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Fully self-consistent $GW$ and quasi-particle self-consistent $GW$ for molecules

Two self-consistent schemes involving Hedin's $GW$ approximation are studied for a set of sixteen different atoms and small molecules. We compare results from the fully self-consistent $GW$ approximation (SC$GW$) and the quasi-particle self-consistent $GW$ approximation (QS$GW$) within the same numerical framework. Core and valence electrons are treated on an equal footing in all the steps of the calculation. We use basis sets of localized functions to handle the space dependence of quantities and spectral functions to deal with their frequency dependence. We compare SC$GW$ and QS$GW$ on a qualitative level by comparing the computed densities of states (DOS). To judge their relative merit on a quantitative level, we compare their vertical ionization potentials (IPs) with those obtained from coupled-cluster calculations CCSD(T). Our results are futher compared with "one-shot" $G_0W_0$ calculations starting from Hartree-Fock solutions ($G_0W_0$-HF). Both self-consistent $GW$ approaches behave quite similarly. Averaging over all the studied molecules, both methods show only a small improvement (somewhat larger for SC$GW$) of the calculated IPs with respect to $G_0W_0$-HF results. Interestingly, SC$GW$ and QS$GW$ calculations tend to deviate in opposite directions with respect to CCSD(T) results. SC$GW$ systematically underestimates the IPs, while QS$GW$ tends to overestimate them. $G_0W_0$-HF produces results which are surprisingly close to QS$GW$ calculations both for the DOS and for the numerical values of the IPs.