{"title":"十二面体的顶点和面着色及其所有不可约表示的畸变:对动态手性、五边形、扬-泰勒和拉长畸变的见解","authors":"Krishnan Balasubramanian","doi":"10.1007/s10910-025-01704-1","DOIUrl":null,"url":null,"abstract":"<div><p>We present combinatorial cum group theoretical generating function methods for vertex and face colorings of a dodecahedron for all irreducible representations of the icosahedral group. We demonstrate the usefulness of the Mȍbius inversion method. We consider several types of distortions arising from the highly symmetric dodecahedron, in particular, to an elongated dodecahedron and a pyrithohedron (T<sub>h</sub>). Elaborate combinatorial enumerations and tables of combinatorial numbers are explicitly constructed for all irreducible representations of both the distorted and the parent undistorted structures. It is shown that the combinatorial cum computational techniques provide new insights into the dynamic chirality arising from such distortions which include the pentagonal distortions, elongated distortions and so forth. We point out applications to the dynamic NMR and ESR spectroscopies as well to the dynamic stereochemistry of topological metamorphosis through a combination of combinatorics and group theory.</p></div>","PeriodicalId":648,"journal":{"name":"Journal of Mathematical Chemistry","volume":"63 4","pages":"982 - 1034"},"PeriodicalIF":1.7000,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Vertex and face colorings of dodecahedron and its distortions for all irreducible representations: insights into dynamic chirality, pentagonal, Jahn–Teller and elongated distortions\",\"authors\":\"Krishnan Balasubramanian\",\"doi\":\"10.1007/s10910-025-01704-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We present combinatorial cum group theoretical generating function methods for vertex and face colorings of a dodecahedron for all irreducible representations of the icosahedral group. We demonstrate the usefulness of the Mȍbius inversion method. We consider several types of distortions arising from the highly symmetric dodecahedron, in particular, to an elongated dodecahedron and a pyrithohedron (T<sub>h</sub>). Elaborate combinatorial enumerations and tables of combinatorial numbers are explicitly constructed for all irreducible representations of both the distorted and the parent undistorted structures. It is shown that the combinatorial cum computational techniques provide new insights into the dynamic chirality arising from such distortions which include the pentagonal distortions, elongated distortions and so forth. We point out applications to the dynamic NMR and ESR spectroscopies as well to the dynamic stereochemistry of topological metamorphosis through a combination of combinatorics and group theory.</p></div>\",\"PeriodicalId\":648,\"journal\":{\"name\":\"Journal of Mathematical Chemistry\",\"volume\":\"63 4\",\"pages\":\"982 - 1034\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2025-02-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Chemistry\",\"FirstCategoryId\":\"92\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10910-025-01704-1\",\"RegionNum\":3,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Chemistry","FirstCategoryId":"92","ListUrlMain":"https://link.springer.com/article/10.1007/s10910-025-01704-1","RegionNum":3,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Vertex and face colorings of dodecahedron and its distortions for all irreducible representations: insights into dynamic chirality, pentagonal, Jahn–Teller and elongated distortions
We present combinatorial cum group theoretical generating function methods for vertex and face colorings of a dodecahedron for all irreducible representations of the icosahedral group. We demonstrate the usefulness of the Mȍbius inversion method. We consider several types of distortions arising from the highly symmetric dodecahedron, in particular, to an elongated dodecahedron and a pyrithohedron (Th). Elaborate combinatorial enumerations and tables of combinatorial numbers are explicitly constructed for all irreducible representations of both the distorted and the parent undistorted structures. It is shown that the combinatorial cum computational techniques provide new insights into the dynamic chirality arising from such distortions which include the pentagonal distortions, elongated distortions and so forth. We point out applications to the dynamic NMR and ESR spectroscopies as well to the dynamic stereochemistry of topological metamorphosis through a combination of combinatorics and group theory.
期刊介绍:
The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches.
Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.