Quasi-classic theory

The quasi-classical theory of spectral shape is justified for sufficiently high pressures, when the rotational structure is not resolved. For isotropic Raman spectra the corresponding criterion is given by inequality (3.2). At lower pressures the well-resolved rotational components are related to the quantum number j of quantized angular momentum. At very low pressure each of the components may be considered separately and its broadening is qualitatively the same as of any other isolated line in molecular or atomic spectroscopy. 

In conclusion, we have probed the local density of states in the vicinity of various N-S structures [19, 23] at very low temperature. The samples with a lateral geometry (and a reduced gap) showed a good agreement between the experiment and the quasi-classical theory. In contrast, N-S bilayers with a... 

For weak fields of amplitude <electron through the Coulomb barrier and the survival probability P t) of the ground state follows approximately an exponential law. The order of magnitude of the width T of the resonance is given approximately by the quasi-classic theory [37] ... 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

As the process of rotational relaxation is close to a correlated one (y 1) for both gases, according to (3.74) the aE cross-section is twice as large as oj. This result agrees with experiment and it appears that quasi-classical impact theory may be applied to description of rotational relaxation in moderately dense gases. 

Firstly, we are going to demonstrate how branch interference may be taken into account within the quasi-classical impact theory. Then we shall analyse a quasi-static case, when the exchange frequency between branches is relatively small. An alternative case, when exchange is intensive and the spectrum collapses, has been already considered in Chapter 2. Now it will be shown how the quasi-static spectrum narrows with intensification of exchange. The models of weak and strong collisions will be compared with each other and with experimental data. Finally, the mutual agreement of various theoretical approaches to the problem will be considered. 

Let us consider the quasi-classical formulation of impact theory. A rotational spectrum of ifth order at every value of co is a sum of spectral densities at a given frequency of all J-components of all branches... 

Quite recently Raes (1985) applied the classical theory of homogenous nucleation originally developed by Bricard et al (1972) to atmospheres containing SO2, H2O and 218Po ions. Depending on the H2O and SO2 concentrations, ions could grow to a quasi-stable cluster which would evaporate upon electrical neutralization, or to a larger size which would survive neutralization. 

From the beginning, London s theory was recognized as an expedient, but somewhat arbitrary, device to simplify numerical evaluations and recover quasi-classical interpretations of selected long-range contributions to the total intermolecular interaction in the words of a classic text,25... 

The activated dissociation of H2 (D2) on Cu(l 11) and other single crystal Cu surfaces has played a special role in the development of reactive gas-surface dynamics. Early experiments and theory by Cardillo and collaborators [217-219] first demonstrated the power of molecular beam techniques to probe activated adsorption and the theoretical methodology developed by them (6D quasi-classical dynamics on a model PES) only differs from modem treatments in the use of DFT based PES. 

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

Mass spectra of dibenzo[/Af]thiepine 32 and dibcnzo[/),e]thiepine 5,5 -dioxide 33 were studied via the classical approximation of the quasi-equilibrium theory. A good agreement was achieved between calculated and experimental data <1998RRC849>. 

Truhlar, D.G. and Muckerman, J.T. (1979). Reactive scattering cross sections III Quasi-classical and semiclassical methods, in Atom-Molecule Collision Theory, ed. R.B. Bernstein (Plenum Press, New York). 

Table 6.3 A comparison of different theoretical approaches to the evaluation of the thermal rate constant for the F + H2 —> HF + H reaction at T = 300 K. TST is transition-state theory (Example 6.2), QCT is the quasi-classical trajectory method [Chem. Phys. Lett. 254, 341 (1996)], and QM is (exact) quantum mechanics [J. Phys. Chem. 102, 341 (1998)].

A further analytical approximation to Eq. (369), proposed by Miller and coworkers [84-86], demonstrates how the above semiclassical reaction rate theory approaches a quasi-classical reaction rate theory. Specifically, consider the... 

Equation (374) can be regarded as a quasi-classical extension of classical reaction rate theory. 

Lattice dynamics calculations on the plastic /3-nitrogen phase are relatively scarce because, obviously, the standard (quasi-) harmonic theory cannot be applied to this phase. Classical Monte Carlo calculations have been made by Gibbons and Klein (1974) and Mandell (1974) on a face-centered cubic (a-nitrogen) lattice of 108 N2 molecules, while Mandell has also studied a 32-molecule system and a system of 96 N2 molecules on a hexagonal close-packed (/3-nitrogen) lattice. Gibbons and Klein used 12-6... 

Like many other quasi-classical methods applied within the domain of quantum mechanics, the free electron theory explained the gross features of the phenomenon and provided an attractively simple physical picture. The method did not of course, suggest any parallels with the quantum mechanically based HMO theory (which predicted an alternation in 7c-electron properties among the annulenes). 

---