Calculation of Two-Center Nuclear Attraction Integrals over Integer and Noninteger n-Slater Type Orbitals in Nonlined-Up Coordinate Systems

Two-center nuclear attraction integrals over Slater type orbitals with integer and noninteger principal quantum numbers in nonlined up coordinate systems have been calculated by means of formulas in our previous work (T. Özdoan and M. Orbay, Int. J. Quant. Chem. 87 (2002) 15). The computer results for integer case are in best agreement with the prior literature. On the other hand, the results for noninteger case are not compared with the literature due to the scarcity of the literature, but also compared with the limit of integer case and good agreements are obtained. The proposed algorithm for the calculation of two-center nuclear attraction integrals over Slater type orbitals with noninteger principal quantum numbers in nonlined-up coordinate systems permits to avoid the interpolation procedure used to overcome the difficulty introduced by the presence of noninteger principal quantum numbers. Finally, numerical aspects of the presented formulae are analyzed under wide range of quantum numbers, orbital exponents and internuclear distances.     

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 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.     

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

Two‐center overlap integrals over Slater type orbitals (STOs) have been expressed in terms of the well‐known Mulliken's integrals B_n(pt) using 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 ≤ n,n′ ≤ 3, 0 ≤ l,l′ ≤ 2, ?2 ≤ m,m′ ≤ 2. Numerical results of this work, for some quantum sets, have also been compared with prior literature and best agreement achieved with recent works of Barnett while some discrepancies were obtained with works of Öztekin et al. and Guseinov et al.     

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     

Evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals with integer and noninteger principal quantum numbers

A general formula has been established for the expansion of the product of two normalized associated Legendre functions centered on the nuclei a and b. This formula has been utilized for the evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals (STOs) with integer and noninteger principal quantum numbers. The formulas given in this study for the evaluation of two‐center overlap and nuclear attraction integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, orbital exponents, and internuclear distances. © 2001 Wiley Periodicals, Inc. Int J Quantum Chem, 2001     

Efficiency of the algorithms for the calculation of Slater molecular integrals in polyatomic molecules

The performances of the algorithms employed in a previously reported program for the calculation of integrals with Slater-type orbitals are examined. The integrals are classified in types and the efficiency (in terms of the ratio accuracy/cost) of the algorithm selected for each type is analyzed. These algorithms yield all the one- and two-center integrals (both one- and two-electron) with an accuracy of at least 12 decimal places and an average computational time of very few microseconds per integral. The algorithms for three- and four-center electron repulsion integrals, based on the discrete Gauss transform, have a computational cost that depends on the local symmetry of the molecule and the accuracy of the integrals, standard efficiency being in the range of eight decimal places in hundreds of microseconds.     

Calculation of molecular electric and magnetic multipole moment integrals of integer and noninteger n Slater orbitals using overlap integrals

Closed formulas are established for the magnetic multipole moment integrals of integer and noninteger n Slater‐type orbitals (ISTOs and NISTOs) in terms of electric multipole moment integrals for which the analytic expressions through the overlap integrals with ISTOs and NISTOs are derived. The overlap integrals are evaluated by the use of auxiliary functions. Using the derived expressions the multipole moment integrals, and therefore the electric and magnetic properties of molecules, can be evaluated most efficiently and accurately. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Analytical Evaluation of Multicenter Multielectron Integrals of Central and Noncentral Interaction Potentials over Slater Orbitals Using Overlap Integrals and Auxiliary Functions

Using expansion formulas for central and noncentral interaction potentials (CIPs and NCIPs, respectively) in terms of Slater type orbitals (STOs) obtained by the author (I.I. Guseinov, J. Mol. Model., in press), the multicenter multielectron integrals of arbitrary interaction potentials (AIPs) are expressed through the products of overlap integrals with the same screening parameters and new auxiliary functions. For auxiliary functions, the analytic and recurrence relations are derived. The relationships obtained for multicenter multielectron integrals of AIDs are valid for the arbitrary quantum numbers, screening parameters and location of orbitals.     

Evaluation of the auxiliary function $${G_{-ns}^q}$$ using basic overlap integrals over Slater type orbitals

This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs) for the evaluation of auxiliary function which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function is completely general for p _a ≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function is compared with other method.     

