Quantitative analysis of Fermi resonances by harmonic derivatives of perturbation theory corrections

Abstract

Vibrational perturbation theory has proven to be a highly accurate and efficient method for extending the harmonic approximation in the treatment of polyatomic molecular vibrations. Unfortunately, accidental near-degeneracies of the harmonic vibrational levels can lead to resonance and a breakdown of the perturbation approximation. These resonances can be resolved by the diagonalization of a small effective Hamiltonian derived from either of the usual Rayleigh–Schrödinger or van Vleck perturbation theories. However, the proper choice of states for inclusion in the effective Hamiltonian is crucial to the accuracy of the results, and is not often clearly evident. It is proposed that the analytical partial derivatives of the anharmonic vibrational correction with respect to the various harmonic frequencies, called ‘Harmonic Derivatives’ in this work, can be used as a tool to quantitatively assess the existence and strength of first-order, or Fermi, resonances. These derivatives are shown to concisely and clearly reflect the quality of the perturbation approximation and the effect of its breakdown on the computed vibrational levels.