One-electron spectra and susceptibilities of the three-dimensional electron gas from self-consistent solutions of Hedin's equations

A few approximate schemes to solve the Hedin equations self-consistently introduced in Phys. Rev. B 94, 155101 (2016) are explored and tested for the three-dimensional (3D) electron gas at metallic densities. We calculate one-electron spectra, dielectric properties, compressibility, and correlation energy. Considerable reduction in the calculated bandwidth (as compared to the self-consistent $GW$ result) has been found when vertex correction was used for both polarizability and self-energy. Generally, it is advantageous to obtain the diagrammatic representation of polarizability from the definition of this quantity as a functional derivative of the electronic density with respect to the total field (external plus induced). For self-energy, the first-order vertex correction seems to be sufficient for the range of densities considered. Whenever it is possible, we compare the accuracy of our vertex-corrected schemes with the accuracy of the self-consistent quasiparticle $GW$ approximation (QSGW), which is less expensive computationally. We show that the QSGW approach performs poorly and we relate this poor performance with an inaccurate description of the screening in the QSGW method (with an error comprising a factor 2--3 in the physically important range of momenta).