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This lack of conservation arises not so much from the hopping direction (which is easily adjusted) but more generally from propagating adiabatic dynamics along surfaces that are not time reversible. We show that one solution to this problem is to run along eigenvalues of phase-space electronic Hamiltonians H(R, P) (i.e., electronic Hamiltonians that depend on both nuclear position and momentum) with an electronic\u2013nuclear coupling \u0393\u2009\u00b7\u2009P [see Eq. (25)], and we delineate the conditions that must be satisfied by the operator \u0393. The present results should be extremely useful as far as developing new semiclassical approaches that can treat systems where the nuclear, electronic orbital, and electronic spin degrees of freedom altogether are all coupled together, hopefully including systems displaying the chiral-induced spin selectivity effect.<\/jats:p>","DOI":"10.1063\/5.0179599","type":"journal-article","created":{"date-parts":[[2024,1,11]],"date-time":"2024-01-11T11:24:14Z","timestamp":1704972254000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":14,"title":["Linear and angular momentum conservation in surface hopping methods"],"prefix":"10.1063","volume":"160","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9140-2782","authenticated-orcid":false,"given":"Yanze","family":"Wu","sequence":"first","affiliation":[{"name":"Department of Chemistry, University of Pennsylvania 1 , Philadelphia, Pennsylvania 19104, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9681-3775","authenticated-orcid":false,"given":"Jonathan","family":"Rawlinson","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Manchester 2 , Manchester M13 9PL, United Kingdom"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4438-0952","authenticated-orcid":false,"given":"Robert G.","family":"Littlejohn","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California 3 , Berkeley, California 94720, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2170-2020","authenticated-orcid":false,"given":"Joseph E.","family":"Subotnik","sequence":"additional","affiliation":[{"name":"Department of Chemistry, University of Pennsylvania 1 , Philadelphia, Pennsylvania 19104, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2024,1,11]]},"reference":[{"key":"2024011111240789500_c1","doi-asserted-by":"publisher","first-page":"2653","DOI":"10.1038\/srep02653","article-title":"Ultrafast intersystem-crossing in platinum containing \u03c0-conjugated polymers with tunable spin\u2013orbit coupling","volume":"3","year":"2013","journal-title":"Sci. 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