Effklent Computation of Two-Electron-Repulsion Integrals and Thelr nth-Order Derivatives Using Contracted Gaussian Basis Sets

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 ket) 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. We show how these recurrence relations (and their analogues for kets) may be used to compute ERI nth derivatives from easily calculated one-center Hermite integrals. The five recurrence relations are overcomplete in the sense that there is generally more than one path through them by which a given bra can be constructed. We have written a computer program that selects an efficient path to any necessary bra or ket. We present comparative FLOP counts and timings which demonstrate that, for calculations using contracted basis sets, the new methodology (the BRAKET algorithm) is very competitive, both theoretically and practically, with all previous approaches.