BioPHYSICAL chemistry

simple representation of electron transfer, an electron on a donor must

make a transition to the acceptor. Both the donor and acceptor can be

considered to be local regions with stable potentials (Figure 7.9). In order

to make the transition, the electron must travel through regions that

have varying potentials, but are generally less favorable, and therefore

less stable, than those associated with the donor and acceptor. The prob-

lem then can be represented as calculating the probability of an electron

making a transition through a series of potential barriers. The presence

of the barriers is modeled in terms of the coupling constant V. For each

barrier, there will be a decrease in the transmission of the electron, and

this is modeled as giving rise to a decrease in the coupling strength. The

effect of the multiple barriers is to give rise to a product of terms, each of

which, identified as εI, contributes to the overall strength of the coupling

with a proportionality constant A:

(10.53)

The value of each contribution εIcan be estimated using two different

possible approaches. In one approach, each of the steps is considered to

be of equal strength, εI=εa 9 for all steps i. In this case, the coupling can

be written in terms of the average barrier for each step and expressed

in terms of the physical separation between the donor and acceptor,

rDA:

(10.54)

Thus, the prediction is that the coupling, and hence the electron-transfer

rate, will decay exponentially with increasing separation between the

donor and acceptor. The alternative approach is that each step has a dis-

tinctive effect on the coupling. In this scheme, the individual coupling

is determined by the nature of the space that the electron is traversing.

In general, the steps can be classified into three groups, with the best

coupling arising for travel through covalent bonds, weaker coupling for

travel through hydrogen bonds, and the weakest coupling being for

travel through space. The net coupling is calculated as the product of these

three contributions:

(10.55)

VA

i I j j

(^22)
=∏

⎡

⎣⎢

⎤

⎦⎥

εε()(covalent bonds∏ bonds)(through spacehydrogen b)

⎡

⎣

⎢

⎤

⎦

⎥∏

⎡

⎣⎢

⎤

k k ⎦⎥

ε

VA Ae

N

a R

N R

a

222 2==εβε−β =−^2 ln ≈ 1

99 DA

DA

Å−−^1

VA

i I

2 2 2

=∏()ε

216 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY