  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.
