Reflection-Equivariant Diffusion for 3D Structure Determination from Isotopologue Rotational Spectra in Natural Abundance

Structure determination is necessary to identify unknown organic molecules, such as those in natural products, forensic samples, the interstellar medium, and laboratory syntheses. Rotational spectroscopy enables structure determination by providing accurate 3D information about small organic molecules via their moments of inertia. Kraitchman analysis uses these moments to determine 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 $+/-$ signs make it a hard computational problem to determine the actual structure from just the substitution coordinates. To tackle this inverse problem, we develop KREED (Kraitchman REflection-Equivariant Diffusion), a diffusion generative model which infers a molecule's all-atom 3D structure conditioned on the molecular formula, moments of inertia, and unsigned substitution coordinates of carbon and other heavy atoms. KREED's top-1 predictions identify the correct 3D structure with $>$98\% accuracy on the QM9 and GEOM datasets when provided with substitution coordinates of all heavy atoms with natural isotopic abundance. When substitution coordinates are restricted to only a subset of carbons, accuracy is retained at 91\% for QM9 and 32\% for GEOM. On a test set of experimentally measured substitution coordinates gathered from the literature, KREED can identify the correct all-atom 3D structure in 25 of 33 cases, demonstrating experimental applicability for context-free 3D structure determination with rotational spectroscopy.