  We calculate single-particle excitation energies for a series of 33 molecules
using fully selfconsistent GW, one-shot G$_0$W$_0$, Hartree-Fock (HF), and
hybrid density functional theory (DFT). All calculations are performed within
the projector augmented wave (PAW) method using a basis set of Wannier
functions augmented by numerical atomic orbitals. The GW self-energy is
calculated on the real frequency axis including its full frequency dependence
and off-diagonal matrix elements. The mean absolute error of the ionization
potential (IP) with respect to experiment is found to be 4.4, 2.6, 0.8, 0.4,
and 0.5 eV for DFT-PBE, DFT-PBE0, HF, G$_0$W$_0$[HF], and selfconsistent GW,
respectively. This shows that although electronic screening is weak in
molecular systems its inclusion at the GW level reduces the error in the IP by
up to 50% relative to unscreened HF. In general GW overscreens the HF energies
leading to underestimation of the IPs. The best IPs are obtained from one-shot
G$_0$W$_0$ calculations based on HF since this reduces the overscreening.
Finally, we find that the inclusion of core-valence exchange is important and
can affect the excitation energies by as much as 1 eV.
