Essentials of Computational Chemistry

15.3 TRANSITION-STATE THEORY 531
secondary KIEs can be ‘inverse’, which is to say that the light atom rate over the heavy
atom rate can be less than one. In this case, no particular simplifications of Eq. (15.33) are
general, and each partition function may play a role in addition to those of the ZPVEs.
This is particularly true because different vibrational modes may cancel one another in the
secondary KIE. That is, one mode may lead to a large normal KIE but be canceled by another
mode that leads to a large inverse KIE, such that more subtle effects associated with, say,
rotational motion, may be made manifest.
One caveat that must be observed when comparing computed and experimental isotope
effects is that experimental measurements can sometimes be for multistep reactions. When a
particular elementary reaction is not rate-determining, that is, it is not the bottleneck in the
overall process, then it does not matter whether or not that reaction has associated with it a
large KIE; it will not influence the observed overall rate. A separate caveat with light atoms
at low to moderate temperatures is that tunneling effects may play a significant role.
15.3.2 Variational Transition-state Theory
Canonical TST defines the free energy of the activated complex based on the TS structure.
This is convenient because, as it is a stationary point, we can use the machinery of the
rigid-rotor-harmonic-oscillator approximation to compute the necessary partition functions
to define its (reduced-dimensionality) free energy. However, it is by no means guaranteed
that the free energy associated with the TS structure really is the highest free energy of any
point along the MEP – it is only guaranteed that it is the highest point of potential energy
along the MEP. As a simple example, it might be the case that the potential energy wells
associated with some normal modes tighten up after the TS structure is reached, even though
the bottoms of those wells are at a point on the MEP slightly below the energy of the TS
structure. The increase in ZPVE resulting from those tighter potentials may exceed the drop
in bottom-of-the-well energy such that the free energy of the non-stationary point is higher
than that of the TS structure.
Variational transition-state theory (VTST), as its name implies, variationally moves the
reference position along the MEP that is employed for the computation of the activated
complex free energy, either backwards or forwards from the TS structure, until the rate
constant is minimized. Notationally
k VTST (T , s) = min
s
kBT
h
Q ‡ (T , s)
QR
Q o R
Q ‡,o e−V ‡ (s)/kBT
(15.35)
where s is a position on the MEP at which k VTST is evaluated. By convention, s = 0 refers to
the saddle point, and negative and positive values are displaced to the reactant and product
sides of the saddle point, respectively.
To compute the r.h.s. of Eq. (15.35), we need to define how we compute the partition
function (and the ZPVE) for the non-stationary point s. In this case, we simply continue to
take advantage of our decision to treat the activated complex as a species having 3N − 7
bound degrees of freedom. In order to define this space for an arbitrary point on the MEP,