Location of Stationary Points

This section is concerned with some numerical methods which are of interest when searching for minimizers and/or saddle points of an energy functional. 

These assumptions are basic for the procedures considered in this section. Since 2) is convex, the conditions (Al) and (A2) ensure the 

An energy functional defined on (overall translations/rotations of the molecular system are admited) can never satisfy the 

We start with a short discussion on the both ways that allow a determination of stationary points. Firstly, stationary points can be determined by solving Eg.(4). But, since the solution set of this equation contains the minimizers, the maximizers as well as the saddle points of E, an additional examination is necessary to determine the type of the solution when an unspecific (Newton-like) method is used, 

Alternatively, one can use an auxiliary functional, namely the defect functional 

---

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. 

TflUe 7. Procedures for location of stationary points on en hypcrsurface... 

The importance of analytic derivative methods in quantum chemistry cannot be overstated. Analytic methods have been demonstrated to be more efficient than are corresponding finite difference techniques. Calculation of the first derivatives of the energy with respect to the nuclear coordinates is perhaps the most common these provide the forces on the nuclei and facilitate the location of stationary points on the potential energy hypersurface. Differentiating the electronic energy with respect to a parameter x (which may be, but is not required to be, a nuclear coordinate), leads to the well-known expression... 

It is evident from the superimposed data points in Figure 4.12 that they are veiy closely aligned with the predictions of the NRTL model. The experimental results indicate that the profiles in fact move past the stable node predicted by the UNIQUAC model (highlighted by the curved arrow), and do not pinch at this point as one would expect if using this model. The predictions of the location of stationary points made by the UNIQUAC model are therefore false and those made by the NRTL are much better suited to this particular system. 

Analytic gradient methods became widely used as a result of their implementation for closed-shell self-consistent field (SCF) wavefunctions by Pulay, who has reviewed the development of this topic. Since then, these methods have been extended to deal with all types of SCF wavefunctions, - as well as multi-configuration SCF (MC-SCF), - " configuration-interaction (Cl) wavefunctions, and various non-variational methods such as MoUer-Plesset (MP) perturbation theory - - and coupled-cluster (CC) techniques. - In short, it is possible to obtain analytic energy derivatives for virtually all the standard ab initio approaches. The main use of analytic gradient methods is, and will remain, the location of stationary points on a potential energy siuface, to obtain equilibrium and transition-state geometries. However, there is a specialized use in the calculation of quantities such as dipole derivatives. 

So far, discussing distillation trajectories and their bundles, we proceeded from the fact, that separation stages are equilibrium ( theoretical plates). In real separation process at plates of distillation columns equilibrium is not achieved and the degree of nonequilibrium is different for different components. That leads to decrease of difference between compositions at neighboring plates and to change of curvature of distillation trajectories (Castillo Towler, 1998), but does not influence the location of stationary points of distillation trajectory bundles because in the vicinity of stationary points equilibrium and nonequilibrium trajectories behave equally. Therefore, implemented above analysis of the structure and of evolution of section trajectory bundles is also valid for nonequilibrium trajectory bundles. 

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 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. 

An important outcome of these simulations is the location of HB points (largely ignored in previous work), which is important for the development of extinction theory. In particular, the turning point E lies on a locally unstable stationary solution branch and does not coincide with the actual extinction, as previously thought. The actual extinction point is the termination point of oscillations. Thus, local stability analysis is essential to properly analyze flame stability and develop extinction theory. 

To find the minimum of G( ), we must locate the stationary point dG/d )T P = 0. For this purpose we employ (8.10), with help from (8.5) ... 

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. 

Transitions between the stabdity regions I—V of the exceptional points location can correlate with variations in the value of controding para meter a. In a typical diagram (Figure 3.5A), the coordinates of stationary point y are plotted along the axis and controding parameter a (the system remoteness from the initial equdibrium) is plotted along the abscissa. 

At stationary points there are GF. curves leading along the direction of all Hessian eigenvectors, and by tracing out GE paths it is possible to locate many stationary points... 

From our short summary it comes out that the calculation of energy derivatives with respect to the nuclear coordinates is an essential point in the characterization of stationary points. Actually, the calculation of derivatives is also a decisive tool in the search for the location of these stationary points. There is a large, and still fast growing, number of reviews surveying the formal and computational aspects of this problem (Schlegel, 1987 Bernardi and Robb, 1987 Dunning 1990 Schlick 1992 McKee and Page, 1993). 

---