Combining the local spin density approximation $(\mathrm{LSDA})+U$ and an analysis of superexchange interactions beyond density functional theory, we describe the magnetic ground state of Cr-doped ${\mathrm{TiO}}_2$, an intensively studied and debated dilute magnetic oxide. In parallel, we correct our $\mathrm{LSDA}+U$ ($+$ superexchange) ground state through $GW$ corrections ($GW@\mathrm{LSDA}+U$) that reproduce the position of the impurity states and the band gaps in satisfying agreement with experiments. Because of the different topological coordinations of Cr-Cr bonds in the ground states of rutile and anatase, superexchange interactions induce either ferromagnetic or antiferromagnetic couplings of Cr ions. In Cr-doped anatase, this interaction leads to a new mechanism which stabilizes a (nonrobust) ferromagnetic ground state, in keeping with experimental evidence, without the need to invoke $F$-center exchange. The interplay between structural defects and vacancies in contributing to the superexchange is also unveiled.

• Received 15 February 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.136402