Topological saddles

When the equilibrium state is topologically saddle, condition (C.2.8) distinguishes between the cases of a simple saddle and a saddle-focus. However, when the equilibrium is stable or completely unstable, the presence of complex characteristic roots does not necessarily imply that it is a focus. Indeed, if the nearest to the imaginary axis (i.e. the leading) characteristic root is real, the stable (or completely imstable) equilibrium state is a node independently of what other characteristic roots are. 

Points on the zero-flux surfaces that are saddle points in the density are passes or pales. Should the critical point be located on a path between bonded atoms along which the density is a maximum with respect to lateral displacement, it is known as a pass. Nuclei behave topologically as peaks and all of the gradient paths of the density in the neighborhood of a particular peak terminate at that peak. Thus, the peaks act as attractors in the gradient vector field of the density. Passes are located between neighboring attractors which are linked by a unique pair of trajectories associated with the passes. Cao et al. [11] pointed out that it is through the attractor behavior of nuclei that distinct atomic forms are created in the density. In the theory of molecular structure, therefore, peaks and passes play a crucial role. 

The electron density is a continuous function that is experimentally observable, hence uniquely defined, at all points in space. Its topology can be described in terms of the distribution of its critical points, i.e. the points at which the electron density has a zero gradient in all directions. There are four kinds of critical point which include maxima (A) usually found near the centres of atoms, and minima (D) found in the cavities or cages that lie between the atoms. In addition there are two types of saddle point. The first (B) represents a saddle point that is a maximum in two directions and a minimum in the third, the second (C) represents a saddle point that is a minimum in two direction and a maximum in the third. One can draw lines of steepest descent connecting the maxima (A) to the minima (D), lines whose direction indicates the direction in which the electron density falls off most rapidly. Of the infinite number of lines of steepest descent that can be drawn there exists a unique set that has the property that, in passing from the maximum to the minimum, each line passes successively through a B and a C critical point. This set forms a network whose nodes are the critical points and whose links are the lines of steepest descent connecting them. 

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... 

Fig. 4. The Petersen graph for the topology of the G 0 (g 0 h) JT surfaces for preferential H coupling by Ceulemans and Fowler [29]. The numbered 10 vertices of the graph represent the 10 trigonal minima (ft orbits). The centers of the edges represent the D2 saddles (8 orbits). There are two types of tunneling between wells one between adjacent wells (minima connected with edges in the graph) the other between non-adjacent wells.

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. 

Figure 4.27 shows residue curve maps for the reactive reboiler at three different Damkohler numbers. In the nonreactive case (Da = 0 Fig. 4.27(a)), the map topology is structured by one unstable node (pure B), one saddle point (pure C), and one stable node (pure A). Since pure A is the only stable node of nonreactive distillation, this is the feasible bottom product to be expected in a continuous distillation process. 

Residue curve maps of the THF system were predicted for reactive distillation at different reaction conditions (Fig. 4.29). The topology of the map at nonreactive conditions (Da = 0) is structured by a binary azeotrope (unstable node) between water and THF. Pure water and pure THF are saddle nodes, while the 1,4-BD vertex is a stable node. 

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