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

## Abstract

**:**

_{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.

## 1. Introduction

_{sym}, is projected from H by symmetry averaging, summing over all transformations and dividing by the order of the group:

_{R}is the remainder. If H

_{sym}is nonzero and the remainder H

_{R}is sufficiently small in some sense, perturbation theory may give acceptable approximations to the eigenvalues and eigenvectors of H. A number of groups may be used for any particular system, but the optimum choice for analysis would generally be the largest group yielding a nonzero H

_{sym}. In general, the larger the group, the more the reduction in analytical requirements.

_{R}has no identity component and provides a relationship between reduced matrix elements of H

_{sym}and matrix elements of H.

_{sym}] will be reduced although [H] will not be. By a corollary to the symmetry-generation theorem [4,5,6], the reduced matrix elements of [H

_{sym}] are simple averages of certain elements of [H] making the construction of [H

_{sym}] much more efficient than the computationally demanding projection. Therefore, even though H

_{sym}may not be the Hamiltonian of a real molecule [H

_{sym}] is readily determined.

## 2. Symmetry-averaging

^{α}are commuting orthogonal idempotents:

_{sym}commutes with all elements of the group. The original operator is expressed as the sum of H

_{sym}and a remainder:

## 3. Symmetry-adapted basis

_{a}]

^{α}, ρ distinguishes repeated irreducible representations, and ω identifies the complete space. The representation of the group on a symmetry-adapted basis is completely reduced. A symmetry-adapted basis is suitably-conditioned if all of the |ω;ραr〉, ρ = 1,…, f(ω;α), transform according to identical matrices [G

_{a}]

^{α}. Here, f(ω;α) is the number of times the α irreducible representation occurs, given by the usual character formula. Matrix elements of an operator H are expressed on this basis as:

^{ω}on some defining basis: {|ωi〉, i = 1,…,f(ω)}, it is necessary to perform the transformation to the symmetry-adapted basis:

_{sym}, that commutes with G satisfy the relation [9]:

_{3v}contains one A

_{1}and two E irreducible representations:

_{a}is:

_{sym}is obtained by symmetry-averaging of H, as in equation (10), and if the basis is suitably conditioned, then the trace of a block in matrix (17) is the sum of traces of corresponding transformed bocks of the matrix of H: ${\left[{G}_{a}\right]}^{\alpha}{\left[H\right]}^{\rho \alpha ,{\rho}^{\prime}\alpha}{\left[{G}_{a}^{-1}\right]}^{\alpha}$. Then as a corollary to the symmetry-generation theorem the reduced matrix elements of H

_{sym}on a suitably conditioned symmetry-adapted basis are given by:

## 4. Two dimensional matrix

_{sym}] and the more appropriate this choice of symmetry. A symmetry-adapted basis is:

_{sym}on this basis is:

_{R}:

_{R}] makes only a second order correction.

_{sym}] the zero-order eigenvalues are −40.0 and −30.0. With the second order correction, these become −42.5 and −27.5.

## 5. Three-vertex examples

**S**

_{2}Averaging**S**

_{3}Averaging_{3}symmetry averaged matrix is:

_{3v}names of the irreducible representations of have been used. On this basis, the matrix is:

_{sym}]

^{ω}are then:

## 6. Conclusion

_{2}treatment of the three vertex example the exact eigenvalues separate into two groups similar to the approximate result. In the second S

_{3}treatment, the smaller difference term gives an indication of how symmetry averaging can be useful. In this case, the approximate eigenvalues are within 1% or less of the exact values.

_{R}would then be the sum of several irreducible tensorial operators. The Wigner-Eckart theorem can then be used to evaluate the matrix elements of these operators. This approach is being developed.

## References

- Cotton, F. A. Chemical Applications of Group Theory, 2nd ed.; Wiley: New York, USA, 1971. [Google Scholar]
- Wigner, E. Group Theory and its Applications to Atomic Spectra; Academic Press: New York, USA, 1959. [Google Scholar]
- Hamermesh, M. Group Theory and its Applications to Physical Problems; Addison-Wesley, Reading: Massachusetts, USA, 1962. [Google Scholar]
- Ellzey, M. L., Jr. Finite group theory for large systems. 1. Symmetry-adaptation. J. Chem. Inf. Comput. Sci.
**2003**, 43, 178–181. [Google Scholar] [CrossRef] [PubMed] - Ellzey, M. L., Jr.; Villagran, D. Finite group theory for large systems. 2. Generating relations and irreducible representations for the icosahedral point group, I
_{h}. J. Chem. Inf. Comput. Sci.**2003**, 43, 1763–1770. [Google Scholar] [CrossRef] [PubMed] - Ellzey, M. L., Jr. Finite group theory for large systems. 3. Symmetry-generation of reduced matrix elements for icosahedral C
_{20}and C_{60}molecules. J. Computational Chem.**2007**, 28, 811–817. [Google Scholar] [CrossRef] [PubMed] - Littlewood, D. E. The Theory of Group Characters; Oxford University Press: Oxford, UK, 1950. [Google Scholar]
- Weyl, H. The Theory of Groups and Quantum Mechanics; Dover: New York, 1931; (H. P. Robertson, trans.). [Google Scholar]
- Ellzey, M. L., Jr. Normalized irreducible tensorial matrices and the Wigner-Eckart theorem for unitary groups: A superposition Hamiltonian constructed from octahedral NITM. Int. J. Of Quantum Chem.
**1992**, 41, 653–665. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Ellzey, M.L., Jr.
Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging. *Symmetry* **2009**, *1*, 10-20.
https://doi.org/10.3390/sym1010010

**AMA Style**

Ellzey ML Jr.
Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging. *Symmetry*. 2009; 1(1):10-20.
https://doi.org/10.3390/sym1010010

**Chicago/Turabian Style**

Ellzey, Marion L., Jr.
2009. "Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging" *Symmetry* 1, no. 1: 10-20.
https://doi.org/10.3390/sym1010010