Essentials of Computational Chemistry

536 15 ADIABATIC REACTION DYNAMICS
where V ‡ is the zero-point-including potential energy difference between the TS structure
and the reactants, and V is 0 for an exoergic reaction and the (positive) zero-point-including
potential energy difference between reactants and products for an endoergic reaction. In the
case where α ≤ β, the corresponding expression is
κ(T) = β
e
β − α
[(β−α)(V ‡
−V)]
− 1
(15.41)
An inspection of the power series expansion for the exponential in Eqs. (15.40) and (15.41)
indicates that neither expression diverges as α and β become arbitrarily close to equal (an
analogous consideration of the power series expansion for the sine function in Eq. (15.40)
indicates the first term on the r.h.s. to be similarly free from singularities).
A still more sophisticated approach involves fitting the reaction coordinate to a so-called
Eckart potential (Eckart 1930). The Eckart potential permits an exact, analytic solution of
the probability of tunneling through the barrier (and of non-classical reflection) from the
time-independent Schrödinger equation for systems of fixed energy E. When that result
is numerically integrated over all energies, weighted by the Boltzmann probability of the
reacting system having a particular energy at a given temperature T , a very good estimate of κ
in the limit of tunneling along a single dimension is obtained. [Note that when transition state
theory is formulated for a system of constant energy, as opposed to constant temperature, it is
called microcanonical TST (µTST) or Rice–Ramsperger–Kassel–Marcus (RRKM) theory
for the unimolecular case; a microcanonical variational TST (µVTST) can be applied in a
fashion analogous to VTST, with the choice of dividing surface location s now potentially
different at each energy E.]
It should be noted, however, that even the best one-dimensional tunneling estimate is still
likely to underestimate the full tunneling contribution, since tunneling may occur through
dimensions of the PES other than the reaction coordinate. Multi-dimensional tunneling approximations
are sufficiently complex, however, that they will not be further discussed here.
Another important point that must be borne in mind is that failure to account for tunneling,
or to recognize its contribution in the first place, can lead to significant errors in the interpretation
of experimental data. For example, Watson (1990) analyzed an Eyring plot of
apparent rate constants for methane metathesis by methyllutetiocene (Figure 15.5) to infer
CH 4
Lu CH3 Lu
– CH4 CH3 H
CH3 Figure 15.5 Transition-state structure for rate-determining hydrogen atom transfer in the methane
metathesis reaction of methyllutetiocene. Note that the kinetics for this narcissistic reaction may be
followedbyusinga 13 C label either in the reacting methane or in the methyl group of the starting
organometallic