MODULE 27_03
ELECTRON TRANSFER REACTIONS
Some Basic Principles
Processes involving the transfer of electrons occur widely in the physical and life sciences. They
range from simple exchange reactions in chemistry
2
3+
Feaq
 Feaq
3+
2
Feaq
 Feaq
to processes that drive energy storage and respiration in biological systems, such as the
cytochrome-c/cytochrome oxidase couple which are both heme proteins.
Cytochrome
oxidase
Cytochrome-c
e
Fe
Fe
oxidase
Fe
Fe
In recent years, photochemical scientists have become keenly interested in electron transfer
reactions because:
(i)
they occur as primary events in many photoprocesses.
(ii)
they can be conveniently studied using photophysical techniques
(iii)
photophysical methods offer excellent ways of testing the theories.
Kinetic Aspects of Bimolecular Reactions
An electron transfer reaction between individual molecules freely diffusing in a mobile liquid
has characteristics of all such bimolecular reactions:
kf
D+A
D+ + A -
kr
The two independent entities diffuse together, react, and two different entities diffuse apart. The
overall process presumably proceeds via a collision complex, one or more reaction intermediates,
or a transition state. Thus, micro-reversibility applies, and the process occurs on a continuous
potential energy surface. No excited states have yet been invoked, so we can imagine the
1
process to be adiabatic (no crossings to other PE surfaces), although this concept is not always
true.
We can define rate constants for the forward and reverse processes, kf and kr, and an equilibrium
constant, Keq. Students of electron transfer reactions are interested in determining values of these
parameters and understanding the factors that influence them. The overall reaction, like all
chemical reactions, will have characteristic G0, 0, and S0 parameters. In addition, since this
is specifically electron transfer, we can relate G0 to reduction potentials, viz.,
G0   nFE 0   nF ( EA0 / A  ED0  / D )
(27.1)
However, thermodynamic constants are useful for describing equilibrium states only, not
providing information on the mechanistic details. Experiment shows that for exoergic electron
transfer processes, rate constant values occur over a wide range, up to the limit imposed by
diffusion. It is of great interest to understand this variation. Let us consider a detailed scheme
for the overall bimolecular process shown above.
D+ + A-
D+A
kd
k-d
k-d
e
kn
[DA]
Precursor
complex
[DA]
k-n
kd
k’-n
#
+
- #
[D+A-]
[D A ]
e
k’n
Reorganized
precursor
complex
Successor
complex
2
Reorganized
successor
complex
A treatment using steady state approximation leads to
1
kobs
 (kd  ka ) / kd ka
(27.2)
where
ka 
under the condition that
kd kn ve
2ve  k n
(27.3)
k n  k n
When ka >> kd, we see that kobs = kd, i.e., the measured rate constant approaches the limit set by
diffusion (see Module 24).
Let us examine equation (27.3) in more detail
(i) Under conditions when ve  k n i.e., the electron moves to A more rapidly than the
reorganized complex relaxes, then
1
1
ka  kd kn  kd k n K n
2
2
1
 kd k n exp (Gn† / RT )
2
(27.4)
(27.5)
where k-n is the rate of relaxation of vibrationally excited precursor complex, and Gn† is the
energy barrier to nuclear reorganization. Thus ka is independent of e, the electron-hopping rate.
(ii) On the other hand, when k n  ve
ka  kd ve Kn
(27.6)
 kd ve exp (Gn† / RT )
(27.7)
Therefore in both cases (and all others), the nuclear reorganization process is a barrier to electron
transfer and imposes an activation step.
So we see that the overall (measured) rate constant is a combination of diffusion-dependent (kd)
and activation-dependent (ka) terms:
1
kobs
 ka1  kd1
(27.8)
As we saw above, when ka >> kd, then kobs = kd. This is a simplifying situation, but it means that
our kinetic measurements can provide no information on the activation-dependent process
because everything is limited by diffusion. However, the central segment of the sequence of
processes
3
 ( D / A)
kn
( D / A)
ve
†
k n


