Why Does the <i>GW</i> Approximation Give Accurate Quasiparticle Energies? The Cancellation of Vertex Corrections Quantified

Hedin's <i>GW</i> approximation to the electronic self-energy has been impressively successful in calculating 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 the <i>GW</i> approximation. This claim is mostly based on past calculations using vertex corrections within the crude local-density approximation. Here, we explore a wide variety of nonlocal 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 for obtaining realistic electronic properties. We finally consider increasingly large clusters and extrapolate that our conclusions about the compensation of the two vertices would hold for extended systems.