The Prism Algorithm for Two - Electron Integrals

A new approach to the evaluation of two-electron repulsion integrals over contracted Gaussian basis functions is developed. The new scheme encompasses 20 distinct, but interrelated, paths from simple shell-quartet parameters to the target integrals, and, for any given integral class, the path requiring the fewest floating-point operations (FLOPS) is that used. Both theoretical (FLOP counting) and practical (CPU timing) measures indicate that the method represents a substantial improvement over the HGP algorithm. Introduction Because of their large number, the evaluation and manipulation of twoelectron integrals is the major difficulty in a Hartree-Fock calculation. A. Szabo and N. S. Ostlund [l] Over the decades, this realization has been the single most important driving force in formulating improvements to practical implementations of the HartreeFock self-consistent field (SCF) method. By 1980, several ingenious algorithms [2-41 for the evaluation of two-electron repulsion integrals (EMS) were available, and since (in a conventional SCF calculation) the ERIS need be computed only once (after which they are stored and retrieved on each iteration of the SCF procedure), there appeared to be little to be gained by further improving the ERI evaluation algorithms. The balance, however, was shifted substantially when, in 1982, Almlof and coworkers introduced the “direct SCF” method [5] in which ERIS are recomputed on each iteration of the SCF. Recently, too, the “direct” approach has been extended to M P ~ calculations [5,6]. Direct methods allow very large calculations to be performed without prohibitively large disk requirements, but, naturally, they cost much more than do their conventional analogs. Indeed, the cost of a direct SCF or direct M P ~ calculation is essentially some multiple of the cost of the associated ERI evaluation. Clearly, within such a framework, it is crucial that highly efficient methods for ERI computation be utilized. After lying dormant for several years, the study of novel ERI algorithms was invigorated by the discovery of the Obara-Saika (0s) recurrence relation [7] (which had been implicit in earlier work [8] by Schlegel) in 1986. Two years later, Head-Gordon and Pople (HGP) suggested [9] that the 0s methodology is improved