Beyond Quasi-Particle Self-Consistent <i>GW</i> for Molecules with Vertex Corrections

We introduce the Σ^BSE@<i>L</i>^BSE self-energy in the quasi-particle self-consistent <i>GW</i> (qs<i>GW</i>) framework (qsΣ^BSE@<i>L</i>^BSE). Here, <i>L</i> is the two-particle response function, which we calculate by solving the Bethe-Salpeter equation with the static, first-order <i>GW</i> kernel. The same kernel is added to Σ directly. For a set of medium organic molecules, we show that including the vertex both in <i>L</i> and Σ is crucial. This approach retains the good performance of qs<i>GW</i> for predicting first ionization potentials and fundamental gaps, while it greatly improves the description of electron affinities. Its good performance places qsΣ^BSE@<i>L</i>^BSE among the best-performing electron propagator methods for charged excitations. Adding the vertex in <i>L</i> 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<i>GW</i> and qsΣ^BSE@<i>L</i>^BSE for neutral charge-transfer excitation and find both methods to perform similar. We conclude that Σ^BSE@<i>L</i>^BSE is a promising approximation to the electronic self-energy beyond <i>GW</i>. 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.