Graph theory and the Jahn-Teller theorem
The Jahn-Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory fo...
Gespeichert in:
Veröffentlicht in: | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2012-04, Vol.468 (2140), p.971-989 |
---|---|
Hauptverfasser: | , , , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
container_end_page | 989 |
---|---|
container_issue | 2140 |
container_start_page | 971 |
container_title | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences |
container_volume | 468 |
creator | Ceulemans, A. Lijnen, E. Fowler, P. W. Mallion, R. B. Pisanski, T. |
description | The Jahn-Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group. |
doi_str_mv | 10.1098/rspa.2011.0508 |
format | Article |
fullrecord | <record><control><sourceid>jstor_cross</sourceid><recordid>TN_cdi_crossref_primary_10_1098_rspa_2011_0508</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>41511046</jstor_id><sourcerecordid>41511046</sourcerecordid><originalsourceid>FETCH-LOGICAL-c549t-1aba18fdfbfdafbdb07a184ef573a5ac2cfb88112773b1b9ce7e7be0d5cca6083</originalsourceid><addsrcrecordid>eNp9j81Lw0AQxYMoWKtXb0KPXhJnsrvZzbFUrUpB0doel02yS9OvhN1UjH-9iZGCiJ5mht-8N_M87xwhQIjFlXWlCkJADICBOPB6SDn6YUyjw6YnEfUZhHjsnTi3BICYCd7zLsdWlYtBtdCFrQdqm7Xt4EEttv5Ur9fadkhvTr0jo9ZOn33Xvvd6ezMd3fmTx_H9aDjxU0bjykeVKBQmM4nJlEmyBHgzU20YJ4qpNExNIgRiyDlJMIlTzTVPNGQsTVUEgvS9oPNNbeGc1UaWNt8oW0sE2eaUbU7Z5pRtzkZAOoEt6uaxIs11VctlsbPbZvxbtfpP9fzyNHyjkchDpCCbfQROYxLKj7zsrBooc-d2Wn6t_LT_fe2iu7Z0VWH3iSgyRKBRw_2O567S73uu7EpGnHAmZ4LKOZvBdH49kRH5BLymk8A</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Graph theory and the Jahn-Teller theorem</title><source>JSTOR Mathematics & Statistics</source><source>JSTOR Archive Collection A-Z Listing</source><source>Alma/SFX Local Collection</source><creator>Ceulemans, A. ; Lijnen, E. ; Fowler, P. W. ; Mallion, R. B. ; Pisanski, T.</creator><creatorcontrib>Ceulemans, A. ; Lijnen, E. ; Fowler, P. W. ; Mallion, R. B. ; Pisanski, T.</creatorcontrib><description>The Jahn-Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group.</description><identifier>ISSN: 1364-5021</identifier><identifier>EISSN: 1471-2946</identifier><identifier>DOI: 10.1098/rspa.2011.0508</identifier><language>eng</language><publisher>The Royal Society Publishing</publisher><subject>Automorphisms ; Chemical bonding ; Eigenvalues ; Eigenvectors ; Electronic/spectral Degeneracy ; Electrons ; Group Theory ; Hückel Theory ; Jahn Teller effect ; Molecules ; Spectral Graph Theory ; Symmetry ; Vertices</subject><ispartof>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, 2012-04, Vol.468 (2140), p.971-989</ispartof><rights>COPYRIGHT © 2012 The Royal Society</rights><rights>This journal is © 2011 The Royal Society</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c549t-1aba18fdfbfdafbdb07a184ef573a5ac2cfb88112773b1b9ce7e7be0d5cca6083</citedby><cites>FETCH-LOGICAL-c549t-1aba18fdfbfdafbdb07a184ef573a5ac2cfb88112773b1b9ce7e7be0d5cca6083</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/41511046$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/41511046$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,780,784,803,832,27924,27925,58017,58021,58250,58254</link.rule.ids></links><search><creatorcontrib>Ceulemans, A.</creatorcontrib><creatorcontrib>Lijnen, E.</creatorcontrib><creatorcontrib>Fowler, P. W.</creatorcontrib><creatorcontrib>Mallion, R. B.</creatorcontrib><creatorcontrib>Pisanski, T.</creatorcontrib><title>Graph theory and the Jahn-Teller theorem</title><title>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences</title><addtitle>Proc. R. Soc. A</addtitle><addtitle>Proc. R. Soc. A</addtitle><description>The Jahn-Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group.