Saddle point definition

Equation (5.86c), written as a strict equality, may also be taken to define the NRT transition state as an alternative to (and slightly different from) the usual definitions based on energetic, saddle-point-curvature, or density-of-states criteria. Note that this NRT alternative definition can be employed for non-IRC choices of reaction coordinate, and remains valid even in the case of barrierless processes (such as many ion-molecule or radical-recombination reactions) for which the reaction profile does not exhibit an energy maximum as in Fig. 5.52. The NRT definition is practically identical to the usual saddle-point definition of the transition state in the present examples. 

Figure 2.7. Schematic illustration of reaction coordinate and saddle point definition of the transition...

A saddle point approximation to the above integral provides the definition for optimal trajectories. The computations of most probable trajectories were discussed at length [1]. We consider the optimization of a discrete version of the action. 

Imaginary frequencies are listed in the output of a frequency calculation as negative numbers. By definition, a structure which has n imaginary frequencies is an n order saddle point. Thus, ordinary transition structures are usually characterized by one imaginary frequency since they are first-order saddle points. 

An IRC calculation examines the reaction path leading down from a transition structure on a potential energy surface. Such a calculation starts at the saddle point and follows the path in both directions from the transition state, optimizing the geometry of the molecular system at each point along the path. In this way, an IRC calculation definitively connects two minima on the potential energy surface by a path which passes through the transition state between them. 

In particular, the TS trajectory remains bounded for all times, which satisfies the general definition. The constants c and c in Eq. (39) depend on the specific choice of the TS trajectory. Because the saddle point of the autonomous system becomes a fixed point for large positive and negative times, one might envision an ideal choice to be one that allows the TS trajectory to come to rest at the saddle point both in the distant future and in the remote past. However, this is impossible in general because the driving force will transfer energy into or out of the bath modes in such a way that... 

Steepest descent can terminate at any type of stationary point, that is, at any point where the elements of the gradient of /(x) are zero. Thus you must ascertain if the presumed minimum is indeed a local minimum (i.e., a solution) or a saddle point. If it is a saddle point, it is necessary to employ a nongradient method to move away from the point, after which the minimization may continue as before. The stationary point may be tested by examining the Hessian matrix of the objective function as described in Chapter 4. If the Hessian matrix is not positive-definite, the stationary point is a saddle point. Perturbation from the stationary point followed by optimization should lead to a local minimum x. ... 

In both solvents, the variational transition state (associated with the free energy maximum) corresponds, within the numerical errors, to the dividing surface located at rc = 0. It has to be underlined that this fact is not a previous hypothesis (which would rather correspond to the Conventional Transition State Theory), but it arises, in this particular case, from the Umbrella Sampling calculations. However, there is no information about which is the location of the actual transition state structure in solution. Anyway, the definition of this saddle point has no relevance at all, because the Monte Carlo simulation provides directly the free energy barrier, the determination of the transition state structure requiring additional work and being unnecessary and unuseful. 

If we examine a potential energy surface there are several features which play an important role in the interpretation of kinetic processes. These are minima (stable configurations of all the atoms), valleys (separate stable groups of atoms which we identify as reactants and products) and saddle points (transition states). However, before we give a more formal definition of these features we have to consider the coordinate system that is used. 

It may not at first be obvious that the Jahn-Teller theorem applies to transition states (40). The proof rests on the fact that the matrix element of the distortion gives a first-order change in energy and hence is linear in Q. In other words there must be a non-zero slope in some direction and this is incompatible with the definition of a transition point as a saddle point on the potential energy surface. 

This section presents first the formulation and basic definitions of constrained nonlinear optimization problems and introduces the Lagrange function and the Lagrange multipliers along with their interpretation. Subsequently, the Fritz John first-order necessary optimality conditions are discussed as well as the need for first-order constraint qualifications. Finally, the necessary, sufficient Karush-Kuhn-Dicker conditions are introduced along with the saddle point necessary and sufficient optimality conditions. 

This section presents the basic definitions of a saddle point, and discusses the necessary and sufficient saddle point optimality conditions. 

Remark 1 From this definition, we have that a saddle point is a point that simultaneously minimizes the function 6 with respect to x for fixed y = y and maximizes the function 0 with respect toy for fixed jc = jc. Note also that no assumption on differentiability of 0(jc,y) is introduced. 

Definition 3.2.8 (Karush-Kuhn-Dicker saddle point) Let the Lagrange function of problem... 

It is difficult to determine if a particular minimum is the global minimum, which is the lowest energy point where force is zero and second derivative matrix is positive definite. Local minimum results from the net zero forces and positive definite second derivative matrix, and saddle point results from the net zero forces and at least one negative eigenvalue of the second derivative matrix. 

A point x is called a stationary point of fti g(x ) = 0 but H(x ) is not necessarily positive-definite. Thus, local and global minima are stationary points, but there are more general stationary points, such as saddle points, which are neither local nor global minima. Special techniques are needed for detection of saddle points, which are often related to structural transitions in molecular applications. 

With suitable definitions of search functions, EA methods can also be used to locate more features on the PES than just low-energy local and global minima. Chaudhury et al. [144,145] have implemented methods for finding first-order saddle points and reaction paths, applying them to LJ clusters up to n=30. It remains to be tested, however, if these method can be competitive with deterministic exhaustive searches for critical points for small systems [146], on the one hand, and with the large arsenal of methods for finding saddles and reaction paths between two known minima for larger systems [63], on the other hand. 

In the coordinate-space treatment of TST, certain assumptions must be made concerning the nature of the Hamiltonian of the system. First, it must be assumed that it can be partitioned into the sum of two terms, the kinetic and the potential energy. Furthermore, one must also assume that the kinetic energy is positive definite and is quadratic in the momenta. With these assumptions, then the point of stationary flow in phase space and the saddle point of the potential energy... 

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