Why does the $GW$ approximation give accurate quasiparticle energies? The cancellation of vertex corrections quantified

Abstract: Hedin's $GW$ approximation to the electronic self-energy has been impressively successful to calculate quasiparticle energies, such as ionization potentials, electron affinities, or electronic band structures. The success of this fairly simple approximation has been ascribed to the cancellation of the so-called vertex corrections that go beyond $GW$. This claim is mostly based on past calculations using vertex corrections within the crude local-density approximation. Here, we explore a wide variety of non-local vertex corrections in the polarizability and the self-energy, using first-order approximations or infinite summations to all orders. In particular, we use vertices based on statically screened interactions like in the Bethe-Salpeter equation. We demonstrate on realistic molecular systems that the two vertices in Hedin's equation essentially compensate. We further show that consistency between the two vertices is crucial to obtain realistic electronic properties. We finally consider increasingly large clusters and extrapolate that our conclusions would hold for extended systems.