It is demonstrated that U can be constructed as a function of the interatomic distance variables rij = |qi − qj |, given that the potential U meets certain symmetry assumptions, by considering the interatomic potential U(q), where q = [q1, q2,..., qN ] ∈ R 3N is a vector defining locations, qi ∈ R 3. Furthermore, if N > 5, the potential U can be expressed as a function of a suitable subset of the distance variables rij, where the number of distance variables varies in a linear fashion with N, the number of atoms.