Analogous saddle points

A geographical analogy can be a helpful way to illustrate many of the concepts we shall encounter in this chapter. In this analogy minimum points correspond to the bottom of valleys. A minimum may be described as being in a long and narrow valley or a flat and featureless plain. Saddle points correspond to mountain passes. We refer to algorithms taking steps uphill or downhill. ... 

With the identification of the TS trajectory, we have taken the crucial step that enables us to carry over the constructions of the geometric TST into time-dependent settings. We now have at our disposal an invariant object that is analogous to the fixed point in an autonomous system in that it never leaves the barrier region. However, although this dynamical boundedness is characteristic of the saddle point and the NHIMs, what makes them important for TST are the invariant manifolds that are attached to them. It remains to be shown that the TS trajectory can take over their role in this respect. In doing so, we follow the two main steps of time-independent TST first describe the dynamics in the linear approximation, then verify that important features remain qualitatively intact in the full nonlinear system. 

The four maxima and the saddle point are critical points in the function L(r) analogous to the maxima and saddle points in p(r) discussed in Chapter 6. Every point on the sphere of maximum charge concentration of a spherical atom is a maximum in only one direction, namely, the radial direction. In any direction in a plane tangent to the sphere, the function L does not change therefore the corresponding curvatures are zero. When an atom is part of... 

In siunmary, although the application of detailed chemical kinetic modeling to heterogeneous reactions is possible, the effort needed is considerably more involved than in the gas-phase reactions. The thermochemistry of surfaces, clusters, and adsorbed species can be determined in a manner analogous to those associated with the gas-phase species. Similarly, rate parameters of heterogeneous elementary reactions can be estimated, via the application of the transition state theory, by determining the thermochemistry of saddle points on potential energy surfaces. 

According to the MNDO calculations (88MI4), the planar structure of the trigermacyclopropenyl cation (296) also corresponds to a minimum on the PES, whereas the analogous structure of (SnH)3+ (297) is a third-order saddle point. The MNDO calculation of the ISE (291) [isodesmic reaction (88)], with a correction for the strain energy determined from isodesmic reaction (89), shows the aromatic stabilization of (291) to be insignificant (1.8 kcal/mol) ... 

Fig. 4. A section of the energy surface, analogous to Fig. 3, for an isomeric transition. The values of the configurational coordinates about a and b correspond to the two isomeric forms of the molecule which are stable against small atomic displacements. The electronic energies for the two forms, namely Ea and E0, are assumed to be the same, although this need not necessarily be the case. This section of the energy surface is assumed to pass through the saddle point in the potential range separating the two minima a and b. The energy of the saddle point is E. ...

Transition state theory (Chapter 2, section A) was derived for chemical bonds that obey quantum theory. An equation analogous to that for transition state theory may be derived even more simply for protein folding because classical low energy interactions are involved and we can use the Boltzmann equation to calculate the fraction of molecules in the transition state i.e., = exp(— AG -D/RT), where A G D is the mean difference in energy between the conformations at the saddle point of the reaction and the ground state. Then, if v is a characteristic vibration frequency along the reaction coordinate at the saddle point, and k is a transmission coefficient, then... 

Fig. 5.41 The distribution of the electron density (charge density) p for a homonuclear diatomic molecule X2. One nucleus lies at the origin, the other along the positive z-axis (the z-axis is commonly used as the molecular axis). The xz plane represents a slice through the molecule along the z-axis. The —p = f(x, z) surface is analogous to a potential energy surface E = /(nuclear coordinates), and has minima at the nuclei (maximum value of p) and a saddle point, corresponding to a bond critical point, along the z axis (midway between the two nuclei since the molecule is homonuclear)...

This transition-state-like point is called a bond critical point. All points at which the first derivatives are zero (caveat above) are critical points, so the nuclei are also critical points. Analogously to the energy/geometry Hessian of a potential energy surface, an electron density function critical point (a relative maximum or minimum or saddle point) can be characterized in terms of its second derivatives by diagonalizing the p/q Hessian([Pg.356]

In analogous manner, residue curve maps of the reactive membrane separation process can be predicted. First, a diagonal [/e]-matrix is considered with xcc = 5 and xbb = 1 - that is, the undesired byproduct C permeates preferentially through the membrane, while A and B are assumed to have the same mass transfer coefficients. Figure 4.28(a) illustrates the effect of the membrane at nonreactive conditions. The trajectories move from pure C to pure A, while in nonreactive distillation (Fig. 4.27(a)) they move from pure B to pure A. Thus, by application of a C-selective membrane, the C vertex becomes an unstable node, while the B vertex becomes a saddle point This is due to the fact that the membrane changes the effective volatilities (i.e., the products xn a/a) of the reaction system such that xcc a. ca > xbbO-ba-... 

Some crucial aspects of studying the GHF wave functions are connected with the relationship between the GHF and UHF methods. First of all, it is evident that the RHF and UHF wave functions are particular solutions also to the GHF problem In this case the components p " and p" of the Fock-Dlrac density matrix are zero, and the GHF equations separate into two sets of equations for the orbitals of spins a and p, respectively. The system of equations obtained in this way is identical to that of the ordinary UHF scheme. We note that the two sets of equations are still coupled through the components p++ and p". The situation is in some way analogous to the case of the UHF equations for a closed-shell system, for which the RHF functions always provide a particular solution. Similarly to the RHF versus UHF case, the UHF (or RHF) solution can, in principle, represent either a true (local) minimum or a saddle point for the GHF problem. 

To appreciate the special role of a conical intersection as a transition point between the excited and the ground state in a photochemical reaction, it is useful to draw an analogy with a transition state associated with the barrier in a potential energy surface in a thermally activated reaction (Figure 6.6). In the latter, one characterizes the transition state with a single vector that corresponds to the reaction path through the saddle point. The transition structure is a minimum in all coordinates except the one that corresponds to the reaction path. In contrast, a conical intersection provides two possible linearly independent reaction path directions. 

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