On the evaluation of derivatives of Gaussian integrals

Summary We show that by a suitable change of variables, the derivatives of molecular integrals over Gaussian-type functions required for analytic energy derivatives can be evaluated with significantly less computational effort than current formulations. The reduction in effort increases with the order of differentiation.Dedicated to Prof. Klaus Ruedenberg     

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.     

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.     

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     

Calculation of multicenter electronic attraction,electric field and electric field gradient integrals of Coulomb potential over integer and noninteger <Emphasis Type="Italic">n</Emphasis> Slater orbitals

With the help of complete orthonormal sets of - ETOs, where = 1,0, – 1, – 2, ... a large number of series expansion formulas for the multicenter electronic attraction (EA), electric field (EF) and electric field gradient (EFG) integrals of integer and noninteger n Slater type orbitals (ISTOs and NISTOs) is established through the overlap integrals with the same screening constants and the new central and noncentral interaction potentials depending on the coordinates of the nuclei of a molecule are introduced. The convergence of the series is tested by calculating concrete cases for arbitrary quantum numbers, screening constants and location of ISTOs and NISTOs.     

应用Gauss积分法计算STO双中心重叠积分

本文将Gauss求积法引进STO双中心重叠积分的计算中.结果表明此法计算简便快速,具有相当高的精确度,并大大地扩展了STO双中心重叠积分的适用范围。     

Direct quantum chemical integral evaluation

Performing quantum chemical integral evaluation directly, without recursion and without direct coupling of angular momenta according to the rotation group is analyzed. The rotation group limits the structure of these closed‐form expressions. The result of all cross differentiation is a rotational invariant. Closed‐form expressions are obtained for the general three‐ and four‐center Gaussian integral. The solid harmonic addition formula can be used to express these integrals as sums of products of an exponent‐independent (angular) factor and a molecular‐orientation‐independent (exponential) factor in a variety of ways. The results are products of two such factors summed over the set of distinct, relevant polynomials of the exponents. The coefficients of these polynomials, angular factors, are complicated but common to all n‐center matrix elements and independent of any type of contraction. Derivatives must be obtained using the product rule. An implementation in the Solid Spherical Harmonic Gaussian (SSHG) computer code is outlined and preliminary comparison is made. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 373–383, 2001     

A semiempirical model for the two-center repulsion integrals in the NDDO approximation

A self-consistent formalism is proposed for the two-center electron repulsion integrals in the NDDO approximation, based on their expansion in terms of multipole-multipole interactions and free from adjustable parameters.     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

A rigorous and optimized strategy for the evaluation of the Boys function kernel in molecular electronic structure theory

This work is focused on the efficient evaluation of the Boys function located at the heart of Coulomb and exchange type electron integrals. Different evaluation strategies for individual orders and arguments of the Boys function are used to achieve a minimal number of floating‐point operations. Based on previous work of other groups, two similar algorithms are derived that are compared based on both accuracy and efficiency: The first algorithm combines the work of Gill et al. (Int. J. Quantum Chem. 1991, 40, 745) and Kazuhiro Ishida (Int. J. Quantum Chem. 1996, 59, 209 and following work), amplifying the benefits of the two strategies. © 2015 Wiley Periodicals, Inc.     

应用Monte Carlo平均值法计算STO双中心重迭积分

把Monte Carlo方法引进STO双中心重叠积分的计算中,结果表明,它不仅计算简便、快速、很容易在计算机上实现,而且具有较高的精确度,有望推广应用于更复杂的多中心分子积分中.     

Bounds to electron-repulsion energies

A review of rigorous bounds to electron-repulsion integrals for atoms and molecules is presented. Inequalities involving direct (classical) as well as indirect (quantal) Coulomb energies are discussed. This is followed by an account of two-electron integrals in a Hartree Fock context over Gaussian basis-sets. Novel rigorous bounds to these integrals are derived and tested for some organic molecules. Connections are established with the density-based inequalities presented earlier. The present results are expected to enhance the efficiency of a generalab initio Gaussian program and yet have a sound theoretical footing.     

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     

Rapid evaluation of molecular integrals with ETOs

In this article, we discuss a way in which the theory of hyperspherical harmonics may be used for rapid evaluation of difficult molecular integrals when exponential‐type orbitals (ETOs) are used as a basis. One of us (J.E.A.) has implemented the method, and programs are available for general use. As a byproduct of this work, we are also able to evaluate generalized scattering factors for ETOs which allow first‐order density matrices to be measured experimentally using high‐quality X‐ray diffraction data. © 2015 Wiley Periodicals, Inc.     

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.     

Convergence accelerator approach for the evaluation of some three‐electron integrals containing explicit rij factors

A simplified analysis is presented for the evaluation of the three‐electron one‐center integrals of the form ∫rrrrrred r ₁d r ₂d r ₃, for the cases i, j, k, ≥−2, l=−2, m≥−1, n≥−1. These integrals arise in the calculation of lower bounds for energy levels and certain relativistic corrections to the energy when Hylleraas‐type basis sets are employed. Convergence accelerator techniques are employed to obtain a reasonable number of digits of precision, without excessive CPU requirements. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 93–99, 1999     

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.     

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 angular integrals by harmonic projection

Summary Angular momentum and angular integrations are discussed from the standpoint of the theory of harmonic polynomials. General formulae are developed which provide alternatives to the usual group theoretical approach. These formulae are illustrated by applications to the calculation of molecular electrostatic potentials, Fourier transforms of charge densities, and multipole expansions.It is a pleasure for us to dedicate this paper to Madame Alberte Pullman, one of the great pioneers of quantum biochemistry. Her work has illuminated many important aspects of chemical reactivity. Among these is the role of excess charge density, and the electrostatic potential which it generates, in determining biochemical reactivity and specificity [1–9]. We are happy to be able to discuss some aspects of this problem in the present paper, using the methods of harmonic projection.