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These spin-adapted cumulants enable rigorous treatment of both S\u0302z and S\u03022 symmetries in quantum systems, providing a foundation for spin-resolved electronic structure methods. We show that complete spin adaptation\u2014referred to as completeS-representability\u2014can be enforced by constraining the variances of S\u0302z and S\u03022, which require the 2-RDM and 4-RDM, respectively. Importantly, the cumulants of RDMs scale linearly with system size\u2014size-extensive\u2014making them a natural object for incorporating spin symmetries in scalable electronic structure theories. The developed formalism is applicable to density-based methods, one-particle RDM functional theories, and two-particle RDM methods. We further extend the approach to spin\u2013orbit-coupled systems via total angular momentum adaptation. Beyond spin, the framework enables the adaptation of RDM theories to additional symmetries through the construction of suitable irreducible tensor operators.<\/jats:p>","DOI":"10.1063\/5.0282007","type":"journal-article","created":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T16:53:44Z","timestamp":1753289624000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Spin adaptation of the cumulant expansions of reduced density matrices"],"prefix":"10.1063","volume":"163","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4530-4900","authenticated-orcid":false,"given":"Julia","family":"Liebert","sequence":"first","affiliation":[{"name":"Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen 1 , Theresienstrasse 37, 80333 M\u00fcnchen,","place":["Germany"]},{"name":"Munich Center for Quantum Science and Technology (MCQST) 2 , Schellingstrasse 4, 80799 M\u00fcnchen,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6781-4111","authenticated-orcid":false,"given":"Christian","family":"Schilling","sequence":"additional","affiliation":[{"name":"Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen 1 , Theresienstrasse 37, 80333 M\u00fcnchen,","place":["Germany"]},{"name":"Munich Center for Quantum Science and Technology (MCQST) 2 , Schellingstrasse 4, 80799 M\u00fcnchen,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9938-3886","authenticated-orcid":false,"given":"David A.","family":"Mazziotti","sequence":"additional","affiliation":[{"name":"Department of Chemistry and The James Franck Institute, The University of Chicago 3 , Chicago, Illinois 60637,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2025,7,23]]},"reference":[{"volume-title":"Reduced Density Matrices in Quantum Chemistry","year":"2012","key":"2025072312533745300_c1"},{"volume-title":"Many-Electron Densities and Reduced Density Matrices","year":"2000","key":"2025072312533745300_c2"},{"key":"2025072312533745300_c3","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1021\/ar050029d","article-title":"Quantum chemistry without wave functions: Two-electron reduced density matrices","volume":"39","year":"2006","journal-title":"Acc. 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