An analytic procedure for inverting ab initio cohesive energy vs. volume curves to obtain a parameter-free interatomic potential is presented. The procedure determines the radial functions in a cluster potential, assuming an angular dependence. The method is a nonlinear generalization of the ab initio pair potential formula of Carlsson, Gelatt and Ehrenreich to higher orders of cluster expansion [1]. We demonstrate our method by deriving an ab initio cluster potential for silicon with the Stillinger-Weber angular dependence [2]. A novel property of the potential is that bond-bending forces are not monotonic in the bond length, as is the case with most empirical potentials, but rather are peaked at the first neighbor distance. The validity of the inverted potential is discussed along with its use in the development of new interatomic potentials.