What an SO(3) orientation flow can and cannot learn in molecular-crystal structure prediction
- 1Purdue University
Abstract
Rigid-body factorization is an appealing route to molecular-crystal structure prediction: freezing each molecule’s conformer reduces a crystal to a lattice, a set of fractional centroids on the torus T3, and a per-molecule orientation on SO(3). Whether the orientation field is actually learnable on real molecular crystals has not been examined. We introduce SymMC-Flow, a space-groupconditioned rigid-body flow on lattice×T3×SO(3), used as a diagnostic study on 1,127 crystals from the Cambridge Structural Database. The lattice and centroid flows learn normally, but the absolute per-molecule orientation flow remains at its predict-zero floor. We trace this to a decomposition of the orientation target, Rm = rot(gm) · Rasym, into a space-group-determined relative rotation between symmetry copies and the asymmetric unit’s gauge-arbitrary free orientation, which is not learnable from composition and packing; the relative part is. Re-gauging to cancel the free part lifts held-out non-reference orientation loss by 27% (versus ∼0% for the absolute target) and reconstructs 13.7% (mean over three seeds) of held-out multi-copy packings exactly under StructureMatcher, against 0% for the predict-floor baseline. Conditioning on a discrete space-group coset identity raises the learnable signal to roughly 48% (three-seed mean), showing the residual ceiling is set by the difficulty of inferring the symmetry operation from packing rather than by network capacity. End-to-end joint generation does not yet reconstruct packings exactly, placing this as a characterization rather than a finished generator. The result maps what a rigid-body SO(3) flow can and cannot contribute to molecular-crystal generation, and suggests such generators should condition on crystal symmetry for the relative orientation and sample, rather than regress, the remaining free pose.
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BibTeX
@misc{cai2026what,
title = {{What an SO(3) orientation flow can and cannot learn in molecular-crystal structure prediction}},
author = {Frank Cai},
howpublished = {ChemRxiv},
year = {2026},
month = sep,
doi = {10.26434/chemrxiv.15008279/v1},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15008279/v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - What an SO(3) orientation flow can and cannot learn in molecular-crystal structure prediction
AU - Frank Cai
PY - 2026
DA - 2026/09/03
PB - ChemRxiv
DO - 10.26434/chemrxiv.15008279/v1
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15008279/v1
AB - Rigid-body factorization is an appealing route to molecular-crystal structure prediction: freezing each molecule’s conformer reduces a crystal to a lattice, a set of fractional centroids on the torus T3, and a per-molecule orientation on SO(3). Whether the orientation field is actually learnable on real molecular crystals has not been examined. We introduce SymMC-Flow, a space-groupconditioned rigid-body flow on lattice×T3×SO(3), used as a diagnostic study on 1,127 crystals from the Cambridge Structural Database. The lattice and centroid flows learn normally, but the absolute per-molecule orientation flow remains at its predict-zero floor. We trace this to a decomposition of the orientation target, Rm = rot(gm) · Rasym, into a space-group-determined relative rotation between symmetry copies and the asymmetric unit’s gauge-arbitrary free orientation, which is not learnable from composition and packing; the relative part is. Re-gauging to cancel the free part lifts held-out non-reference orientation loss by 27% (versus ∼0% for the absolute target) and reconstructs 13.7% (mean over three seeds) of held-out multi-copy packings exactly under StructureMatcher, against 0% for the predict-floor baseline. Conditioning on a discrete space-group coset identity raises the learnable signal to roughly 48% (three-seed mean), showing the residual ceiling is set by the difficulty of inferring the symmetry operation from packing rather than by network capacity. End-to-end joint generation does not yet reconstruct packings exactly, placing this as a characterization rather than a finished generator. The result maps what a rigid-body SO(3) flow can and cannot contribute to molecular-crystal generation, and suggests such generators should condition on crystal symmetry for the relative orientation and sample, rather than regress, the remaining free pose.
ER -