</description><subject>Automorphisms</subject><subject>Chemical bonding</subject><subject>Eigenvalues</subject><subject>Eigenvectors</subject><subject>Electronic/spectral Degeneracy</subject><subject>Electrons</subject><subject>Group Theory</subject><subject>Hückel Theory</subject><subject>Jahn Teller effect</subject><subject>Molecules</subject><subject>Spectral Graph Theory</subject><subject>Symmetry</subject><subject>Vertices</subject><issn>1364-5021</issn><issn>1471-2946</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNp9j81Lw0AQxYMoWKtXb0KPXhJnsrvZzbFUrUpB0doel02yS9OvhN1UjH-9iZGCiJ5mht-8N_M87xwhQIjFlXWlCkJADICBOPB6SDn6YUyjw6YnEfUZhHjsnTi3BICYCd7zLsdWlYtBtdCFrQdqm7Xt4EEttv5Ur9fadkhvTr0jo9ZOn33Xvvd6ezMd3fmTx_H9aDjxU0bjykeVKBQmM4nJlEmyBHgzU20YJ4qpNExNIgRiyDlJMIlTzTVPNGQsTVUEgvS9oPNNbeGc1UaWNt8oW0sE2eaUbU7Z5pRtzkZAOoEt6uaxIs11VctlsbPbZvxbtfpP9fzyNHyjkchDpCCbfQROYxLKj7zsrBooc-d2Wn6t_LT_fe2iu7Z0VWH3iSgyRKBRw_2O567S73uu7EpGnHAmZ4LKOZvBdH49kRH5BLymk8A</recordid><startdate>20120408</startdate><enddate>20120408</enddate><creator>Ceulemans, A.</creator><creator>Lijnen, E.</creator><creator>Fowler, P. W.</creator><creator>Mallion, R. B.</creator><creator>Pisanski, T.</creator><general>The Royal Society Publishing</general><general>The Royal Society</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20120408</creationdate><title>Graph theory and the Jahn-Teller theorem</title><author>Ceulemans, A. ; Lijnen, E. ; Fowler, P. W. ; Mallion, R. B. ; Pisanski, T.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c549t-1aba18fdfbfdafbdb07a184ef573a5ac2cfb88112773b1b9ce7e7be0d5cca6083</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Automorphisms</topic><topic>Chemical bonding</topic><topic>Eigenvalues</topic><topic>Eigenvectors</topic><topic>Electronic/spectral Degeneracy</topic><topic>Electrons</topic><topic>Group Theory</topic><topic>Hückel Theory</topic><topic>Jahn Teller effect</topic><topic>Molecules</topic><topic>Spectral Graph Theory</topic><topic>Symmetry</topic><topic>Vertices</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ceulemans, A.</creatorcontrib><creatorcontrib>Lijnen, E.</creatorcontrib><creatorcontrib>Fowler, P. W.</creatorcontrib><creatorcontrib>Mallion, R. B.</creatorcontrib><creatorcontrib>Pisanski, T.</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><jtitle>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ceulemans, A.</au><au>Lijnen, E.</au><au>Fowler, P. W.</au><au>Mallion, R. B.</au><au>Pisanski, T.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Graph theory and the Jahn-Teller theorem</atitle><jtitle>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences</jtitle><stitle>Proc. R. Soc. A</stitle><addtitle>Proc. R. Soc. A</addtitle><date>2012-04-08</date><risdate>2012</risdate><volume>468</volume><issue>2140</issue><spage>971</spage><epage>989</epage><pages>971-989</pages><issn>1364-5021</issn><eissn>1471-2946</eissn><abstract>The Jahn-Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group.</abstract><pub>The Royal Society Publishing</pub><doi>10.1098/rspa.2011.0508</doi><tpages>19</tpages><oa>free_for_read</oa></addata></record> |
fulltext | fulltext |
identifier | ISSN: 1364-5021 |
ispartof | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, 2012-04, Vol.468 (2140), p.971-989 |
issn | 1364-5021 1471-2946 |
language | eng |
recordid | cdi_crossref_primary_10_1098_rspa_2011_0508 |
source | JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; Alma/SFX Local Collection |
subjects | Automorphisms Chemical bonding Eigenvalues Eigenvectors Electronic/spectral Degeneracy Electrons Group Theory Hückel Theory Jahn Teller effect Molecules Spectral Graph Theory Symmetry Vertices |
title | Graph theory and the Jahn-Teller theorem |
url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-25T03%3A16%3A46IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Graph%20theory%20and%20the%20Jahn-Teller%20theorem&rft.jtitle=Proceedings%20of%20the%20Royal%20Society.%20A,%20Mathematical,%20physical,%20and%20engineering%20sciences&rft.au=Ceulemans,%20A.&rft.date=2012-04-08&rft.volume=468&rft.issue=2140&rft.spage=971&rft.epage=989&rft.pages=971-989&rft.issn=1364-5021&rft.eissn=1471-2946&rft_id=info:doi/10.1098/rspa.2011.0508&rft_dat=%3Cjstor_cross%3E41511046%3C/jstor_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_id=info:pmid/&rft_jstor_id=41511046&rfr_iscdi=true |