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 $S_N \downarrow C_m$, 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 $XeOF_4$, $SF_4$, 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.