Essentials of Computational Chemistry

G
l
A − + B
15.5 NON-ADIABATIC DYNAMICS 543
q
(a)
A + B −
∆G° = 0
l/4
G
G
q
(b)
q
(c)
−∆G° = l
−∆G° > l
Figure 15.8 Electron-transfer reaction coordinate diagrams used in Marcus theory. Diagram (a) refers
to a case with no net free energy of reaction, in which case the intersection of the two curves occurs
at λ/4 above the minima and is taken as the barrier to the electron transfer (a barrier associated with
solvent reorganization in the simplest limit). When the overall driving force is equal in magnitude to
λ (b), the two curves cross at the equilibrium solvent configuration of the first state, and reaction is
barrierless. However, when the driving force becomes still greater (c), the crossing of the two curves
proceeds to the left on the reaction coordinate, and occurs at higher energy than the minimum of
the reactant curve. This situation creates the inverted region where rate decreases with increasing
exergonicity
above the two equal minima. This situation in illustrated in the first reaction coordinate
diagram of Figure 15.8, and rationalizes the denominator of the exponential in Eq. (15.50):
if Go AB is zero, then the argument of the exponential is λ/4RT which is indeed the ‘barrier’
for reaction in the system with no thermochemical driving force in either direction.
Note that Marcus theory in the form of Eq. (15.50) makes a rather surprising prediction.
If Go AB is equal to λ in magnitude but of opposite sign, which is to say the exergonicity
of the electron transfer exactly cancels the reorganization energy, than the argument of the
exponential is zero and the rate is predicted to be diffusion-controlled. However, if the
driving force becomes greater still, then the argument of the exponential returns to positive,
and the rate is predicted to decrease (Figure 15.8). This corresponds to the so-called inverted
region of Marcus theory. That is, as one of a pair of reactants in an electron-transfer reaction
is varied so that the reaction becomes more and more favorable in a free-energy sense, the
rate is predicted to reach a maximum and then decrease. Experimental verification of this
prediction did not occur until many years after the initial publication of the theory, in part
because the required driving force is so high and in part because of the technical challenges
associated with measuring very large rate constants. Nevertheless, an inverted region has