Phase space surfaces

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

As a result, a trajectory generated by the dynamics of (58) will not sample the entire phase space, but instead will sample a subspace of the entire phase space surface determined by the intersection of the hypersurfaces ylfc(x) = Cfc, where Ck is a set of constants. The microcanonical distribution function, that is generated by these systems, can be constructed from a product of 5-functions that represent these conservation laws ... 

Figure 10 Phase space surfaces of section at r, = r, pt > 0 for trajectories of the two-mode hydrazoic acid model at a total energy of 0.0610 au. Each frame is for a different family of initial states (a) p2 = 0 au (b) p2 = —8.905 au. Each number indicates the passage of a trajectory through this surface i.e., a 2 indicates the second pass of that trajectory. Note that trajectories cluster together by passage number.

Figure 12 Phase-space surface of section p, versus r, at r2 = r" at a total energy of 0.0610 au. The dashed line is the total energy boundary. The asterisks indicate the points where the trajectories of the perimeter family pass through this surface after leaving the transition state. The solid line is the constraint determined by Eq. (24) for a fixed E2 = 0.0700 au. Equation 24 accurately predicts both the location and distribution of the trajectories as they pass through this surface and thus describes the location of the reactive cylinder in phase space.

Table 5.1 Table of phase space surfaces influencing reaction dynamics and their representations in normal form coordinates on an energy surface of energy greater than the energy of the saddle equilibrium point... 

First, consider the case of trapped motion within a single isomer. The phase space of 2 is (always) an ellipse, which has the same topology as a onedimensional sphere (which a mathematician would name S ). However, the phase space of is also elliptical and has the same topology (S ). The topology of the two-dimensional phase-space surface on which the dynamics lies is the Cartesian product of these two, which is a two-dimensional torus, or a phase-space doughnut (T = SI X The toroidal geometry is shown in... 

Figure 8 Representative phase space surfaces for uncoupled two-state isomerization in two degrees of freedom. A reactive torus labeled spans both isomers, while...

In N degrees of freedom, a hierarchy of N integral invariants exists. For an arbitrary phase space surface S with symplectic projections consisting of 2 -dimensional volumes the th member of this hierarchy is of the form... 

In a series of studies our group demonstrated that not only could reactions be monitored online [76] and these data used for very rapid optimization [77], but also that this information could be used to plot local phase space in detail rarely attempted before. Several thousand data points per minute can be generated using appropriate in-line sampling and control techniques, allowing detailed phase space surfaces to be plotted [78]. This represents an increase in speed of over four orders of magnitude when compared to conventional methods. 

Cullen, C.J., et al. Rapid phase space surface generation using an integrated microfabricated device reaction detection system and automated control, in Micro Total Analytical Systems 2006 The 10th International Conference on Miniaturized Systems for Chemistry and Life Sciences, 2006, Tokyo Society for Chemistry and Micro-Nano Systems. 

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the FES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schtddinger equation should, however, be kept in mind. 

B(A) is the probability of observing the system in state A, and B(B) is the probability of observing state B. In this model, the space is divided exactly into A and B. The dividing hyper-surface between the two is employed in Transition State Theory for rate calculations [19]. The identification of the dividing surface, which is usually assumed to depend on coordinates only, is a non-trivial task. Moreover, in principle, the dividing surface is a function of the whole phase space - coordinates and velocities, and therefore the exact calculation of it can be even more complex. Nevertheless, it is a crucial ingredient of the IVansition State Theory and variants of it. 

One property of the exact trajectory for a conservative system is that the total energy is a constant of the motion. [12] Finite difference integrators provide approximate solutions to the equations of motion and for trajectories generated numerically the total energy is not strictly conserved. The exact trajectory will move on a constant energy surface in the 61V dimensional phase space of the system defined by. 

Another problem with microcononical-based CA simulations, and one which was not entirely circumvented by Hermann, is the lack of ergodicity. Since microcanoriical ensemble averages require summations over a constant energy surface in phase space, correct results are assured only if the trajectory of the evolution is ergodic i.e. only if it covers the whole energy surface. Unfortunately, for low temperatures (T << Tc), microcanonical-based rules such as Q2R tend to induce states in which only the only spins that can flip their values are those that are located within small... 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

This is exactly the autonomous linearized Hamiltonian (7), the dynamics of which was discussed in detail in Section II. One therefore finds the TS dividing surface and the full set of invariant manifolds described earlier one-dimensional stable and unstable manifolds corresponding to the dynamics of the variables A<2i and APt, respectively, and a central manifold of dimension 2N — 2 that itself decomposes into two-dimensional invariant subspaces spanned by APj and AQj. However, all these manifolds are now moving manifolds that are attached to the TS trajectory. Their actual location in phase space at any given time is obtained from their description in terms of relative coordinates by the time-dependent shift of origin, Eq. (42). 

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