First-principles modeling of systems with localized $d$ states is currently a great challenge in condensed-matter physics. Density-functional theory in the standard local-density approximation (LDA) proves to be problematic. This can be partly overcome by including local Hubbard $U$ corrections $\left(\text{LDA}+U\right)$ but itinerant states are still treated on the LDA level. Many-body perturbation theory in the $GW$ approach offers both a quasiparticle perspective (appropriate for itinerant states) and an exact treatment of exchange (appropriate for localized states), and is therefore promising for these systems. $\text{LDA}+U$ has previously been viewed as an approximate $GW$ scheme. We present here a derivation that is simpler and more general, starting from the static Coulomb-hole and screened exchange approximation to the $GW$ self-energy. Following our previous work for $f$-electron systems [H. Jiang, R. I. Gomez-Abal, P. Rinke, and M. Scheffler, Phys. Rev. Lett. 102, 126403 (2009)] we conduct a systematic investigation of the $GW$ method based on $\text{LDA}+U\left(GW@\text{LDA}+U\right)$, as implemented in our recently developed all-electron $GW$ code FHI-gap (Green’s function with augmented plane waves) for a series of prototypical $d$-electron systems: (1) ScN with empty $d$ states, (2) ZnS with semicore $d$ states, and (3) late transition-metal oxides (MnO, FeO, CoO, and NiO) with partially occupied $d$ states. We show that for ZnS and ScN, the $GW$ band gaps only weakly depend on $U$ but for the other transition-metal oxides the dependence on $U$ is as strong as in $\text{LDA}+U$. These different trends can be understood in terms of changes in the hybridization and screening. Our work demonstrates that $GW@\text{LDA}+U$ with “physical” values of $U$ provides a balanced and accurate description of both localized and itinerant states.

• Received 20 April 2010

DOI:https://doi.org/10.1103/PhysRevB.82.045108

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