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 within 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 G₀W₀Γ₀ ⁽¹⁾ 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 G₀W₀Γ₀ ⁽¹⁾ 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 nine 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 G₀W₀Γ₀ ⁽¹⁾, 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 indicates the best amount of EXX in the SP for G₀W₀Γ₀ ⁽¹⁾ calculations is independent of MO types, and this is in contrast with the situation in G₀W₀ 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 G₀W₀Γ₀ ⁽¹⁾ faces difficulties in describing the energy separations between certain states of interest, worsening the already underestimated G₀W₀ predictions.