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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) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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