  We investigate static correlation and delocalization errors in the
self-consistent GW and random-phase approximation (RPA) by studying molecular
dissociation of the H_2 and LiH molecules. Although both approximations contain
topologically identical diagrams, the non-locality and frequency dependence of
the GW self-energy crucially influence the different energy contributions to
the total energy as compared to the use of a static local potential in the RPA.
The latter leads to significantly larger correlation energies which allow for a
better description of static correlation at intermediate bond distances. The
substantial error found in GW is further analyzed by comparing spin-restricted
and spin-unrestricted calculations. At large but finite nuclear separation
their difference gives an estimate of the so-called fractional spin error
normally determined only in the dissociation limit. Furthermore, a calculation
of the dipole moment of the LiH molecule at dissociation reveals a large
delocalization error in GW making the fractional charge error comparable to the
RPA. The analyses are supplemented by explicit formulae for the GW Green's
function and total energy of a simplified two-level model providing additional
insights into the dissociation limit.
