If the Hamiltonian in the time independent Schrödinger equation, *H*Ψ = *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

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If the Hamiltonian in the time independent Schrödinger equation,

*H*Ψ =

*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.

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