Two - Electron Repulsion Integrals Over Gaussian s Functions

We present an efficient scheme to evaluate the [O](m) integrals that arise in many ab initio quantum chemical two-electron integral algorithms. The total number of floating-point operations (FLOPS) required by the scheme has been carefully minimized, both for cases where multipole expansions of the integrals are admissable and for cases where this is not so. The algorithm is based on the use of a modified Chebyshev interpolation formula to compute the function exp(-7') and the integral F,,,(T) = J&'"exp(-Tu2) du very cheaply. Introduction In retrospect, the 1986 paper [l] by Obara and Saika (0s) appears to have heralded a renaissance of interest in the efficient computation of the many twoelectron repulsion integrals (ERIS) that are needed in most ab initio calculations of electronic structure. 0s introduced an eight-term recurrence relation by which the ERIS between Gaussian basis functions of arbitrarily high angular momentum may be generated recursively from auxiliary s-type integrals, and in the years following, a number of algorithms [2-61 have been developed that successfully extend this approach by using other recurrence relations in addition to, or instead of, the original one. The generation of ERIS using such algorithms involves two disjoint tasks. First, a small number of auxiliary s-type integrals ([ss I ssIcm) or [O]'")) are formed, and, second, these are suitably combined together using the available recurrence relations. However, although considerable effort has recently focused on the optimization of the second of these tasks, the first has received scant attention. The predictable consequence of this neglect, as Hamilton and Schaefer have noted [3], is that the point has been reached at which the computation of integrals of comparatively low angular momentum, for example, highly contracted ( p p 1 pp) , is now dominated by the formation of the requisite auxiliary s-type integrals. In the present paper, we address this problem and describe a highly optimized scheme for computing [O]'"' integrals. A Statement of the Problem Suppose that we have a primitive shell of Cartesian Gaussian functions on each of four centers A, B, C, and D and that their exponents and contraction co