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

摘要

If the Hamiltonian in the time independent Schr?dinger equation, HΨ = EΨ, is invariant 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.