Using group theory to obtain eigenvalues of nonsymmetric systems by symmetry averaging (Q2446109)

Summary: If the Hamiltonian in the time independent Schrödinger equation, \(H\Psi=E\Psi\), 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 \(H_{\mathrm{sym}}\) projected from \(H\) by the process of symmetry averaging. In this case \(H=H_{\mathrm{sym}}+H_R\) 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 \([H_{\mathrm{sym}}]\) 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 \(\mathrm C_s\) and in the second case to the nonabelian \(\mathrm C_{3v}\). These examples illustrate key aspects of the symmetry-averaging process.