Fermi bifurcation

This study [14] has shown that a period-doubling bifurcation associated with the Fermi resonance occurs in this subsystem at the energy E = 3061.3 cm-1 (with Egp = 0). Below the Fermi bifurcation, there exist edge periodic orbits of normal type, which are labeled by ( i, 22 -)normai- At the Fermi bifurcation, a new periodic orbit of type (2,1°, ->Fermi appears by period doubling around a period of 2T = 100 fs. This orbit is surrounded by an elliptic island that forms a region of local modes in phase space. Therefore, another family of edge periodic orbits of local type are bom after the Fermi bifurcation that may be labeled by the integers (n n , -)iocai- They are distinct... 

Ezra G S 1996 Periodic orbit analysis of molecular vibrational spectra-spectral patterns and dynamical bifurcations in Fermi resonant systems J. Chem. Phys. 104 26... 

It has been shown recently that the vibrational spectra of HCP [33-36], HOCl [36-39], and HOBr [40,41] obtained from quantum mechanical calculations on global ab initio surfaces can be reproduced accurately in the low to intermediate energy regime (75% of the isomerization threshold for HCP, 95% of the dissociation threshold for HOCl and HOBr) with an integrable Fermi resonance Hamiltonian. Based on the analysis of this Hamiltonian, this section proposes an interpretation of the most salient feature of the dynamics of these molecules, namely the first saddle-node bifurcation, which takes place in the intermediate energy regime. 

To conclude this analysis based on the Fermi resonance Hamiltonian, let us mention that HOCl, behaves very much like HOBr. Indeed, Fig. 10b of Ref. 36 shows that for this molecule the saddle-node bifurcation SNl takes place at P = 21.8, (for v = 0), the period-doubling bifurcation PD occurs at P = 24.6, and the second saddle-node bifurcation SN2 takes place at around P = 38, very close to the dissociation threshold. In contrast, the dynamics of HCP is somewhat simpler, in the sense that the first saddle-node bifurcation SNl is indeed observed at P = 14.3, but PD and SN2 do not take place (see Fig. 13 of Ref. 35 or Fig. 10a of Ref. 36). 

Most of the bifurcations, which take place in the high-energy regime, are however not reproduced by the Fermi resonance Hamiltonian, essentially because they result from the superposition of the 1 2 Fermi-resonance and higher-order ones. In order to gain information on the dynamics close to the reaction threshold, one therefore has to analyze the dynamics on the PES by... 

It turns out that the language of normal and local modes that emerged from the bifurcation analysis of the Darling-Dennison Hamiltonian is not sufficient to describe the general Fermi resonance case, because the bifurcations are qualitatively different from the normal-to-local bifurcation in figure Al.2.10. For example, in 2 1 Fermi systems, one type of bifurcation is that in which resonant collective modes are bom [54]. The resonant collective modes are illustrated in figure A12.11 their difference from the local modes of the Darling-Dennison system is evident. Other types of bifurcations are also possible in Fermi resonance systems a detailed treatment of the 2 1 resonance can be found in [44]. 

Nonetheless, it is still possible to perform the bifurcation analysis on the multiresonance Hamiltonian. In fact, the existence of the polyad number makes this almost as easy, despite the presence of chaos, as in the case of an isolated single Fermi or Darling-Dennison resonance. It is found [ ] that most often (though not always), the same qualitative bifurcation behaviour is seen as in the single resonance case, explaining why the simplified individual resonance analysis very often is justified. The bifurcation analysis has now been performed for triatomics with two resonances [60] and for C2H2 with a number of resonances [ ]. 

See, for example, the following and references contained therein E. L. Sibert 111, W. P. Reinhardt, and J. T. Hynes, /. Chem. Phys., 81, 1115 (1984). Intramolecular Vibrational Relaxation and Spectra of CH and CD Overtones in Benzene and Perdeuterobenzene. S. P. Neshyba and N. De Leon,. Chem. Phys., 86, 6295 (1987). Qassical Resonances, Fermi Resonances, and Canonical Transformations for Three Nonlinearly Coupled Oscillators. S. P. Neshyba and N. De Leon,. Chem. Phys., 91, 7772 (1989). Projection Operator Formalism for the Characterization of Molecular Eigenstates Application to a 3 4 R nant System. G. S. Ezra, ]. Chem. Phys., 104, 26 (1996). Periodic Orbit Analysis of Molecular Vibrational Spectra Spectral Patterns and Dynamical Bifurcations in Fermi Resonant Systems. Also see Ref. 6. 

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