{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T19:42:44Z","timestamp":1792784564210,"version":"4.5.0"},"reference-count":51,"publisher":"American Chemical Society (ACS)","issue":"19","license":[{"start":{"date-parts":[[2026,9,17]],"date-time":"2026-09-17T00:00:00Z","timestamp":1789603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000171","name":"Division of Human Resource Development","doi-asserted-by":"publisher","award":["HRD-1547380"],"award-info":[{"award-number":["HRD-1547380"]}],"id":[{"id":"10.13039\/100000171","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Physics Department, Brooklyn College","id":[{"id":"https:\/\/ror.org\/019k4jq75","id-type":"ROR","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,10,13]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The potential energy landscape (PEL) formalism is a powerful tool within statistical mechanics to study the thermodynamic behavior of classical and quantum liquids as well as non-annealing glasses. In the quantum case, this theoretical framework is based on the path-integral formulation of statistical mechanics where the liquid\/non-annealing glass is mapped onto a classical system of ring-polymers. In previous studies, and motivated by path-integral (PI) computer simulations, the PEL formalism was developed assuming that the ring-polymers were discrete, i.e., each ring-polymer was composed of a finite number of beads, nb. In this work, we revisit the PEL formalism for the case where the ring-polymers of the system are continuous, i.e., nb \u2192 \u221e (as required in the PI formulation of statistical mechanics). We find that in this limit, the PEL of the quantum liquid is unique and well-defined. The main PEL properties of the system, including the average inherent structure energy EIS, configurational entropy SIS, and vibrational Helmholtz free energy Fvib, are also well-defined (i.e., they converge). Notably, in the harmonic approximation of the PEL, Fvib can be expressed analytically; it is identical to the free energy of a multidimensional quantum harmonic oscillator with the corresponding frequencies given by the normal mode frequencies of the classical liquid counterpart (accessible from classical computer simulations). When applied to non-annealing glasses and crystals, this new version of the PEL formalism reduces to the so-called quasi-harmonic approximation, commonly used in the study of solids. Overall, the PEL formalism for nb \u2192 \u221e provides an alternative approach to its previous version (finite nb); it is also more elegant and intuitive, and simpler to apply. For a Gaussian and harmonic PEL, this formalism (nb \u2192 \u221e) allows one to obtain the Helmholtz free energy of the quantum liquid using classical computer simulations, avoiding the more expensive PI computer simulations.<\/jats:p>","DOI":"10.1021\/acs.jctc.6c01077","type":"journal-article","created":{"date-parts":[[2026,9,17]],"date-time":"2026-09-17T18:02:19Z","timestamp":1789668139000},"page":"9963-9975","source":"Crossref","is-referenced-by-count":0,"title":["The Potential Energy\nLandscape Formalism for Quantum\nLiquids in the Continuous Ring-Polymer Limit"],"prefix":"10.1021","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0142-1507","authenticated-orcid":true,"given":"Yang","family":"Zhou","sequence":"first","affiliation":[{"name":"The Graduate Center of the City University of New York , , , ,","place":["New York, New York, United States, 10016"]},{"name":"Brooklyn College of the City University of New York , , , ,","place":["Brooklyn, New York, United States, 11210"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1660-547X","authenticated-orcid":true,"given":"Gustavo E.","family":"Lopez","sequence":"additional","affiliation":[{"name":"The Graduate Center of the City University of New York , , , ,","place":["New York, New York, United States, 10016"]},{"name":"Lehman College of the City University of New York , , , ,","place":["Bronx, New York, United States, 10468"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1149-0693","authenticated-orcid":true,"given":"Nicolas","family":"Giovambattista","sequence":"additional","affiliation":[{"name":"The Graduate Center of the City University of New York , , , ,","place":["New York, New York, United States, 10016"]},{"name":"Brooklyn College of the City University of New York , , , ,","place":["Brooklyn, New York, United States, 11210"]},{"name":"The Graduate Center of the City University of New York , , , ,","place":["New York, New York, United States, 10016"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"316","published-online":{"date-parts":[[2026,9,17]]},"reference":[{"key":"2026101310080189500_cit1","doi-asserted-by":"publisher","first-page":"978","DOI":"10.1103\/PhysRevA.25.978","article-title":"Hidden Structure in Liquids","volume":"25","author":"Stillinger","year":"1982","journal-title":"Phys. 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