Molecular Symmetry Decomposition by Graph Automorphism: Fast, Symmetry-Aware RMSD, Structure Alignment, and Canonical Atom Ranks
Abstract
The root-mean-square deviation (RMSD) is among the most widely used metrics in computational chemistry. However, computing a chemically meaningful RMSD requires solving a combinatorial “atom mapping” problem: finding the RMSD-minimizing and chemically acceptable correspondence between atoms in two structures. Existing methods force a trade-off. Graph-isomorphism enumeration is exact but can be computationally prohibitive. The faster (cubic scaling with molecular size) Hungarian assignment algorithm disregards the bonding topology and can return chemically invalid mappings and spuriously low RMSD values. We eliminate this trade-off by recasting the assignment problem in the language of graph automorphism, where the bond-preserving permutations form a group whose internal structure (orbits, stabilizers, and product decompositions) mirrors molecular topology. This yields a complete classification of molecular graph symmetry into coupled core and independent peripheral symmetry (falling into exactly two graph-theoretic classes), reducing the search from a product of local symmetries to a sum. The resulting algorithm, snapRMSD, produces assignments that are bond-preserving by construction and recovers the global RMSD minimum under mild conditions satisfied by most molecular structures. We demonstrate that snapRMSD is significantly faster than existing methods, often by orders of magnitude. As a byproduct, the same machinery yields exact canonical atom ranks, a rigorous alternative to the refinement heuristics used since the 1960s.
Related
- On the effective rank of canonical polyadic decomposition of electron repulsion integrals
- Comparing Symmetry-Adapted Perturbation Theory to Intermolecular Energy Decomposition Analysis Based on Absolutely Localized Molecular Orbitals
- Methyl-TROSY NMR at 1.2 GHz: exploiting resolution and magnetic field alignment for structure and dynamics of large biomolecular assemblies
- Graph Laplacian spectral descriptors for alignment-free molecular shape and electrostatic comparison
- PDBe-SIFTS: An open-source tool for Structure Integration with Function, Taxonomy, and Sequences, featuring improved alignment, scoring scheme, and accelerated search.
BibTeX
@misc{hicks2026molecular,
title = {{Molecular Symmetry Decomposition by Graph Automorphism: Fast, Symmetry-Aware RMSD, Structure Alignment, and Canonical Atom Ranks}},
author = {Colton B. Hicks and Todd J. Martinez},
howpublished = {ChemRxiv},
year = {2026},
month = oct,
doi = {10.26434/chemrxiv.15007012/v1},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15007012/v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - Molecular Symmetry Decomposition by Graph Automorphism: Fast, Symmetry-Aware RMSD, Structure Alignment, and Canonical Atom Ranks
AU - Colton B. Hicks
AU - Todd J. Martinez
PY - 2026
DA - 2026/10/12
PB - ChemRxiv
DO - 10.26434/chemrxiv.15007012/v1
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15007012/v1
AB - The root-mean-square deviation (RMSD) is among the most widely used metrics in computational chemistry. However, computing a chemically meaningful RMSD requires solving a combinatorial “atom mapping” problem: finding the RMSD-minimizing and chemically acceptable correspondence between atoms in two structures. Existing methods force a trade-off. Graph-isomorphism enumeration is exact but can be computationally prohibitive. The faster (cubic scaling with molecular size) Hungarian assignment algorithm disregards the bonding topology and can return chemically invalid mappings and spuriously low RMSD values. We eliminate this trade-off by recasting the assignment problem in the language of graph automorphism, where the bond-preserving permutations form a group whose internal structure (orbits, stabilizers, and product decompositions) mirrors molecular topology. This yields a complete classification of molecular graph symmetry into coupled core and independent peripheral symmetry (falling into exactly two graph-theoretic classes), reducing the search from a product of local symmetries to a sum. The resulting algorithm, snapRMSD, produces assignments that are bond-preserving by construction and recovers the global RMSD minimum under mild conditions satisfied by most molecular structures. We demonstrate that snapRMSD is significantly faster than existing methods, often by orders of magnitude. As a byproduct, the same machinery yields exact canonical atom ranks, a rigorous alternative to the refinement heuristics used since the 1960s.
ER -