  We present theoretical calculations of quasiparticle energies in closed-shell
molecules using the GW method. We compare three different approaches: a
full-frequency $G_0W_0$ (FF-$G_0W_0$) method with density functional theory
(DFT-PBE) used as a starting mean field; a full-frequency $GW_0$ (FF-$GW_0$)
method where the interacting Green's function is approximated by replacing the
DFT energies with self-consistent quasiparticle energies or Hartree-Fock
energies; and a $G_0W_0$ method with a Hybertsen-Louie generalized plasmon-pole
model (HL GPP-$G_0W_0$). While the latter two methods lead to good agreement
with experimental ionization potentials and electron affinities for methane,
ozone, and beryllium oxide molecules, FF-$G_0W_0$ results can differ by more
than one electron volt from experiment. We trace this failure of the
FF-$G_0W_0$ method to the occurrence of incorrect self-energy poles describing
shake-up processes in the vicinity of the quasiparticle energies.