Use of Filter-Steinborn B and Guseinov $${Q_{ns}^q }$$ auxiliary functions in evaluation of two-center overlap integrals over Slater type orbitals

An efficient method for computing overlap integral over Slater type orbitals based on the B Filter-Steinborn and Guseinov auxiliary functions is presented. The final results are expressed through the binomial coefficients with the help of which the overlap integrals can be evaluated efficiently and accurately. The results of calculation are in good agreement with those obtained by other method for arbitrary principal quantum numbers and different screening constants. An erratum to this article can be found at     

Fast evaluation of molecular auxiliary functions <Emphasis Type="Italic">A</Emphasis><Subscript>α</Subscript> and <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> by analytical relations

Analytical relations through the initial values are derived for the molecular auxiliary functions A _α (x) and B _n (x), where α =n+ɛ, 0⩽ ɛ < 1 and n=0,1,2,.... These relations are useful in the fast calculation of multicenter molecular integrals over integer and noninteger n Slater type orbitals. It is shown that the formulas obtained are numerically stable for all values of n,ɛ and x.PACS No: 31.15.+q, 31.20.EjAMS subject classification: 81-V55, 81-V45     

Analytical evaluation for two-center nuclear attraction integrals over slater type orbitals by using Fourier transform method

In this study, we shall suggest analytical expressions for two-center nuclear attraction integrals over STO’s with a one-center charge distribution by using Fourier transform method. The derivation is based on partial-fraction decompositions and Taylor expansions of rational functions. Analytical expressions obtained by this method are expressed in terms of Gegenbauer, and binomial coefficients and linear combinations of STO’s. Finally, it is relatively easy to express the Fourier integral representations of two-center nuclear attraction integrals with a one-center charge distribution mentioned above as finite and infinite of series of STO’s and irregular solid harmonics which may be considered to be limiting cases of STO’s.     

Expansion Formulae for Two-center Integer and Noninteger <Emphasis Type="Italic">n</Emphasis> STO Charge Densities and their Use in Evaluation of Multi-center Integrals

Using complete orthonormal sets of Ψ^α-exponential type orbitals (Ψ^α-ETOs, α =1, 0, −1, −2, ...) introduced by the author, the series expansion formulae are derived for the two-center integer and noninteger n STO (ISTO and NISTO) charge densities in terms of integer n STOs at a third center. The expansion coefficients occurring in these relations are presented through the two-center overlap integrals between STOs with integer and noninteger principal quantum numbers. The general formulae obtained for the STO charge densities are utilized for the evaluation of two-center Coulomb and hybrid integrals of NISTOs appearing in the Hartee–Fock–Roothaan approximation. The final results are expressed in terms of both the overlap integrals and the one-center basic integrals over integer n STOs. It should be noted that the result for the multi-center multielectron integrals with two-center noninteger n STO charge densities presented in this paper were not appeared in our past publications.     

Three‐center Coulomb repulsion integrals with Slater functions

The translation method for products of two Slater functions is improved and combined with the short–long range separation in order to develop a robust and efficient algorithm for the evaluation of three‐center Coulomb integrals with Slater functions. Several tests are carried out showing that the algorithm reported here yields integrals with an absolute error below 10^?12 hartree and a computational cost of a few microseconds per integral. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Calculation of Negative Ions of B,C, N,O and F Using Noninteger n Slater Type Orbitals

Combined Hartree‐Fock‐Roothaan calculations have been performed using noninteger n Slater type orbitals for the ground states of the lowest electron configurations 1s²2s²2pⁿ (2 ≤ n ≤ 6) for negative ions of B, C, N, O and F. These results are compared with the corresponding results obtained from the use of integer n Slater type orbitals. All of the nonlinear parameters are fully optimized. The results of calculation of coupling‐projection coefficients, orbital and total energies and virial ratios are presented. It is shown that the noninteger n Slater type orbitals, in general, improve the orbital energies.     

Calculation of electric multipole moment integrals with the different screening parameters via the Fourier transform method

Three‐center electric multipole moment integrals over Slater‐type orbitals (STOs) can be evaluated by translating the orbitals on one center to the other and reducing the system to an expansion of two‐center integrals. These are then evaluated using Fourier transforms. The resulting expression depends on the overlap integrals that can be evaluated with the greatest ease. They involve expressions for STO with different screening parameters that are known analytically. This work gives the overall expressions analytically in a compact form, based on Gegenbauer polynomials. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

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.