Efficient and Rapid Evaluation of Three-Center Two Electron Coulomb and Hybrid Integrals Using Nonlinear Transformations

Abstract

Among the two electron integrals occurring in the molecular context, the three-center Coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy. The analytical and numerical difficulties arise mainly from the presence of the spherical Bessel function and hypergeometric series in these integrals. The present work pursues the acceleration of convergence for three-center two electron Coulomb and hybrid integrals. We have proven that the hypergeometric function can be expressed as a finite expansion and that the integrand involving this series and a product of Bessel functions satisfies a linear differential equation with coefficients having a power series expansion in the reciprocal of the variable suitable for application of the nonlinear D- and D-transformations. These transformations depend strongly on the order of the differential equation that the integrand of interest satisfies. This work concentrates on reduction of this order to two, exploiting properties of spherical and reduced Bessel functions, leading to greatly simplified calculations to evaluate the integrals precisely by reducing the order of linear systems to be solved. It also avoids the long and difficult implementations of successive derivatives of the integrands. The numerical results section illustrates clearly the reduction of the calculation time we obtained for a high predetermined accuracy.