Semiclassical quantization using

To calculate N (E-Eq), the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The fomier approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Hamionic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by perfomiing an appropriate nomial mode analysis as a fiinction of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to detemiine anliamionic energy levels for die transitional modes [27]. 

Given a potential energy curve, it is possible to locate (iteratively) the vibrational energy levels using the semiclassical quantization condition... 

The preceding normal-mode/rigid-rotor sampling assumes the vibrational-rotational levels for the polyatomic reactant are well described by separable normal modes and separability between rotation and vibration. However, if anharmonicities and mode-mode and vibration-rotation couplings are important, it may become necessary to go beyond this approximation and use the Einstein-Brillouin-Keller (EBK) semiclassical quantization conditions [32]... 

The adiabatic switching semiclassical quantization method [60-62] may also be used to choose initial conditions for polyatomic reactants. This approach does not require an explicit determination of the topologically independent paths Ct and actions Jt for Eq. (3.36) and, in principle, may be more easily applied to larger polyatomics than the EBK semiclassical quantization approach described above. However, what is required is a separable zero-order Hamiltonian H0 that gives rise to the same kind of intramolecular motion as does the complete Hamiltonian [63,64]. 

EBK) semiclassical quantization condition given by Eq. (2.72). In contrast to the RKR method for diatomics, a direct method has not been developed for determining potential energy surfaces from experimental anharmonic vibrational/rotational energy levels of polyatomic molecules. Methods which have been used are based on an analytic representation of the potential energy surface (Bowman and Gazdy, 1991). At low levels of excitation the surface may be represented as a sum of quadratic, cubic, and quartic normal mode coordinates (or internal coordinate) terms, that is,... 

Computational studies have indicated that chaotic behavior is expected in classical mechanical descriptions of the motion of highly excited molecules. As a consequence, intramolecular dynamics relates directly to the fundamental issues of quantum vs classical chaos and semiclassical quantization. Practical implications are also clear if classical mechanics is a useful description of intramolecular dynamics, it suggests that isolated-molecule dynamics is sufficiently complex to allow a statistical-type description in the chaotic regime, with associated relaxation to equilibrium, and a concomitant loss of controlled reaction selectivity. 

Also, it is possible to use semiclassical quantization techniques to select the vibrational states of triatomic molecules taking part in the chemical reactions. Good examples of QCT applications are the reactions... 

The obvious defect of classical trajectories is that they do not describe quantum effects. The best known of these effects is tunnelling tln-ough barriers, but there are others, such as effects due to quantization of the reagents and products and there are a variety of interference effects as well. To circumvent this deficiency, one can sometimes use semiclassical approximations such as WKB theory. WKB theory is specifically for motion of a particle in one dimension, but the generalizations of this theory to motion in tliree dimensions are known and will be mentioned at the end of this section. More complete descriptions of WKB theory can be found in many standard texts [1, 2, 3, 4 and 5, 18]. 

The periodic-orbit quantization can be used to calculate not only the resonances but also the full shape of the photoabsorption cross section using (2.26) and (2.27). This semiclassical formula for the cross section separates in a natural way the smooth background from the oscillating structures due to the periodic orbits. In this way, the observation of emerging periodic orbits by the Fourier transform of the vibrational structures on top of the continuous absorption bands can be explained. 

This method of numerical quantization can be also used in the case of low barriers, for which the semiclassical approximation is no longer valid. 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

We have also used the periodic reduction method to predict with good accuracy the 3D structure of vibrationally bonded molecules. It should be stressed though, that in principle it is not necessary to use periodic reduction. As shown in Fig. 9 the RPO s of the IHI system are stable also in 3D, one can find bound quasi-periodic orbits and quantize them semiclassically directly without resorting to periodic reduction. 

Semiclassical transition state theory based on second-order perturbation theory (89) provides another way to assign quantized energy levels of the transition state, and an application (90) to the H + H2 reaction yielded encouraging results in comparison to the full quantum (8) calculations. One difference in assignments (8,90) was later explained (88), using the resonance theory reformulation of variational transition state theory, as a consequence of the inadequacy of second-order perturbation theory. 

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