$GW$+2SOSEX self-energy made positive semi-definite

TL;DR

This paper introduces a positive semi-definite extension to the $GW$+2SOSEX self-energy, ensuring mathematical consistency and improving accuracy in quasiparticle energy predictions for molecules.

Contribution

The authors develop a $GW$+2SOSEX-psd method that guarantees positive semi-definiteness of the self-energy, addressing a key challenge in vertex correction schemes.

Findings

The $GW$+2SOSEX-psd self-energy is positive semi-definite.

The method accurately predicts quasiparticle energies for molecules.

The approach cancels problematic energy poles in the self-energy.

Abstract

The formulation of vertex corrections beyond the $GW$ approximation within the framework of perturbation theory is a subtle and challenging task, which accounts for the wide variety of schemes proposed over the years. Exact self-energies are required to satisfy the mathematical condition of positive semi-definiteness. The $GW$ self-energy fulfills this property, but the vast majority of the vertex-corrected self-energy approximations do not. In this study, we devise a positive semi-definite extension to the $GW$+2SOSEX self-energy that we name $GW$+2SOSEX-psd. To reach this goal, we demonstrate the cancellation of the bare energy poles that are contained in the fully dynamic second-order in $W$ self-energy ($G3W2$). We then demonstrate on molecular examples the correct positive semi-definiteness of the proposed self-energy approximation and its good accuracy in predicting accurate quasiparticle energies for valence and core states.