Efficient evaluation of the geometrical first derivatives of three-center Coulomb integrals

The calculation of the geometrical derivatives of three-center electron repulsion integrals (ERIs) over contracted spherical harmonic Gaussians has been optimized. We compared various methods based on the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys polynomial algorithms using Cartesian, Hermite, and mixed Gaussian integrals for each scheme. The latter ERIs contain both Hermite and Cartesian Gaussians, and they combine the advantageous properties of both types of basis functions. Furthermore, prescreening of the ERI derivatives is discussed, and an efficient approximation of the Cauchy-Schwarz bound for first derivatives is presented. Based on the estimated operation counts, the most promising schemes were implemented by automated code generation, and their relative performances were evaluated. We analyzed the benefits of computing all of the derivatives of a shell triplet simultaneously compared to calculating them just for one degree of freedom at a time, and it was found that the former scheme offers a speedup close to an order of magnitude with a triple-zeta quality basis when appropriate prescreening is applied. In these cases, the Obara-Saika method with Cartesian Gaussians proved to be the best approach, but when derivatives for one degree of freedom are required at a time the mixed Gaussian Obara-Saika and Gill-Head-Gordon-Pople algorithms are predicted to be the best performing ones.