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Nuclear-spin statistical weights from Young diagrams
Eric Kubischta1,2, Ian Teixeira3
1Quantum Initiative, Florida State University, Tallahassee, Florida 32306, USA.
The Journal of Chemical Physics
|May 21, 2026
Summary
This study introduces a combinatorial method to determine nuclear-spin symmetry species and statistical weights for molecules with cyclic and dihedral symmetry. The approach simplifies analysis for rovibrational spectroscopy and molecular symmetry.
Area of Science:
- Molecular Spectroscopy
- Quantum Chemistry
- Symmetry Analysis
Background:
- Nuclear-spin statistical weights and selection rules are crucial in molecular spectroscopy.
- Existing methods often rely on proper subgroups of the symmetric group for rigid molecules.
Purpose of the Study:
- To extend the Schur-Weyl framework to cyclic and dihedral permutation subgroups relevant to molecular geometries.
- To develop a fully combinatorial method for determining nuclear-spin symmetry species and statistical weights.
Main Methods:
- Combining Schur-Weyl decomposition of n-spin Hilbert space with major-index branching rules.
- Utilizing standard Young tableaux and their major indices to determine cyclic characters and dihedral symmetry.
Main Results:
- A combinatorial method for nuclear-spin symmetry analysis is established.
- The method applies uniformly to various molecular point groups, including XeOF4, SF4, benzene, and deuterated benzene.
- Established statistical weights are reproduced, providing transparent spin-resolved structures.
Conclusions:
- The developed approach offers a practical and automatable framework for nuclear-spin symmetry analysis in polyatomic molecules.
- It provides essential symmetry information for rovibrational line-intensity modeling.
- Eliminates the need for projection operators or case-specific constructions.
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