Essentials of Computational Chemistry

542 15 ADIABATIC REACTION DYNAMICS
them is Marcus theory (Marcus 1964). The full scope of Marcus theory is very broad, and we
consider here only the simplest application of the model. We will take the generic electron
transfer reaction
A − + B → A + B −
(15.49)
For this simple case, Marcus theory predicts the rate constant for electron transfer to be
kET = ZABe −(Go
AB +λ)2 /4λRT
(15.50)
where ZAB is the collision frequency for the reactants (typically in the range of 109 to
10 10 sec−1 for reactions in non-viscous liquids at ambient temperatures), Go AB is the free
energy change for the electron transfer, λ is the so-called reorganization energy, R is the
universal gas constant, and T is the temperature.
The reorganization energy term derives from the solvent being unable to reorient on the
same timescale as the electron transfer takes place. Thus, at the instant of transfer, the bulk
dielectric portion of the solvent reaction field is oriented to solvate charge on species A,
and not B, and over the course of the electron transfer only the optical part of the solvent
reaction field can relax to the change in the position of the charge (see Section 14.6). If the
Born formula (Eq. (11.12)) is used to compute the solvation free energies of the various
equilibrium and non-equilibrium species involved, one finds that
λ = (q) 2
1
−
ε∞
1
1
+
ε0 2rA
1
−
2rA
1
(15.51)
rAB
where q is the amount of charge transferred (1 for the reaction of Eq. (15.49)), ε∞ is the
fast dielectric constant (sometimes called the optical dielectric constant, equal to the square
of the index of refraction – around 2 for typical solvents), ε0 is the slow, or bulk, dielectric
constant, rA and rB are the radii of species A and B, respectively, and rAB is the distance
between them at reaction. The quantity in Eq. (15.51) is sometimes called λo because it
considers only ‘outer-sphere’, which is to say solvent, reorganization. More sophisticated
approaches can be used when inner-sphere reorganization is also important, e.g., for ligated
metal systems where the metal–ligand bond lengths might vary significantly as a function of
charge. In such instances, inner-sphere reorganization energies can often be estimated from
calculations of relaxation energies when the geometry of the species for the initial charge
state is allowed to relax to the final charge state.
The exact form of Eq. (15.50) is made more intuitive by considering the simple reaction
coordinate diagrams of Figure 15.8. In these cases, we consider two parabolic potential
energy surfaces corresponding to the two sides of Eq. (15.49). The reaction coordinate may,
in particularly simple instances, be thought of as a generalized solvent coordinate. Thus, when
the solvent is optimally configured for A − + B, the energy of the curve for state A + B −
is quite high. If the free energies of the left and right sides of Eq. (15.49) are the same
(which would happen if A and B were different isotopes of the same metal, for instance),
the separation of the two curves at either minimum is exactly λ. From the mathematics of
parabolae, this requires the intersection of the two curves to take place at λ/4 energy units