Periodic orbits classical dynamics

The classical counterpart of resonances is periodic orbits [91, 95, 96, 97 and 98]. For example, a purely classical study of the H+H2 collinear potential surface reveals that near the transition state for the H+H2 H2+H reaction there are several trajectories (in R and r) that are periodic. These trajectories are not stable but they nevertheless affect strongly tire quantum dynamics. A study of tlie resonances in H+H2 scattering as well as many other triatomic systems (see, e.g., [99]) reveals that the scattering peaks are closely related to tlie frequencies of the periodic orbits and the resonance wavefiinctions are large in the regions of space where the periodic orbits reside. 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS IN INTRAMOLECULAR AND DISSOCIATION DYNAMICS... 

Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos, Tlie application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. 

The interrelations between the propagator, the resolvent, and the level density will be central to our discussion. In particular, the trace formulas referred to in Section I represent semiclassical approximations to the quantities (2.6) or (2.7) and turn out to involve the periodic orbits of the classical dynamics. 

Figure 2 depicts the vibrogram corresponding to the dynamics on the ground state of iodine, modeled by a Morse potential with the equilibrium distance r = 2.67 A and the dissociation energy D = 12,542 cm 1 [14, 108]. The periodic orbit and its repetitions clearly appear in the vibrogram. The classical... 

In tetra-atomic molecules that are linear, the number of degrees of freedom is F =7, which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. 

Other classically chaotic scattering systems have been shown to have repellers described by a symbolic dynamics similar to (4.10). One of them is the three-disk scatterer in which a point particle undergoes elastic collisions on three hard disks located at the vertices of an equilateral triangle. In this case, the symbolic dynamics is dyadic (M = 2) after reduction according to C)V symmetry. Another example is the four-disk scatterer in which the four disks form a square. The C4 symmetry can be used to reduce the symbolic dynamics to a triadic one based on the symbols 0,1,2), which correspond to the three fundamental periodic orbits described above [14]. 

On the other hand, we should mention that, at the level of classical mechanics, periodic-orbit analysis provides a topological characterization of the system in terms of a symbolic dynamics, which appears as a common feature for a given class of systems. 

The interpretation of a spectrum from a dynamical point of view can also be applied to a spectrum containing a broad feature associated with direct and/or indirect dissociation reactions. From such spectra dynamics of a dissociating molecule can also be extracted via the Fourier transform of a spectrum. An application of the Fourier transform to the Hartley band of ozone by Johnson and Kinsey [3] demonstrated that a small oscillatory modulation built on a broad absorption feature contains information of the classical trajectories of the vibrational motion on PES, so-called unstable periodic orbits, at the transition state of a unimolecular dissociation. 

The classical picture of how the periodic orbits influence the dissociation dynamics goes as follows The majority of trajectories that we start randomly in the FC region immediately dissociate. Some of them, however, set out very close to a periodic orbit and therefore they stay in its vicinity for at least one full round. If a trajectory were accidentally to begin exactly on a periodic orbit, it would follow this orbit forever, or until numerical inaccuracies accumulate and ultimately throw it out of the periodic orbit ... 

Fig. 8.11. (cont.) the unstable periodic orbit, represented by the solid line, influences the dissociation dynamics all direct trajectories, which fragment immediately without any recurrence, are discarded. The times range from 0 fs in (a) to 50.8 fs in (h). The arrows schematically indicate the evolution of the classical wavepacket and the heavy dot marks the equilibrium of the R-state potential energy surface. Adapted from Weide, Kiihl, and Schinke (1989). 

Further insights into reaction dynamics can be obtained by analyzing the stability of classical trajectories. Presumably, stable periodic orbits will be restricted to KAM tori and therefore be nonreactive and unstable periodic orbits will provide information about the location of resonances and therefore some quahtative features of the intramolecular energy flow. 

The purpose of this chapter is to review some properties of isomerizing (ABC BCA) and dissociating (ABC AB + C) prototype triatomic molecules, which are revealed by the analysis of their dynamics on precise ab initio potential energy surfaces (PESs). The systems investigated will be considered from all possible viewpoints—quanmm, classical, and semiclassical mechanics—and several techniques will be applied to extract information from the PES, such as Canonical Perturbation Theory, adiabatic separation of motions, and Periodic Orbit Theory. 

The studies of Wiesenfeld [28] and Lai et al. [43] on the classical dynamics of a one-electron atom in a sinusoidal external field provide a physically realistic example in which the presence of KAM tori surrounding stable periodic orbits leads to deviations from the generic behaviour characteristic of a hyperbolic scattering system as discussed in Sect. 2. Although this system (10) seems simple, further studies illuminating the mathematical structures behind the scattering process, e.g. calculation of the Liapunov exponents of the unstable trapped orbits and the fractal dimension of the trapped set, have yet to be performed. 

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