Fast Computation of Exact Symmetry-Corrected RMSD of Conformers

This work introduces the Hierarchical Neighborhood of Atoms (HNA) partitioning, a recursive classification of chemically equivalent atoms that enables ultrafast computation of exact symmetry-corrected atomic correspondence and RMSD for molecular conformers. By decomposing the assignment problem hierarchically, the algorithm reduces the number of evaluated combinations from the product of branch possibilities to their sum, yielding reductions of up to 16 orders of magnitude for systems such as myoglobin. Topology-unaware approaches based on linear assignment produce chemically invalid atomic correspondences in over 89% of protein conformer pairs, while topology-aware graph isomorphism methods time out on 51-66% of biologically relevant molecules. In contrast, the proposed method achieves 100% topologically correct assignments without timeouts across all 1.4 million pairs tested, with millisecond-scale mean execution times (1.3-3.8 ms for the CCD data set, 2.7-16.5 ms for the BIRD data set) and 11-42× speedups over polynomial-time methods on protein conformer ensembles. Beyond RMSD, the HNA framework can be used for symmetry-consistent comparison of atomic properties, canonical atom labeling, and force field parametrization.