Models degrees of freedom

The vibrational dynamics of this system can be adequately studied by a two degrees of freedom model, with the C-N distance kept frozen at its equilibrium value of re = 2.186 a.u. The vibrational (total angular momentum J = 0) Hamiltonian in scattering or Jacobi coordinates is given by... 

Messina et al. [25] test the time-dependent Hartree reduced representation with a simple two-degree-of-freedom model consisting of the h vibration coupled to a one-harmonic-oscillator bath. The objective function is a minimum-uncertainty wavepacket on the B state potential curve of I2. Figure 12, which displays a typical result, shows that this approximate representation gives a rather good account of the short-time dynamics of the system. 

We focus on the nonlinear dynamics for the collinear configuration which we treat as part of the full multidimensional system. This is to be contrasted with two-degree-of-freedom models where the molecule is assumed to be frozen in some angular configuration, such that the bending degree of freedom is excluded from the dynamics. In our analysis, bending is taken into account in terms of linearized dynamics, which allows us to extend the results for the collinear situation to the full three-dimensional system. The restriction we must be aware of is that the three-dimensional system may have periodic orbits that are not of collinear type. 

Note Frequencies and energies in reciprocal centimeters and times in femtoseconds. "Collinear two-degree-of-freedom model by Zewail et al. [151] by quantization around the equilibrium point of linear geometry (bending ignored). 

Three-degree-of-freedom model fitted to experimental data for the overall band structure by Schinke et al. [159] (nonlinear equilibrium geometry). 

Collinear two-degree-of-freedom model with the Karplus-Porter surface by Pechukas et al. [ 143] (linear equilibrium geometry). 

The behavior of the dielectric spectra for the two-rotational-degree-of-ffeedom (needle) model is similar but not identical to that for fixed-axis rotators (one-rotational-degree-of-fireedom model). Here, the two- and one-rotational-degree-of-freedom models (fractional or normal) can predict dielectric parameters, which may considerably differ from each other. The differences in the results predicted by these two models are summarized in Table I. It is apparent that the model of rotational Brownian motion of a fixed-axis rotator treated in Section IV.B only qualitatively reproduces the principal features (return to optical transparency, etc.) of dielectric relaxation of dipolar molecules in space for example, the dielectric relaxation time obtained in the context of these models differs by a factor 2. 

States with lifetimes of up to 0.1 sec are found. The paper ends with presentation of preliminary adiabatic semiclassical estimates of resonance energies for BOD In two and three degrees-of-freedom models. 

Table II As In Table I, but for mass ratio 1 64 1, appropriate to a two degrees of freedom model of H2O.

One can Immediately ask, what If there Is not such an EBK torus for the coupled system The empirical result Is that the method may well work anyway Table 111 shows results (26) obtained by adiabatic quantization of the Hase ( ) HCC two-degrees-of-free-dom problem. Away from the 5 2 resonances (see Fig. 5) the adiabatic and Hase results are In accord, but the adiabatic method also successfully quantizes the resonance zones. Another Illustration Is given In Table IV where a two-degrees-of-freedom model of HOD Is adlabatlcally quantized above the classical dissociation threshold... 

Figure 6. The density of adiabatlcally switched trajectories Is plotted vs. the final energy for HOD In a two-degrees-of-freedom model Initially In the (8,8) resonance state. The results are obtained from 83 randomly selected trajectories using a switching time of 30,000 a.u. The position Is the energy of the (8,8) state obtained using the Hamiltonian H. ...

Table IV. Adlabatlcally quantized semlclasslcal results for energies of doubly excited states of HOD, In a two degrees of freedom model, as obtained by Skodje and Reinhardt (27) using the technique of Refs. (24), (25). All of these states are above the classical dissociation threshold. AE Is the uncertainty In the adiabatic quantization, as Illustrated In Fig. 6. Eo Is the "unperturbed energy, and E the result of the quantization procedure In this weakly coupled system.

Example 1.6 The single degree of freedom model of Example 1.4 has the energy as a first integral. The system is therefore integrable. 

Sketch the level sets of energy for a single degree of freedom model with energy E q, q) = q f 2 + fuiq), where Lennard-Jones potential. 

In order to use a one degree-of-freedom model on an infinite domain in the canonical setting, we would need to incorporate a restraining potential, slightly modifying for example the Lennard-Jones potential to U q) = q — q + sq (Fig. 6.2). 

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