( D . / A. ) † 
ve
occurs independently of how the sequence is initiated and if we wish to investigate its role in the
sequence we need to circumvent the diffusion limiting problem. Later we will see how this can
be done. For now, we assume that it can and proceed to examine ka.
From the foregoing, it has emerged that the parameters k-n, e, and Gn† are important in
determining the magnitude of ka. Theoreticians have examined these using classical mechanics
(Marcus, Sutin, Hush) or semi-classical/quantum methods (Jortner, Levich).
Gn† ;The Transition State Picture
The partial scheme is
 ( D / A)
precursor
kn
( D / A)
k n
†
reorganized
precursor
ve
ve
( D  / A ) † 
successor
The steps prior to the formation of (D/A) and those after ( D  / A ) are ignored in this argument.
D and A may be polyatomic molecules, aquated metal ions, etc., and the reaction above proceeds
with changes in bonding coordinates in D and A, and solvation around the complex
( D / A)† and ( D  / A )† have identical nuclear configurations; they differ only in that a single
electron has moved.
The situation resembles a Franck-Condon type event, or even a
radiationless transition between two states.
Thinking of electron transfer in terms of a
radiationless transition, one can write the expression
k
2
 e2  n
(27.9)
where  is an average density of states in the acceptor. In electron transfer theory, this term
usually appears as a Franck-Condon weighted density of states (FCWD). The electronic matrix
2
element term,  e , contains the operator which drives the process.
The classical picture due to Marcus (1956) is less rigorous and simpler, but provides useful
physical insights. It also provides a comparison to the transition state theory (TST) of kinetics.
Marcus chose to represent the complex multidimensional PE surfaces of polyatomic reactant
pairs as a parabolic energy curve in "nuclear configuration space", see Figure 27.1.
4
20
PRODUCT STATE
REACTANT STATE
15
10
G

5
G#n
FIG. 30.1
0
4
2
0A
2
4
6
8
Nuclear configuration
space
The quantity  is the energy required to move the electron in the reactant state (D/A) to the


product state ( D / A ) without prior nuclear reorganization. In this it resembles a Franck-
Condon event. The Gn# quantity represents the energy required to reconfigure the precursor
5
complex to a non-equilibrium nuclear configuration in which the electron transfer can occur and
the system can switch from the reactant state curve to the product state surface. Note that
Gn#   . At the curve crossing, the electron can hop from one curve to another with some
probability (rate).
The situation shown in the schematic is for G 0  0 , i.e., an isoergic process. Analytical
geometry of intersecting parabolas showed Marcus that
Gn#   / 4
in the isoergic condition. When
(27.10)
G 0  0 , then
Gn#  (G 0   ) 2 / 4
(27.11)
#
Thus the energy barrier (Gn ) to electron transfer depends on G 0 and  in a quadratic
manner. Figure 27.2 schematically demonstrates the effect on G#n as the value of G0 becomes
increasingly negative. As the product parabola is lowered with respect to the red reactant curve,
the nuclear reorganization barrier first becomes less and then increases again (the only change is
in the overall driving force, there are no shape changes, nor does the curve minimum shift to left
or right). 
l( x)  k( x)  25
20
0
G0
20
FIG. 27.2
40
10
5
0
5
6
10
15
†
†
Earlier we showed ka  exp(Gn / RT ) or ln ka   Gn / RT . Thus, Marcus theory predicts that for
weakly exoergic reactions log ka increases as G 0 increases, maximizes at G 0   , and
decreases again as G 0 increases beyond . This is a remarkable result, flying in the face of
expectation and it led to Marcus’s Nobel Prize in Chemistry in the early 1990s. Figure 27.3
 G E  
shows the graph of log ka vs. inkvolts
E . a exp  

