No abstract

Spectral patterns associated with recently proposed assignments of Fermi resonance systems are investigated with specific application to the 2:1 Fermi resonance fitting Hamiltonian. It is shown that the spectrum of a pair of resonant modes has characteristic patterns associated with the structure of the corresponding classical phase space. In particular, when a spectral fitting Hamiltonian has a separatrix structure in its classical phase space, the quantum Hamiltonian has an emblematic spectral pattern, a ‘‘dip’’ in the level spacings. This basic pattern is the starting point for an investigation of level patterns corresponding to the bifurcation and catastrophe map classification and associated dynamically based assignments of Fermi resonance Hamiltonians. The 2:1 Fermi resonance Hamiltonian is investigated in detail as a typical system. There are distinctive patterns for polyads from different zones of the catastrophe map classification of the 2:1 system. Conversely, when these patterns occur in an experimental spectrum, then in order to reproduce them in a reasonably behaved spectral fit, it is necessary and sufficient to invoke a resonant coupling term in the fitting Hamiltonian. Spectral fitting therefore gives reliable information about the phase space structure of a molecule. These considerations are used to address the interpretation of recent experimental and theoretical investigations of H2CO and benzophenone vibrational spectra.

The classical phase space structure of a spectroscopic Hamiltonian for two coupled vibrational modes is analyzed using bifurcation theory, classified on catastrophe maps, for a variety of higher order resonances (3:2, 4:2, 5:2, 6:2 and 4:4, 5:4, 6:4), cases not considered in previous work. A type of bifurcation not encountered for lower resonance orders, based on oVerlap of separatrices rather than change in behavior of fixed points, is analyzed, and a procedure is developed to augment the catastrophe map. Energy level patterns are associated with the new resonances, in analogy with the patterns of adjacent level spacings considered earlier for the 2:1 resonance.