Vertex effects in describing the ionization energies of the first-row transition-metal monoxide molecules

The $GW$ approximation is considered to be the simplest approximation with Hedin's formulation of many-body perturbation theory. It is expected that some of the deficiencies of the $GW$ approximation can be overcome by adding the so-called vertex corrections. In this work, the recently implemented $G_0W_0Γ^{(1)}_0$ scheme, which incorporates the vertex effects by adding the full second-order self-energy correction to the $GW$ self-energy, is applied to a set of first-row transition-metal monoxide (TMO) anions. Benchmark calculations show that results obtained by $G_0W_0Γ^{(1)}_0$ on top of the B3LYP hybrid functional starting point (SP) are in good agreement with experiment data, giving a mean absolute error of 0.13 eV for a testset comprising the ionization energies (IEs) of 27 outer valence molecular orbitals (MOs) from 9 TMO anions. A systematic SP-dependence investigation by varying the ratio of the exact exchange (EXX) component in the PBE0-type SP reveals that, for $G_0W_0Γ^{(1)}_0$, the best accuracy is achieved with $20\%$ EXX. Further error analysis in terms of the orbital symmetry characteristics (i.e, $σ$, $π$, or $δ$) in the testset indicate the best amount of EXX in the SP for $G_0W_0Γ^{(1)}_0$ calculations is independent of MO types, and this is in contrast with the situation of $G_0W_0$ calculations where the best EXX ratio varies for different classes of MOs. Despite its success in describing the absolute IE values, we however found that $G_0W_0Γ^{(1)}_0$ faces difficulties in describing the energy separations between certain states of interest, worsening the already underestimated $G_0W_0$ predictions.