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Many-body $GW$ calculations have emerged as a standard for the prediction of band gaps, band structures, and optical properties for main-group semiconductors and insulators, but it is not well established how predictive the $GW$ method is in general for transition metal (TM) compounds. Surveying the series of 3$d$ oxides within a typical $GW$ approach using the random-phase approximation reveals mixed results, including cases where the calculated band gap is either too small or too large, depending on the oxidation states of the TM (e.g., FeO/Fe${}_{2}$O${}_{3}$, Cu${}_{2}$O/CuO). The problem appears to originate mostly from a too high average $d$-orbital energy, whereas the splitting between occupied and unoccupied $d$ symmetries seems to be reasonably accurate. It is shown that augmenting the $GW$ self-energy by an attractive (negative) and occupation-independent on-site potential for the TM $d$ orbitals with a single parameter per TM cation can reconcile the band gaps for different oxide stoichiometries and TM oxidation states. In Cu${}_{2}$O, which is considered here in more detail, standard $GW$ based on wave functions from initial density or hybrid functional calculations yields an unphysical prediction with an incorrect ordering of the conduction bands, even when the magnitude of the band gap is in apparent agreement with experiment. The correct band ordering is restored either by applying the $d$-state potential or by iterating the wave functions to self-consistency, which both have the effect of lowering the Cu-$d$ orbital energy. While it remains to be determined which improvements over standard $GW$ implementations are needed to achieve an accurate ab initio description for a wide range of transition metal compounds, the application of the empirical on-site potential serves to mitigate the problems specifically related to $d$ states in $GW$ calculations.

physical review. b, condensed matter and materials physics

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Band-structure calculations for the 3dtransition metal oxides inGW