Self-consistent vertex corrected $GW$ with static and dynamic screening using tensor hypercontraction: assessment of molecular charged excitations

Abstract

We investigate self-consistent vertex corrections to the $GW$ self-energy for ionization potentials (IPs) and electron affinities (EAs). We benchmark IPs against $\hat{I}”$CCSD(T) references in the $G_0{W}_0\hat{I}“$29 and GW100 sets and compare GW100 EAs with EOM-CCSD references. Because many anions are metastable, these addition energies are model quantities and should not be interpreted as experimental EAs. Tensor hypercontraction (THC) of the Coulomb integrals enables efficient self-consistent $GW\hat{I}{“}_{\hat{I}}\pounds$ implementations, where vertex corrections are included only in the self-energy. We establish a hierarchy of vertex-corrected self-energies relative to sc$GW$, ordered from least to most negative as SOX $>$ SOSEX $>$ G3W2 $>$ 2SOSEX $>$ sc$GW$. Equivalently, the absolute magnitude increases along this sequence. This trend follows an effective-screening picture, in which increasing screening progressively reduces exchange contributions. Static and dynamic variants show consistent differences due to the frequency dependence of the screened interaction. Across all methods, vertex corrections act as an approximately frequency-uniform self-energy renormalization rather than altering its spectral structure. In terms of accuracy, sc$GW\hat{I}{“}_{\hat{I}}\pounds$ does not uniformly improve IPs or the EA model quantities over sc$GW$. For IPs, SOX and SOSEX usually degrade performance, whereas 2SOSEX and G3W2 remain close to sc$GW$, with only marginal improvements for selected systems at higher cost. Although the tested variants reduce EA MAEs, this should not be interpreted as a general improvement for physical anions because many nominal EA states are metastable. These results indicate that generic vertex insertions are insufficient to outperform sc$GW$; systematic improvements require designed diagrammatic approximations combined with efficient tensor factorization.