We perform GW calculations on atoms and diatomic molecules at different levels of self-consistencyand investigate the effects of self-consistency on total energies, ionization potentials, and 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 iscompared to the fully self-consistent GW results and to the GW_0and the G_0W_0approximations. Total energies, ionization potentials, and two-electron removal energies obtained with our partiallyself-consistent GW approximation are in excellent agreement with fully self-consistent GW resultswhile requiring only a fraction of the computational effort. We also find that self-consistent andpartially self-consistent schemes provide ionization energies of similar quality as the G_0W_0valuesbut yield better total energies and energy differences.
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