Nuclear-spin statistical weights from Young diagrams

Abstract

Nuclear-spin statistical weights and selection rules in molecular spectroscopy are governed by the permutation symmetry of identical nuclei, although rigid and semi-rigid molecules typically realize only a proper subgroup of the full symmetric group. Building on the Schur–Weyl framework of Schmiedt, Jensen, and Schlemmer, we extend this approach to molecular geometries whose rovibrational motion realizes cyclic and dihedral permutation subgroups. By combining the Schur–Weyl decomposition of the n-spin Hilbert space with the Kraśkiewicz–Weyman–Adin–Roichman major-index branching rule for the restriction SN ↓ Cm, we obtain a fully combinatorial method for determining nuclear-spin symmetry species and statistical weights. Each standard Young tableau corresponds to a definite cyclic character determined by its major index, with reflections yielding the associated dihedral symmetry, so that nuclear-spin species follow directly from tableau data without projection operators or case-specific constructions. Applications to XeOF4, SF4, benzene, deuterated benzene, and the tropylium cation demonstrate that the method applies uniformly to realistic molecular point groups and reproduces established statistical weights while providing a transparent spin-resolved structure. The approach supplies the symmetry information required for rovibrational line-intensity modeling and provides a practical, automatable framework for nuclear-spin symmetry analysis in polyatomic molecules.