Improved local-field corrections to the G₀W approximation in jellium: Importance of consistency relations

We study the effects of local vertex corrections to the self-energy of the electron gas. We find that a vertex derived from time-dependent density-functional theory can give accurate self-energies, provided, however, a proper decay at large momentum transfer (large q) is built into the vertex function. (The local-density approximation for the vertex fails badly.) Total energies are calculated from the Galitskii-Migdal formula, and it is shown that a proper large-q behavior results in a close consistency between the chemical potentials derived from these energies and those obtained directly from the self-energy. We show that this internal consistency depends critically on including the same vertex correction in both the self-energy and the screening function. In addition the total energies become almost as accurate as those from elaborate Monte Carlo calculations. This as well as previous works show that self-energy corrections are important for properly describing electron propagation at energies around and above the plasmon energy. For easy use in calculations of photoemission and x-ray extended fine structure spectra, we parametrize our calculated self-energies in terms of a simple analytical expression.