GW density matrix in solids as an estimate of self-consistent GW total energies

The GW approximation is a well-established method for calculating ionization potentials and electron affinities in solids and molecules. For numerous years, obtaining self-consistent GW total energies in solids has been a challenging objective that has yet to be accomplished. However, it was shown recently that the linearized GW density matrix permits a reliable prediction of the self-consistent GW total energy for molecules [F. Bruneval et. al. J. Chem. Theory Comput. 17, 2126 (2021)] for which self-consistent GW energies are available. Here we implement, test, and benchmark the linearized GW density matrix for several solids. We focus on the total energy, lattice constant, and bulk modulus obtained from the GW density matrix and compare our findings to more traditional results obtained within the random phase approximation (RPA). We conclude on the improved stability of the total energy obtained from the linearized GW density matrix with respect to the mean-field starting point. We bring compelling clues that the RPA and the GW density matrix total energies are certainly close to the self-consistent GW total energy in solids if we use hybrid functionals with enriched exchange as a starting point.