Bound States of One-Dimensional Helium on the Half-Line

Bound States of One-Dimensional Helium on the Half-Line | 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Research Article Bound States of One-Dimensional Helium on the Half-Line Dmitry Konovalov This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9602948/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 We report a non-relativistic solution of the one-dimensional helium atom confined to the half-line r > 0. Using a variational product basis of Laguerre functions with dimension up to N = 45, we obtained convergence of the first four bound-state energies to six significant digits: -2.10839, -2.05037, -2.02901, and -2.01883 atomic units. Despite the theoretical spin and exchange-symmetry degeneracy, we observed a numerical anomaly in which the symmetric (singlet) states converge drastically more slowly than the antisymmetric (triplet) states, potentially requiring basis sizes exceeding N = 1000 to reach the same triplet energies. With N = 45, the symmetric energies converged only to the first four significant digits. The energy spectrum and wave function plots revealed a clear Rydberg-like ladder of two-electron configurations, characterized by a (1s, ns) nodal structure. Atomic and Molecular Physics 1-dimensional systems Helium Electronic structure Full Text Additional Declarations The authors declare no competing interests. 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-9602948","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":633757069,"identity":"978ff5e8-db95-4ef3-9520-ba04df9fb98c","order_by":0,"name":"Dmitry Konovalov","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0002-3320-8938","institution":"James Cook University","correspondingAuthor":true,"prefix":"","firstName":"Dmitry","middleName":"","lastName":"Konovalov","suffix":""}],"badges":[],"createdAt":"2026-05-04 00:08:12","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9602948/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9602948/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108976231,"identity":"95f9a8e3-2e6e-4ae6-aa1c-8bffcd7c3f08","added_by":"auto","created_at":"2026-05-11 11:00:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3918317,"visible":true,"origin":"","legend":"","description":"","filename":"mainv250802preprint260504.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9602948/v1_covered_b6893698-7642-4afb-b2be-1afe05f18e58.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eBound States of One-Dimensional Helium on the Half-Line\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"James Cook University","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":"1-dimensional systems, Helium, Electronic structure","lastPublishedDoi":"10.21203/rs.3.rs-9602948/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9602948/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe report a non-relativistic solution of the one-dimensional helium atom confined to the half-line r \u0026gt; 0. Using a variational product basis of Laguerre functions with dimension up to N = 45, we obtained convergence of the first four bound-state energies to six significant digits: -2.10839, -2.05037, -2.02901, and -2.01883 atomic units. Despite the theoretical spin and exchange-symmetry degeneracy, we observed a numerical anomaly in which the symmetric (singlet) states converge drastically more slowly than the antisymmetric (triplet) states, potentially requiring basis sizes exceeding N = 1000 to reach the same triplet energies. With N = 45, the symmetric energies converged only to the first four significant digits. The energy spectrum and wave function plots revealed a clear Rydberg-like ladder of two-electron configurations, characterized by a (1s, ns) nodal structure.\u003c/p\u003e","manuscriptTitle":"Bound States of One-Dimensional Helium on the Half-Line","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-05 09:34:38","doi":"10.21203/rs.3.rs-9602948/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":"0cf47c2d-c84f-4cdf-9620-1ce3a3e3ff23","owner":[],"postedDate":"May 5th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":67448225,"name":"Atomic and Molecular Physics"}],"tags":[],"updatedAt":"2026-05-05T09:34:38+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-05 09:34:38","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9602948","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9602948","identity":"rs-9602948","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}