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This is achieved using the second-order transition density, which can be obtained by solving a set of auxiliary equations similar to time-dependent density functional theory (TD-DFT). The proposed formulation therefore leads to no increase in the formal computational complexity when compared to the corresponding ground state properties. We further outline the calculation of fully relaxed spin\u2013orbit coupling matrix elements within the GW\u2013BSE method, allowing us to include perturbative corrections for spin\u2013orbit coupling in aforementioned properties. These corrections are also extended to TD-DFT. Excited state absorption and perturbative spin\u2013orbit coupling corrections within GW\u2013BSE are evaluated for a selected set of molecular systems, yielding promising results.<\/jats:p>","DOI":"10.1063\/5.0244254","type":"journal-article","created":{"date-parts":[[2024,12,23]],"date-time":"2024-12-23T11:39:03Z","timestamp":1734953943000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Excited state properties from the Bethe\u2013Salpeter equation: State-to-state transitions and spin\u2013orbit coupling"],"prefix":"10.1063","volume":"161","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-1066-3209","authenticated-orcid":false,"given":"Paula","family":"Himmelsbach","sequence":"first","affiliation":[{"name":"Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology , Kaiserstra\u00dfe 12, 76131 Karlsruhe,","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8234-260X","authenticated-orcid":false,"given":"Christof","family":"Holzer","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology , Kaiserstra\u00dfe 12, 76131 Karlsruhe,","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2024,12,23]]},"reference":[{"key":"2025040716120030500_c1","doi-asserted-by":"publisher","first-page":"29993","DOI":"10.1039\/d3cp04467j","volume":"25","year":"2023","journal-title":"Phys. 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