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( a $^{)}$Data taken from Ref. 59. 

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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 

where ψ 0 n and ε 0 n are eigenstates and eigenvalues of an approximate single-particle Hamiltonian (often the LDA Hamiltonian), and 

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$_{0}$ ) − $^{1}$. The quasiparticle equation (14) relies on the assump- 

tion that off-diagonal matrix elements, 〈 ψ 0 n | Σ$_{GW}$( ε 0 $_{n}$) − 

## A. Linearized quasiparticle equation 

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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 

## B. G$_{0}$ -dependence 

