Sternheimer shielding using various approximations

Abstract

Sternheimer shielding functions $\gamma \mathrm{}\left(r\right)\mathrm{}$ are derived by means of Sternheimer's procedure for a series of elements (Li, Na, K, Rb, F, Cl, Br, I, Cu, Fe, Ag, and Pr) with varying electronic configurations, with and without valence electrons, which are generally included in molecular-orbital calculations with a limited basis set. Direct and exchange contributions to $R=\frac{〈\gamma \mathrm{}\left(r\right)\mathrm{}{r}^{-3\mathrm{}}〉}{〈r^{-3\mathrm{}}〉}$ are presented, and the various contributions to $\gamma \mathrm{}\left(r\right)\mathrm{}$ due to angular and radial excitations are discussed. The discrepancy in $R$ values for halides compared to Sternheimer's values is due mainly to an exchange contribution to $R$ which has been omitted by Sternheimer. Self-consistent calculations of $\gamma \mathrm{}\left(r\right)\mathrm{}$ have been performed with and without $X\alpha$ exchange interaction. The general tendency of self-consistent ${\gamma }_{\infty } [={\lim }_{r\to \infty }\gamma \mathrm{}(r\left)\right]$ results is that direct Coulomb interaction leads to ${\gamma }_{\infty }$ values more positive than those derived from Sternheimer's procedure (noniterative and excluding exchange interaction), while exchange interaction partly balances this effect. The overall effect in the case of iron is that Sternheimer's value is nearly identical to our self-consistent ${\gamma }_{\infty }$ value including exchange contributions. Self-consistency was obtained only for neutral atoms and positive ions. However, no such self-consistency was obtained for negative ions. Thus the $X_{\alpha }$ approximation is not very appropriate for the present purpose of calculating $\gamma \mathrm{}\left(r\right)\mathrm{}$ for elements with varying electronic configurations.