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

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Abstract
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.