Chaotic classical dynamics

In coimection with the energy transfer modes, an important question, to which we now turn, is the significance of classical chaos in the long-time energy flow process, in particnlar the relative importance of chaotic classical dynamics, versus classically forbidden processes involving dynamical tuimelling . 

The question of non-classical manifestations is particularly important in view of the chaos that we have seen is present in the classical dynamics of a multimode system, such as a polyatomic molecule, with more than one resonance coupling. Chaotic classical dynamics is expected to introduce its own peculiarities into quantum spectra [29, 77]. In Fl20, we noted that chaotic regions of phase space are readily seen in the classical dynamics corresponding to the spectroscopic Flamiltonian. Flow important are the effects of chaos in the observed spectrum, and in the wavefiinctions of tire molecule In FI2O, there were some states whose wavefiinctions appeared very disordered, in the region of the... 

One-electron atoms subjected to a time-dependent external field provide physically realistic examples of scattering systems with chaotic classical dynamics. Recent work on atoms subjected to a sinusoidal external field or to a periodic sequence of instantaneous kicks is reviewed with the aim of exposing similarities and differences to frequently studied abstract model systems. Particular attention is paid to the fractal structure of the set of trapped unstable trajectories and to the long time behavior of survival probabilities which determine the ionization rates of the atoms. Corresponding results for unperturbed two-electron atoms are discussed. 

The ultrafast initial decay of the population of the diabatic S2 state is illustrated in Fig. 16 for the first 30 fs. Since the norm of the semiclassical wave function is only approximately conserved, the semiclassical results are displayed as rough data (dashed line) and normalized data (dotted line) [i.e. pnorm P2/ Pi + P2)]. The normalized results for the population are seen to match the quantum reference data quantitatively. It should be emphasized that the deviation of the norm shown in Fig. 16 is not a numerical problem, but rather confirms the common wisdom that a two-level system as well as its bosonic representation is a prime example of a quantum system and therefore difficult to describe within a semiclassical theory. Nevertheless, besides the well-known problem of norm conservation, the semiclassical mapping approach clearly reproduces the nonadiabatic quantum dynamics of the system. It is noted that the semiclassical results displayed in Fig. 16 have been obtained without using filtering techniques. Due to the highly chaotic classical dynamics of the system, therefore, a very large number of trajectories ( 2 x 10 ) is needed to achieve convergence, even over... 

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... 

The highly excited and reactive dynamics, the details of which have been made accessible by recently developed experimental techniques, are characterized by transitions between classically regular and chaotic regimes. Now molecular spectroscopy has traditionally relied on perturbation expansions to characterize molecular energy spectra, but such expansions may not be valid if the corresponding classical dynamics turns out to be chaotic. This leads us to a reconsideration of such perturbation techniques and provides the starting point for our discussion. From there, we will proceed to discuss the Gutzwiller trace formula, which provides a semiclassical description of classically chaotic systems. 

The analysis of the classical dynamics shows a transition to chaotic motion leading to diffusion and ionization [6]. In the quantum case, interference effects lead to localization and the quantum distribution reaches a steady state that is exponentially localized (in the number of photons) around the initially excited state. As a consequence, ionization will take place only when the localization length is large enough to exceed the number of photons necessary to reach the continuum. 

In Figure 6.15 we present a representative example of phase control in this system. In the chosen parameter region the underlying classical dynamics of rota- tional excitation is strongly chaotic [231 ] and the excitation is far off-resonance, with many levels excited. We choose j = 1 and j2 =2 to create the initial superposition y state ( 1, 0) 2, 0))/V2, that is, a = 7t/4 and p = 0, % in Eq. (6.69). (Such states 1 can be prepared experimentally by, for example, STIRAP, a technique discussed in detail in Section 9.1.) The results, shown in Figure 6.15, display striking phase... 

