Essentials of Computational Chemistry

534 15 ADIABATIC REACTION DYNAMICS
reactants to products in a classical system is a Heaviside function of the energy, as illustrated
in Figure 15.4. In a quantum system, however, the transmission probability is sigmoidal in
shape, reflecting the phenomena of tunneling and non-classical reflection. Tunneling refers to
the ability of a quantum system having an energy below the saddle point to ‘tunnel’ through
the barrier to the products side, while non-classical reflection refers to the possibility that a
quantum system above the saddle-point energy will suffer from destructive interference in
a way that prevents it from crossing to products. This situation is compared to the classical
one in Figure 15.4.
Insofar as tunneling increases the rate constant by allowing lower-energy systems to be
reactive and non-classical reflection decreases the rate constant by reducing the reactivity of
higher-energy systems, one might imagine that the two could safely be assumed to cancel.
However, a thermally equilibrated Boltzmann population has a much larger percentage of
P
1
0
Temperature-dependent
Boltzmann distribution
tunneling
non-classical
reflection
∆V ‡
Reactant energy
dP
dE
Figure 15.4 Probabilities of reaction (P ) for systems moving towards a parabolic barrier for a reaction
with a zero-point-including potential energy of activation V ‡ . Classical systems ( )below
the barrier height have zero probability of reaction and above the barrier height have unit probability
(i.e., the ‘curve’ describes a Heaviside function). Quantum systems (------), on the other hand,
have increasingly non-zero probabilities as the barrier energy is approached from below because of
tunneling and increasingly less than unit probabilities as the barrier energy is approached from above
because of non-classical reflection. Note that because of the Boltzmann distribution of energies in a
thermalized population of reacting systems (-· -· -· -· referenced to the right ordinate), typically many
more molecules have energies in the region where tunneling can increase the reaction rate than have
energies in the region where non-classical reflection can reduce the reaction rate. As a result, the
former is the more quantitatively important of the two quantum phenomena