arXiv2019
The $GW$ approximation to many-body perturbation theory is a reliable tool for describing charged electronic excitations, and it has been successfully applied to a wide range of extended systems for several decades using a plane-wave basis. However, the $GW$ approximation has been used to test limited spectral properties of a limited set of finite systems (e.g. frontier orbital energies of closed-shell $sp$ molecules) only for about a decade using a local-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 them to experiments to evaluate the performance of the $GW$ approximation on the electronic structure of small molecules containing 3$d$ transition metals. We find that the $G$-only eigenvalue self-consistent $GW$ scheme with $W$ fixed to the PBE level ($G_{n}W_{0}$@PBE), which gives the best compromise between accuracy 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 these systems reasonably well is likely due to the fortuitous cancellation effect between the overscreening of the Coulomb interaction by PBE and the underscreening by the neglect of vertex corrections. Together with the absence of the self-consistent field convergence error (e.g. spin contamination in open-shell systems) and the $GW$ multi-solution issue, the $G_{n}W_{0}$@PBE scheme gives the possibility to predict the electronic structure of complex real systems (e.g. molecule-solid and $sp$-$d$ hybrid systems) accurately and efficiently.