Numerical calculation of overlap and kinetic integrals in prolate spheroidal coordinates. II

Abstract

ABSTRACT:The efficient algorithm calculating the overlap and the kinetic integrals for the numerical atomic orbitals is presented. On the basis of the prolate spheroidal coordinates, the overlap and the kinetic integral are reduced to the integral over the rectangular domain. The integration over the rectangular domain is performed by the adaptive integration scheme. The developed algorithm is applied to calculate the integrals for the pairs of hydrogen and gallium eigenfunctions. It is demonstrated that high accuracy can be obtained for small number of integrand evaluations what guarantees the efficiency of the presented algorithm.

On the fast evaluation of three-center nuclear attraction integrals using one-range addition theorems for Slater functions

“…The convergence can be described by As an application, the convergence properties of the series expansion relation for integral I ac,b 300,211 (9.8, 5.3; R ac , R ab ) as a function of N for given values of α are shown in figure 1. This figure shows that the best values of three-center nuclear attraction integrals (14) with respect to the convergence and accuracy of series expansion relations are obtained for α = 0. Figure 1 shows that the series expansion relations display the most rapid convergence to the numerical results with five digits stable for N = 16.…”

“…It is well known that the evaluation of molecular integrals over χ-STOs has attracted continuous attention [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. To our knowledge, although many improvements have been made in previous years by the use of computers, an efficient program for the calculation of molecular integrals, particularly for the three-center nuclear attraction integral of χ-STOs, is not yet available in the literature.…”

Molecular calculations for HeH+ with two-center correlated orbitals

“…It is a challenge to use the two-center orbitals for molecules with more than two nuclei. Recent progress in numerical calculations of the overlap and the kinetic integrals for the numerical atomic orbitals in the prolate spheroidal coordinates [16] might shed light on the application of the two-center orbitals to multiple-nuclei systems. Moreover, one-electron, and many-center orbitals, which in concept are similar to the two-center orbitals, have been obtained analytically by Shibuya and Wulfman in terms of integral equations on the unit hypersphere of the 4-dimensional momentum space [17] .…”

Analytical evaluation of relativistic molecular integrals. II: Method of computation for molecular auxiliary functions involved

“…This may be because practical multi−precision libraries and the symbolic programming languages were not available when Allouche's paper was published. Even if they were available it would still be laborious (Please see [40][41][42][43] where numerical three−dimensional adaptive integration procedure used for calculation of two−center integrals with Slater−type functions (STOs). The principal quantum numbers restricted to be integers yet, even with lowest values of quantum numbers the results are insufficient).…”