Comparing Self-Consistent <i>GW</i> and Vertex-Corrected <i>G</i>₀<i>W</i>₀ (<i>G</i>₀<i>W</i>₀Γ) Accuracy for Molecular Ionization Potentials

We test the performance of self-consistent <i>GW</i> and several representative implementations of vertex-corrected <i>G</i>₀<i>W</i>₀ (<i>G</i>₀<i>W</i>₀Γ). These approaches are tested on benchmark data sets covering full valence spectra (first ionization potentials and some inner valence shell excitations). For small molecules, when comparing against state-of-the-art wave function techniques, our results show that full self-consistency in the <i>GW</i> scheme either systematically outperforms vertex-corrected <i>G</i>₀<i>W</i>₀ or gives results of at least comparative quality. Moreover, <i>G</i>₀<i>W</i>₀Γ results in additional computational cost when compared to <i>G</i>₀<i>W</i>₀ or self-consistent <i>GW</i>. The dependency of <i>G</i>₀<i>W</i>₀Γ on the starting mean-field solution is frequently more dominant than the magnitude of the vertex correction itself. Consequently, for molecular systems, self-consistent <i>GW</i> performed on the imaginary axis (and then followed by modern analytical continuation techniques) offers a more reliable approach to make predictions of ionization potentials.