Example of a dissociation reaction

Let us again look at the dissociation reaction

for which the reaction coordinate is appropriate. The transformation to center of mass and relative coordinate is:

The inverse of this transformation is

Next, the vector is transformed to spherical polar coordinates according to

and the derivative of the potential with respect to r can be easily worked out using the chain rule:

The Jacobian is clearly

so that

Thus, the free energy derivative is

which is expressed completely in terms of Cartesian quantities and can be calculated straightforwardly.

Once is known, it can be integrated to obtain the full free energy profile . This is another use of thermodynamic integration.

It is always interesting to see what A(r) looks like compared to the bare potential. Suppose the dissociation reaction is governed by a pair potential describing the interaction of the dissociating molecule with a solvent:

The potential might look like:

  
Figure 2:

If, at a given temperature T, the solvent assists in the dissociation process, then we might expect A(r) to have a lower dissociation threshold and perhaps a slightly longer effective minimum bond length:

  
Figure 3:

If, on the other hand, at a given temperature, T, the solvent hinders dissociation, by causing the molecule to bury itself in a cavity, for example, then we might expect A(r) to appear as follows:

  
Figure 4:

with a higher dissociation threshold energy and slightly shorter effective minimum bond length. Such curves will, of course, be temperature dependent and depend on the specific nature of the interactions with the solvent.