With the aim of identifying universal trends, we compare fully self-consistent electronic spectra and total energies obtained from the
GW approximation with those from an extended $\mathrm{GW}\ensuremath{\Gamma}$ scheme that includes a nontrivial vertex function and the fundamentally distinct Bethe-Goldstone approach based on the T matrix. The self-consistent Green's function
G, as derived from Dyson's equation, is used not only in the self-energy but also to construct the screened interaction W for a model system. For all approximations we observe a similar deterioration of the spectrum, which is not removed by vertex corrections. In particular, satellite peaks are systematically broadened and move closer to the chemical potential. The corresponding total energies are universally raised, independent of the system parameters. Our results, therefore, suggest that any improvement in total energy due to self-consistency, such as for the electron gas in the
GW approximation, may be fortuitous.
The approach to equilibrium of a system of electrons in a periodic system is studied by deriving a kinetic equation in the self-consistent field approximation. The derivation based on the introduction of an hierarchy of equations for the s-body density matrices and employing a truncation scheme, valid in the self-consistent field approximation. The irreversibility is introduced by the adiabatic hypothesis of Bogoliubov. The kinetic equation takes proper account of the collective aspects of the electron system and includes the dynamic shielding.