The construction of U as a function of the interatomic distance variables rij = |qi − qj | is established, provided that the potential U adheres to specific symmetry assumptions. This is achieved by examining the interatomic potential U(q), where q = [q1, q2,..., qN ] ∈ R 3N represents a vector that defines locations, with qi ∈ R 3. Additionally, when N exceeds 5, the potential U can be formulated as a function of an appropriate subset of the distance variables rij, with the count of distance variables changing linearly in relation to N, the total number of atoms.

ABSTRACT: The construction of U as a function of the interatomic distance variables rij = |qi − qj | is established, provided that the potential U adheres to specific symmetry assumptions.