Phase-space transition states examples

As discussed in section A3.12.2. intrinsic non-RRKM behaviour occurs when there is at least one bottleneck for transitions between the reactant molecule s vibrational states, so drat IVR is slow and a microcanonical ensemble over the reactant s phase space is not maintained during the unimolecular reaction. The above discussion of mode-specific decomposition illustrates that there are unimolecular reactions which are intrinsically non-RRKM. Many van der Waals molecules behave in this maimer [4,82]. For example, in an initial microcanonical ensemble for the ( 211 )2 van der Waals molecule both the C2H4—C2H4 intennolecular modes and C2H4 intramolecular modes are excited with equal probabilities. However, this microcanonical ensemble is not maintained as the dimer dissociates. States with energy in the intermolecular modes react more rapidly than do those with the C2H4 intramolecular modes excited [85]. 

One of the problems with VMC is that it favors simple states over more complicated states. As an example, consider the liquid-solid transition in helium at zero temperature. The solid wave function is simpler than the liquid wave function because in the solid the particles are localized so that the phase space that the atoms explore is much reduced. This biases the difference between the liquid and solid variational energies for the same type of trial function, (e.g. a pair product form, see below) since the solid energy will be closer to the exact result than the liquid. Hence, the transition density will be systematically lower than the experimental value. Another illustration is the calculation of the polarization energy of liquid He. The wave function for fully polarized helium is simpler than for unpolarized helium because antisymmetry requirements are higher in the polarized phase so that the spin susceptibility computed at the pair product level has the wrong sign ... 

The transitional mode Hamiltonian is given by the last four terms in equation (7.36). In the work of Wardlaw and Marcus, the phase space volume for the transitional modes was calculated versus the center of mass separation R, so that R is assumed to be the reaction coordinate. In recent work, Klippenstein (1990, 1991) has considered a more complex reaction coordinate. The multidimensional phase space volume for the transitional modes can not be determined analytically, but must be evaluated numerically, for example, by a Monte Carlo method of integration (Wardlaw and Marcus, 1984). The density of states is then obtained by dividing the phase space volume by h", where n is the dimensionality of the integral, and differentiating with respect to the energy. The total sum of states of the transition state is obtained by convoluting the density of the transitional modes with the sum of the conserved modes, N(E,J) so that... 

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... 

Oped a so-called reactive island theory the reactive islands are the phase-space areas surrounded by the periodic orbits in the transition state, and reactions are interpreted as occurring along cylindrical invariant manifolds through the islands. Fair et al. [29] also found in their two- and three-dof models of the dissociation reaction of hydrazoic acid that a similar cylinderlike structure emerges in the phase space as it leaves the transition state. However, these are crucially based on the findings and the existence of (pure) periodic orbits for all the dof, at least in the transition states. Hence, some questions remain unresolved, for example, How can one extract these periodic orbits from many-body dof phase space and How can the periodic orbits persist at high energies above the saddle point, where chaos may wipe out any of them ... 

This section opened with an example of the macroscopic theory which is based, of course, on the conservation laws. The "mesoscopic" description (a term due to VAN KAMPEN [2.93) permits knowledge not only of the average behavior of an aerosol but also of its stochastic behavior through so-called master equations. However, this mesoscopic level of description may require (in complex systems) some physical assumptions as to the transition probabilities between states describing the system. Finally, the microscopic approach attempts to develop the theory of an aerosol from "first principles"—that is, through study of the dynamics of molecular motion in a suitable phase space. Master equations and macroscopic theory appear from the microscopic theory by the reduction of the complete dynamical description of the system in a suitable phase space to small subsets of chosen variables. 

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