Saddle regions defined

In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. All the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet... 

Here, the center manifold is defined by the equation y = 0. The surfaces x = constant are the leaves of the strong-stable invariant foliation In particular, x = 0 is the equation of the strong-stable manifold of O. At fi — Oj the function g (nonlinear part of the map on W ) has a strict extremum at X = 0. For more definiteness, we assume that it is a minimum, i.e. y(x, 0) > 0 when X 0. Thus, the saddle region on the cross-section corresponds to x > 0, and the node region corresponds to x < 0. Since the saddle-node disappears when /Lt > 0, it follows that y(x,/x) > 0 for all sufficiently small x and for all small positive //. 

The transition state concept, once understood in static terms only, as the saddle point separating reactants and products, may be fruitfully expanded to encompass the transition region, a landscape in several significant dimensions, one providing space for a family of trajectories and for a significant transition state lifetime. The line between a traditional transition structure and a reactive intermediate thus is blurred The latter has an experimentally definable lifetime comparable to or longer than some of its vibrational periods. 

Figure 19. Scattering resonances of 2F collinear models of the dissociation of H3 obtained by Manz et al. [143] on the Karplus-Porter surface and by Sadeghi and Skodje on a DMBE surface [132], The energy is defined with respect to the saddle point. The dashed lines mark the bifurcations of the transition to chaos 1, 5, and 3 are the numbers of shortest periodic orbits, or PODS, in each region.

Type II trajectories start at a point p in the internuclear region between two bonded atoms and end at one of the two nuclei in question. There are just two trajectories per bond, which together define a path of maximum electron density (MED path) that is visible in the perspective drawing of p r) shown in Figure 9. Each lateral displacement from the MED path leads to a decrease of p(r). The point p corresponds to the minimum of p(r) along the path and to a saddle point of p(r) in three dimensions. 

Manz and Romelt (1981). Rm and 7 hi are the two I-H bond distances. The heavy point marks the saddle point and the shaded area indicates schematically the Franck-Condon region in the photodetachment experiment. The arrow along the symmetric stretch coordinate (f Hi = -Rih) illustrates the early motion of the wavepacket and the two heavy arrows manifest dissociation into the two identical product channels, (b) The same PES as in (a) but represented in terms of hyperspherical coordinates (p, i9) defined in (7.33). The horizontal and the vertical arrows illustrate symmetric and anti-symmetric stretch motions, respectively, as indicated by the two insets. 

Figure 8.2 depicts a typical potential energy surface (PES) for a symmetric molecule ABA with intramolecular bond distances R and R2] the ABA bond angle is assumed to be 180° (collinear configuration). The PES is symmetric with respect to the line defined by Ri = R2 it has a saddle point at short distances and decreases monotonically from the saddle point out into the two identical product channels A + BA and AB + A (see also Figure 7.18). The shaded area indicates the Franck-Condon (FC) region accessed via photon absorption and the two arrows illustrate the main dissociation paths for the quantum mechanical wavepacket or, equivalently, a swarm of classical trajectories. Because no barrier obstructs dissociation, the majority of trajectories immediately evanesce in either one of the two product channels without ever returning to the vicinity of the FC point.

A time-independent adiabatic approximation, based on the local separability of symmetric and anti-symmetric stretch motion in the region of the saddle point, provides a complementary picture (Pack 1976). Within the adiabatic limit the eigenenergies of the symmetric stretch motion on top of the potential ridge are defined through the one-dimensional... 

The transition region around the saddle point has the width / and the time for the passage of one complex will then be l/ vx)+ (i.e., the lifetime of the complex, the time it takes to change from reactants to products). So the number of passages per second will therefore be the reciprocal of that time, that is, the frequency with which the complexes pass over the barrier is (vx)+/l. In order to get the rate of the reaction which is defined as a change in concentration per unit time, [(AB) ] is multiplied by this frequency or, equivalently, divided by the lifetime. Thus,... 

In the first step, we define the relevant activated complexes as microcanonical transition states having a total energy H = E and a value for the reaction coordinate qi that lies between q and q + dq. The separation of the reaction coordinate from the other degrees of freedom in the saddle-point region implies that the Hamiltonian in this region can be written as... 

Except for uphill procedures, the initial geometry must lie within the descent region of the transition structure, i.e., the region from which a certain stationary point can be reached. The size of the descent region depends on the method used however, it is difficult to define it explicitly for a multidimensional surface. There is still another difficulty in the search for saddle points. In systems with possible internal rotations or inversions, saddle points associated with this type of motion are often found instead of the saddle point of the reaction under study (134,135). This difficulty can so far be overcome only by forcing the search direction on the basis of rather vague chemical intuition. 

Contour plots of two-dimensional functions help illustrate these concepts. In general, the equation f(x) — y defines a surface in R"+I. When n - 2, the plane curves corresponding to various values of y generate contour plots (or maps) of a function. Figure 4 shows the contour plots for the two-dimensional functions discussed before. Note, for the first, the two stationary points corresponding to a minimum and saddle point. For the second, note the region of weak local minima. The contour plots are shaded so that darker areas correspond to higher function values. 

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