Electron-pair logarithmic convexity and interelectronic moments in atoms: Application to heliumlike ions.

Abstract

The electron-pair function [ital h]([ital u]) of a finite many-electron system is not monotonic, but the related quantity [ital h]([ital u])/[ital u][sup [alpha]], [alpha][gt]0, is not only monotonically decreasing from the origin but also convex for the values [alpha][sub 1] and [alpha][sub 2], respectively, as has been recently found. Here, it is first argued that this quantity is also logarithmically convex for any [alpha][ge][alpha][prime] with [alpha][prime]=max[l brace][minus][ital u][sup 2][ital d2][ln[ital h]([ital u])]/[ital du][sup 2][r brace]. Then this property is used to obtain a general inequality which involves three interelectronic moments [l angle][ital u][sup [ital t]][r angle]. Particular cases of this inequality involve relevant characteristics of the system such as the number of electrons and the total electron-electron repulsion energy. Second, the logarithmic-convexity property of [ital h]([ital u]) as well as the accuracy of this inequality are investigated by the optimum 20-term Hylleraas-type wave functions for two-electron atoms with nuclear charge [ital Z]=1, 2, 3, 5, and 10. It is found that (i) 14[lt][alpha][prime][lt]20 for these atomic cases (we remark that [alpha][prime][much gt][alpha][sub 2][much gt][alpha][sub 1]) and (ii) the accuracy of the inequality which involves moments of contiguous orders oscillates between 62.4% and 96.7% according to the specific He-like atom andmore » the moments involved. Finally, the importance of the logarithmic-convexity effects on the interelectronic moments relative to those coming from other monotonicity properties of [ital h]([ital u])/[ital u][sup [alpha]] are analyzed in the same numerical Hylleraas framework.« less