becomes a semiconductor, as the defect state is separated from the conduction band. However, using only one parameter U (for the d electrons of Ti) it is not possible to fit both the experimenta l band gap and the position of the defect state inside the gap (which is at 0.8 eV from the conduction band). It has been shown for reduced rutile TiO$_{2}$ surfaces that a discontinuity appears for this property as a function of U at 4 eV and that a larger value is required to ensure the semiconductor character of the system as well as the correct position of the gap state$^{23}$. At this point it is important to remark that in the present work we consider a large concentration of vacancies in the interfaces, with the defect state developing into an impurity band, and therefore the exact position of the defect state is not so relevant. Turning now to vacancies at the interface, Fig. 6 shows the densities of states for the super-cells A and B of Fig. 3 when all oxygen atoms on the anatase side of the interface have been removed, calculated with LSDA and with LSDA+U. This extreme case is very interesting, as the total energy of systems A and B are equal within the calculation error (1 mRy) but system B is magnetic while system A is not. The only difference between the two structures lies on the second neighbor layer of the interface, as shown by a dotted line in Fig. 3. The local distortion is similar in both cases, producing a ripple in the inter-facial AlO$_{2}$ layer. The magnetic moment in case B is mostly located at the inter-facial Ti, as can be seen in Fig. 7. However, the total occupation of the d orbitals in this Ti atom is very similar in cases A and B, both can be considered Ti 3 $^{+}$, calculated with LSDA and LSDA+U. In both approximations, the defect level widens to form a band that almost fills the band gap compl etely, due to the large number of oxygen vacancies. This wide feature inside the band gap is also found in XPS experiments$^{24}$. The results shown in the previous section call our attention to the following fact: oxygen va- cancies lead to a change of valence in a nearby Ti atom turning it into Ti 3 $^{+}$, but this does not necessarily imply that the system will have a magnetic solution. Trying to understand this un- expected result, we resort to the well known Stoner criterion. The main idea behind it is that hybridization and band splitting are competing mechanisms to lower the energy of the system. For this purpose we performed non-magnetic calculations of super-lattices A and B, in the unrelaxed structures. The results are shown in Figures 8, 9 and 10. The stronger hybridization in case A is quite 5 IV. DISCUSSION AND CONCLUSIONS