{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T20:42:52Z","timestamp":1775076172755,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:00:00Z","timestamp":1760140800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The second virial coefficient (SVC) of the Lennard-Jones fluid is a cornerstone of molecular theory, yet its calculation has traditionally relied on the complex integration of the pair potential. This work introduces a fundamentally different approach by reformulating the problem in terms of ordinary differential equations (ODEs). For the classical component of the SVC, we generalize the confluent hypergeometric and Weber\u2013Hermite equations. For the first quantum correction, we present entirely new ODEs and their corresponding exact-analytical solutions. The most striking result of this framework is the discovery that these ODEs can be transformed into Schr\u00f6dinger-like equations. The classical term corresponds to a harmonic oscillator, while the quantum correction includes additional inverse-power potential terms. This formulation not only provides a versatile method for expressing the virial coefficient through a linear combination of functions (including Kummer, Weber, and Whittaker functions) but also reveals a profound and previously unknown mathematical structure underlying a classical thermodynamic property.<\/jats:p>","DOI":"10.3390\/e27101059","type":"journal-article","created":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T08:10:31Z","timestamp":1760343031000},"page":"1059","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Exact ODE Framework for Classical and Quantum Corrections for the Lennard-Jones Second Virial Coefficient"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-9175-521X","authenticated-orcid":false,"given":"Zhe","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Resources and Materials, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4692-8800","authenticated-orcid":false,"given":"Alfredo","family":"Gonz\u00e1lez-Calder\u00f3n","sequence":"additional","affiliation":[{"name":"SECIHTI\u2014Department of Geomatic and Hydraulic Engineering, University of Guanajuato, Guanajuato 36000, Gto, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1961-151X","authenticated-orcid":false,"given":"Jorge Adri\u00e1n","family":"Perera-Burgos","sequence":"additional","affiliation":[{"name":"SECIHTI\u2014Department of Mining, Metallurgy and Geology Engineering, University of Guanajuato, Guanajuato 36020, Gto, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0864-1262","authenticated-orcid":false,"given":"Antonio","family":"Estrada","sequence":"additional","affiliation":[{"name":"SECIHTI\u2014Centro de Ingenier\u00eda y Desarrollo Industrial, Quer\u00e9taro 76125, Qro, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5433-0986","authenticated-orcid":false,"given":"Horacio","family":"Hern\u00e1ndez-Anguiano","sequence":"additional","affiliation":[{"name":"Department of Geomatic and Hydraulic Engineering, University of Guanajuato, Guanajuato 36000, Gto, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7132-4916","authenticated-orcid":false,"given":"Celia","family":"Mart\u00ednez-L\u00e1zaro","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Universidad Aut\u00f3noma de Guerrero, Chilpancingo 39087, Gro, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6988-6776","authenticated-orcid":false,"given":"Yanmei","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mining, Metallurgy and Geology Engineering, University of Guanajuato, Guanajuato 36020, Gto, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,11]]},"reference":[{"key":"ref_1","first-page":"463","article-title":"On the determination of molecular fields. II. From the equation of state of a gas","volume":"106","author":"Jones","year":"1924","journal-title":"R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2400115","DOI":"10.1002\/andp.202400115","article-title":"On the History of the Lennard\u2013Jones Potential","volume":"536","author":"Lenhard","year":"2024","journal-title":"Ann. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"A405","DOI":"10.1103\/PhysRev.136.A405","article-title":"Correlations in the Motion of Atoms in Liquid Argon","volume":"136","author":"Rahman","year":"1964","journal-title":"Phys. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3379","DOI":"10.1021\/acs.jctc.4c00135","article-title":"100 Years of the Lennard-Jones Potential","volume":"20","author":"Schwerdtfeger","year":"2024","journal-title":"J. Chem. Theory Comput."