Potential energy minima and saddle points

Let us denote (/ ) s V. The most interesting points of the hypersurface V are its critical points, i.e. the points for which the gradient VK is equal to zero  

There are several types of critical points. Each type can be identified after considering the Hessian, i.e. the matrk with elements 

It is advisable to construct the above mentioned analytical functions following some theoretical arguments. These are supplied by intermolecular interaction theory (see Chapter 13). 

Intermolecular Motion of Electrons and Nuclei Chemical Reactions 

The eigenvalues were obtained by diagonalization of the Hessian. Such diag-onalization corresponds to a rotation of the local coordinate system (cf. p. 297). Imagine a two-dimensional surface that at the minimum could be locally approximated by an ellipsoidal valley. The diagonalization means such a rotation of the coordinate system x,y that both axes of the ellipse coincide with the new axes x, y (Chapter 7). On the other hand, if our surface locally resembled a cavalry saddle, diagonalization would lead to such a rotation of the coordinate system that one axis would be directed along the horse, and the other across.  

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Potential Energy Minima and Saddle Points Distinguished Reaction Cowdinate (DRO Steepest Descent Path (SDP)... 

Using the coordinates of special geometries, minima, and saddle points, together with the nearby values of potential energy, you can calculate spectroscopic properties and macroscopic therm ody-riatriic and kinetic parameters, sncfi as enthalpies, entropies, and thermal rate constants. HyperChem can provide the geometries and energy values for many of these ealeulatiori s. 

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. 

Maxima, minima and saddle points are stationary points on a potential energy surface characterized by a zero gradient. A (first-order) saddle point is a maximum along just one direction and in general this direction is not known in advance. It must therefore be determined during the course of the optimization. Numerous algorithms have been proposed, and I will finish this chapter by describing a few of the more popular ones. 

There are numerous algorithms of different kinds and quality in routine use for the fast and reliable localization of minima and saddle points on potential energy surfaces (see 47) and refs, therein). Theoretical data about structure and properties of transition states are most interesting due to a lack of experimental facts about activated complexes, whereas there is an abundance of information about educts and products of a reaction. 

The optimization of minima and saddle points of a function in many variables is reviewed. Emphasis is on methods applicable to the calculation of electronic wave functions (ground and excited states) and the optimization of minima and transition states of molecular potential energy surfaces. 

Both minima and saddle points are of interest. In the case of wave functions, the ground state is a minimum and the excited states are saddle points of the electronic energy function.6 On potential surfaces minima and first-order saddle points correspond to equilibrium geometries and transition states. Higher-order saddle points on potential energy surfaces are of no interest. 

Fig. 15.9. Potential energy along the intrinsic reaction coordinate (IRC) for OH + CH3F —> CH3OH + F s is the distance along the IRC. This figure shows the structures at the potential energy minima and at the saddle point barrier. Adapted from Ref. [121].

The determination of minima and saddle points on the potential-energy surface of a molecule plays an important role (Schaefer and Miller, 1977, Chapter 4) in describing the electronic structure and chemical reactivity of molecules. In this section, we show how such stationary points on a molecule s potential energy surface may be found by using an approach similar to that employed in Section 5.B. We first consider how the electronic Hamiltonian changes when the nuclear positions are changed from an initial set of positions, to R, i.e., R - R + u. The electron-nuclear interaction is the only term in the Hamiltonian that depends explicitly on the nuclear position. Performing a Taylor expansion of this potential about the point R, we obtain... 

Figure 1.1. Prototypical potential energy surface of a simple system (a) and of a complex system (b). In a simple, low-dimensional system, dynamical bottlenecks for transitions between long-lived stable states most often coincide with saddle points on the potential energy surface. Locating these stationary points reveals the reaction mechanism. In a typical complex system, the potential energy surface is rugged and has countless local minima and saddle points. Nevertheless, there can be well-defined long-lived stable states and rare transitions between them. Such transitions can occur via a multitude of different transition pathways.

Fig. 9. The gradient norm squared for the model potential energy surface used in Figs. 4, S and 7. The gradient norm squared is a local minimum in many places in addition to the minima and saddle point on the potential.

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