Essentials of Computational Chemistry

15.2 REACTION PATHS AND TRANSITION STATES 523
It is worth digressing for a moment to note that following an MEP is often crucial to
understanding the nature of a TS structure. Sometimes, when a molecule has a single imaginary
frequency, visualization of the corresponding normal mode does not necessarily make
it obvious what the reaction coordinate is. It can often happen that the TS structure that
has been located corresponds to some process other than the one of interest, e.g., a TS
structure for the internal rotation of a methyl group may be found when the desired TS
structure was for some bond-making or bond-breaking process. In such a case, following
the MEP will lead, in each direction, to the ultimate minimum energy structures connected
by the TS structure. On complex potential energy surfaces, such connections can be critical
to understanding the overall topology of the PES (see, for instance, Gustafson and Cramer
1995).
Although the MEP and its connection to TS structure(s) is tremendously useful as a
conceptual tool, it can also be somewhat misleading to the extent that it focuses analysis on
the PES itself. It should always be kept in mind that the equilibria and kinetics of reacting
systems are nearly always governed by the free energy of populations of molecules, and not
the potential energy of single molecules. To the extent the free energy describes a thermal
distribution of particles composing the reacting system, one may think of the system as a
cloud hovering over the PES, with the density of the cloud thinning as it rises according to
Boltzmann statistics. Within the cloud, individual molecules may be exchanging energy with
one another to rise and fall relative to the PES, but the net distribution remains dictated by
temperature. A reacting system may be thought of as a cloud over the PES headed towards
a mountain pass whose saddle point is the TS structure. However, the passage of the cloud
over the pass need by no means take place directly over the TS structure. Depending on
how wide the pass is and how tall the cloud is, many cloud particles may be able to pass
arbitrarily far to the left and right of the TS structure (when the pass is very narrow one
says that the reaction has an entropic bottleneck, meaning that little variation in degrees of
freedom other than the reaction coordinate is permitted).
So, while the TS structure, by virtue of being a stationary point on the PES, can be
informative about the height of the pass, and local topology (by Taylor expansion of the
surface about the stationary point), it is only one representative of the population of molecules
passing from reactants to products. As such, one should be rather careful not to confuse the
TS structure, which is the stationary point, with the transition state, which may be somewhat
more rigorously defined for an N-atom system as a surface having 3N − 7degreesof
freedom (i.e., one less than the reactants) through which the reactive flux is maximized. That
is, the ratio of the number of molecules crossing the surface in the direction reactants →
products to the number crossing in the opposite direction in a given time interval is maximal.
To make the distinction between the TS structure and the transition state more clear, it is
helpful to return to a somewhat older term for the latter, namely, the ‘activated complex’.
The remainder of this chapter will hew to this distinction as closely as possible.
Returning to kinetics, while theory can be advantageously used to decompose a complex
system into its constituent series of elementary reactions, we have not yet described any
relationship between a theoretical quantity associated with the individual elementary reactions
and their forward and reverse rate constants. It is axiomatic that reactions with high-energy