Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging
If the Hamiltonian in the time independent Schrödinger equation, HΨ = 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 gro...
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| Format: | Article |
| Language: | English |
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MDPI AG
2009-08-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/1/1/10/ |
| _version_ | 1811301111792926720 |
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| author | Marion L. Ellzey |
| author_facet | Marion L. Ellzey |
| author_sort | Marion L. Ellzey |
| collection | DOAJ |
| description | If the Hamiltonian in the time independent Schrödinger equation, HΨ = 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 Hsym projected from H by the process of symmetry averaging. In this case H = Hsym + HR where HR 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 [Hsym] 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 Cs and in the second case to the nonabelian C3v. These examples illustrate key aspects of the symmetry-averaging process. |
| first_indexed | 2024-04-13T07:03:01Z |
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| id | doaj.art-4c67aa80822347c9b77d8d0935905d22 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T07:03:01Z |
| publishDate | 2009-08-01 |
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| series | Symmetry |
| spelling | doaj.art-4c67aa80822347c9b77d8d0935905d222022-12-22T02:57:03ZengMDPI AGSymmetry2073-89942009-08-0111102010.3390/sym1010010Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry AveragingMarion L. EllzeyIf the Hamiltonian in the time independent Schrödinger equation, HΨ = 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 Hsym projected from H by the process of symmetry averaging. In this case H = Hsym + HR where HR 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 [Hsym] 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 Cs and in the second case to the nonabelian C3v. These examples illustrate key aspects of the symmetry-averaging process.http://www.mdpi.com/2073-8994/1/1/10/Hamiltonian symmetrygroup theorysymmetry-adapted basisreduced matrix elementssymmetry-averaging |
| spellingShingle | Marion L. Ellzey Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging Symmetry Hamiltonian symmetry group theory symmetry-adapted basis reduced matrix elements symmetry-averaging |
| title | Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging |
| title_full | Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging |
| title_fullStr | Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging |
| title_full_unstemmed | Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging |
| title_short | Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging |
| title_sort | using group theory to obtain eigenvalues of nonsymmetric systems by symmetry averaging |
| topic | Hamiltonian symmetry group theory symmetry-adapted basis reduced matrix elements symmetry-averaging |
| url | http://www.mdpi.com/2073-8994/1/1/10/ |
| work_keys_str_mv | AT marionlellzey usinggrouptheorytoobtaineigenvaluesofnonsymmetricsystemsbysymmetryaveraging |