Quasiperiodic trajectories

Wolf R J and Hase W L 1980 Quasiperiodic trajectories for a multidimensional anharmonic classical Hamiltonian excited above the unimolecular threshold J. Chem. Phys. 73 3779-90... 

Figure 9.14 Nested invariant tori and a surface of section. Each dotted circle corresponds to a quasiperiodic trajectory at different E. When the radius of the circle collapses to zero, the trajectory is periodic. The dotted trajectory shown looping about the outermost invariant torus, will eventually fill the surface of that torus and generate a solid circle on the surface of section (from Child, 1991).

The fixed points on the phase space diagrams or phase spheres in Fig. 9.13 are labeled A, B, Ca, and C. Each corresponds to a periodic orbit that is said to organize the surrounding region of phase space that is filled with topologically similar quasiperiodic trajectories. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

Each trajectory is launched at chosen initial values of Jb and ipb and at fra = 0. Since any point on the 3-dimensional energy shell may be specified by three linearly independent coordinates, selection of initial values J , ip%, and ip°, implies a definite value of J°. Thus trajectories are launched at various [Jfi, ip%, ip° = 0, J°(J , ipl, ip°,) ] initial values until all of the qualitatively distinct regions of phase space are represented on the surface of section by either a family of closed curves (quasiperiodic trajectories) that surround a fixed point (a periodic trajectory that defines the qualitative topological nature of the neighboring quasiperiodic trajectories) or an apparently random group of points (chaos). Often, color is used to distinguish points on the surface of section that belong to different trajectories. 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

Figure 27 The torus attractor generated by quasiperiodic trajectories. In (a) a quasiperiodic trajectory is shown making slightly more than one pass around the torus. As can be seen on the front left side of the torus attractor, the quasiperiodic trajectory does not exactly match up with its first pass the eventual result is that the surface of the torus will be covered completely by the quasiperiodic trajectory. TTie Poincare surface of section shown in (b) results in a circular or ellipsoidal cross section (c).

FIGURE 11 Sample quasiperiodic trajectory in a two-degrees-of-freedom system as it moves on the surface of a torus in phase space. The trajectory shown is actually periodic in general, the trajectory will fill the entire torus surface. 

The previous analysis is confirmed when one measures the quasiclassical probability Pn(t) for remaining in the initial state (n, 0) for an ensemble of two hundred trajectories defined with initial conditions J = n + 1/2, = random J = 1/2, = random. The decay is close -fo exponential when the central fixed point is unstable and on the contrary it is distinctively non exponential with prominent oscillations ("beats") when the fixed point is stable. Thus, the sensitivity of short time relaxation to potential energy coupling derives from the drastic effect of a change from instability to stability on the stretch-bend energy flow in quasiperiodic trajectories. In this way, we have shown that the short time overtone decay dynamics of the two mode model exhibit the same sensitivity to potential energy coupling as does the trajectory calculations for the full planar benzene Hamiltonian of Hase and coworkers(10). 

The fundamental question of what distinguishes systems with simple dynamics from systems with chaotic dynamics can only be answered if we can correspond certain types of trajectories to physically observable processes. We began the classification with the study of quasiperiodic trajectories (Chap. 4 in the first part of this book). Even though these trajectories are non-rough, they were shown to model adequately such phenomena as beats and modulations. 

Quasiperiodic trajectories are a special case of Poisson-stable trajectories. The latter plays one of the leading roles in the theory of dynamical systems as they form a large class of center motions in the sense of Birkhoff (Sec. 7.2). Birkhoff had partitioned the Poisson-stable trajectories into a number of subclasses. This classification is schematically presented in Sec. 7.3. Having chosen this scheme as his base, as early as in the thirties, Andronov had undertaken an attempt to collect and correlate all known types of dynamical motions with those observable from physical experiments. Since his arguments were based on the notion of stability in the sense of Lyapunov for an individual trajectory, Andronov had soon come to the conclusion that all possible Lyapunov-stable trajectories are exhausted by equilibrium states, periodic orbits and almost-periodic trajectories (these are quasiperiodic and limit-quasiperiodic motions in the finite-dimensional case). 

The structure of the minimal set of a limit-quasiperiodic trajectory is a fractal. In other words, it is characterized locally as a direct product of an m-dimensional disk and a zero-dimensional Cantor set K. Obviously, in the limit-periodic case, it has the form of a direct product of an interval and K. 

In recent work we have found that it is probably not necessary to use small step sizes for unimolecular trajectories. This is because an ensemble of stochastic trajectories yields the same result regardless of whether the trajectories are backward integrable. In fact it is the stochastic (unstable) trajectories which require a small step size (e,g., as small as 5 x 10 " s) to achieve backward integration. Quasiperiodic trajectories, on the other hand, "feel a separable Hamiltonian and a large step size of about 1/20 of the shortest vibrational period a step size of 5 x 10 s for a... 

In section III the model H-C-C H + C=G potential energy surfaces and the methodology of the classical trajectory calculations are described. The trajectory unimolecular lifetime distributions and rate constants for the different surfaces are compared in section IV. Also discussed in section IV are the observed quasiperiodic trajectories for some of the potential energy surfaces. The calculated product energy distributions are presented in section V along with a dynamical model which explains the product energy partitioning. 

A very good correspondence is found between the nature of the unimolecular lifetime distribution and the fraction of trajectories that are quasiperiodic. Surface VA, which has the most intrinsically non-RRKM lifetime distribution, also contains the largest fraction of quasiperiodic trajectories. The fraction of quasiperiodic trajectories is negligibly small for the surfaces with intrinsically RRKM lifetime distributions. A summary of our findings is given in Table 4. The A and B surfaces are the ones with the largest number of quasiperiodic trajectories, and these surface types are most similar to the ethyl radical potential energy surface. 

In conclusion, the work presented here has illustrated the effect of various potential energy surface properties on unimolecular dynamics. The calculations show that the energy distributions of the unimolecular products contain very little information about the intramolecular dynamics of the species undergoing unimolecular dissociation. It was also found that anharmonic potential energy surfaces may contain large fractions of quasiperiodic trajectories above the unimolecular threshold. 

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