Calculation of integrals over ab initio pseudopotentials

Abstract

An approach is presented for the evaluation of the tw distinct types of one-electron integrals arising from the ab initio fseudopotentials introduced by Kahn and Goddard. The integrals art shown to reduce to a sum over products of angular and radial integrals, the latter being approximated by power and asymptotic series combined with appropriate recursion relations. The method is valid for arbitt iry angular monvnta of both the pseudopotential and the Cartesian Gd sian basis functions. • 99 f y 9 exp(-p y)dy exp(-x)dx Part 1 (Mapleton House, Brooklyn, N.Y., 1948).

Valence quantum Monte Carlo with a b i n i t i o effective core potentials

“…(17) has been lumped into the single term, This procedure would have to be carried out for each choice of importance function and geometry.…”

Fast computation of analytical second derivatives with effective core potentials: Application to Si8C12, Ge8C12, and Sn8C12

“…The first is the original implementation by Kahn 8 and has since been improved by several groups. 9 However, a third method, proposed by Kolar, 10 when combined with the original method of Kahn, provides a very efficient method which easily scales to higher angular momenta.…”

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

“…As has been noted previously 23,28,29 , there are a number of possible recurrences on the Bessel functions that can be used to try and evaluate the primitive radial integrals. However, for this approach to be feasible, care has to be taken in choosing not only which relations to use, but also the order to use them in, as this will have a significant impact on the numerical stability of the algorithm.…”

“…Here we briefly summarise the expansion of the ECP matrix elements in terms of angular and radial integrals, as described in more detail in other sources 23,25 . As noted above, taking the matrix element over equation 1 results in two types of integral.…”

“…Of particular early significance were the methods due to Kahn 22 and McMurchie 23 . The exposition of the latter will be summarised in the next section.…”