interaction is removed in PBE0. However, the fact that the HF results are significantly closer to experiments in- dicates that the 25% Fock exchange included in the PBE0 is not sufficient to cure the erroneous description of (oc- cupied) molecular orbitals. On the other hand PBE0 gives good results for band gaps in semi-conductors and insulators where in contrast full Hartree-Fock does not perform well. 14–16 We conclude that the amount of Fock exchange to be used in the hybrid functionals to achieve good quasiparticle energies is highly system dependent. A similar problem is encountered with self-interaction corrected exchange-correlation functionals. 13 Accordingly the quasiparticle energy can be written as the bare HF energy and a correction due to the energy- dependent part of the GW self-energy (the dynamical screening term) In Fig. 2 the line y = x corresponds to Δ GW n = 0, and the vertical displacement from the line thus represents the effect of screening on the calculated HOMO energy. We first notice that the effect of screening is to shift the HOMO level upwards in energy, i.e. to reduce the ion- ization potential. This can be understood by recalling that the Hartree-Fock eigenvalue represents the energy cost of removing an electron from the HOMO when or- bital relaxations in the final state are neglected (Koop- mans’ theorem $^{16}$). In Ref. 37 we showed, on the basis of GW and exact calculations for semi-empirical models of conjugated molecules, that Δ GW n mainly describes the orbital relaxations in the final state and to a lesser extent accounts for the correlation energy of the initial and final states. This explains the negative sign of Δ GW n because the inclusion of orbital relaxation in the final state lowers the energy cost of removing an electron. We note that this is different from the situation in extended, periodic systems where orbital relaxations vanish and the main ef- fect of the GW self-energy is to account for correlations in the initial and final states. In Table I we list the calculated HOMO energy for each of the 33 molecules. In addition to selfconsistent GW we have performed one-shot G$_{0}$W$_{0}$ calculations based on the HF and PBE Green’s function, respectively. The best agreement with experiment is obtained for G$_{0}$W$_{0}$[HF]. This is because the relatively large Hartree-Fock HOMO- LUMO gap reduces the (over-)screening described by the resulting GW self-energy. There are not many GW cal- culations for molecules available in the litterature. Below Table I we list the few we have found. As can be seen they all compare quite well with our results given the differences in the implementation of the GW approxima- tion. As can be seen from Fig. 1, GW performs better than Hartree-Fock for the HOMO energy yielding a mean ab- solute error with respect to experiments of 0.5 eV com- pared to 0.81 eV with Hartree-Fock. As expected the dif- ference between HF and GW is not large on an absolute scale (around 1 eV on average, see Table II) illustrating the fact that screening is weak in small molecules. On a relative scale selfconsistent GW improves the agreement with experiments by almost 30% as compared to HF. For comparison we have included the HOMO energy predicted by second order Møller-Plesset theory (MP2) [taken from Ref. 59] with a Gaussian 6-311G ∗∗ basis set. These are generally very close to our calculated HF values, with a tendency to lower energies which worsens the agreement with experiment slightly as compared to HF. We have also calculated the DFT-PBE total energy dif- ference between the neutral and cation species, E ( N ) − E ( N − 1), see last column of Table I. This procedure leads to IPs in very good agreement with the experimen- tal values (MAE of 0.24 eV). We stress that although this method is superior to the GW method for the IP of the small molecules studied here, it can yield only the HOMO and LUMO levels while higher excited states are inaccessible. Moreover it applies only to isolated systems and cannot be directly used to probe QP levels of e.g. a molecule on a surface. To gain more insight into the influence of screening on the orbital energies, we compare in Fig. 2 the deviation of the HF and GW energies from IP$_{exp}$. The GW self- energy can be split into the bare exchange potential and an energy-dependent correlation part FIG. 2: The deviation of the calculated HOMO energy from the experimental ionization potential in GW and HF, respec- tively. The vertical displacement of points from the line x = y gives the difference between the GW and HF energies and rep- resents the effect of screening. Notice that the GW correction is always negative (corresponding to higher HOMO energy) and that it generally overcorrects the HF energies. Also no- tice that the GW correction is larger for molecules where HF presents the largest overestimation of the ionization potential. ε QP n = ε HF n + Δ GW n . (13) Σ$_{GW}$( r , $^{$_{r}$′}$; ε ) = v$_{x}$ ( r , $^{$_{r}$′}$) + Σ$_{corr}$ ( r , $^{$_{r}$′}$; ε ) (12) 5