  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. 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 $+/-$ 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's complete 3D structure from its molecular formula, moments of
inertia, and unsigned substitution coordinates of 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% on QM9 and 32% on
GEOM. 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 applicability for context-free 3D
structure determination with rotational spectroscopy.
