Quasiparticle self-consistent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:math> with effective vertex corrections in the polarizability and the self-energy applied to MnO, Fe

Through quasiparticle self-consistent $GW$, we investigate the electronic structure of the antiferromagnetic ground state of four transition-metal monoxides: MnO, FeO, CoO, and NiO. In addition to the random-phase approximation, we consider two different schemes incorporating effective vertex corrections. The first scheme includes in the polarizability a vertex function derived from the solution of the Bethe-Salpeter equation (BSE), whereas the second scheme includes in both the polarizability and self-energy a vertex function, which carries a long-range part satisfying the Ward identity and a short-range part derived from the adiabatic local density approximation. Our results include fundamental band gaps, macroscopic dielectric constants, and local magnetic moments, emphasizing the role of vertex corrections in the description of these key electronic properties. We provide quasiparticle band structures and projected densities of states allowing us to establish a connection with the Mott insulator picture and to highlight the overall good agreement with experimental photoemission spectra. The inclusion of the vertex in the self-energy preserves the overall shape of the band structures but generally produces larger rigid energy shifts for $d$ bands than for $sp$ bands. Through the solution of the Bethe-Salpeter equation, we determine excitonic effects and obtain dielectric response functions. Despite the experimental uncertainties, which appear sizable in the case of FeO, it is possible to discern a good general correspondence between the calculated and measured spectra, particularly for the absorption onsets and the peak positions. Furthermore, deep excitons are found to occur in the absorption spectra of FeO, CoO, and NiO, in qualitative agreement with the available characterization. Our work indicates that many-body perturbation theory including effective vertex corrections is able to provide a quantitatively accurate description of key electronic-structure features of transition-metal monoxides.