Title: Oxygen-vacancy driven electron localization and itinerancy in rutile-based TiO$_2$,
Abstract: Oxygen-deficient TiO$_2$ in the rutile structure as well as the Ti$_3$O$_5$
Magn{é}li phase is investigated within the charge self-consistent combination
of density functional theory (DFT) with dynamical mean-field theory (DMFT). It
is shown that an isolated oxygen vacancy (V$_{\rm O}$) in titanium dioxide is
not sufficient to metallize the system at low temperatures. In a semiconducting
phase, an in-gap state is identified at $\varepsilon_{\rm IG}^{\hfill}\sim
-0.75\,$eV\, in excellent agreement with experimental data. Band-like impurity
levels, resulting from a threefold V$_{\rm O}$-Ti coordination as well as
entangled $(t_{2g},e_g)$ states, become localized due to site-dependent
electronic correlations. Charge localization and strong orbital polarization
occur in the V$_{\rm O}$-near Ti ions, which details can be modified by a
variation of the correlated subspace. At higher oxygen vacancy concentration, a
correlated metal is stabilized in the Magn{é}li phase. A V$_{\rm O}$-defect
rutile structure of identical stoichiometry shows key differences in the
orbital-resolved character and the spectral properties. Charge
disproportionation is vital in the oxygen-deficient compounds, but obvious
metal-insulator transitions driven or sustained by charge order are not
identified.