Minima on a Potential Energy Surface

Hessian matrix A matrix of second-order partial derivatives that determine curvature used in calculations to test for minima on a potential energy surface. 

The transition state scaling relations imply scaling relations for the activation energy of a surface chemical reaction (see Fig. 6.7). Let X and Y define two minima on a potential energy surface if both E and AE scale with a set of adsorption energies, then E will as well. 

For a molecule that has no observable tiumelling between minima on the potential energy surface (i.e., for a... 

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. 

A minimum on a potential energy surface represents an equilibrium stracture. There will invariably be a number of such local minima, and we can imagine a number of paths on the surface that connect one particular minimum to another. If the highest-energy point on each path is considered, the transition structure can be defined as the lowest of these maxima. The reaction path is the lowest-energy route between two minima. 

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. 

The calculations were performed into two basis sets, with full geometiy optimization except for the torsional angles a and 6. Two non planar conformations were considered, which correspond to minima on the potential energy surface into the GVB approximation [21]. In these conformations, the molecule adopts a pyramidal conformation, as in methanal. In addition, the hydroxilic group is rotated up or down the OCO plane. 

These methods are best when determining conformers in local minima on the potential energy surface. The number of rotatable bonds the molecule of interest possesses dictates the failure limit value accordingly, molecules with few rotatable bonds require a lower failure value than molecules with many rotatable... 

Theoretical calculations on the cycloaddition reactions of a range of 1,3-dipoles to ethene in the gas phase have been carried out (85) with optimization of the structures of these precursor complexes and the transition states for the reactions at the B3LYP/6-31G level. Calculated vibration frequencies for the orientation complexes revealed that they are true minima on the potential energy surface. The dipole-alkene bond lengths in the complexes were found to be about twice that in the final products and binding was relatively weak with energies <2 kcal mol . Calculations on the cycloaddition reactions of nitrilium and diazonium betaines to ethene indicate that the former have smaller activation energies and are more exothermic. 

Stationary Point. A point on a Potential Energy Surface for which all energy first derivatives with respect to the coordinates are zero. Local Minima and Transition States are stationary points. 

There are three important minima on the potential energy surfaces calculated for [M(bidentate)5]. At b x 1.1, isomers I and II correspond to two of the possible ways of arranging bidentate ligands around a bicapped square antiprism, whereas isomer III is a sphenocorona (Figure 102). 

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