## Transition States

### Definitions and properties

One of the important techniques in organic chemistry is the consideration of the structure of the system as the reaction proceeds. Each configuration of the atoms during the process of changing from reactants to products has an associated energy. This defines the energy map. Since reactions generally involve bringing the reactants close together and breaking bonds, these sructures generally have higher energies than the isolated reactants. That is, as the reactants approach each other and start to undergo the molecular changes that will eventually result in the products, the energy of the system increases. As the reaction encounter continues, the potential energy continues to increase until the system reaches a structure of maximum energy. This is the Transition State. Thereafter the changes that result in the final products continue, but the structures represent lower and lower energy until the products are fully formed.

A transition states is always a first-order saddle point in the energy map. Energy rises in all directions but one. If energy would have lowered in two directions or more, then it is obvious that there would be an other path between reactants and products where every point is lower than this one (because the energy map is a continous surface).

### Methods to locate the transition states

The method which is decribed below is a generalisation. It is equally applicable to energy maps computed using ab initio  methods, such as Gaussian 94, semi-empirical packages, such as MOPAC 93, or mechanics calculations . Since it is impractical to calculate every point on the surface, we would like to follow a sensible path across it, and to be able to do this, we must be able to calculate the gradient at any given point. Localization of the transition states

The reaction coordinate is generally not a simple geometrical coordinate (like bond length, bond angle or dihedral angle). Therefore we cannot follow the reaction pathway easily.
1. First we have to find a set of geometrical coordinates which are close to the reaction coordinate. For instance in the Diels-Alder reaction, these are the two forming bond lengths.
2. We have to lock these geometrical parameters in a point close to the transition state. This is point 1 on the scheme.
3. We minimize the energy, moving on a surface (yellow line) perpendicular to the geometry locked (green line) and find point 2 which is very close to the reaction pathway.
4. If our initial guess was good enough, we are not far from the saddle point. We minimize the gradient (its norm) and find the transition state 3. If our initial guess was not so good, there is an inflexion point between point 2 and the Transition State and the gradient minimization leads us either to the reactants or to the products.