Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging

Abstract

If the Hamiltonian in the time independent Schrödinger equation, H E    , isinvariant under a group of symmetry transformations, the theory of group representations can help obtain the eigenvalues and eigenvectors of H. A finite group that is not a symmetry group of H is nevertheless a symmetry group of an operator H sym projected from H by the process of symmetry averaging. In this case H = H sym + H R where H R is the nonsymmetric remainder. Depending on the nature of the remainder, the solutions for the full operator may be obtained by perturbation theory. It is shown here that when H is represented as a matrix [H] over a basis symmetry adapted to the group, the reduced matrix elements of [H sym ] are simple averages of certain elements of [H], providing a substantial enhancement in computational efficiency. A series of examples are given for the smallest molecular graphs. The first is a two vertex graph corresponding to a heteronuclear diatomic molecule. The symmetrized component then corresponds to a homonuclear system. A three vertex system is symmetry averaged in the first case to C s and in the second case to the nonabelian C 3v . These examples illustrate key aspects of the symmetry-averaging process.

Electronic Structure and Symmetry in Conjugated π‐Electron Systems

“…Note that the procedure used here to obtain a continuous symmetry measure for the Hamiltonian is strongly related to the symmetry averaging procedure proposed by Ellzey to obtain eigenvalues for nonsymmetric systems. [32] For any point group all the symmetry elements must pass through the center of the system. It is however not always trivial to define the center of a matrix and there is no general way of doing it.…”

“…where H † is the conjugate transpose of H and the maximization procedure refers to the orientation of the symmetry group. Note that the procedure used here to obtain a continuous symmetry measure for the Hamiltonian is strongly related to the symmetry averaging procedure proposed by Ellzey to obtain eigenvalues for nonsymmetric systems 32…”