The classical dynamics of a system can also be analyzed on the so-caUed Poincare surface of section (PSS). Hamiltonian flow in the entire phase space then reduces to a Poincare map on a surface of section. One important property of the Poincare map is that it is area-preserving for time-independent systems with two DOFs. In such systems Poincare showed that all dynamical information can be inferred from the properties of trajectories when they cross a PSS. For example, if a classical trajectory is restricted to a simple two-dimensional toms, then the associated Poincare map will generate closed KAM curves, an evident result considering the intersection between the toms and the surface of section. If a Poincare map generates highly erratic points on a surface of section, the trajectory under study should be chaotic. The Poincare map has been a powerful tool for understanding chemical reaction dynamics in few-dimensional systems. 

As shown above, classical unimolecular reaction rate theory is based upon our knowledge of the qualitative nature of the classical dynamics. For example, it is essential to examine the rate of energy transport between different DOFs compared with the rate of crossing the intermolecular separatrix. This is also the case if one attempts to develop a quantum statistical theory of unimolecular reaction rate to replace exact quantum dynamics calculations that are usually too demanding, such as the quantum wave packet dynamics approach, the flux-flux autocorrelation formalism, and others. As such, understanding quantum dynamics in classically chaotic systems in general and quantization effects on chaotic transport in particular is extremely important. 

With chaos in classical dynamical systems well established, the question arises whether quantum systems axe able to display exponential sensitivity and chaos. The answer is that most quantum systems do not. Not even if their classical counterparts axe chaotic. We can say that chaos is suppressed on the quantum level. An example of this suppression... 

The salient features of the dynamics of our model molecule are best exhibited with the help of Poincare sections that have already proved useful in the analysis of the double pendulum presented in Section 3.2. Fig. 4.7 shows the rpp projection of an x = 0 surface of section of a trajectory for a = 0.1, uq = 10 and E = 4 started at 0 = 0.957T, x = sin(0), y = 0, z = cos(0) and tj = 1.42. The resulting y-p Poincare section clearly shows chaotic features. This indicates that the classical dynamics of the skeleton of the model molecule is chaotic. But the most striking feature of the model molecule is its fully chaotic quantum dynamics. This is proved by Fig. 4.8, which shows the chaotic quantum fiow of the molecule on the southern hemisphere of the Bloch sphere. Fig. 4.8 was produced in the following way. First we defined the Poincar6 section by p = 0, dp/dt > 0. Then, we ran 40 trajectories in x,y,z,r],p) space for a = 0.1, Uo = 10 and E = Q starting at the 40 different initial conditions... 

If the classical dynamics is ergodic, with a chaotic sampling of all phase space on a timescale that is short compared to the dissociation timescale, then the rate coefficient is independent of time. Furthermore, the rate coefficient may then be expressed as an average, over the statistical distribution Pstatiq,p), of the flux through an arbitrary dividing surface (S=0) separating reactants (S<0) from products (S>0)... 

Initiated by the work of Bunker [323,324], extensive trajectory simulations have been performed to determine whether molecular Hamiltonians exhibit intrinsic RRKM or non-RRKM behavior. Both types have been observed and in Fig. 43 we depict two examples, i.e., classical lifetime distributions for NO2 [271] and O3. While Pd t) for NO2 is well described by a single-exponential function — in contrast to the experimental and quantum mechanical decay curves in Fig. 31 —, the distribution for ozone shows clear deviations from an exponential decay. The classical dynamics of NO2 is chaotic, whereas for O3 the phase space is not completely mixed. This is in accord with the observation that the quantum mechanical wave... 

Comparisons between state-specific quantum mechanical and classical calculations have been made for four systems, HO2 [60], NO2 [271], HNO [39], and HCO [51]. For the first three systems the quantum dynamics is statistical state-specific and the classical dynamics is in essence irregular above the dissociation threshold HCO is an example of mode-specific quantum mechanical behavior and the classical phase space is certainly not completely chaotic. 

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