The pendulum is the simplest zero-order model for an isomerizing vibrational mode (one which passes through a saddle point). We utilize the classical cation/angle theory of the pendulum, for which new results re given in the appendix, to determine generic scaling laws between the quantum mechanical pendulum eigenvalue distribution and the coupling matrix elements. These scaling rules are more appropriate for isomerizing vibrational modes than are the usual harmonic oscillator scaling rules, encoded in traditional spectroscopic effective Hamiltonians, which break down catastrophically at a saddle point. As a simple example of resonant quantum dynamics in the vicinity of a saddle point, we analyze a system consisting of a pendulum model for bend/internal rotor motion, anharmonically coupled to a stretching harmonic oscillator, in qualitatie agreement with the known dynamic of HCP. The dominance of just two of the infinite number of resonance, 2:1 and 4:1, at all energies including that of the saddle point, is related to the scaling properties of the zero-order pendulaum model.
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