 R T 
110
 13
13 1 10
1 10
12
normal
region
inverted
region
ka
k( E  0.5) 1 10
11

1 10
10
Figure 27.3
1 10
9
1 10
9
2
0
3
2
E
4
4
In the foregoing development we
20
have assumed that the product
state is formed in its zeroth
10
vibrational state. However, this
is a simplification and in fact the
0
formation of product species in
vibrational
10
above
the
zeroth is very possible.
FIG. 27.4
20
states
10
5
In Figure 27.4 the family of
0
5
10
15
dashed curves represent four
vibrational
7
energies
of
the
product state. Whereas the red dashed curve (v = 0) crosses the reactant curve in the inverted
region, the higher vibrational modes do not. The rate constant observed will be a weighted sum
of the contributions from all the modes, and it will be larger than that if the v = 0 mode was the
only contributor. Thus the inverted region will be less pronounced than otherwise; the parabola
will depart from the symmetrical form shown in Figure 27.3.
The Marcus approach, being geometrical, assumes symmetrical sets of pure parabolas, i.e., there
is only very weak interaction at the crossing point. Thus the reaction is by necessity
non-adiabatic, because curve crossing must occur. Figure 27.5 shows the adiabatic/non-adiabatic
situation
FIG. 30.5
We can see that at the crossing point the two curves are degenerate and under such a condition
time dependent perturbation theory instructs us that the probability of transferring to the product
curve will be proportional to sin 2 V t . The vibrational motion of the nuclei in the reactant state
will be such that it spends only a short time in the crossing region. Thus for states where V is
small (< kBT), the probability of reaction will be small and the system will continue on the
reactant surface for many passages through the crossing region. On the other hand, when the
perturbation is large (> kBT), as for the adiabatic case, the argument of the sin2 term is large and
the oscillatory frequency will be sufficient to ensure easy passage into product space.
The reorganization energy, 
The barrier to electron transfer, according to Marcus' model, is manifest as a free energy term
composed of G 0 and  components. The former is a state property, which defines the overall
8
free energy changes in going from reactants to products. The quantity  is an energy term that
describes the reorganization of nuclear configurations that are needed to allow the electron to
hope form one parabola to another. Energy requirements due to structural adjustments within the
molecular frameworks of the D and A species (inner sphere) and readjustment of solvent dipoles
to accommodate the charge shift (outer sphere) are included in  , thus
  s  0
where
(27.12)
s depends on solvent motions and 0 depends on internal nuclear adjustments (bond
lengths, and so on.)
Marcus (1959) arrived at an expression for s on the idea that a solvent behaves as dielectric
continuum (no local structure; hard sphere molecules).
s  e2[1/ 2rD 1/ 2rA 1/ RDA ]( Dop1  Ds1 )
(27.13)
where rD , rA are the radii of donor and acceptor, RDA is the separation of D and A, and Dop , DS
are the optical and static dielectric constants of the solvent
For polar organic liquids ( CH 3CN )
s 0.75 eV
For non-polar organic liquids ( C6 H12 )
s 0.15 eV
The value of
0 is not easily calculated, but it is usually estimated from considerations of the
force constants of normal mode vibrations in the reactant and product species. A typical value
for is ca 0.4 eV.
The ve parameter:
ve has
been used to represent the frequency (or rate) at which an electron can shift between
(D/A)# and (D+ /A- )# . This can be broken down
ve  vn  el
vn
(27.14)
is the frequency with which the reorganized nuclear configuration is reached and  el is the
probability of electron transferring from one PE curve to another in the reorganized
configuration.
9
In TST terms, vn can be regarded as being similar to an entropy-controlled term, viz.,
vn 
k BT
exp (S # / R )
h
(27.15)
The maximum value of n is kBT/h, which is approximately 1013 s-1 when S# = 0.
 el is similar to the transmission coefficient of TST and can take values between zero and one.
It represents the effectiveness (probability) of the electron switching its molecular identity with
the concomitant change of the system from the reactant surface to the product surface. Thus, it
depends on the interaction energy between the two curves (surfaces) at the crossing region-see
Figure 27.5 and the preceding discussion. Thus  el  V , where V is the electronic coupling
2
matrix element (equivalent to  e used earlier).
Molecular wave functions decrease exponentially with distance from their maximum amplitude.
Thus, the overlap increases as the distance between the reacting species decreases (Figure 27.6)
FIG. 27.6
It is found that V  exp (  r RDA ) , where  r (not to be confused with  e ) is a multiplier of
RDA having dimensions of m-1 and can be thought of as being equivalent to a molecular
resistance. If  r is small for a given
effectively; if
RDA , then V
is large and electron transfer occurs
 r is large for a given RDA , then V is less and the transfer efficiency is reduced.
Thus, we see that the overall rate constant for electron transfer in the activated case (ka) depends
on
(i)
distance between the participants
(ii)
the frequency with which nuclei reorganize
(iii)
a reorganization barrier
10
Overall the activated rate constant can be described by
ka  vn  el exp[(G 0   ) 2 / 4 RT ]
(27.16)
The maximum value of n is approximately 1013 s-1. The electronic coupling factor, Kel, depends
on RDA and on the relative orientation between the dipoles in the donor and acceptor. The
parameter is the reorganization energy, which depends on solvent reorientation and on changes
Figure 27.7 shows a 3-D plot of ln k vs  and G
internal to the molecules.
FIG.27.7
20
18
ln k
16
14
12
10
8
30

20
10
-10
-5
0
5
10
15
20
25
30
35
40
G
Note that nothing in the foregoing has required photoexcitation. The development has been quite
independent of the electronic state of the reacting species.
Reviews and Literature Entries
1. G. L. Closs and J. R. Miller. Science (1988), 240, 440-447
2. G. McLendon. Acc. Chem. Res. (1988), 21, 160-167
3. R. A. Marcus and N. Sutin. Biochem. Biophys. Acta (1984), 811, 265-322.
11
4. M. D. Newton and N. Sutin. Ann. Rev. Phys. Chem. (1984), 35, 437-80.
12