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Math."],"published-print":{"date-parts":[[2026,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We present a geometric reformulation of the H\u00fcckel molecular orbital\/tight-binding (HMO\/TB) method that integrates tools from algebraic graph theory and statistical mechanics. By extending classical notions of charge density and bond order to incorporate bond orbital energies, we construct a probabilistic framework that embeds molecular graphs into Euclidean (pseudometric) space. This embedding introduces new geometric descriptors\u2014such as internodal distances, node angles, and electronic propagation efficiency\u2014that correlate with chemical properties including bond lengths, reactivity, and molecular stability. Analytical results for linear and cyclic polyenes as well as polycyclic aromatic hydrocarbons illustrate the utility of these descriptors and suggest a pathway toward closer integration of HMO\/TB with ab initio methods and machine learning approaches in molecular modeling.<\/jats:p>","DOI":"10.1137\/25m1791883","type":"journal-article","created":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T12:36:50Z","timestamp":1784896610000},"page":"1882-1905","source":"Crossref","is-referenced-by-count":0,"title":["The Geometry of the H\u00fcckel\/Tight-Binding Method for Modeling Molecular Systems"],"prefix":"10.1137","volume":"86","author":[{"given":"Marisol","family":"Berm\u00fadez-Monta\u00f1a","sequence":"first","affiliation":[{"name":"Tecnologico de Monterrey, Escuela de Ingenier\u00eda y Ciencias, 52926 Estado de M\u00e9xico, M\u00e9xico."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3066-7418","authenticated-orcid":true,"given":"Ernesto","family":"Estrada","sequence":"additional","affiliation":[{"name":"Institute of Cross-Disciplinary Physics and Complex Systems (IFISC), UIB-CSIC, Palma de Mallorca, Spain."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,7,24]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.5155\/eurjchem.6.3.261-269.1239"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cplett.2025.142276"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/24M167038X"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1002\/wcms.1493"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s10825-009-0287-x"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.mtquan.2024.100001"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1021\/ja00768a058"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.91.025003"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1107\/S0108270188005840"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2006.04.006"},{"key":"ref11","first-page":"1","volume":"28","author":"Colombo L.","year":"2005","journal-title":"La Rivista del Nuovo Cimento"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1039\/C4CP00908H"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1016\/j.ccr.2023.215112"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2003.11.003"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.molap.2019.100054"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1039\/D1CP00244A"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2012.01.017"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-023-02571-9"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1137\/141000555"},{"key":"ref20","first-page":"203","volume":"77","author":"Estrada E.","year":"2004","journal-title":"Croat. 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