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

Abstract

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 $\overline{\lambda }$, 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 $\overline{\lambda }$ 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 $\overline{\lambda }$ can be further found and checked against the condition $\overline{\lambda }\le 1$. A comparative analysis of selected resonances for some molecules (H${}_2$O, HDO, H${}_2$S and H${}_2$CO) revealed a threshold value of ${\mathrm{\Xi }}^{\ast }\approx 0.076$ that is equivalent to the RSPT parameter $\overline{\lambda }\le 1$. The fundamentally different approach based on the polyad analysis of the resonance vectors spanning the ($3N_{at}-7$)-dimensional subspace for the ethylene molecule led to nearly the same value for the resonance criterion.

GRAPHICAL ABSTRACT

Keywords:

• Anharmonic vibrations
• vibrational resonance
• resonance criteria
• Rayleigh-Schrödinger perturbation theory
• canonical Van Vleck perturbation theory
• resummation of divergent series
• Padé-Hermite approximant
• vibrational polyads

Acknowledgements

We wish to acknowledge the support of the authors by Prof. Norman C. Craig from Oberlin College (Oberlin/OH, USA) and Prof. Oleg L. Polyansky from University College London (UK) for reading the manuscript and helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).