Prescreening and efficiency in the evaluation of integrals over ab initio effective core potentials

New, efficient schemes for the prescreening and evaluation of integrals over effective core potentials (ECPs) are presented. The screening is shown to give a rigorous, and close bound, to within on average 10% of the true value. A systematic rescaling procedure is given to reduce this error to approximately 0.1%. This is then used to devise a numerically stable recursive integration routine that avoids expensive quadratures. Tests with coupled clusters with single and double excitations and perturbative triple calculations on small silver clusters demonstrate that the new schemes show no loss in accuracy, while reducing both the power and prefactor of the scaling with system size. In particular, speedups of roughly 40 times can be achieved compared to quadrature-based methods.