},{"key":"ref_5","first-page":"441","article-title":"On the determination of molecular fields. I. From the variation of the viscosity of a gas with temperature","volume":"106","author":"Jones","year":"1924","journal-title":"R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.molstruc.2014.04.006","article-title":"Analytical treatment of second virial coefficient over Lennard-Jones (2n \u2212 n) potential and its application to molecular systems","volume":"1068","author":"Mamedov","year":"2014","journal-title":"J. Mol. Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1103\/RevModPhys.25.831","article-title":"Virial coefficients and models of molecules in gases","volume":"25","author":"Kihara","year":"1953","journal-title":"Rev. Mod. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1320","DOI":"10.1063\/1.1748040","article-title":"Low Temperature Second Virial Coefficients for a 6\u201312 Potential","volume":"19","author":"Epstein","year":"1951","journal-title":"J. Chem. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1670","DOI":"10.1063\/1.1700265","article-title":"Low Temperature Second Virial Coefficients for a Lennard-Jones Gas","volume":"20","author":"Epstein","year":"1952","journal-title":"J. Chem. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1063\/1.1744287","article-title":"On the Virial Coefficients for a Lennard-Jones Gas","volume":"28","author":"Nosanow","year":"1958","journal-title":"J. Chem. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1063\/1.1761851","article-title":"Low-Temperature Quantum Corrections to the Second Virial Coefficient","volume":"9","author":"Michels","year":"1966","journal-title":"Phys. Fluids"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"084505","DOI":"10.1063\/1.4961653","article-title":"The second virial coefficient and critical point behavior of the Mie Potential","volume":"145","author":"Heyes","year":"2016","journal-title":"J. Chem. Phys."},{"key":"ref_13","unstructured":"Eu, B.C. (2009). Exact analytic second virial coefficient for the Lennard-Jones fluid. arXiv."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1088\/0305-4470\/13\/1\/037","article-title":"A closed form for the second virial coefficient of the Lennard-Jones gas","volume":"13","author":"Garrett","year":"1980","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Santos, A. (2013). Playing with marbles: Structural and thermodynamic properties of hard-sphere systems. arXiv.","DOI":"10.31338\/uw.9788323517399.pp.203-298"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Santos, A. (2016). A Concise Course on the Theory of Classical Liquids, Springer.","DOI":"10.1007\/978-3-319-29668-5"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/S0375-9601(02)00814-9","article-title":"Second virial coefficient for a Lennard-Jones (2n \u2212 n) system in d dimensions and confined to a nanotube surface","volume":"300","author":"Glasser","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"034305","DOI":"10.1063\/1.4905663","article-title":"Second virial coefficient of a generalized Lennard-Jones potential","volume":"142","year":"2015","journal-title":"J. Chem. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/S0378-4371(00)00362-9","article-title":"Second virial coefficient for the Lennard-Jones potential","volume":"290","author":"Vargas","year":"2001","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1103\/RevModPhys.27.412","article-title":"Virial Coefficients and Models of Molecules in Gases. B","volume":"27","author":"Kihara","year":"1955","journal-title":"Rev. Mod. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"57","DOI":"10.6028\/jres.075A.007","article-title":"Quantum Corrections to the Second Virial Coefficient for the Lennard-Jones (m-6) potential","volume":"75A","author":"Boyd","year":"1971","journal-title":"J. Res. Natl. Bur. Stand."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"012010","DOI":"10.1088\/1742-6596\/766\/1\/012010","article-title":"Evaluation of Quantum Corrections to Second Virial Coefficient with Lennard-Jones (12-6) Potential","volume":"766","author":"Mamedov","year":"2016","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"074504","DOI":"10.1063\/1.5041320","article-title":"Second virial coefficient properties of the n\u2013m Lennard-Jones\/Mie potential","volume":"149","author":"Sadus","year":"2018","journal-title":"J. Chem. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7757","DOI":"10.1021\/acs.jpcb.8b05725","article-title":"Intermolecular potential-based equations of state from molecular simulation and second virial coefficient properties","volume":"122","author":"Sadus","year":"2018","journal-title":"J. Phys. Chem. