Fully self-consistent $\mathrm{GW}$ self-energy of the electron gas

We present fully self-consistent results for the self-energy of the electron gas within the $\mathrm{GW}$ approximation. This means that the self-consistent Green’s function $G,$ as obtained from Dyson’s equation, is used not only for obtaining the self-energy but also for constructing the screened interaction $W$ within the random-phase approximation. Such a theory is particle and energy conserving in the sense of Kadanoff and Baym. We find an increase in the weight of the quasiparticle as compared to ordinary non-self-consistent calculations but also to calculations with partial self-consistency using a fixed $W.\mathrm{}$ The quasiparticle bandwidth is larger than that of free electrons and the satellite structure is broad and featureless; both results clearly contradict the experimental evidence. The total energy, though, is as accurate as that from quantum Monte Carlo calculations, and its derivative with respect to particle number agrees with the Fermi energy as obtained directly from the pole of the Green’s function at the Fermi level. Our results indicate that, unless vertex corrections are included, non-self-consistent results are to be preferred for most properties except for the total energy.