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PreprintCC BYChemRxiv

Symmetry Numbers: A Flexible Approach for Molecules and Transition States

  • Maxim Papusha1,2
  • Andreas V. Copan3,4,5
  • Brandon Rotavera3,4,5
  • Kai Leonhard1,2
5 affiliations
  1. 1Institute of Technical Thermodynamics
  2. 2RWTH Aachen University
  3. 3College of Engineering
  4. 4University of Georgia
  5. 5Department of Chemistry
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Abstract

Symmetry numbers are essential corrections in thermochemistry and chemical kinetics, but their reliable assignment remains difficult in automated workflows, especially for stereochemical, fluxional, cyclic, and transition-state structures. We present a stereochemistry-aware automorphism (SAA) approach for computing symmetry numbers using the StereoMolGraph (SMG) framework. In this representation, local stereodescriptors encode rigid local geometries directly in the molecular graph, so the symmetry number is obtained as the order of the automorphism group that preserves both connectivity and encoded stereochemistry. This avoids the need for a post hoc “label stereocenter” correction and provides a direct graph-theoretical analogue of permutation-based symmetry number definitions. Model dependence is handled explicitly by choosing which molecular features are treated as fixed or fluxional, enabling consistent treatment of rigidity, free rotation, umbrella inversion, and pseudorotation assumptions. Beyond overall molecular and transition-state symmetry numbers, the same framework yields external and internal symmetry numbers for hindered-rotor treatments, including an effective external symmetry number for cases where the overall symmetry number is non-separable. Benchmark calculations reproduce the expected symmetry numbers for all supported benchmark molecules and resolve cases where existing rule-based or corrected-connectivity approaches fail, including stereochemically distinct structures with identical connectivity. The SAA approach is highly robust, making it suitable for automated thermochemical and kinetic modeling of molecules and reactions.

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Cite

BibTeXRIS
BibTeX
@misc{papusha2026symmetry,
  title = {{Symmetry Numbers: A Flexible Approach for Molecules and Transition States}},
  author = {Maxim Papusha and Andreas V. Copan and Brandon Rotavera and Kai Leonhard},
  howpublished = {ChemRxiv},
  year = {2026},
  month = oct,
  doi = {10.26434/chemrxiv.15006158/v2},
  url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15006158/v2},
  note = {CC BY},
}
RIS
TY  - UNPB
TI  - Symmetry Numbers: A Flexible Approach for Molecules and Transition States
AU  - Maxim Papusha
AU  - Andreas V. Copan
AU  - Brandon Rotavera
AU  - Kai Leonhard
PY  - 2026
DA  - 2026/10/02
PB  - ChemRxiv
DO  - 10.26434/chemrxiv.15006158/v2
UR  - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15006158/v2
AB  - Symmetry numbers are essential corrections in thermochemistry and chemical kinetics, but their reliable assignment remains difficult in automated workflows, especially for stereochemical, fluxional, cyclic, and transition-state structures. We present a stereochemistry-aware automorphism (SAA) approach for computing symmetry numbers using the StereoMolGraph (SMG) framework. In this representation, local stereodescriptors encode rigid local geometries directly in the molecular graph, so the symmetry number is obtained as the order of the automorphism group that preserves both connectivity and encoded stereochemistry. This avoids the need for a post hoc “label stereocenter” correction and provides a direct graph-theoretical analogue of permutation-based symmetry number definitions. Model dependence is handled explicitly by choosing which molecular features are treated as fixed or fluxional, enabling consistent treatment of rigidity, free rotation, umbrella inversion, and pseudorotation assumptions. Beyond overall molecular and transition-state symmetry numbers, the same framework yields external and internal symmetry numbers for hindered-rotor treatments, including an effective external symmetry number for cases where the overall symmetry number is non-separable. Benchmark calculations reproduce the expected symmetry numbers for all supported benchmark molecules and resolve cases where existing rule-based or corrected-connectivity approaches fail, including stereochemically distinct structures with identical connectivity. The SAA approach is highly robust, making it suitable for automated thermochemical and kinetic modeling of molecules and reactions.
ER  -