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"084506","DOI":"10.1063\/5.0133412","article-title":"Equation of state for the Mie (\u03bbr, 6) fluid with a repulsive exponent from 11 to 13","volume":"158","author":"Pohl","year":"2023","journal-title":"J. Chem. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"115217","DOI":"10.1063\/1.5123613","article-title":"Critical temperatures of real fluids from the extended law of corresponding states","volume":"9","author":"Luis","year":"2019","journal-title":"AIP Adv."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"164508","DOI":"10.1063\/5.0007583","article-title":"Effective Hardness of Interaction from Thermodynamics and Viscosity in Dilute Gases","volume":"152","author":"Bell","year":"2020","journal-title":"J. Chem. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Woodcock, L.V. (2018). Thermodynamic fluid equations-of-state. Entropy, 20.","DOI":"10.3390\/e20010022"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"469","DOI":"10.3390\/e22040469","article-title":"Equation of state of four-and five-dimensional hard-hypersphere mixtures","volume":"22","author":"Santos","year":"2020","journal-title":"Entropy"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1534","DOI":"10.3390\/nano13091534","article-title":"Influence of Lennard-Jones parameters in the temperature dependence of real gases diffusion through nanochannels","volume":"13","year":"2023","journal-title":"Nanomaterials"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"042810","DOI":"10.1103\/PhysRevA.102.042810","article-title":"Second virial coefficients for helium-4 and helium-3 from accurate relativistic interaction potential","volume":"102","author":"Czachorowski","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2212","DOI":"10.1021\/acs.jced.3c00300","article-title":"Cross Second Virial Coefficients of the H2O\u2013H2 and H2S\u2013H2 Systems from First-Principles","volume":"68","author":"Hellmann","year":"2023","journal-title":"J. Chem. Eng. Data"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"052111","DOI":"10.1063\/5.0147306","article-title":"Influence of molecular anisotropy and quadrupolar moment on evaporation","volume":"35","author":"Homes","year":"2023","journal-title":"Phys. Fluids"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"119419","DOI":"10.1016\/j.molliq.2022.119419","article-title":"Molecular models for O2 and N2 from the second virial coefficient","volume":"360","year":"2022","journal-title":"J. Mol. Liq."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.jct.2019.06.015","article-title":"The comparative analytical evaluation of quantum corrections to the second virial coefficient with Morse potential and its applications to real systems","volume":"138","author":"Cacan","year":"2019","journal-title":"J. Chem. Thermodyn."},{"key":"ref_36","unstructured":"NIST Digital Library of Mathematical Functions (2025, August 29). Chapter 12: Parabolic Cylinder Functions, Available online: https:\/\/dlmf.nist.gov\/12."},{"key":"ref_37","unstructured":"NIST Digital Library of Mathematical Functions (2025, August 29). Chapter 13: Confluent Hypergeometric (Kummer) Functions, Available online: https:\/\/dlmf.nist.gov\/13."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s11075-023-01515-y","article-title":"Computation of the confluent hypergeometric function U(a,b,x) and its derivative for positive arguments","volume":"94","author":"Gil","year":"2023","journal-title":"Numer. Algorithms"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/S0031-8914(36)80346-2","article-title":"The quantum theory of the non-ideal gas I. Deviations from the classical theory","volume":"3","author":"Uhlenbeck","year":"1936","journal-title":"Physica"},{"key":"ref_40","unstructured":"Abramowitz, M., and Stegun, I.A. (1964). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover Publications."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Zaitsev, V.F., and Polyanin, A.D. (2002). Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press.","DOI":"10.1201\/9781420035339"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1119\/1.1526131","article-title":"The one-dimensional harmonic oscillator in the presence of a dipole-like interaction","volume":"71","author":"Palma","year":"2003","journal-title":"Am. J. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/10\/1059\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T06:15:36Z","timestamp":1760508936000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/10\/1059"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,11]]},"references-count":42,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["e27101059"],"URL":"https:\/\/doi.org\/10.3390\/e27101059","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,11]]}}}