Reaction coordinates

Often simple chemical processes can be described in terms of a one-dimensional coordinate, which is not one of the Cartesian coordinates in a system but a generalized coordinate, q. Such a coordinate is, in general, a function of the Cartesian coordinates:

In some cases, a set of reaction coordinates is required.

Examples 1: Dissociation reactions.

Consider a dissociation reaction of the form

A reaction coordinate for such a process is the distance r between the two atoms in the diatomic. If and are the Cartesian positions of atoms A and atom B, respectively, then

What set of generalized coordinates contains r explicitly? Let us transform to center-of-mass and relative coordinate:

Then, let be transformed to spherical polar coordinates

Example 2: Proton transfer reactions.

Consider proton transfer through a hydrogen bond according to:

which is illustrated schematically in the cartoon below:

  
Figure 1:

The two heavy atoms are assumed to be of the same type (e.g. oxygen atoms). A reaction coordinate describing this process is the difference in the two distances and :

What generalized coordinate system contains explicitly?

To see what coordinate system this is, consider transforming to center of mass and two relative coordinates as follows:

Now the six degrees of freedom are transformed to 6 new degrees of freedom, which are the spherical polar coordinates of and the confocal elliptic coordinates for the position of the proton:

Then, the coordinate is the reaction coordinate .