Overlap must explicitly be included when empirical interatomic potentials for use in environments of low symmetry are derived using tight-binding theory. We derive a simplified real-space total-energy expression that takes into account the first and second moments of the density of states including overlap. It uses the extended Hückel theory and Taylor and Padé truncations for the inverse of the overlap matrix. Our approximate expression is tested for an s-band nearest-neighbor model by carrying out numerical integrations over exact densities of states for many lattices of different dimensionality and coordination number. Within a two-body simplification the inclusion of overlap leads to a new functional form for use in developing embedded-atom functionals. Quantitative accuracy is improved, however, when the three-body Padé-approximation-derived form is used.