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

Department of Chemistry, The University of Texas at El Paso, El Paso, Texas 79968-0513, USA

Received: 10 July 2009 / Revised: 5 August 2009 / Accepted: 6 August 2009 / Published: 6 August 2009

(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)

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# Abstract

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*projected from

_{sym}*H*by the process of symmetry averaging. In this case

*H*=

*H*+

_{sym}*H*where

_{R}*H*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

_{R}*H*is represented as a matrix [

*H*] over a basis symmetry adapted to the group, the reduced matrix elements of [

*H*] are simple averages of certain elements of [

_{sym}*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.

*Keywords:*Hamiltonian symmetry; group theory; symmetry-adapted basis; reduced matrix elements; symmetry-averaging

This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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