# $GW$ vertex corrected calculations for molecular systems

## TL;DR

This paper implements vertex-corrected $GW$ calculations for small molecules, demonstrating improved accuracy and a new simplified approximation that avoids double counting, aligning well with more expensive wavefunction methods.

## Contribution

It introduces a $GW ext{}\Gamma$ scheme with vertex corrections at multiple levels and proposes a simplified $GW^{
m{tc-tc}}$ approximation that maintains accuracy.

## Key findings

- Good agreement with wavefunction methods.
- Vertex corrections improve $GW$ accuracy.
- Simplified $GW^{
m{tc-tc}}$ avoids double counting.

## Abstract

Hedin's scheme is solved with the inclusion of the vertex function ($GW\Gamma$) for a set of small molecules. The computational scheme allows for the consistent inclusion of the vertex both at the polarizability level and in the self-energy. A diagrammatic analysis shows that the self-energy formed with this four-point vertex does not lead to double counting of diagrams, that can be classified as direct "bubbles" and exchange diagrams. By removing the exchange diagrams from the self-energy, a simpler approximation is obtained, called $GW^{\rm{tc-tc}}$. Very good agreement with expensive wavefunction-based methods is obtained for both approximations.