Phase space separatrix

With this brief overview of classical theories of unimolecular reaction rate, one might wonder why classical mechanics is so useful in treating molecular systems that are microscopic, and one might question when a classical statistical theory should be replaced by a corresponding quantum theory. These general questions bring up the important issue of quantum-classical correspondence in general and the field of quantum chaos [27-29] (i.e., the quantum dynamics of classically chaotic systems) in particular. For example, is it possible to translate the above classical concepts (e.g., phase space separatrix, NHIM, reactive islands) into quantum mechanics, and if yes, how What is the consequence of... 

The Davis-Gray theory teaches us that by retaining the most important elements of the nonhnear reaction dynamics it is possible to accurately locate the intramolecular bottlenecks and to have an exact phase space separatrix as the transition state. Unfortunately, even for systems with only two DOFs, there may be considerable technical difficulties associated with locating the exact bottlenecks and the separatrix. Exact calculations of the fluxes across these phase space structures present more problems. For these reasons, further development of unimolecular reaction rate theory requires useful approximations. 

The fundamental difference between the ARRKM theory and the traditional RRKM theory is that the former utilizes a phase-space separatrix rather than a... 

Figure Al.2.12. Energy level pattern of a polyad with resonant collective modes. The top and bottom energy levels conespond to overtone motion along the two modes shown in figure Al.2.11. which have a different frequency. The spacing between adjacent levels decreases until it reaches a minimum between the third and fourth levels from the top. This minimum is the hallmark of a separatrix [29, 45] in phase space.

There are also two local bifurcations. The first one takes place for r 13.926..., when a homoclinic tangency of separatrixes of the origin O occurs (it is not shown in Fig. 20) and a hyperbolic set appears, which consists of a infinite number of saddle cycles. Beside the hyperbolic set, there are two saddle cycles, L and L2, around the stable states, Pi and P2. The separatrices of the origin O reach the saddle cycles Li and L2, and the attractors of the system are the states Pi and P2. The second local bifurcation is observed for r 24.06. The separatrices do not any longer reach to the saddle cycles L and L2. As a result, in the phase space of the system a stable quasihyperbolic state appears— the Lorenz attractor. The chaotic Lorenz attractor includes separatrices, the saddle point O and a hyperbolic set, which appears as a result of homoclinic tangency of the separatrices. The presence of the saddle point in the chaotic... 

Figure 21. Structure of the phase space of the Lorenz system. An escape trajectory measured by numerical simulation is indicated by the filled circles. The trajectory of the Lorenz attractor is shown by a thin line the separatrixes T and T2 by dashed lines [182].

Gray, Rice, and Davis [12] developed an alternative RRKM (ARRKM) theory in an attempt to simplify the Davis-Gray theory for van der Waals predissociation reactions. Specifically, they replaced the exact separatrix with an approximate phase space dividing surface by dropping a number of small terms in the system Hamiltonian, and they replaced the exact mapping that defines the flux across the tme separatrix with an analytic treatment of the flux across the approximate separatrix. This simplification is schematically presented in Fig. 18. 

The separation of the phase-space inside the separatrix into a region inside and a region outside the intramolecular energy transfer dividing surface defines the initial conditions N 0) and A 2(0), and leads us to write... 

Clearly, the A and B isomer states should be inside the separatrix, and the state C should be in the phase-space region outside of the separatrix but inside the energy boundary. A schematic diagram of this three-state isomerization model is presented in Fig. 20. From the results of previous analyses of predissociation we expect that within the A and B domains there are, in general, intramolecular bottlenecks to energy transfer. However, these bottlenecks are... 

Figure 22. Schematic mechanism of reaction including intramolecular energy transfer. The phase-space of state A is partitioned into Aj and A2. Si is a representation of an intramolecular energy transfer dividing surface, and S2 is the A-state separatrix.

There are a number of open issues associated with statistical descriptions of unimolecular reactions, particularly in many-dimensional systems. One fundamental issue is to find a qualitative criterion for predicting if a reaction in a many-dimensional system is statistical or nonstatistic al. In a recent review article, Toda [17] discussed different aspects of the Arnold web — that is, the network of nonlinear resonances in many-dimensional systems. Toda pointed out the importance of analyzing the qualitative features of the Arnold web— for example, how different resonance zones intersect and how the intersections further overlap with one another. However, as pointed out earlier, even in the case of fully developed global chaos it remains challenging to define a nonlocal reaction separatrix and to calculate the flux crossing the separatrix in a manydimensional phase-space. 

As mentioned earlier, Komatsuzaki and Berry [26] have recently developed a promising approach to analyzing many-dimensional reacting systems. By seeking appropriate canonical transformations that yield local approximate constants of motion associated with the reactive mode, they were able to transform the conventional dividing surface in configuration space to a manydimensional separatrix in phase-space. Specifically, suppose the original phase-space variables are denoted by = q, q2,..., qn ,P, Pi, , Pn) the... 

With these properties, if locally separates phase space, as illustrated in the scheme (Figure 1). It is very important to note that even if if has codimension 1 and is locally a separatrix, it does mean in n DOFs that if neither has a simple geometry, because it is subject to stretching and folding because of chaos [24-26], nor separates globally (see Ref. 27). Let us now make a summary of the... 

As for the dynamics of JC under the unperturbed Hamiltonian Hq x,I), we assume that the reaction coordinate jc has a saddle X I) = (Q I),P I)). Its location, in general, depends on the action variables 7. Suppose that the saddle X I) has a separatrix orbit JCo(t,7) connecting it with itself. See Fig. 9 for a schematic picture of the phase space jc = (q,p) under the unperturbed Hamiltonian Hq x,I). Here, we show the saddle X and the separatrix orbit on the two-dimensional phase space jc = (q,p). 

Fig. 5.7. Qualitative sketch of the pendulum phase space indicating libration and rotation regions. The separatrix, in this case a one-dimensional curve, separates the two regions.

These calculations reveal another interesting property. As the complexity of the plots increases, chaotic trajectories appear, which grow (as expected) from the separatrix outwards. The presence of such orbits heralds new paths for ionisation, which may eventually dominate the whole of phase space. 

Although in classical mechanics E may be varied continuously, the trajectories are displayed only for the values of E that are quantum eigen-energies. The dotted /z(V0 line is a separatrix, which is a dividing surface that no trajectory can cross and which divides the accessible phase space into qualitatively distinct regions (exhibiting normal vs. local mode behavior) filled with qualitatively different 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... 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

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