{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T07:29:36Z","timestamp":1788334176156,"version":"build-2803163510"},"reference-count":105,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T00:00:00Z","timestamp":1746576000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["22233001"],"award-info":[{"award-number":["22233001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fundamental Research Funds for the Central Universities","award":["22233001"],"award-info":[{"award-number":["22233001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Molecules"],"abstract":"<jats:p>We present an economical approach to treat spin\u2013orbit coupling (SOC) in the state-averaged driven similarity renormalization group second-order perturbation theory (SA-DSRG-PT2). The electron correlation is first introduced by forming the SA-DSRG-PT2 dressed spin-free Hamiltonian. This Hamiltonian is then augmented with the Breit\u2013Pauli Hamiltonian and diagonalized using spin-pure reference states to obtain the SOC-corrected energy spectrum. The spin\u2013orbit mean-field approximation is also assumed to reduce the cost associated with the two-electron spin\u2013orbit integrals. The resulting method is termed BP1-SA-DSRG-PT2c, and it possesses the same computational scaling as the non-relativistic counterpart, where only the one- and two-body density cumulants are required to obtain the vertical transition energy. The accuracy of BP1-SA-DSRG-PT2c is assessed on a few atoms and small molecules, including main-group diatomic molecules, transition-metal atoms, and actinide dioxide cations. Numerical results suggest that BP1-SA-DSRG-PT2c performs comparably to other internally contracted multireference perturbation theories with SOC treated using the state interaction scheme.<\/jats:p>","DOI":"10.3390\/molecules30092082","type":"journal-article","created":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T11:46:19Z","timestamp":1746618379000},"page":"2082","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Cost-Effective Treatment of Spin\u2013Orbit Couplings in the State-Averaged Driven Similarity Renormalization Group Second-Order Perturbation Theory"],"prefix":"10.3390","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7543-0262","authenticated-orcid":false,"given":"Meng","family":"Wang","sequence":"first","affiliation":[{"name":"Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University, Beijing 100875, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0059-4174","authenticated-orcid":false,"given":"Chenyang","family":"Li","sequence":"additional","affiliation":[{"name":"Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University, Beijing 100875, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1218","DOI":"10.1063\/1.462209","article-title":"Second-order perturbation theory with a complete active space self-consistent field reference function","volume":"96","author":"Andersson","year":"1992","journal-title":"J. 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