Locating a stationary point

At both minima and saddle points, the first derivative of the energy, known as the gradient, is zero. Since the gradient is the negative of the forces, the forces are also zero at such a point. A point on the potential eneigy surface where the forces are zero is called a stationary point All successful optimizations locate a stationary point, although not always the one that was intended. 

The initial step in locating a stationary point is defining the internal coordinate system. An effort should be made to choose a coordinate system where the parameters are not interdependent (coupled). As shown below, there is a coupling (i.e., the bond length will affect the optimum bond angle) between the CH distance, rj, and the HCC angle, aj. The coupling can be... 

Schiitz et al. were unable to locate a. stationary point of the form (uuud) in their calculations. In this case the (uupd) and (uudp) transition states actually mediate asynchronous double flips, as illustrated in Figure 4. In Figure 4(a) the second flip occurs between the last two frames in Figure 4(b) the first flip occurs between the first and fourth frames. 

Tlie function to be optimized, and its derivative(s), are calculated with a finite precision, which depends on the computational implementation. A stationary point can therefore not be located exactly, the gradient can only be reduced to a certain value. Below this value the numerical inaccuracies due to the finite precision will swamp the true functional behaviour. In practice the optimization is considered converged if the gradient is reduced below a suitable cut-off value. It should be noted that this in some cases may lead to problems, as a function with a very flat surface may meet the criteria without containing a stationary point. 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

This equation provides a prescription for the location of stationary points. In principle, starting from an arbitrary structure having coordinates q , one would compute its gradient vector g and its Hessian matrix H, and then select a new geometry q( +0 according to Eq. (2.41). Equation (2.40) shows that the gradient vector for this new structure will be the 0 vector, so we will have a stationary point. 

In terms of the gradient of the stationary-state locus, the condition expressed by eqn (7.55) locates a singular point as dx/drres - 0/0. Loosely, we can think of the locus simultaneously having turning points both horizontally and vertically, as indicated in Fig. 7.7. This corresponds to situations such as (a) an isolated point or (b) the transcritical touching of two branches. 

To develop a method to locate saddle points by selectively following one eigenvector we note that at a stationary point all n components of the gradient in the diagonal representation are zero ... 

Unfortunately, constraint functions g,y are not necessarily monotonic. However, each gti has at most one stationary point with respect to flow rates w , and Wj (Saboo et al., 1987b). Standard nonlinear programming can be used to locate each stationary point. If the stationary point of g,y lies outside the uncertainty range or if the stationary point is a maximum, then gii.min occurs at a corner point. 

Any trajectory can end when p - I at a stationary point (SP), in which all the right-hand parts of equations (5.2) equal zero. In the case of the terminal model (2.8) all such SPs are those solutions of the non-linear set of the algebraic equations (4.13) which have a physical meaning. Inside m-simplex one can find no more than one SP, the location of which is determined by the solution of the linear equations (4.14). In addition to such an inner azeotrope of the m-simplex, azeotropes can also exist on its boundaries which are n-simplexes (2 S n m - 1). For each of these boundary azeotropes (m — n) components of vector X are equal to zero, so it is found to be an inner azeotrope in the system of the rest n monomers. Moreover, the equations (4.13) always have m solutions x( = 8is (where 8js is the Cronecker Delta-symbol which is equal to 1 when i = s and to 0 when i =(= s) corresponding to each of the homopolymers of the monomers Ms (s = 1,. ..,m). Such solutions together with all azeotropes both inside m-simplex and on its boundaries form a complete set of SPs of the dynamic system (5.2). 

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