Exposition of atomic pair correlation functions and associated energies obtained by analysis of correlated wave functions

Using Sinanoğlu's method of successive partial orthogonalizations, pair correlation functions have been extracted from the energetically accurate configuration-interaction wave functions of Weiss for a series of two-, three-, and four-electron ions. For selected systems, the pair correlation functions are examined by constructing contour diagrams which explicitly reveal the relative corrections to the Hartree-Fock wave functions brought about by inclusion of pairwise correlation interactions. For each pair function, an associated pair correlation energy is evaluated. The characteristics of both the pair functions and the pair energies are examined together with (i) their $Z$ dependence for a constant number of electrons and (ii) their dependence upon the number of electrons for constant $Z$. The small differences which occur between the $K$ shells for ions of a constant $Z$ have been rationalized in terms of an orthogonality requirement, while the slight but smooth variation of the $K$-shell pair energies with $Z$ has enabled us to comment on the stability of the ion ${\mathrm{He}}^{-}$. Our pictorial representation has highlighted differences between the $L$- and $K$-shell correlation effects within the Be-like ions. In particular, the large degree of angular correlation in the $L$ shell, brought about by the near degeneracy of the $2s$ and $2p$ orbitals, is clearly apparent from the contour diagrams. Also revealed is a nearly linear variation of the $L$-shell pair energies with $Z$. Comparison of the pair energies for the four-electron ions with the results of other workers shows that the wave functions analyzed here provide a good description of the correlation effect within the $K$- and $L$-shells, the description of the latter being superior to that of the former. The intershell correlation behavior, however, is accounted for only very approximately. For the Be-like series as a whole, the $L$-shell pair energies are expected to be highly reliable.