  The Co-doped anatase TiO$_{2}$, a recently discovered room-temperature
ferromagnetic insulator, has been studied by the first-principles calculations
in the pseudo-potential plane-wave formalism within the local-spin-density
approximation (LSDA), supplemented by the full-potential linear augmented plane
wave (FP-LAPW) method. Emphasis is placed on the dependence of its electronic
structures and linear optical properties on the Co-doping concentration and
oxygen vacancy in the system in order to pursue the origin of its
ferromagnetism. In the case of substitutional doping of Co for Ti, our
calculated results are well consistent with the experimental data, showing that
Co is in its low spin state. Also, it is shown that the oxygen vacancy enhances
the ferromagnetism and has larger effect on both the electronic structure and
optical properties than the Co-doping concentration only.
