New program for molecular calculations with Slater‐type orbitals

A new program for computing all the integrals appearing in molecular calculations with Slater‐type orbitals (STO) is reported. This program follows the same philosophy as the reference pogram previously reported but introduces two main changes: Local symmetry is profited to compute all the two‐electron integrals from a minimal set of seed integrals, and a new algorithm recently developed is used for computing the seed integrals. The new code reduces between one and two orders of magnitude the computational cost in most polyatomic systems. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 148–153, 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     

Molecular integrals for Gaussian and exponential‐type functions: Shift operators

Basis functions with arbitrary quantum numbers can be attained from those with the lowest numbers by applying shift operators. We derive the general expressions and the recurrence relations of these operators for Cartesian basis sets with Gaussian and exponential radial factors. In correspondence, the expressions of molecular integrals involving functions with arbitrary quantum numbers can be obtained by applying these operators on the integrals with the lowest quantum numbers. Since the original form of the shift operators is not appropriate to deal with integrals, we give their representation in terms of derivatives with respect to the parameters on which these integrals explicitly depend. Moreover, we translate the recurrence relations to the new representation and, finally, we analyze the general expressions ot the molecular integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 137–145, 2000     

Analytic evaluation of two‐center STO electron repulsion integrals via ellipsoidal expansion

Extant analytic methods for evaluating two‐center electron repulsion integrals in a Slater‐type orbital (STO) basis using ellipsoidal coordinates and the Neumann expansion of 1/r₁₂ have problems of numerical stability that are analyzed in detail using computer‐assisted algebraic techniques. Some of these problems can be eliminated by use of procedures known in this field 40 years ago but seemingly forgotten now. Others can be removed by use of a formulation suitable for small values of the STO screening parameter. A recent attempt at such a formulation is corrected and extended in a way permitting its practical use. The main functions encountered in the integrations over the ellipsoidal coordinate of the range 1 … ∞ are Bessel functions or generalizations thereof, as pointed out here for the first time. This fact is used to motivate the derivation of recurrence relations additional to those previously known. Novel techniques were devised for using these recurrence relations, thereby providing new ways of calculating the quantities that enter the ellipsoidal expansion. The convergence rate of this expansion and the numerical characteristics of several computational strategies are reported in enough detail to identify the ranges where various schemes can be used. This information shows that recent discussions of the “convergence characteristics of [the] ellipsoidal coordinate expansion” are in fact not that, but are instead discussions of an inability to make accurate calculations of the individual terms of the expansion. It is also seen that the parameter range suitable for use of Kotani's well‐known recursive scheme is more limited than seems generally believed. The procedures discussed in this work are capable of yielding accurate two‐center electron repulsion integrals by the ellipsoidal expansion method for all reasonable STO screening parameters, and have been implemented in illustrative public‐domain computer programs. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Use of analytical relations in evaluation of exponential integral functions

Analytical formulas through the initial values suitable for numerical computation are developed for the exponential integral functions E_n (x). The relationships obtained are numerically stable for all values of n and for x < 1. Numerical results are also given.PACS No: 31.15.+q, 31.20.Ej AMS subject classification:81-V55, 81V45     

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     

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.     

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     

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 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.     

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     

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     

Master formulas for two‐ and three‐center one‐electron integrals involving Cartesian GTO,STO, and BTO

As a first application of the shift operators method we derive master formulas for the two‐ and three‐center one‐electron integrals involving Gaussians, Slater, and Bessel basis functions. All these formulas have a common structure consisting in linear combinations of polynomials of differences of nuclear coordinates. Whereas the polynomials are independent of the type (GTO, BTO, or STO) of basis functions, the coefficients depend on both the class of integral (overlap, kinetic energy, nuclear attraction) and the type of basis functions. We present the general expression of polynomials and coefficients as well as the recurrence relations for both the polynomials and the whole integrals. Finally, we remark on the formal and computational advantages of this approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 83–93, 2000     

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     

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     

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     

Auxiliary functions for molecular integrals with Slater‐type orbitals. I. Translation methods

Many types of molecular integrals involving Slater functions can be expressed, with the ζ‐function method in terms of sets of one‐dimensional auxiliary integrals whose integrands contain two‐range functions. After reviewing the properties of these functions (including recurrence relations, derivatives, integral representations, and series expansions), we carry out a detailed study of the auxiliary integrals aimed to facilitate both the formal and computational applications of the ζ‐function method. The usefulness of this study in formal applications is illustrated with an example. The high performance in numerical applications is proved by the development of a very efficient program for the calculation of two‐center integrals with Slater functions corresponding to electrostatic potential, electric field, and electric field gradient. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

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