Gustavson’s procedure and the dynamics of highly excited vibrational states

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Abstract
The well-known Birkhoff–Gustavson canonical perturbation theory has been used so far to obtain a reasonable approximation of model systems near the bottom of the well. It is argued in the present work that Gustavson’s calculation procedure is also a powerful tool for the study of the dynamics of highly excited vibrational states, as soon as the requirement that the transformed Hamiltonians be in Birkhoff’s normal form is dropped. Mathematically, this amounts to modifying the content of Gustavson’s null space. Physically, the transformed Hamiltonians are of the single or multiresonance type instead of just trivial Dunham expansions, even though no exact resonance condition is fulfilled. This idea is checked against 361 recently calculated levels of HCP up to 22 000 cm−1 above the bottom of the well and involving up to 30 quanta in the bending degree of freedom. Convergence up to 13th order of perturbation theory and an average absolute error as low as 2.2 cm−1 are reported for a two-resonance Hamiltonian, whereas the Dunham expansion converges only up to 4th order at an average error of 215 cm−1. The principal advantages of the resonance Hamiltonians compared to the exact one rely on its remaining good quantum numbers and classical action integrals. Discussions of the limitations of the method and of the connections to other canonical perturbation theories, like Van Vleck or Lie transforms, are also presented.