Saddle point diagram

Figure 1.1 Transition-state saddle point diagram. Schematic representation of potential energy as a function of reaction coordinate.

Graphical representation of the saddle point (here marked with an X) for the transfer of atom B as the substance A-B reacts with another species, C. Potential energy is plotted in the vertical direction. Note also that the surface resembles a horse saddle, with the horn of the saddle closest to the observer. As drawn here, the dissociation to form three discrete species (A + B J- C) requires much more energy than that needed to surmount the path that includes the saddle point. A two-dimensional "slice" through a saddle point diagram is typically called a reaction-coordinate diagram or potential-energy profile. 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

Figure 5. Potential-energy diagram including zero-point energy for the HCC0 + 02 reaction. Energies of reactants and products ignore differences between and Intermediates species are denoted by Roman numerals, saddle points by Arabic numerals, and reactions paths are labeled A-F. Reproduced from [47] by permission of the PCCP Owner Societies.

The saddle-point equation leads to the momentum dependent dynamical quark mass Mf(k) = MfF2(k). Mf here is a function of current mass mf (M.M. Musakhanov, 2002). It was found that that M[m] is a decreasing function and for the strange quark with ms = 0.15 GeV Ms 0.5 Mu>d. This result in a good correspondence with (P. Pobylitsa, 1989), where another method was completely applied - direct sum is of planar diagrams. 

Fig. 11. Potential energy diagram for a hypothetical exothermic reaction, showing the transition state as the saddle point.

Reactant conversion into its mirror image, NARCISSISTIC REACTION REACTING BOND RULES REACTING ENZYME CENTRIFUGATION REACTION COORDINATE DIAGRAM POTENTIAL ENERGY DIAGRAM SADDLE POINT... 

Fig. 9. Linear stability diagram illustrating (21). The bifurcation parameter B is plotted against the wave number n. (a) and (b) are regions of complex eigenvalues < > (b) shows region corresponding to an unstable focus (c) region corresponding to a saddle point. The vertical lines indicate the allowed discrete values of n. A = 2, D, = 8 103 D2 = 1.6 10-3.

Figure 8.8(a) shows a typical example of the phase plane for a system with three stationary solutions, chosen such that there are two stable states and the middle saddle point. The trajectories drawn on to the diagram indicate the direction in which the concentrations will vary from a given starting point. In some cases this movement is towards the state of no conversion (ass = 1, j8ss = 0), in others towards the stable non-zero solution. Only two trajectories approach the saddle point these divide the plane into two and separate those initial conditions which move to one stable state from those which move to the other. These two special trajectories are known as the separatrices of the saddle point. 

We assume that the equations (7.200) have a simple hysteresis type static bifurcation as depicted by the solid curves in Figures 10 to 12 (A-2). The intermediate static dashed branch is always unstable (saddle points), while the upper and lower branches can be stable or unstable depending on the position of eigenvalues in the complex plane for the right-hand-side matrix of the linearized form of equations (7.198) and (7.199). The static bifurcation diagrams in Figures 10 to 12 (A-2) have two static limit points which are usually called saddle-node bifurcation points. 

Values of and r2 for configurations of the same PE give a contour diagram of curves of constant PE (Figure 4.20).The activated complex is again at the col or the saddle point. 

Answer. Along AB, rAB = rBC. The activated complex lies at the saddle-point lying on AB. The PE contour diagram is symmetrical about the critical configuration. The activated intermediate lies in the well of the elliptical contours and at configurations where rBC > rAB. There is a further activated complex as marked along the line PQ. 

Figure 2.6 The traditional diagram of energy versus reaction coordinate is in fact only a slice of the multidimensional space. This slice represents just one of the degrees of freedom of the system. Going one dimension higher, we see that the transition state is not a maximum but a saddle point.

Fig. 2 Energy surface for a two-dimensional reaction diagram. 19.1 kcal/mol is the free energy of activation (saddle point - initial comer). Reproduced with permission from ref. 9.

The phase plane plot of Figure 2 illustrates the behavior of the concentrations of X and Y within this region. Initial concentrations of X and Y corresponding to a point above the broken line will evolve in time to the limit cycle. The broken line represents the separatrix of the middle unstable steady state which has the stability characteristics of a saddle point. Initial values for X and Y corresponding to a point below the separatrix will evolve to the stable state to the right of the diagram. 

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