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The $GW$ approximation to many-body perturbation theory is a reliable toolfor describing charged electronic excitations, and it has been successfullyapplied to a wide range of extended systems for several decades using aplane-wave basis. However, the $GW$ approximation has been used to test limitedspectral properties of a limited set of finite systems (e.g. frontier orbitalenergies of closed-shell $sp$ molecules) only for about a decade using alocal-orbital basis. Here, we calculate the quasiparticle spectra of closed-and open-shell molecular anions with partially and completely filled 3$d$shells (shallow and deep 3$d$ states, respectively), ScO$^{-}$, TiO$^{-}$,CuO$^{-}$, and ZnO$^{-}$, using various levels of $GW$ theory, and compare themto experiments to evaluate the performance of the $GW$ approximation on theelectronic structure of small molecules containing 3$d$ transition metals. Wefind that the $G$-only eigenvalue self-consistent $GW$ scheme with $W$ fixed tothe PBE level ($G_{n}W_{0}$@PBE), which gives the best compromise betweenaccuracy and efficiency for solids, also gives good results for both localized($d$) and delocalized ($sp$) states of 3$d$-transition-metal oxide molecules.The success of $G_{n}W_{0}$@PBE in predicting electronic excitations in thesesystems reasonably well is likely due to the fortuitous cancellation effectbetween the overscreening of the Coulomb interaction by PBE and theunderscreening by the neglect of vertex corrections. Together with the absenceof the self-consistent field convergence error (e.g. spin contamination inopen-shell systems) and the $GW$ multi-solution issue, the $G_{n}W_{0}$@PBEscheme gives the possibility to predict the electronic structure of complexreal systems (e.g. molecule-solid and $sp$-$d$ hybrid systems) accurately andefficiently.

Practical $GW$ scheme for electronic structure of 3$d$-transition-metal monoxide anions: ScO$^{-}$, TiO$^{-}$, CuO$^{-}$, and ZnO$^{-}$