the present paper are performed fixing the unit cell size in th e direction normal to the stacking to that of cubic LAO (lattice constant = 3.8 ˚ A) and relaxation is only allowed in the direction of the super-lattice. For the Density Functional Theory (DFT) 18 calculations we have used the Wien2k code, 19 that is an implementation of the FPLAPW (full potential linear augmented plane waves) method, in which the space is divided into muffin tin spheres around the a toms and an interstitial region. Plane waves are used to describe the region outside the spheres. The number of plane waves in the interstitial region is set by the cut-off parameter RK$_{max}$. In this work we use RK$_{max}$=7 that corresponds to an energy cutoff of 340 eV. We consider both the local spin density approximation LSDA 20 and the LSDA+U 21 one. The calculation is scalar relativistic and the atomic sphere radii are taken to be small, so that the atomic spheres do not overlap when relaxing the structure. They are 1.7 a.u. for Ti and Al, 1.4 a.u. for O and 2.5 a.u. for La. The number of k-points in the Brillouin zone is 50 or 100 for the relaxation procedure and increased to 200 k-points for the relaxed structure. Figure 4 shows the densities of states calculated within LSDA for the component systems and for the super-lattice. There are no magnetic solutions for any of these cases. The gap obtained for anatase is 2.2 eV, to be compared with the experimental 3.2 eV, the gap for LAO is 4.3 eV to be compared with the 5.6 eV experimental value. This feature of the LSDA approximation, that gives small band gaps, can be corrected using the LSDA+U method or hybrid functionals$^{22}$. The LSDA+U approach introduces an additional term based on a simple Hubbard model for electron on-site repulsion. This removes the self-interaction error by energetically penalising partial oc- cupation of the relevant electronic states, at the price of introducing an empirical parameter in an ab-initio calculation. Before studying the effect of oxygen vacancies at the interface, we start with bulk TiO$_{2}$. Figure 5 shows the densities of states of bulk TiO$_{2}$ anatase with one oxygen vacancy in the unit cell, for two different concentrations and both methods of calculation, relaxing all structures separately. No magnetic solutions are found for any of these cases. A defect state appears inside the band gap, too close to the conduction band in the LSDA approximation, thus making the system metallic. In the LSDA+U approximation, using U = 5 eV, the band gap increases by about 1eV and the system 4 III. METHOD OF CALCULATION AND RESULTS