Computational Chemistry

distribution) will have the extra energy needed to traverse the higher-energy regions
of this flatter saddle, away from the exact TS point, than in the case of A 2 !B 2 ; if
the saddle were curved infinitely steeply,nomolecules could stray outside the
reaction path. Thus reaction 1 must be faster than reaction 2, although they have
identical computed free energies of activation; the rate constant for reaction 1 must be
bigger than that for reaction 2. The difficulty of obtaining good rate constants from
accurate calculations on just two PES points, the reactant and the TS, is mitigated by
the fact that the vibrational frequencies of the TS sample the curvature of the saddle
region both along the reaction path (this curvature is represented by the imaginary
frequency) and at “right angles” to the reaction path (represented by the other
frequencies). High frequencies correspond to steep curvature. So when we use the
TS frequencies in the partition function equation for the rate constant we are, in a
sense, exploring regions of the PES saddle other than just the stationary point. The
role of the curvature of the PES in affecting reaction rates is nicely alluded to by
Cramer, who also shows the place of partition functions in rate equations [ 208 ].
Another way to calculate rates is bymolecular dynamics[ 209 ]. Molecular
dynamics calculations use the equations of classical physics to simulate the motion
of a molecule under the influence of forces; the required force fields can be
computed by ab initio methods or, for large systems, semiempirical methods
(Chapter 6). In a molecular dynamics simulation of the reaction A!B, molecules
of A are “shaken” out of their potential well, and some pass through the saddle
region. A shaken mechanical model with a molded surface and ball-bearing mole-
cules would represent a good analogue of the computer simulation. At a given
“temperature” (corresponding to the vigor of shaking), the rate of passage of
molecules (or ball bearings) through the saddle region will depend on the height

PE surface curved gently upward

in the transition state region

Many molecules pass

per second through the

TS region Only relatively few molecules pass

per second through the

TS region

PE surface curved steeply upward

in the transition state region

(^12)
intrinsic reaction coordinate
TS
TS
Free energy
reactant reactant
Fig. 5.29 Possible potential energy surfaces for two reactions with the same calculated free
energy of activation. Reaction 1 is nevertheless faster than reaction 2 because its transition state
region is flatter. As a result, in a given time more molecules can stray from the intrinsic reaction
coordinate and pass through the transition state region to the product
324 5 Ab initio Calculations