Frank E. Harris

Department of Physics, University of Utah

Quantum Theory Project, University of Florida

(submitted to Int. J. Quantum Chem., September 2001)

Extant analytical methods for evaluating two-center electron repulsion integrals in a Slater-type orbital (STO) basis using ellipsoidal coordinates and the Neumann expansion of 1/r_{12} 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 range 1 ... infinity are Bessel functions or generalizations thereof, as is 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.