A new algorithm to evaluate bielectronic integrals with 1s Slater‐type orbitals obtained by the integral transforms

This article presents a variation of the integral transform method to evaluate multicenter bielectronic integrals (12|34), with 1s Slater‐type orbitals. It is proved that it is possible to define, out of the expression of (12|34) given by the integral transform method, a function F(q) that has the property of having a unique Q, such that F(Q) = (12|34). Therefore, F(q) may be used to calculate (12|34). It is shown that the evaluation of F(Q) turns out to be simpler than the three‐dimensional integral involved in the calculation of (12|34), and an algorithm is presented to calculate Q. The results show that relative errors on the order of 10^?3 or lower are obtained very efficiently. In addition, it is shown that the proposed algorithm is very stable. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

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.     

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     

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     

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.     

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     

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     

Evaluation of two-center overlap integrals using slater type orbitals in terms of bessel type orbitals

Using the definition of STOs in terms of BTOs, we have presented analytical formula for two-center overlap integrals. The obtained formula contains generalized binomial coefficients and Mulliken integrals A_k and B_k. Taking into account the recent advances on the efficient calculation of Mulliken integrals (Harris, Int. J. Quantum Chem., 100 (2004) 142), we have obtained many more satisfactory results for two-center overlap integrals, for arbitrary quantum numbers, scaling parameters, and location of atomic orbitals.PACS No: 31.15.+qAMS Subject Classification: 81V55, 81–08     

Evaluation of two‐center,three‐ and four‐electron integrals over Slater‐type orbitals in elliptical coordinates

An algorithm for evaluation of two‐center, three‐electron integrals with the correlation factors of the type rr and rrr as well as four‐electron integrals with the correlation factors rrr and rrr in the Slater basis is presented. This problem has been solved here in elliptical coordinates, using the generalized and modified form of the Neumann expansion of the interelectronic distance function r for k ≥ ?1. Some numerical results are also included. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

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.     

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     

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

The multicenter charge‐density expansion coefficients [I. I. Guseinov, J Mol Struct (Theochem) 417 , 117 (1997)] appearing in the molecular integrals with an arbitrary multielectron operator were calculated for extremely large quantum numbers of Slater‐type orbitals (STOs). As an example, using computer programs written for these coefficients, with the help of single‐center expansion method, some of two‐electron two‐center Coulomb and four‐center exchange electron repulsion integrals of Hartree–Fock–Roothaan (HFR) equations for molecules were also calculated. Accuracy of the results is quite high for the quantum numbers, screening constants, and location of STOs. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 146–152, 2000     

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     

Evaluation of integrals over STOs on different centers and the complementary convergence characteristics of ellipsoidal‐coordinate and zeta‐function expansions

One‐electron integrals over three centers and two‐electron integrals over two centers, involving Slater‐type orbitals (STOs), can be evaluated using either an infinite expansion for 1/r₁₂ within an ellipsoidal‐coordinate system or by employing a one‐center expansion in spherical‐harmonic and zeta‐function products. It is shown that the convergence characteristics of both methods are complimentary and that they must both be used if STOs are to be used as basis functions in ab initio calculations. To date, reports dealing with STO integration strategies have dealt exclusively with one method or the other. While the ellipsoidal method is faster, it does not always converge to a satisfactory degree of precision. The zeta‐function method, however, offers reliability at the expense of speed. Both procedures are described and the results of some sample calculation presented. Possible applications for the procedures are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 1–13, 1999     

Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater‐type orbitals with noninteger principal quantum numbers

The expansion formula has been presented for Slater‐type orbitals with noninteger principal quantum numbers (noninteger n‐STOs), which involves conventional STOs (integer n‐STOs) with the same center. By the use of this expansion formula, arbitrary multielectron multicenter molecular integrals over noninteger n‐STOs are expressed in terms of counterpart integrals over integer n‐STOs with a combined infinite series formula. The convergence of the method is tested for two‐center overlap, nuclear attraction, and two‐electron one‐center integrals, due to the scarcity of the literature, and fair uniform convergence and great numerical stability under wide changes in molecular parameters is achieved. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Implementation of atomic basis set composed of 1s Gaussian and 1s Slater-type orbitals to carry out quantum mechanics molecular calculations

A mixed atomic basis set formed with ls Slater-type orbitals and 1s floating spherical Gaussian orbitals is implemented. Evaluation of multicenter integrals is carried out using a method based on expansion of binary products of atomic basis functions in terms of a complete basis set, and a systematic analysis is performed. The proposed algorithm is very stable and furnishes fairly good results for total energy and geometry. An LCAO-SCF test calculation is carried out on LiH. The trends observed show that there are some combinations of mixed orbitals that are appropriate to describe the system. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 604–609, 1999     

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.     

Analytical evaluation of three-center nuclear-attraction integrals over s-Slater orbitals for asymmetrical conformations of the centers

Analytical formulas for three-center nuclear-attraction integrals over Slater orbitals are given for any location of the three atomic centers. In the mathematical derivations the Neumann expansion has been used and new general auxiliary integrals which depend on the elliptical coordinates of one of the centers are defined. The orbital exponents within the integrals may be different.     

Theory of complete orthonormal relativistic vector wave function sets and Slater type relativistic vector orbitals in coordinate,momentum and four-dimensional spaces

The new analytical relations for the relativistic vector wave functions and Slater type relativistic vector orbitals in coordinate, momentum and four-dimensional spaces are derived using the properties of quasirelativistic vector spherical harmonics introduced by the author in previous paper (I.I. Guseinov, J. Math. Chem., 44, 197 (2008)) and complete orthonormal scalar basis sets of nonrelativistic ψ ^α -exponential type orbitals (ψ ^α -ETO), -momentum space orbitals (-MSO) and z ^α -hyperspherical harmonics (z ^α -HSH). The 6-component relativistic vector wave functions obtained are complete without the inclusion of the continuum. The relativistic vector wave function sets and Slater type relativistic vector orbitals are expressed through the corresponding quasirelativistic vector wave functions and Slater vector orbitals, respectively. The analytical formulas are also derived for overlap integrals over Slater type relativistic vector orbitals with the same screening constants in coordinate space.