Coulomb holes and correlation coefficients for electronic shells: A comparative analysis of several wave functions for Be

Abstract

A partitioning technique used previously to examine correlation trends in individual electronic shells for a series of ions has been extended and applied to a detailed comparison of four well-correlated wave functions for the Be atom. The present analysis of a correlated two-particle density, generalized for any N-electron system, retained all contributions from products of all terms in the wave function up to and including the pair-correlation effects. For each correlated description of Be, Coulomb holes have been evaluated and compared for the K(1S), L(1S), KL(1S), and KL(3S) shells. The inverted nature of the inter-shell holes, relative to the intra-shell effect, has been examined and rationalized in terms of the 2s–2p near-degeneracy which exists in Be. The total Coulomb holes for the two energetically best wave functions showed a previously unseen structure which was directly attributable to the inter-shells. Several one- and two-particle expectation values are reported and these were used to determine various angular and radial correlation coefficients τ. Thus, for each wave function, a global assessment could be obtained of the correlation effects within different regions of the intra- and inter-electronic shells.