  Doping is one of the most common strategies for improving the photocatalytic
and solar energy conversion properties of TiO$_2$, hence an accurate
theoretical description of the electronic and optical properties of doped
TiO$_2$ is of both scientific and practical interest. In this work we use
many-body perturbation theory techniques to investigate two typical n-type
dopants, Niobium and Hydrogen, in TiO$_2$ rutile. Using the GW approximation to
determine band edges and defect energy levels, and the Bethe Salpeter equation
for the calculation of the absorption spectra, we find that the defect energy
levels form non-dispersive bands %associated with localized states lying
$\simeq 2.2 eV$ above the top of the corresponding valence bands ($\simeq 0.9
eV$ below the conduction bands of the {\it pristine} material). The defect
states are also responsible for the appearance of low energy absorption peaks
that enhance the solar spectrum absorption of rutile. The spatial distributions
of the excitonic wavefunctions associated with these low energy excitations are
very different for the two dopants, suggesting a larger mobility of
photoexcited electrons in Nb-TiO$_2$.
