-4 -2 0 2 4 0 5 10 15 20 25 30 LSDA LSDA+U -4 -2 0 2 4 E - E$_{F}$ (eV) 0 20 40 60 80 100 DOS Ti$_{4}$O$_{7}$ Ti$_{16}$O$_{31}$ FIG. 5: Calculated densities of states of bulk anatase, with two different concentrations of oxygen vacancies in the LSDA and LSDA+U approximations (U = 5 eV). The structure is relaxed separately in each case. 13