Close to the interface between a nematic liquid crystal (NLC) and another medium the elastic constants are expected to become functions of distance z from the interface and of angle theta between the director n and the unit vector k orthogonal to the interface. Furthermore, due to symmetry breaking, new elastic contributions that are absent in the bulk may become important close to the interface. In this paper we consider a simple microscopic model based on van der Waals induced dipole-induced dipole interactions under the special assumption of perfect orientational order (S = 1). By using this model, we calculate the free energy density and the excess of surface free energy. Some elastic constants are obtained by using a numerical procedure. In particular, a new elastic contribution which is linear in the director gradients is obtained for the first time. Close to the interface, these elastic constants greatly depend on distance z and are simple functions of the scalar product n . k.