The Solution of Problems Involving Permutation Degeneracy

Abstract

Matrix methods are developed for calculating the energy levels of many electron systems. These methods are based on the intimate relation between the energy matrix and the irreducible representations of the symmetric permutation group. Part I deals with the ``pure matrix method,'' which involves explicit calculation of the matrices representing the permutations. The permutation matrices are given for states of various multiplicities arising from configurations of eight or fewer electrons. From these we obtain a very convenient orthogonal representation of the energy matrix. In particular, for symmetrical polyatomic molecules, it is shown that correct labelling of the orbits leads directly to a partial factorization of the secular determinant. A systematic procedure is given for determining just how this factorization can be achieved. Part II takes up the ``algebraic method,'' which makes use of algebraic relations satisfied by the permutations. Explicit representations of the permutations are not required if this method is employed. The procedure is illustrated by the four elecron problem, including non‐orthogonality corrections. The interactions between configurations can also be calculated by the algebraic method. For symmetrical polyatomic molecules, the group of symmetry operators which commute with the Hamiltonian function can be used to obtain the secular equation in a factored form. The factorization is complete in the group theory sense. A number of illustrative examples are worked out, the energy levels of all multiplicities being found for the hexagonal (benzene) configuration, the configuration of eight similar orbits at cube corners, and the body‐centered cube. The singlet levels of the tetrahedral (methane) configuration are also given.