The relativistic Hartree-Fock and random phase approximations for many-electron atoms

Abstract

A recent treatment by us of energy eigenvalues and radiative transition matrix elements for non-relativistic atomic systems is extended to the relativistic case. The starting point is QED, from which approximate models are generated by restricting the electron-electron interaction to the instantaneous limit of 1-photon exchange kernels. As in the non-relativistic case, the equation for the 1-electron propagator with such a kernel leads to the relativistic Hartree-Fock (RHF) approximation. The equation for the electron/electron-hole propagator leads to the relativistic Bethe-Salpeter or random phase approximation (RRPA). Gaugeinvariant (GI) 1-photon transition amplitudes are obtained in both approximations in a manner quite analogous to the non-relativistic case. The essential changes from the non-relativistic case are in the definitions of the current density (and, thus, the charge operator), and the corresponding consistent definition of the equal-time limit of the 1-electron propagator. There is a need for renormalization in the relativistic formalism, unlike in the non-relativistic case. After it is carried out, and a consistent approximation of neglecting the residual (higher order) finite radiative corrections is made, the results for RHF and RRPA become formally analogous to the non-relativistic (Hartree-Fock (HF), and random phase (RPA)) results.