7 TABLE II: Mean absolute deviation between the IPs of the 33 molecules calculated with the different methods and experiment. The mean absolute deviation with respect to experiment coincide with the last row in Table I Method Expt. (a) PBE-eig PBE0-eig HF-eig GW G$_{0}$W$_{0}$[HF] MP2 (a) PBE-tot Expt. 0.00 4.35 2.55 0.81 0.5 0.4 0.82 0.24 PBE 4.35 0.00 1.79 4.90 3.9 4.1 4.99 4.27 PBE0 2.55 1.79 0.00 3.11 2.1 2.3 3.20 2.48 HF 0.81 4.90 3.11 0.00 1.0 0.8 0.17 0.80 GW 0.5 3.9 2.1 1.0 0.00 0.3 1.1 0.4 G$_{0}$W$_{0}$[HF] 0.4 4.1 2.3 0.8 0.3 0.00 0.9 0.3 MP2 0.82 4.99 3.20 0.17 1.1 0.9 0.00 0.84 PBE-tot 0.24 4.27 2.48 0.80 0.4 0.3 0.84 0.00 ( a $^{)}$Data taken from Ref. 59. v$_{xc}$ | ψ 0 m 〉 , can be neglected, and that the frequency de- pendence of Σ$_{GW}$ can be approximated by its first or- der Taylor expansion in a sufficiently large neighbor- hood of ε 0 $_{n}$. We have found that these two assumptions are indeed fullfilled for the molecular systems studied here. More precisely, for the GW and G$_{0}$W$_{0}$(HF) self- energies, the QP energies obtained from Eq. (14) are always very close to the peaks in the density of states Eq. (9). We emphasize that this result could well be re- lated to the rather large level spacing of small molecules, and may not hold for extended systems. An example is presented in Fig. 3 which shows the full HF and GW density of states for NH$_{3}$ together with the real part of 〈 ψ 0 HOMO | Σ$_{corr}$( ε ) | ψ 0 HOMO 〉 . As explained in the follow- ing section this is not quite the case for the G$_{0}$W$_{0}$(PBE) calculations. As stated in the previous section the GW and G$_{0}$W$_{0}$(HF) energies can be obtained either from the full spectral function or from the QP equation. In this case, returning to Table I, we see that G$_{0}$W$_{0}$(HF) yields sys- tematically larger IPs than GW. This is easy to under- stand since G$_{HF}$ describes a larger HOMO-LUMO gap than G$_{GW}$ , and therefore produces less screening. When the PBE rather than the HF Green function is used to evaluate the GW self-energy, we find that the spectral function obtained from Eq. (2) does not resemble a simple discrete spectrum. In fact the peaks are signif- icantly broadened by the imaginary part of Σ$_{GW}$ and it becomes difficult to assign precise values to the QP ener- gies. Apart from the spectral broadening, the molecular gap is significantly reduced with respect to its value in the GW and G$_{0}$W$_{0}$(HF) calculations. Both of these effects are due to the very small HOMO-LUMO gap described by G$_{PBE}$ which leads to severe overscreening and QP life- time reductions. A similar effect was observed by Ku and Eguiluz in their comparison of GW and G$_{0}$W$_{0}$(LDA) for Si and Ge crystals $^{40}$. The problems encountered when attempting to solve The quasiparticle equation (14) relies on the assump- tion that off-diagonal matrix elements, 〈 ψ 0 n | Σ$_{GW}$( ε 0 $_{n}$) − Moreover the GW self-energy is evaluated non- selfconsistently from the single-particle Green function, i.e. Σ$_{GW}$ = iG$_{0}$W [ G$_{0}$ ], with G$_{0}$ ( z ) = ( z − H$_{0}$ ) − $^{1}$. where ψ 0 n and ε 0 n are eigenstates and eigenvalues of an approximate single-particle Hamiltonian (often the LDA Hamiltonian), and In the conventional GW method the full Green func- tion of Eq. (2) is not calculated. Rather one obtains the quasiparticle energies from the quasiparticle equation FIG. 3: (Color online) Density of states for the NH$_{3}$ molecule calculated in HF and GW, respectively. Arrows mark the level corresponding to the HOMO in the two calculations. The in- tersection between the line y = ε − ε HF n and the real part of 〈 ψ 0 $_{HOMO}$| Σ$_{corr}$( ε ) | ψ 0 $_{HOMO}$〉 (green curve) determines the posi- tion of the GW level. -50 -40 -30 -20 -10 0  (eV) -10 0 10 20 Energy (eV) DOS (GW) DOS (HF) n QP n HF Re n HF  -13 -12 -11 -10 B. G$_{0}$ -dependence A. Linearized quasiparticle equation ε QP n = ε 0 n + Z$_{n}$ 〈 ψ 0 n | Σ$_{GW}$( ε 0 $_{n}$) − v$_{xc}$ | ψ 0 n 〉 . (14) Z$_{n}$ = [ 1 − ∂ 〈 ψ DFT n | Σ$_{GW}$( ε ) | ψ 0 n 〉 ∂ε ∣ ε 0 n ] − 1 . (15)