Correction Functions to Hartree-Fock Orbitals Derived from Correlated Wave Functions for He-, Li-, and Be-Like Ions

Abstract

Using energetically reliable configuration-interaction and Hartree-Fock wave functions for $\mathrm{He}\left({}_{}{}^1{}_{}{}^{}S \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\right)-$, $\mathrm{Li}\left({}_{}{}^2{}_{}{}^{}S \mathrm{and} {}_{}{}^2{}_{}{}^{}P \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}\right)-$, and $\mathrm{Be}\left({}_{}{}^1{}_{}{}^{}S \mathrm{state}\right)$-like ions, we applied the method of successive partial orthogonalizations to determine orbital correction $f$ functions (introduced by Sinanoğlu in his many-electron theory) which arise from electron-correlation effects. The addition of an $f$ function to an appropriate Hartree-Fock orbital gives an improved orbital representation as far as the one-particle density is concerned. Hence, changes in the corresponding radial density distributions were obtained and their characteristics and $Z$-dependent trends were rationalized in terms of the fluctuation potential proposed by Sinanoğlu. For He-like ions, changes in the $K$-shell density showed strong split-shell characteristics. This also held for the $K$ shell of the Be-like ions; however, as $Z$ increased, the perturbative influence of the $L$ shell became more apparent. In the $L$ shells for the Be-like systems, because of the $2s-2p$ near degeneracy, angular correlation is relatively high and, consequently, the fluctuation potential gave rise to an $f$ effect which produced an over-all density contraction towards the nucleus. These density changes for the $2s$ electrons are considerably larger than the changes associated with the $K$ shells. At a low-$Z$ value for $\mathrm{Li}\left({}_{}{}^2{}_{}{}^{}S\right)$-like ions, spin polarization of the $K$ shell owing to the unpaired $L$-shell electron is small, the changes in density being predominantly intrashell effects. For high $Z$, spin-polarization effects are greatly increased. The $L$-shell electron in the ${}_{}{}^2{}_{}{}^{}S$ state experiences only an intershell fluctuation potential—which is purely attractive. For this state, the density changes for the outer electron are about 25% of the $L$-shell effect in the $\mathrm{Be}\left({}_{}{}^1{}_{}{}^{}S\right)$-like ion and they are almost four times larger than the corresponding $K$-shell effect. In the $\mathrm{Li}\left({}_{}{}^2{}_{}{}^{}P\right)$ series of ions, the $K$-shell density shifts due to the $f$ functions are dominated by intrashell effects for $Z=3\mathrm{}$ whereas, at higher-$Z$ values, an exchange effect becomes the major influence. Our results for the $L$ shell appear less satisfactory and suggest that a further such study of the excited state of these ions would be of interest.