On resonance-type effective vibrational Hamiltonians for CO 2 I. Theoretical background

Abstract

This is the first one of a series of two papers aimed at investigating the relation of resonance-type effective Hamiltonians to molecular potential energy surfaces for linear ABA triatomic molecules, and more precisely for CO 2. Single- or multi-resonance effective vibrational Hamiltonians are much simpler to handle than the exact one, yet retaining all its essential features, at least in the regular energy range. The relations between these effective Hamiltonians and the exact one will be analysed in some details, particularly in connection with the fit of force constants to experimentally observed transition frequencies. Another question, which is also addressed, is that of the underlying coordinates. Indeed, at least two canonical transformations are needed to derive the effective Hamiltonians from the exact one using Birkhoff-Gustavson perturbation theory. It will be shown that the differences between the two sets of coordinates become small, as soon as a second angle ϕ 1 is added to the Fermi resonance ϕ 1 + 2 ϕ 2 dependence. The theoretical background for this study, namely the expansion of the exact expression for kinetic energy up to fifth order in dimensionless coordinates and fourth-order Birkhoff-Gustavson perturbation theory, is to be found in this first paper, whereas the second one is devoted to explicit numerical results for CO 2.