Levels of self-consistency in the GW approximation

We perform $GW$ calculations on atoms and diatomic molecules at different levels of self-consistency and investigate the effects of self-consistency on total energies, ionization potentials and on particle number conservation. We further propose a partially self-consistent $GW$ scheme in which we keep the correlation part of the self-energy fixed within the self-consistency cycle. This approximation is compared to the fully self-consistent $GW$ results and to the $G W_0$ and the $G_0W_0$ approximations. Total energies, ionization potentials and two-electron removal energies obtained with our partially self-consistent $GW$ approximation are in excellent agreement with fully self-consistent $GW$ results while requiring only a fraction of the computational effort. We also find that self-consistent and partially self-consistent schemes provide ionization energies of similar quality as the $G_0W_0$ values but yield better total energies and energy differences.