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This is especially true for systems with tetrahedral and octahedral symmetries, whose electronic states may be threefold degenerate and experience complicated Jahn\u2013Teller and pseudo-Jahn\u2013Teller interactions. To accurately describe these interactions, high quality spin\u2013orbit vibronic Hamiltonian operators are needed. In this study, we present a unified one-electron Hamiltonian formalism for spin\u2013orbit vibronic interactions for systems in all tetrahedral and octahedral symmetries. The formalism covers all spin\u2013orbit Jahn\u2013Teller and pseudo-Jahn\u2013Teller problems in the symmetries with arbitrary types and arbitrary numbers of vibrational modes and generates Hamiltonian expansion formulas of arbitrarily high order.<\/jats:p>","DOI":"10.1063\/5.0090053","type":"journal-article","created":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T20:34:32Z","timestamp":1648758872000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Unified one-electron Hamiltonian formalism of spin\u2013orbit Jahn\u2013Teller and pseudo-Jahn\u2013Teller problems in tetrahedral and octahedral symmetries"],"prefix":"10.1063","volume":"157","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5087-3761","authenticated-orcid":false,"given":"Ekadashi","family":"Pradhan","sequence":"first","affiliation":[{"name":"Department of Chemistry, York University 1 , Toronto, Ontario M3J1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3720-6865","authenticated-orcid":false,"given":"Guoying","family":"Yao","sequence":"additional","affiliation":[{"name":"MOE Laboratory of Bioinorganic and Synthetic Chemistry, Lehn Institute of Functional Materials, School of Chemistry, Sun Yat-sen University 2 , Guangzhou, Guangdong 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6403-8679","authenticated-orcid":false,"given":"Zhenyu","family":"Yang","sequence":"additional","affiliation":[{"name":"MOE Laboratory of Bioinorganic and Synthetic Chemistry, Lehn Institute of Functional Materials, School of Chemistry, Sun Yat-sen University 2 , Guangzhou, Guangdong 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1553-7850","authenticated-orcid":false,"given":"Tao","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of Chemistry, York University 1 , Toronto, Ontario M3J1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2022,8,8]]},"reference":[{"volume-title":"Vibronic Interactions in Molecules and Crystals","year":"1989","key":"2023081006084164100_c1"},{"key":"2023081006084164100_c2","doi-asserted-by":"crossref","DOI":"10.1142\/5406","volume-title":"Conical Intersections: Electronic Structure, Dynamics and Spectroscopy","author":"Domcke","year":"2004"},{"key":"2023081006084164100_c3","doi-asserted-by":"crossref","DOI":"10.1142\/7803","volume-title":"Conical Intersections: Theory, Computation and Experiment","author":"Domcke","year":"2011"},{"key":"2023081006084164100_c4","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1098\/rspa.1937.0142","volume":"161","year":"1937","journal-title":"Proc. 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