Construction of Symmetry-Adapted Functions in the Many-Particle Problem

Abstract

A new method is presented for obtaining many-particle angular momentum eigenfunctions and matrix elements of an invariant Hamiltonian. The same technique can be used to construct symmetry-adapted functions for any group of operators that commute with the Hamiltonian, and to simplify the evaluation of matrix elements in the symmetry-adapted basis. Applied to an arbitrary configuration, the method produces orthonormal functions identical with those that would be obtained by Schmidt orthogonalization of the projections of the original basis functions of the configuration. Because of this relationship, matrix elements of the Hamiltonian are greatly simplified, but the functions are obtained without explicitly constructing the projection operators or their matrix representations. To illustrate the method, it is applied to the spin coupling of configurations with three, four, and five particles outside closed shells, and to the two 2D functions of the atomic configuration d3, in Russell-Saunders coupling. Tables of the coefficients needed to evaluate all independent matrix elements are obtained for these examples, and typical matrix elements are calculated.