Classification of vibrational resonances in the energy spectrum of the formaldehyde molecule and Katz’s branch points

The Rayleigh–Schrödinger perturbation theory of high orders and the algebraic Padé–Hermite approximants are used to determine the singular points of a vibrational energy function of the formaldehyde molecule dependent on a complex perturbation parameter as on the argument. It is shown that the Fermi, Darling–Dennison, and other higher-order vibrational resonances are related to Katz’s points—common branch points on the complex plane of the energy of two vibrational states. Analysis of Katz’s points that connect different vibrational states allows one to reveal essential resonance perturbations, to introduce an additional classification for them, and to determine the polyad structure of an energy spectrum.