Inorganic Frontier Orbital Theory: How Penetration, Shielding and Relativity Reshape the Periodic Table

Inorganic Frontier Orbital Theory: How Penetration, Shielding and Relativity Reshape the Periodic Table | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Inorganic Frontier Orbital Theory: How Penetration, Shielding and Relativity Reshape the Periodic Table Cesar A. Cohen de Mello This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8261906/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract For many decades, persistent irregularities in atomic and molecular electronic structure have resistedany unifying explanation. Classical and modern work by Mulliken, Coulson, Roothaan, Fukui, Hoffmann,Pyykkö, Schwerdtfeger and others mapped emblematic cases: the 4d/5s inversion in Pd, the 6s/5d frontierreordering in Pt and Au, the collapse of the 4f-derived HOMO in Eu/Yb, the 5f/6d competition in U/Pu,and the 𝜋/𝜎 anomalies in CO, NO+ and O2. What remained unclear was whether these “exceptions” sharea common microscopic mechanism or are unrelated curiosities. Here these phenomena are interpreted as manifestations of a single inorganic-chemical mechanism:competition between frontier orbitals with very different penetration, shielding and relativistic response,leading to a curvature-driven reordering of HOMO and LUMO. Instead of the full operator calculus,the analysis focuses on descriptors that can be extracted from modern relativistic electronic-structurecalculations: (i) energy branches of candidate frontier orbitals as functions of effective nuclear charge orchemical environment; (ii) their curvature and near-degeneracy patterns; and (iii) a reduced 2 × 2 mixingproblem that encodes the degree of 𝑠/𝑝–𝑑/ 𝑓 or 𝜎–𝜋 rehybridisation. Within this perspective, anomalies arise whenever two frontier orbitals approach closely in energywhile possessing strongly contrasting radial penetration and angular character. At such points, the morepenetrating or more relativistically stabilised orbital gains control of the frontier region, enforcing anabrupt change of HOMO/LUMO identity. The 4d/5s collapse in Pd, the 6s/5d interplay in Pt/Au, the 4f/6sbalance in Eu/Yb, the 5f/6d covalency in U/Pu, and the 𝜋/𝜎 inversions in CO, NO+ and O2 emerge aschemically transparent instances of the same mechanism. Periodic trends, oxidation-state flexibility, unusual covalencies and small-molecule 𝜋/𝜎 anomaliesthus appear as local projections of a single frontier-orbital manifold. The geometry of competing orbitalchannels provides a spectrally informed, inorganic theory of frontier orbitals across the periodic table. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8261906","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":575871916,"identity":"b62d5267-95b9-4894-a840-35363b5ce788","order_by":0,"name":"Cesar A. Cohen de Mello","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCUlEQVRIiWNgGAWjYDADPnYGxgcQJg+RWtiYGZgNSNbCJkGUFv729oefKxi2ybExMx+r+FFRF80v3XtM8geDTb68A3YtEmfOGEueYbhtzMbMlnaz58zh3JlzzqVJ8zCkWW48gF2LgUQOg2QDw+3ENmYes9uMbQdyN9zIMZNmYDhsYNiAQ4v888c/gVrqQVqKGdvqwFqADsOjRYLBDGRLAhtQCzNjGzNYiwQPUIs8Du9LnMkxs2wwuG3YxsyWLAnxyxljax6DNAMDnCF2/PHNhorb8vzszQc/AEMst1+6x/DmjwobA3kcDoM6D8ViBhYJkIjBAXxaUN3KwPwBROO3ZRSMglEwCkYQAADk1FKIFw9dywAAAABJRU5ErkJggg==","orcid":"","institution":"Cosmo Physics Labs Organization: A Non - Profit Organization","correspondingAuthor":true,"prefix":"","firstName":"Cesar","middleName":"A. Cohen","lastName":"de Mello","suffix":""}],"badges":[],"createdAt":"2025-12-02 14:53:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8261906/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8261906/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100620774,"identity":"b86ca24e-bfd2-4560-a83d-e4e035cf7d30","added_by":"auto","created_at":"2026-01-19 18:24:51","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4419,"visible":true,"origin":"","legend":"","description":"","filename":"f7b45e6f0a7c4472bd69c3a56bb252ef.json","url":"https://assets-eu.researchsquare.com/files/rs-8261906/v1/a588e470af0145e7c5c3597d.json"},{"id":104947391,"identity":"ae65b777-12d9-40ec-bec4-d51075a3a3d0","added_by":"auto","created_at":"2026-03-19 05:41:28","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1309957,"visible":true,"origin":"","legend":"","description":"","filename":"InorganicFrontierOrbitalTheory.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8261906/v1_covered_e48abf7c-7162-4358-a7fc-5f54cf2897ba.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eInorganic Frontier Orbital Theory: How Penetration, Shielding and Relativity Reshape the Periodic Table\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8261906/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8261906/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFor many decades, persistent irregularities in atomic and molecular electronic structure have resistedany unifying explanation. Classical and modern work by Mulliken, Coulson, Roothaan, Fukui, Hoffmann,Pyykkö, Schwerdtfeger and others mapped emblematic cases: the 4d/5s inversion in Pd, the 6s/5d frontierreordering in Pt and Au, the collapse of the 4f-derived HOMO in Eu/Yb, the 5f/6d competition in U/Pu,and the 𝜋/𝜎 anomalies in CO, NO+ and O2. What remained unclear was whether these “exceptions” sharea common microscopic mechanism or are unrelated curiosities.\u003c/p\u003e\n\u003cp\u003eHere these phenomena are interpreted as manifestations of a single inorganic-chemical mechanism:competition between frontier orbitals with very different penetration, shielding and relativistic response,leading to a curvature-driven reordering of HOMO and LUMO. Instead of the full operator calculus,the analysis focuses on descriptors that can be extracted from modern relativistic electronic-structurecalculations: (i) energy branches of candidate frontier orbitals as functions of effective nuclear charge orchemical environment; (ii) their curvature and near-degeneracy patterns; and (iii) a reduced 2 × 2 mixingproblem that encodes the degree of 𝑠/𝑝–𝑑/ 𝑓 or 𝜎–𝜋 rehybridisation.\u003c/p\u003e\n\u003cp\u003eWithin this perspective, anomalies arise whenever two frontier orbitals approach closely in energywhile possessing strongly contrasting radial penetration and angular character. At such points, the morepenetrating or more relativistically stabilised orbital gains control of the frontier region, enforcing anabrupt change of HOMO/LUMO identity. The 4d/5s collapse in Pd, the 6s/5d interplay in Pt/Au, the 4f/6sbalance in Eu/Yb, the 5f/6d covalency in U/Pu, and the 𝜋/𝜎 inversions in CO, NO+ and O2 emerge aschemically transparent instances of the same mechanism.\u003c/p\u003e\n\u003cp\u003ePeriodic trends, oxidation-state flexibility, unusual covalencies and small-molecule 𝜋/𝜎 anomaliesthus appear as local projections of a single frontier-orbital manifold. The geometry of competing orbitalchannels provides a spectrally informed, inorganic theory of frontier orbitals across the periodic table.\u003c/p\u003e","manuscriptTitle":"Inorganic Frontier Orbital Theory: How Penetration, Shielding and Relativity Reshape the Periodic Table","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-19 17:42:24","doi":"10.21203/rs.3.rs-8261906/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"a929e570-2592-440a-b81a-f7ba041fcd84","owner":[],"postedDate":"January 19th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-03-19T05:40:52+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-19 17:42:24","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8261906","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8261906","identity":"rs-8261906","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}