Saddle regions distribution

Figure 3. Distributions of short-term kinetic energies, expressed as effective temperatures, for Ar3, at total energies corresponding to (a) 28.44 K and (b) 30.54 K. The low kinetic energies correspond to trajectory segments in the saddle region the high kinetic energy parts of the distribution are associated with motion above the deep well of the equilibrium geometry. [Reprinted with permission from T. L. Beck, D. M. Leitner, and R. S. Berry, J. Chem. Phys. 89, 1681 (1988). Copyright 1988, American Institute of Physics.]...

The next two figures reveal how these overall cluster properties vary with the size of the cluster, even for quite small clusters. Figure 9 shows the same kind of distributions as Fig. 8, but for the four-particle Lennard-Jones cluster, and Figs. 10 and 11 do the same for the 5-particle cluster, a system with two kinds of saddles (but only one locally stable structure), so two sets of distributions are shown there. Only the distributions over the higher-energy saddle show any detectable differences from the distributions elsewhere on the surface. With still larger clusters, the distinctions between saddle regions and the other parts of the surface essentially disappear. 

Figure 12. The same four distributions as in Fig. 8, now for a six-particle system. These are shown for five stages along the pathway from the capped trigonal bipyramid (top band in each panel) through an approach to the bipyramid-to-octahedron saddle, to the saddle region itself (middle band), then to the exit region toward the octahedron and, finally (at bottom), the behavior in the well around the octahedral global minimum structure. [Reprinted with permission from R. J. Hinde and R. S. Berry, J. Chem. Phys. 99, 2942 (1993). Copyright 1993, American Institute of Physics.]...

Liquid-Fluid Equilibria Nearly all binary liquid-fluid phase diagrams can be conveniently placed in one of six classes (Prausnitz, Licntenthaler, and de Azevedo, Molecular Thermodynamics of Fluid Phase Blquilibria, 3d ed., Prentice-Hall, Upper Saddle River, N.J., 1998). Two-phase regions are represented by an area and three-phase regions by a line. In class I, the two components are completely miscible, and a single critical mixture curve connects their criticsu points. Other classes may include intersections between three phase lines and critical curves. For a ternary wstem, the slopes of the tie lines (distribution coefficients) and the size of the two-phase region can vary significantly with pressure as well as temperature due to the compressibility of the solvent. 

Since the basin of attraction of the CA is bounded by the saddle cycle SI, the situation near SI remains qualitatively the same and the escape trajectory remains unique in this region. However, the situation is different near the chaotic attractor. In this region it is virtually impossible to analyze the Hamiltonian flux of the auxiliary system (37), and no predictions have been made about the character of the distribution of the optimal trajectories near the CA. The simplest scenario is that an optimal trajectory approaching (in reversed time) the boundary of a chaotic attractor is smeared into a cometary tail and is lost, merging with the boundary of the attractor. 

As a consequence of the strong interaction of the two electronic states the shape of the dissociative PES in the saddle point region differs noticeably from the dissociative PES for H20(A) [see Figure 15.6(a)], The saddle is considerably narrower than for H2O. Although the differences are rather subtle, they explain qualitatively the dissimilarity of the final vibrational state distributions. Trajectories starting near the tran-... 

Incidentally we note that Levene, Nieh, and Valentini (1987) measured vibrational state distributions for O2 following the photodissociation of O3 in the Chappuis band which qualitatively resemble those shown in Figure 9.11 for H2S. In order to interpret their results the authors surmised a model PES that actually has the same overall topology in the saddle point region as the one calculated for H2S. 

The steady-state molecular distribution near the saddle point may be written explicitly by invoking two assumptions valid in that region. First, because the reactive and nonreactive modes are nearly uncoupled, the molecular distribution is written as a product of both. Second because near the barrier the reactive flux is directed along the reactive mode coordinate, the distribution function of the n — 1 nonreactive mode system is approximated by its... 

One of these reactions was H + Cl2 (Miller and Light, 1971a, b Light, 1971a, b). In this case the number of open channels is different for reactants and products and a certain number of closed channels must be used with the exponential method in the intermediate region. Four aspects of the surface were varied the path s curvature steepness of the potential position of the barrier and vibrational force constant around the saddle point or col. Vibrational distributions of products were determined, besides the dependence of probabilities on collision energy. 

For the infinitely distributed feed column the feasible region is as large as it can possibly be, bound by the locus of saddle nodes of the internal CSs TTs. Notice that roles of the two product producing CSs have been increasingly dimini ed as the number of feed points are increased. They are almost not needed at all in Figure 6.8 ... 

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