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 , 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
![](/cms/asset/41f1b929-dfa4-4e60-a3d9-423e2b18bd1e/tmph_a_1743887_uf0001_oc.jpg)
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).