Recursion relations for the three‐electron subsidiary integral W (l, m, n; α, β, γ)

Abstract

The subsidiary integral W (l, m, n; a, b, c) is a key integral that appears in the variational calculation of a threeelectron atomic system using Hylleraas coordinates. For the case where the ratio a/(a þ b þ c) $ 1, an important special situation that may occur in the evaluation of the Bethe logarithm, existing approaches for calculating the W integral become impractical due to the problem of slow convergence. In this article, we present a computationally efficient and numerical stable method, in which the W integral can be expressed in terms of either a finite series or a finite recursion relation. Numerical tests are given.

Mathematical and computational science issues in high precision Hylleraas-configuration interaction variational calculations: III. Four-electron integrals

“…In this connection we have found the tables in [2,20] to be useful as an independent check on our W and X integrals. For example, our QD Wfunction Lsums agree with the table 3 [20] Maple MP check calculations up until the Vkl expansion fails for r > 0.95 because of an exceedingly slow convergence.…”

“…Lsums are expensive in terms of floating point operation count (M ranges from 60 to 250 terms or more in equation ( 52)) and it is important to keep them to a minimum in any efficient recursion scheme. An alternate method for calculating the starting functions is the hypergeometric series form used by Frolov and Smith [18] and Drake and Yan [19] in their studies of the W integrals and Frolov [3] and Li et al [20] for the X integrals.…”

“…We have used a Larsson sum in QD (quad-DP) as a check on all of our X auxiliary integrals and we also have checked them where possible against the tables in [2]. In this connection we have found the tables in [2,20] to be useful as an independent check on our W and X integrals. For example, our QD Wfunction Lsums agree with the table 3 [20] Maple MP check calculations up until the Vkl expansion fails for r > 0.95 because of an exceedingly slow convergence.…”

Electronic structure calculations of compressed Li atom using composite technique under Ritz variational framework

“…Even for a two‐electron ion, ab‐initio calculations with electron correlation inside a finite domain is very challenging which has been addressed by several researchers in the past decade [14–16]. To estimate the matrix elements by calculating correlated integrals over inter‐electronic coordinates of “free” ions with more than two electrons have been dealt rigorously by researchers over several years [17–21]. With increasing number of electrons in the system, the correlation factor becomes more important and the level of difficulty to incorporate this in the theoretical methodology increases, both mathematically and computationally.…”

Analytic evaluation of some three- and four- electron atomic integrals involving s STO's and exponential correlation with unlinked $r_{ij}$'s

“…Analytical expressions for the auxialiary functions A, V, W and X (redenoted as A 1 , A 2 , A 3 and A 4 , respectively) along with their highly accurate values have been reported in [53,54]. A computationally efficient and numerically stable method was reported in [4,55] for the highly accurate calculation of auxiliary functions W and X (denoted as W 3 and W 4 ). All the papers cited above in this paragraph do not involve exponential correlation.…”