  Combining LSDA+$U$ and an analysis of superexchange interactions beyond DFT,
we describe the magnetic ground states in rutile and anatase Cr-doped TiO$_2$.
In parallel, we correct our LSDA+$U$ ground state through GW corrections
($GW$@LSDA+$U$) that reproduce the position of 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 ferromagnetic ground state, in
keeping with experimental evidence, without the need to invoke F-center
exchange.
