Hierarchical Fractional Quantization in Molecular Excited States:A Unified Framework from Symmetry Breaking

This work establishes hierarchical fractional quantization (HFQ) as a universal framework for decoding the energy-level structure of quantum systems through successive symmetry breaking. We demonstrate that the total energy of an excited-state multiplet can be decomposed into a sum of contributions from distinct symmetry layers,each characterized by rational energy coefficients derived from group representation theory. The framework is first rigorously demonstrated in the hydrogen atom, where its complete spectrum is reformulated as a hierarchical expansion stemming from the breaking of SO(4) symmetry.The central theorem extends this to molecular systems: for a parent multiplet transforming under a symmetry 𝐺(0), a perturbation with lower symmetry 𝐺(1) induces an average energy shift that is a rational multiple of a fundamental energy scale. Crucially, the rational coefficient is determined a priori by the branching ratios of the group representations, not fitted a posteriori.We validate the theory through a non-circular protocol employing high-precision quantum chemical calculations on four benchmark systems: benzene (𝐷6ℎ), water (𝐶2𝑣), C60 (𝐼ℎ), and an iron(II) complex [Fe(H2O)6]2+(𝑂ℎ).The predicted rational coefficients agree with values extracted from computed energy shifts within statistical uncertainty, passing rigorous 𝜒2 and Bayesian hypothesis tests. For water, the observed Jahn–Teller energy ratio 𝐸JT(D2O)/𝐸JT(H2O) ≈ 4/3 is explained through a symmetry-dimension renormalization model where nuclear statistics impose Hilbert space dimension ratios 𝑑𝐻 : 𝑑𝐷 = 4 : 3.This work unifies atomic and molecular spectroscopy under the principle of symmetry-governed fractional energy quantization, offering a predictive tool for spectral interpretation, molecular quantum materials design, and qubit engineering.