Why Does the <i>GW</i> Approximation Give\nAccurate Quasiparticle Energies? The Cancellation of Vertex Corrections\nQuantified

Hedin’s <i>GW</i> approximation to the\nelectronic\nself-energy has been impressively successful in calculating quasiparticle\nenergies, such as ionization potentials, electron affinities, or electronic\nband structures. The success of this fairly simple approximation has\nbeen ascribed to the cancellation of the so-called vertex corrections\nthat go beyond the <i>GW</i> approximation. This claim is\nmostly based on past calculations using vertex corrections within\nthe crude local-density approximation. Here, we explore a wide variety\nof nonlocal vertex corrections in the polarizability and the self-energy,\nusing first-order approximations or infinite summations to all orders.\nIn particular, we use vertices based on statically screened interactions\nlike in the Bethe–Salpeter equation. We demonstrate on realistic\nmolecular systems that the two vertices in Hedin’s equation\nessentially compensate. We further show that consistency between the\ntwo vertices is crucial for obtaining realistic electronic properties.\nWe finally consider increasingly large clusters and extrapolate that\nour conclusions about the compensation of the two vertices would hold\nfor extended systems.