Chen-Möbius inversion theorem and a structural representation of crystallographic direction families

Abstract

An alternative formula for the inverse calculation of the pairwise interatomic potential from the cohesive energy as a function of the lattice separation is presented. The formula is based on the structural representation of crystallographic direction families for a crystal lattice. The key mathematical treatment comes from the up-to-date Chen-Möbius inversion theorem. Both the case of a pure crystal lattice and the case of a binary superstructure are studied. The solution for the latter is based on a generalized Möbius inversion formula on a unitary semigroup. To make the technique of inverse calculation for pair potentials realistic and practicable for atomic simulations, the inverse calculation of the pair potential function with a volume-dependent correction term is discussed.