Free-rotor model

In fact, these wing-like features are well-reproduced (as shown in Fig. 1.6) by a simple free-rotor model [24], in which the metastable CS2+ is assumed to rotate freely with no interaction with the counterpart S+ ion. In the simulation, the momentum vector imposed on the S+ and CS2+ fragments in the first step (CS —> S+ + CS2+) and psec0nd to the C+ and S+ fragments in the second step (CS2+ —> S+ + C+) are treated as adjustable parameters. 

The two wings observed in the three-dimensional map (Fig. 1.6) are transformed into a pair of curved arms in the two-dimensional momentum map in the A >i2 - O12 plane, as shown in Fig. 1.7a, where Api2 represents the difference between the momentum values of the two S+ ions, i.e., Api2 = Pi - P2-As shown in the two-dimensional momentum map drawn on a logarithmic intensity scale, the observed curved arms are well-reproduced by the classical free-rotor model. 

The rate constants obtained from using a C2 molecule model having a free rotation, instead of the 350 cm.-1 torsion degree of freedom, are shown, by a dashed line in Figures 1 and 2 for complex 2 the free rotor model has an increased value of N (t) (Fig. 6) at all energies, thereby decreasing the rate constant with the effect being most marked at low values of e. 

The free rotor model is not adequate for most molecules as there is in general some potential barrier to internal rotation. As an example, consider ethyl fluoride. Ethyl fluoride, just like ethane, has a predominately threefold potential barrier as shown in Figure 7-3. The potential barrier can be expressed as the following function. 

The Franck-Condon model, which decomposes the initial quasistable state of the wave function in terms of free-rotor states and gives the product rotational distribution if no torques are present, was apphed to Ne CI2 and He CI2 and correctly predicted the low / behavior of the distribution, as well as the invariance of the distributions in the Av = 1 and —2 channels [99, 100]. 

A. Previous models of water (see 1-6 in Section V.A.l) and also the hat-curved model itself cannot describe properly the R-band arising in water and therefore cannot explain a small isotope shift of the center frequency vR. Indeed, in these models the R-band arises due to free rotors. Since the moment of inertia I of D20 molecule is about twice that of H20, the estimated center of the R-band for D20 would be placed at y/2 lower frequency than for H20. This result would contradict the recorded experimental data, since vR(D20) vR(H20) 200 cm-1. The first attempt to overcome this difficulty was made in GT, p. 549, where the cosine-squared (CS) potential model was formally (i.e., irrespective of a physical origin of such potential) applied for description of dielectric response of rotators moving above the CS well (in this work the librators were assumed to move in the rectangular well). The nonuniform CS potential yields a rather narrow absorption band this property agrees with the experimental data [17, 42, 54]. The absorption-peak position Vcs depends on the field parameter p of the model given by... 

Gaspard and Rice [35] also proposed a four-dimensional map in order to smdy Hamiltonian systems with Arnold diffusion. The model Hamiltonian is a free rotor in a Morse-like kicking field and takes the following form ... 

Fig. 6. Free energy (at zero pressure) for a-nitrogen and /3-nitrogen, in different mean field models (closed lines). The dashed free rotor curve has been calculated from the isotropic (/, 12, ly) - (0,0, 0) term of the ab initio potential by adding the free rotor expression for the free energy. The dashed jump model curve has been obtained from the localized mean field solution (with the full anisotropic potential) by adding an entropy term -kBT In 6 (see the text).

We have also shown that some originally proposed (e.g., in GT2) properties of the hat potential turn out to be excessively abstract, so that we disregarded them. We mean here (i) absence of free rotors, performing circular motion, since in the water/ice structure there is no place for such a motion, and (ii) nonexistence of precessors, since, as it appears, the libration of dipoles in a diametric cross section of the hat-like potential well dominates. On the other hand, the well s curvature is important for modeling the spectra, since, unlike the well depth, this curvature strongly depends on temperature and fluid phase state. 

It should be noted that h(co), and Ci(t) depend on a parameter tc and on the free-rotor function C 0>(t). This latter function decays on a time scale determined by the root-mean square angular speed which is proportional to [kBT/I]llZ. Thus we have two characteristic times in this model. 

Fig. 5.18 A three-dimensional free quantum-rotor model and possible nuclear spin-rotational state couplings (Vnt Va) for (a) CH3 and (b) CD3 radicals in the7>3 symmetry. The observed 1 1 1 1 queirtet and 2 2 doublet of CH3 are attributed to four A1 and two doubly degenerated E nuclear spin states coupled with rotational ground 7 = 0 (even) and excited 7=1 (odd) states, respectively. For CD3 only one A2 nuclear spin state is possible in the 7 = 0 state. B stands for the theoretictil rotational constant...

VR Energy Transfer. Free Rotor and Perturbed Rotor Models... 

Another way to generalize the Moore idea is based on the assumption that a mode which induces a vibrational transition in a diatom corresponds to the pure rotation of the latter. This approach was adopted in a free-rotor (FR) model by Billing [28]. It was demonstrated later on that the higher-order anisotropy corrections to the interaction potential contribute much to the transition probability [29] however, the... 

The excitations of the crystal that originate from the MF model may correspond with strongly anharmonic translational vibrations or librations of the molecules they may even correspond with hindered- or free-rotor states. They remain single particle excitations, however, which do not show any dispersion (i.e. wave vector dependence) in their frequencies. The simplest manner to obtain this dispersion is by the so-called Exciton Model or Tamm-Dancoff Approximation [76]. From the crystal ground state, which is a product of (known) MF states ... 

Another approach is due to Steele (18). He approximates the first memory function by the product of the free rotor memory function Mp and a gaussian decay. This may be compared with the memory function for the J diffusion model which is the product of the free rotor memory function and an exponential decay. The... 

Another important mode of rotation in polyatomic molecules is internal modes of rotation. As an example, consider the rotation of a methyl group about the C-C bond axis in ethane. The rotation of the methyl group can be approximated as a free rotor about the < i angle as in the Particle-on-a-Ring model problem (see Section 3.1). From the moment of inertia of the methyl group, the energy of the internal rotational states can be obtained from Equation 3-6. 

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