Symmetries of many-body systems imply distance-dependent potentials

Considering an interatomic potential U(q), where q=[q_{1},q_{2},⋯,q_{N}]∈R^{3N} is a vector describing positions q_{i}∈R^{3}, it is shown that U can be defined as a function of the interatomic distance variables r_{ij}=|q_{i}-q_{j}| provided the potential U satisfies some symmetry assumptions. Moreover, the potential U can be defined as a function of a proper subset of the distance variables r_{ij}, provided N>5, with the number of distance variables used scaling linearly with the number of atoms N.