  We calculate groundstate total energies and single-particle excitation
energies of seven pi conjugated molecules described with the semi-empirical
Pariser-Parr-Pople (PPP) model using self-consistent many-body perturbation
theory at the GW level and exact diagonalization. For the total energies GW
captures around 65% of the groundstate correlation energy. The lowest lying
excitations are overscreened by GW leading to an underestimation of electron
affinities and ionization potentials by approximately 0.15 eV corresponding to
2.5%. One-shot G_0W_0 calculations starting from Hartree-Fock reduce the
screening and improve the low-lying excitation energies. The effect of the GW
self-energy on the molecular excitation energies is shown to be similar to the
inclusion of final state relaxations in Hartree-Fock theory. We discuss the
break down of the GW approximation in systems with short range interactions
(Hubbard models) where correlation effects dominate over screening/relaxation
effects. Finally we illustrate the important role of the derivative
discontinuity of the true exchange-correlation functional by computing the
exact Kohn-Sham levels of benzene.
