Analysis of the atomic fine structure, using a nonrelativistic many-body and a relativistic central-field approach

Abstract

Nonrelativistic many-body calculations of the fine structure of $\mathrm{nd}$ states of sodiumlike systems have been performed for $Z=11\ensuremath{-}42$, taking into account the core polarization to first order in the spin-orbit coupling and to all orders in the Coulomb interaction. The results agree well with the experimental data available, and, in particular, the transition from inverted- to normal-level ordering appears at the right place, around $Z=15$. The alternative approach of evaluating the fine structure by means of relativistic central-field calculations\char22{}introduced by Luc-Koenig\char22{}is analyzed in detail, using the Pauli approximation and a graphical form of perturbation theory. Provided the self-consistent Hartree-Fock potential is used for the core, it can be shown that the relativistic method is (to order ${\ensuremath{\alpha}}^{2}$) equivalent to a nonrelativistic calculation, where the core polarization is included to all orders of the Coulomb interaction. In the calculation of Luc-Koenig a local core potential is used, in which case this equivalence is only partial. In order to be able to make accurate comparisons with our many-body results, we have performed relativistic calculations using the full Hartree-Fock potential of the core.