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Rotational spectroscopy enables structure determination by providing accurate 3D information about small organic molecules via their moments of inertia. Using these moments, Kraitchman analysis determines isotopic substitution coordinates, which are the unsigned |x|, |y|, |z| coordinates of all atoms with natural isotopic abundance, including carbon, nitrogen, and oxygen. While unsigned substitution coordinates can verify guesses of structures, the missing +\/\u2212 signs make it challenging to determine the actual structure from the substitution coordinates alone. To tackle this inverse problem, we develop Kreed (Kraitchman REflection-Equivariant Diffusion), a generative diffusion model that infers a molecule\u2019s complete 3D structure from only its molecular formula, moments of inertia, and unsigned substitution coordinates of heavy atoms. Kreed\u2019s top-1 predictions identify the correct 3D structure with near-perfect accuracy on large simulated datasets when provided with substitution coordinates of all heavy atoms with natural isotopic abundance. Accuracy decreases as fewer substitution coordinates are provided, but is retained for smaller molecules. On a test set of experimentally measured substitution coordinates gathered from the literature, Kreed predicts the correct all-atom 3D structure in 25 of 33 cases, demonstrating experimental potential for de novo 3D structure determination with rotational spectroscopy.<\/jats:p>","DOI":"10.1063\/5.0196620","type":"journal-article","created":{"date-parts":[[2024,3,28]],"date-time":"2024-03-28T09:55:34Z","timestamp":1711619734000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["Determining 3D structure from molecular formula and isotopologue rotational spectra in natural abundance with reflection-equivariant diffusion"],"prefix":"10.1063","volume":"160","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9497-3932","authenticated-orcid":false,"given":"Austin H.","family":"Cheng","sequence":"first","affiliation":[{"name":"Department of Chemistry, University of Toronto 1 , Toronto, Ontario M5S 3H6, Canada"},{"name":"Department of Computer Science, University of Toronto 2 , Toronto, Ontario M5S 2E4, Canada"},{"name":"Vector Institute for Artificial Intelligence 3 , Toronto, Ontario M5G 1M1, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1744-1446","authenticated-orcid":false,"given":"Alston","family":"Lo","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Toronto 2 , Toronto, Ontario M5S 2E4, Canada"},{"name":"Vector Institute for Artificial Intelligence 3 , Toronto, Ontario M5G 1M1, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5121-3853","authenticated-orcid":false,"given":"Santiago","family":"Miret","sequence":"additional","affiliation":[{"name":"Intel Labs, Intel Corporation 4 , Santa Clara, California 95054, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6097-7230","authenticated-orcid":false,"given":"Brooks H.","family":"Pate","sequence":"additional","affiliation":[{"name":"Department of Chemistry, University of Virginia 5 , Charlottesville, Virginia 22904, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8277-4434","authenticated-orcid":false,"given":"Al\u00e1n","family":"Aspuru-Guzik","sequence":"additional","affiliation":[{"name":"Department of Chemistry, University of Toronto 1 , Toronto, Ontario M5S 3H6, Canada"},{"name":"Department of Computer Science, University of Toronto 2 , Toronto, Ontario M5S 2E4, Canada"},{"name":"Vector Institute for Artificial Intelligence 3 , Toronto, Ontario M5G 1M1, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2024,3,28]]},"reference":[{"key":"2024032809552389900_c1","volume-title":"Microwave Molecular Spectra","year":"1984"},{"key":"2024032809552389900_c2","volume-title":"Microwave Spectroscopy","year":"2013"},{"key":"2024032809552389900_c3","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1119\/1.1933338","article-title":"Determination of molecular structure from microwave spectroscopic data","volume":"21","year":"1953","journal-title":"Am. 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