Rigorous vibrational Fermi resonance criterion revealed: two different approaches yield the same result

Molecular vibrational resonances are manifest by abnormally small energy differences in denominators of anharmonic constants derived within the second-order vibrational perturbation theory (VPT2), or in abnormally large coefficients Ξ of unitary transformation generators in the canonical Van Vleck perturbation theory (CVPT). A quantitative measure of the vibrational resonance can be derived by assuming the perturbation parameter to be a complex variable , and the numerical analysis of the diverging high order Rayleigh-Schrödinger perturbation theory (RSPT) expansions for closely spaced states. The location of branch points for complex values of within the unit circle serves as the definitive evidence for a resonance. In practice, RSPT series for diverging resonant states can be treated by resummation using quadratic Padé-Hermite approximants. The critical values of can be further found and checked against the condition . A comparative analysis of selected resonances for some molecules (H O, HDO, H S and H CO) revealed a threshold value of that is equivalent to the RSPT parameter . The fundamentally different approach based on the polyad analysis of the resonance vectors spanning the ( )-dimensional subspace for the ethylene molecule led to nearly the same value for the resonance criterion. GRAPHICAL ABSTRACT