Vibrational resonance analysis of linear molecules using resummation of divergent Rayleigh–Schrödinger perturbation theory series

Abstract

The large order Rayleigh–Schrödinger perturbation theory (RSPT) was applied for calculating vibrational states of linear molecules. Two molecules (CO2 and C2H2) were used as test cases with using of isomorphic Watson Hamiltonian and quartic force fields. For CO2 the Sayvetz condition can remove all degeneracies for purely vibrational states and the non-degenerate perturbation theory can be applied. However, an existence of two degenerate modes in C2H2 requires using the upgraded degenerate version of RSPT that was employed in this context for the first time. The dominating divergent behavior of such series requires the resummation technique that mimics the multivalued nature of the underlying solutions, and the applied quartic Padé–Hermite approximants (QPHA) provided full solution of the problem. Moreover, some mathematical properties of QPHA proved to be an efficient tool for studying resonance effects through the Katz theorem that controls the singular points of the eigenvalues on the complex plane. In the case of C2H2, not only all earlier observed classical resonances were confirmed and quantified, but also subtle interpolyad resonances (K2/55,K3/4555), proposed recently by Herman (2011) were described as well. Following the analysis, we found several novel resonances, of which we proposed one independent interpolyad resonance K2/4444. The complete analysis of such critical points provided the full resonance picture of all studied molecules.