Beyond quasi-particle self-consistent $GW$ for molecules with vertex corrections

TL;DR

This paper introduces a new self-energy method within the qs$GW$ framework that incorporates vertex corrections via the Bethe-Salpeter equation, significantly improving electron affinity predictions for molecules.

Contribution

The authors develop the $ ext{qs}oldsymbol{ extSigma}^{ ext{BSE}}@L^{ ext{BSE}}$ method, integrating vertex corrections in both the response function and self-energy, enhancing accuracy for molecular charged excitations.

Findings

Improved electron affinity and gap predictions for organic molecules.

Vertex inclusion in both $L$ and $oldsymbol{ extSigma}$ is crucial for accuracy.

Method performs well for charged and neutral excitations.

Abstract

We introduce the $\Sigma^{\text{BSE}}@L^{\text{BSE}}$ self-energy in the quasi-particle self-consistent $GW$ (qs$GW$) framework (qs$\Sigma^{\text{BSE}}@L^{\text{BSE}}$). Here, $L$ is the two-particle response function which we calculate by solving the Bethe-Salpeter equation with the static, first-order $GW$ kernel. The same kernel is added to $\Sigma$ directly. For a set of medium organic molecules, we show that including the vertex both in $L$ and $\Sigma$ is crucial. This approach retains the good performance of qs$GW$ for predicting first ionization potentials and fundamental gaps, while it greatly improves the description of electron affinities. Its good performance places qs$\Sigma^{\text{BSE}}@L^{\text{BSE}}$ among the best-performing electron propagator methods for charged excitations. Adding the vertex in $L$ only, as commonly done in the solid state community, leads to devastating results for electron affinities and fundamental gaps. We also test the performance of BSE@qs$GW$ and qs$\Sigma^{\text{BSE}}@L^{\text{BSE}}$ for neutral charge-transfer excitation and find both methods to perform similar. We conclude that $\Sigma^{\text{BSE}}@L^{\text{BSE}}$ is a promising approximation to the electronic self-energy beyond $GW$. We hope that future research on dynamical vertex effects, second-order vertex corrections, and full self-consistency will improve the accuracy of this method, both for charged and neutral excitation energies.