Semiclassical quantization

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]. 

Shirts R B and Reinhardt W P 1982 Approximate constants of motion for classically chaotic vibrational dynamics vague tori, semiclassical quantization, and classical intramolecular energy flow J. Cham. Phys. 77 5204-17... 

Equation (4.46), however, regardless of the phases i/ , does not describe periodic orbits, unless the frequencies oj are commensurate. Thus the first question that is to be answered is, how to semiclassically quantize a separate well. Furthermore, because of symmetry, a tunneling orbit should pass through the point Q = iQ +, Q-)- However, if it sets out from the turning point at... 

Reinhardt, W. P. (1989), Adiabatic Switching A Tool for Semiclassical Quantization and a New Probe of Classically Chaotic Phase Space, Adv. Chem. Phys. 73, 925. 

II. Semiclassical Quantization around Equilibrium Points and Periodic Orbits... 

D. Dissociation on Potentials with a Saddle Semiclassical Quantization... 

We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [19]. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like Hgl2, CO2, and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. 

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal iepeller. As we mentioned in Section II, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodic-orbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. 

The quanta character is even more prominent than in CO2 and the first reported resonances occur at 3863 and 7453 cm 1, respectively, above the transition region (see Fig. 19). Here the zeta function semiclassical quantization with (4.16) should apply. The reported lifetimes are of 33 fs, which is... 

Applying the semiclassical quantization condition to the action around a closed classical orbit yields25-27... 

Main, J., Holle, A., Wiebusch, G., and Welge, K.H. (1987). Semiclassical quantization of three-dimensional quasi-Landau resonances under strong-field mixing, Z. Phys. D 6, 295-302. 

Quantum calculations for a classically chaotic system are extremely hard to perform. If more than just the ground state and a few excited states are required, semiclassical methods may be employed. But it was not before the work of Gutzwiller about two decades ago that a semiclassical quantization scheme became available that is powerful enough to deal with chaos. Gutzwiller s central result is the trace formula which is derived in Section 4.1.3. 

Closed orbits are of special interest for the semiclassical quantization of the helium atom. For a closed orbit F we have... 

Currently available numerical results indicate that the one-dimensional heUum atom is completely chaotic. The best-known semiclassical quantization procedure for completely chaotic systems is Gutzwiller s trace formula (see Section 4.1.3), which is based on classical periodic orbits. Therefore we search for simple periodic orbits of the one-dimensional he-hum atom. Since a two-electron orbit is periodic if the orbits ni t), 0i t)) and (ri2(t), 2( )) of the first and second electron have a common period, the periodic orbits of the one-dimensional model can be labelled with two integers, m and n, which count the 27r-multiplicity of the angle variables 0i and 02 after completion of the orbit. Therefore, if for some periodic orbit... 

Gaspard, P. and Rice, S.A. (1989b). Semiclassical quantization of the scattering from a classically chaotic repellor, J. Chem. Phys. 90, 2242-2254. 

Miller, W.H. (1975). Semiclassical quantization of nonseparable systems A new look at periodic orbit theory, J. Chem. Phys. 63, 996-999. 

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

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