Vibrogram

Section III is devoted to bounded systems. The semiclassical properties relevant for short time scales are discussed and illustrated with so-called vibrograms of l2> Nal, CS2, C2HD, and C2H2. Section III is also concerned... 

Figure 2 depicts the vibrogram corresponding to the dynamics on the ground state of iodine, modeled by a Morse potential with the equilibrium distance r = 2.67 A and the dissociation energy D = 12,542 cm 1 [14, 108]. The periodic orbit and its repetitions clearly appear in the vibrogram. The classical... 

Figure 2. Vibrogram of a Morse model of li(X1E) calculated with the function (3.7) with e = 1000 cm 1. The solid lines are the classical periods given by (3.11).

Figure 3. Experimental vibrogram of Nal obtained by Zewail and co-workers [115]. The triangles give the prime periods and the diamonds their repetitions. The solid line is a fit of the experimental points. (Experimental data from Ref. 115.)...

The experimental vibrogram shows an important recurrence around 160 fs, which may be assigned to the edge periodic orbit (3,2°, -)n0rmai- Recently, the vibrogram analysis has been carried out by Michaille et al. [113] on the basis of another model proposed by Joyeux [118] as well as on an ab initio potential fitted to the experimental data of Pique [119]. Essentially the same classical periodic orbits appear in the different models at low energies. In the same context, let us add that Joyeux has recently applied the Berry-Tabor trace formula to a IF Fermi-resonance Hamiltonian model of CS2 [120] and carried out a classical analysis of several related resonance Hamiltonians [121]. 

The vibrogram has also been obtained by Jost and co-workers in the region of 12,000-15,000 cm 1, where the vibrational spectrum is irregular [111]. The main recurrence occurs at 47 fs and multiples thereof and corresponds to the bending frequency of about 714 cm-1. Thus, the spectrum is dominated by the bending motion in this regime of N02. 

Figure 4. Vibrogram of C2HD calculated with = 2000 cm-1 from all the vibrational energy levels predicted by the Dunham expansion corresponding to the Hamiltonian (3.12) obtained by Herman and co-workers by fitting to high-resolution spectra [112], The periods of the bulk periodic orbits of Table I obtained numerically for the classical Hamiltonian (3.12) are superimposed as circles. On the right-hand side, the main labels (n4,ns) of the periodic orbits are given.

Recurrences of Vibrogram of C2HD and Their Periodic-Orbit Assignment [112]... 

The vibrogram analysis [108] based on the dispersion fluorescence spectrum of acetylene by Solina et al. [125] reveals a recurrence around 50 fs from 0 to 12,000 cm"1, which is similar to the one in the previous analysis. However, an important recurrence appears at 70 fs at higher energies from 4000 to 16,000 cm 1, which is caused either by anharmonic period lengthening or by a transition to a slower regime at higher energies. 

Let us compare these results with the experimental data on CO2. A photoabsorption spectrum of CO2 has been recorded in 1965 by Nakata et al. [156]. The vibrogram as well as direct Fourier transforms of the intense band under discussion reveal main time recurrences around 50 and 60 fs [14]. On the other hand, the above model shows periods corresponding to 0 and 01 or 02 around 40 and 60 fs, respectively. Although there is a qualitative agree-... 

B. A. Hess Dr. Gaspard has introduced the vibrogram as a tool to extract periodic orbits from the spectrum by means of a windowed Fourier transform. This raises the question whether other recent techniques of signal analysis like multiresolution analysis or wavelet transforms of the spectrum could be used to separate the time scales and thus to disentangle the information pertinent to the quasiclassical, semiclassical, and long-time regimes. Has this ever been tried ... 

P. Gaspard As far as I know, the wavelet analysis of spectra has not yet been done and would be very interesting to develop. A remark is that the vibrogram also depends on the width e of the Gaussian window, which may be varied to construct another kind of plot. 

The MQDT calculations we have performed yield a solution to the time-independent Schrodinger equation in principle the time-dependent information can be obtained by Fourier transformation, possibly using the vibrogram method described by Gaspard. 

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