Recursion relations for the four-electron subsidiary integralW4(k,l,m,n;θ,α,β,γ)

Abstract

The subsidiary integral ${W}_{4}(k,l,m,n;\phantom{\rule{0.16em}{0ex}}\ensuremath{\theta},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma})$ plays an essential role in the variational calculation of four-electron atomic systems using Hylleraas coordinates. With respect to the case where the ratio $\ensuremath{\theta}/(\ensuremath{\theta}+\ensuremath{\alpha}+\ensuremath{\beta}+\ensuremath{\gamma})\ensuremath{\sim}1$, an important special situation that may occur in the evaluation of the Bethe logarithm, existing approaches for evaluating the ${W}_{4}$ integral become impractical due to the problem of slow convergence. Based on our recent work for the three-electron subsidiary integral ${W}_{3}(l,m,n;\phantom{\rule{0.16em}{0ex}}\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma})$, we present a computationally efficient and numerically stable method, in which the ${W}_{4}$ integral can be expressed in terms of either a finite series or a finite recursion relation. Numerical experimentation is presented to validate our method.