Locating a saddle point

Locating a saddle point can be a daunting experience. Our usual ideas of covalency and hybridization do not help us as the structure is often not well described by conventional bonding. The best way to locate a saddle point is to start with a good guess. These guesses can come from several sources and include ... 

Muller K and Brown L D 1979 Location of saddle points and minimum energy paths by a constrained simplex optimization procedure Theor. Chim. Acta 53 75... 

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. 

At first sight, the easiest approach is to fit a set of points near the saddle point to some analytical expression. Derivatives of the fitted function can then be used to locate the saddle point. This method has been well used for small molecules (see Sana, 1981). An accurate fit to a large portion of the potential energy surface is also needed for the study of reaction dynamics by classical or semi-classical trajectory methods. 

Surprinslngly, we observe an drastic effect of the concentration on the SRO contribution (figure 2) indeed, in PtaV, the maxima are no longer located at a special point of the fee lattice but the (100) intensity is splltted perpendicularly in the (010) direction and presents a saddle point at (100) position. Notice that these two maxima are not located just above Bragg peaks of the ordered state the A B ground state presents Bragg peaks at ( 00) and equivalent positions whereas the SRO maxima peak between ( 00) and (100). 

As a first step toward a TST treatment of the stochastically driven dynamics, it is crucial to assume, just as in the autonomous case, that the deterministic dynamics has a fixed point that marks the location of an energetic barrier between reactants and products. In the case of Eq. (13), the fixed point is given by a saddle point q0 of the potential U(q). The reaction rate is determined by the... 

Gao et al. located the saddle point of the two-dimensional free energy surface at Rc-n = 1.96 A and Rc-ci = 2.09 A, and used this point to define the transition state in solution. This way, their most striking finding was that on going from the gas phase... 

The top of the profile is maximum (saddle point) and is referred as the transition state in the conventional transition state theory. It is called a saddle point because it is maximum along the orthogonal direction (MEP) while it is minimum along diagonal direction of Fig. 9.12. The minimum energy path can be located by starting at the saddle point and mapping out the path of the deepest descent towards the reactants and products. This is called the reaction path or intrinsic reaction coordinate. 

The Transition State Hypothesis. The general idea that a transition state is located at a saddle point on the PES, as detailed in Section 1.3, is familiar to most organic chemists. However, the original concept of a transition state started out as something rather different. In the development of both transition state and RRKM theory, the transition state was defined as the location of a plane (actually a hyperplane) in phase space, perpendicular to the reaction coordinate. ... 

The contemporary literature is full of studies of transition states, and it would be impossible, even with much more space available, to list all the studies in which the theoretical location of saddle points provided useful insight for inorganic and organometallic chemists. But do such studies ever really prove that a reaction occurs in a particular way Based on the example discussed above, it is interesting to consider this question quite carefully. 

---