Field-theoretical treatment of approximations to radiative transitions in atoms

Starting with a field-theoretical many-body definition of eigenvalues and radiative transition matrix elements for atomic systems, a systematic approach is taken to approximations of the exact results. The guiding principle is the maintenance of gauge invariance (GI) in radiative transition S matrix elements. At the level of the one Coulomb exchange approximation in both the one-electron and the electron-hole propagator kernels, one obtains the well-known Hartree-Fock (HF) and random phase approximation (RPA). A detailed discussion and comparison of various approaches to RPA is made, in the case of both N and N-1 electron shielding (the regular HF and HF with frozen relaxed core-FRC). In the former case, a new and considerably simpler form of the RPA equations are obtained than heretofore proposed equivalent forms. Finally, a different approximation than the usual HF and RPA, involving higher-order correlations, is developed to illustrate how such approximation can be systematically generated