A framework for constructing interatomic potentials is proposed and applied to the case of silicon. The construction relies on a geometrical approach in which we try to reproduce the morphology of the Born-Oppenheimer surface. In particular, we develop a new technique for constructing four body terms. It consists of building invariant coordinate systems on the configuration space of four atoms, thus allowing the choice of the four body term morphology. The potential that we develop has a two and a four body part. It is adjusted to match the only geometrical information known from the Born-Oppenheimer surface which is the position of its minimum of energy and its curvature at that point, as given by the dynamical matrix. The potential is then tested on point defects and surface structures in silicon.