Page 109 - Mechanism and Theory in Organic Chemistry

Figure 2.8 A three-dimensional reaction coordinate diagram. The reaction coordinate is a
path following the lowest altitude line up one valley, over the pass, and down the
other.

over to products. At some configuration the potential passes through a maximum,
and then falls as we proceed to the right, finally reaching a minimum again with
separated products C and D, x = 1. An entirely similar process can be imagined
for a unimolecular reaction. The configuration of atoms corresponding to the
maximum in the reaction coordinate diagram is the transition state, symbolized
by $. It occurs at x = x*.
There are two perhaps obvious but easily overlooked points about the re-
action coordinate diagram that must be stressed. First, it is only a one-dimensional
slice of a 3N - 6 + 1-dimensional surface. (N is the total number of atoms in
A and B.) We can imagine, at each point of the line, motions off the line corre-
sponding to vibrations other than the single one that is carrying the molecules
over to products. These motions are all ordinary vibrations, having nothing to
do (in a first approximation at least) with the reaction, and proceeding quite
independently of it. If we assume that the reaction coordinate corresponds to a
normal mode of the reacting system,39 the reaction coordinate is "perpendicular"
(in 3N - 6-dimensional space) to each of these other normal modes. Our
curve passes along the equilibrium position of each of the other vibrations, so that
if we were to leave the reaction coordinate line and follow the potential energy
surface in the direction of some other mode, the energy would always go up.

39 This assumption is implicit in the transition state theory, although it may not be entirely correct.