  A tight binding supercell approach is used for the calculation of the
electronic structure of the (111) LaAlO$_3$/SrTiO$_3$ interface. The
confinement potential at the interface is evaluated solving a discrete Poisson
equation by means of an iterative method. In addition to the effect of the
confinement, local Hubbard electron-electron terms are included at mean-field
level within a fully self-consistent procedure. The calculation carefully
describes how the two-dimensional electron gas arises from the quantum
confinement of electrons near the interface due to band bending potential. The
resulting electronic sub-bands and Fermi surfaces show full agreement with the
electronic structure determined by angle-resolved photoelectron spectroscopy
experiments. In particular, it is analyzed how the effect of local Hubbard
interactions changes the density distribution over the layers from the
interface to the bulk. Interestingly, the two-dimensional electron gas at
interface is not depleted by local Hubbard interactions which indeed induce an
enhancement of the electron density between the first layers and the bulk.
