This work presents a general approach for accelerating the convergence of these integrals by first demonstrating that the hypergeometric function, involved in the analytical expressions of the integrals of interest, can be expressed as a finite sum and by applying nonlinear transformations for accelerating the convergence of the semi-infinite oscillatory integrals after reducing the order of the differential equation satisfied by the integrand.The convergence properties of the new approach are analysed to show that from the numerical point of view the method corresponds to the most ideal situation.The numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals.     

Evaluation of Orbital- and Ground State Energies of Some Open- and Closed-Shell Atoms over Integer and Noninteger Slater Type Orbitals

Ab initio calculations of the orbital and the ground state energies of some open- and closed-shell atoms over Slater type orbitals with quantum numbers integer and Slater type orbitals with quantum numbers noninteger have been performed. In order to increase the efficiency of these calculations the atomic two-electron integrals were expressed in terms of incomplete beta function. Results were observed to be in good agreement with the literature.