General functions saddle points

Many problems in computational chemistry can be formulated as an optimization of a multidimensional function/ Optimization is a general term for finding stationary points of a function, i.e. points where tlie first derivative is zero. In the majority of cases the desired stationary point is a minimum, i.e. all the second derivatives should be positive. In some cases the desired point is a first-order saddle point, i.e. the second derivative is negative in one, and positive in all other, directions. Some examples ... 

We have outlined how the conceptual tools provided by geometric TST can be generalized to deterministically or stochastically driven systems. The center-piece of the construction is the TS trajectory, which plays the role of the saddle point in the autonomous setting. It carries invariant manifolds and a TST dividing surface, which thus become time-dependent themselves. Nevertheless, their functions remain the same as in autonomous TST there is a TST dividing surface that is locally free of recrossings and thus satisfies the fundamental requirement of TST. In addition, invariant manifolds separate reactive from nonreactive trajectories, and their knowledge enables one to predict the fate of a trajectory a priori. 

From a consideration of either Eqs. (113) or (114) (K3), it is evident that a saddle point is predicted from the fitted rate equation. This could eliminate from consideration any kinetic models not capable of exhibiting such a saddle point, such as the generalized power function model of Eq. (1) and the several Hougen-Watson models so denoted in Table XVI. 

As mentioned in Sec. II the general strategy in saddle point calculations is to take a step that increases the function in some directions while reducing it in others. In the RSO and RF approaches the step may be written... 

To carry out the proof, we note that the LMA equations are equivalent to imposing an extremum on some function of the concentrations—the free energy F for v = const or the thermodynamic potential for p = const. It is in precisely this way, as is well known, that the LMA may be derived from general thermodynamic principles. We will solve the problem if we prove that the surface F or, under the conditions imposed on the concentration and for constant v or p, has one and only one minimum, and does not have either maxima or any other critical points (so-called minimax, or saddle points). 

We first note that Eq. (29), which we derived by taking the limit x — 0 of the result (26) for general x, can also be obtained by a more direct route. In the limit x —> 0, the sizes of particles in the smaller system become independent random variables drawn from n (a) the second phase can be viewed as a reservoir to which the small phase is connected. One writes the moment generating function for V(m) in the small phase as a product of xN independent moment generating functions of n (o) and then evaluates the integral over V m) by a saddle point method [36]. 

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. 

We have introduced the notation qs to signify the value of the coordinate q at the saddle point, and will similarly find it convenient to introduce q to denote the coordinate for the well under consideration. As yet, our statements are general and involve no approximations other than those present in transition state theory itself A standard approximation at this point is to note that in the vicinity of the well it is appropriate to represent the energy surface as a quadratic function in the variable q, and in addition it is asserted that one may make the transcription... 

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