Coulomb holes and correlation coefficients for electronic shells: The Be-like ions

An expression has been obtained for the Coulomb hole associated with any pair of occupied HF spin orbitals for a many-electron system. The required partitioning of the known correlated second-order density matrix was achieved here, up to and including the pair-correlation effects, by using the many-particle theory proposed by Sinanoğlu. Ground-state wavefunctions were then analyzed for the Be-like ions when 3⩽Z⩽8 which, besides validating the partitioning technique, provided insight into the ’’radial’’ and ’’angular’’ components of intrashell correlation effects. The correlation coefficients τϑ, τr and τ1/r, taken from classical statistics, were also determined and their variation was examined with respect to the atomic number Z. These techniques provided a clear-cut means for comparing and contrasting, in a quantitative manner, the role of correlation within individual atomic shells. Moreover, the procedure is applicable to the comparative assessment of different correlated wavefunctions for the same state of a given system.