  We solve the Dyson equation for atoms and diatomic molecules within the GW
approximation, in order to elucidate the effects of self-consistency on the
total energies and ionization potentials. We find GW to produce accurate energy
differences although the self-consistent total energies differ significantly
from the exact values. Total energies obtained from the Luttinger-Ward
functional E_LW[G] with simple, approximate Green functions as input, are shown
to be in excellent agreement with the self-consistent results. This
demonstrates that the Luttinger-Ward functional is a reliable method for
testing the merits of different self-energy approximations without the need to
solve the Dyson equation self-consistently. Self-consistent GW ionization
potentials are calculated from the Extended Koopmans Theorem, and shown to be
in good agreement with the experimental results. We also find the
self-consistent ionization potentials to be often better than the
non-self-consistent G0W0 values. We conclude that GW calculations should be
done self-consistently in order to obtain physically meaningful and unambiguous
energy differences.
