Exactly solvable 1D model explains the low-energy vibrational level structure of protonated methane

A new one-dimensional model is proposed for the low-energy vibrational quantum dynamics of CH5+ based on the motion of an effective particle confined to a 60-vertex graph ${Gamma}_{60}$ with a single edge length parameter. Within this model, the quantum states of CH5+ are obtained in analytic form and are related to combinatorial properties of ${Gamma}_{60}$. The bipartite structure of ${Gamma}_{60}$ gives a simple explanation for curious symmetries observed in numerically exact variational calculations on CH5+.