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The zero-range sticky limit is recovered by letting the well width vanish while keeping the stickiness parameter finite. While Wertheim\u2019s first-order thermodynamic perturbation theory becomes exact in this sticky limit, it fails for finite-range site\u2013site interactions. Because the present model is mathematically equivalent to an exactly solvable one-dimensional nearest-neighbor fluid, we use the exact solution to reformulate the thermodynamics in terms of association-theory variables, including the fraction of unbonded sites and a generalized law of mass action. Finite-range SW sites produce a richer structural behavior than sticky sites, including monotonic and oscillatory asymptotic decay of the pair correlation function, separated by the Fisher\u2013Widom line. In the monotonic regime, the correlation length exhibits an absolute maximum defining the Widom line, while in the oscillatory regime, it may display a local maximum and minimum, whose locus defines the \u201cExtrema of the Correlation length under Oscillatory decay\u201d line. These features disappear in the sticky limit, where the system remains entirely in the oscillatory regime. We also show that the high-pressure behavior of the correlation length changes from \u03be \u223c p2 for finite-range SW sites to \u03be \u223c p3 in the sticky limit.<\/jats:p>","DOI":"10.1063\/5.0344388","type":"journal-article","created":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T12:20:33Z","timestamp":1786450833000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Thermodynamic and structural behavior of one-dimensional divalent patchy hard rods: Wertheim\u2019s first-order thermodynamic perturbation theory vs exact results"],"prefix":"10.1063","volume":"165","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9188-8487","authenticated-orcid":false,"given":"Ana M.","family":"Montero","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidad de Extremadura 1 , E-06006 Badajoz,","place":["Spain"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9564-5180","authenticated-orcid":false,"given":"Andr\u00e9s","family":"Santos","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidad de Extremadura 1 , E-06006 Badajoz,","place":["Spain"]},{"name":"Instituto de Computaci\u00f3n Cient\u00edfica Avanzada (ICCAEx), Universidad de Extremadura 2 , E-06006 Badajoz,","place":["Spain"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5824-5420","authenticated-orcid":false,"given":"P\u00e9ter","family":"Gurin","sequence":"additional","affiliation":[{"name":"Physics Department, Centre for Natural Sciences, University of Pannonia 3 , P.O. Box 158, Veszpr\u00e9m H-8201,","place":["Hungary"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5286-0289","authenticated-orcid":false,"given":"Szabolcs","family":"Varga","sequence":"additional","affiliation":[{"name":"Physics Department, Centre for Natural Sciences, University of Pannonia 3 , P.O. Box 158, Veszpr\u00e9m H-8201,","place":["Hungary"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,8,11]]},"reference":[{"key":"2026081107475609100_c1","doi-asserted-by":"publisher","first-page":"6397","DOI":"10.1039\/c0cp02296a","article-title":"Patchy colloids: State of the art and perspectives","volume":"13","year":"2011","journal-title":"Phys. Chem. Chem. 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