Computing lattice sums for calculating the elastic moduli of bcc metals via cluster decomposition

Symmetry of the Non-Atomic Interactions of N-Atomic Energy and the Atomistic Theory of High-Order Elastic Modules

“…The term which contains sums that depend on the coordinates of two atoms, represents the total irreducible energy of interaction of all pairs of atoms with numbers i and j. The term is the total irreducible potential energy of interaction 2 inside clusters containing all triplets of atoms with numbers i, j, and k. Term is analogously the total irreducible potential energy of interaction within clusters containing quadruplets of atoms with numbers i, j, k, and l… et cetera [27,29].…”

“…The lower indices of vectors represent the number of atoms in the cluster or their coordinates [27,28].…”

“…Any polynomial functions determined for atoms within a given cluster can be written as a polynomial function of the basis invariants that form the integral rational basis of invariants (IRBI) of the representation of group constructed on components of vectors within the complete set of vectors It is clear from the above that the IRBI, which reflects the effect external symmetry has on the form of the dependence must be calculated for the basis vectors of the representation of group Here, is a group of three-dimensional rotations in Euclidian space, and P N is the representation of a group of permutations of N identical atoms in the cluster. As with basis functions, relevant representation must be obtained for a complete set of components of vectors [27,29].…”

“…) by the group of permutations ( ) of numbers of identical atoms inside the cluster: [27,28]. If two clusters partially overlap (i.e., contain several coinciding atoms), direct separation of their potential energies is impossible.…”

“…This must produce a form of the calculated results that would then allow the next step in constructing a microscopic (molecular) theory of macroscopic characteristics of crystals. The second step in constructing the microscopic theory is to find such a way of summing the irreducible potential energies of all clusters so that the final sum is equal to total potential energy of the crystal, even if some clusters contain the same atoms [27,28].…”