BioPHYSICAL chemistry

DA →D+A− (7.29)

In Marcus theory, the transfer is an activated pro-
cess and the electron transfer rate,ket, can be divided
into two components, the electronic coupling, V,
and the Franck–Condon term, which describes the
energetics:

(7.30)

Electron transfer is a quantum-mechanical process
and the coupling, which has factors such as the
distance between the donor and acceptor, will
be considered in Chapter 10. The Franck–Condon
term arises from a consideration of the states DA
and D+A−in terms of harmonic oscillators(Figure
7.9). The two curves are displaced along the reac-
tion coordinate corresponding to changes in the
structure after transfer. These curves share the
crossing point as the single common point where
the two states have a common nuclear configura-
tion. The crossing point represents the nuclear
configuration at which the reactants are poised in
the transitional state.
Electron transfer will depend upon the coupling
and the probability that the product state will
achieve the transition state. As discussed above, the probability of
achieving the transition state can be expressed in terms of probability using
the activation energy. For the specific case of Marcus theory, the activa-
tion energy can be written in terms of the difference in the Gibbs energy
and the reorganization energy,λ:

(7.31)

The value of λrepresents the energy that would need to be added to
the system to force the state DA to have the same nuclear configuration
as D+A−, and hence is a vertical line between the two curves. Substituting
the activation energy into eqn 7.27 and accounting for the various factors
yields:

Franck–Condon= −°+()/ (7.32)

1

4

(^24)
πλ
λ
kT
e
B
ΔGkTB

E

G

A kB

()

=

Δ °+λ^2

4

k
h
et= V(Franck–Condon)

2 π 2

CHAPTER 7 KINETICS AND ENZYMES 145

Gibbs energy

Displacement, q

EA

qR 0 q* qP 0

λ

DA

DA

ΔG^0

Figure 7.9Gibbs

energy surfaces

for a DA pair and

a D+A−pair that

each are parabolas

characteristic of

harmonic oscillators

(Chapter 13).

Electron transfer

usually makes use of

the reorganization

energy λrather

than the activation

energy.