Efficient computation of two-electron - repulsion integrals and their nth-order derivatives using contracted Gaussian basis sets

Abstract

We present an general algorithm for the evaluation of the nth derivatives (with respect to the nuclear Cartesian coordinates) of two-electron-repulsion integrals (ERIs) over Gaussian basis functions. The algorithm is a generalization of our recent synthesis of the McMurchie/Davidson and Head-Gordon/Pople methodologies for ERI generation. Any ERI nth derivative may be viewed as an inner product between a function (which we term a bra) of electron 1 and a function (which we term a key) of electron 2. After defining bras and kets appropriately, we derive five recurrence relations that enable any bra to be constructed recursively from very simple bras which we call p-bras