The rate of electron-transfer reactions in the diffusive limit
A simple electron-transfer reaction is treated in the diffusive limit, in which the motion of the solvent is governed by the Smoluchowski equation. The electronic transition probability is calculated from the Landau-Zener expression. The rate constant of the reaction is calculated as a function o...
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Інститут фізики конденсованих систем НАН України
2001
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | The rate of electron-transfer reactions in the diffusive limit / W. Schmickler // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 773-778. — Бібліогр.: 19 назв. — англ. |
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irk-123456789-1205282017-06-14T17:22:57Z The rate of electron-transfer reactions in the diffusive limit Schmickler, W. A simple electron-transfer reaction is treated in the diffusive limit, in which the motion of the solvent is governed by the Smoluchowski equation. The electronic transition probability is calculated from the Landau-Zener expression. The rate constant of the reaction is calculated as a function of the strength of the electronic interaction between the reactants. For weak interactions, the rate is the same as that obtained form first-order perturbation theory. For strong interactions, solvent dynamics is rate-determining. The calculations presented here bridge these two limits. Проста реакція електронного переносу розглядається в дифузійній границі, в якій рух розчинника описується рівнянням Смолуховського. Імовірність електронного переходу розраховується з виразу Ландау-Зенера. Константа реакції розрахована як функція сили електронної взаємодії між реактантами. Для слабких взаємодій ступінь реакції співпадає з виразом, отриманим в першому наближенні теорії збурень. Для сильних взаємодій динаміка розчинника визначається ступенем реакції. Представлені розрахунки поєднують обидві границі. 2001 Article The rate of electron-transfer reactions in the diffusive limit / W. Schmickler // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 773-778. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 82.30.F, 82.20.Gk DOI:10.5488/CMP.4.4.773 http://dspace.nbuv.gov.ua/handle/123456789/120528 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
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A simple electron-transfer reaction is treated in the diffusive limit, in which
the motion of the solvent is governed by the Smoluchowski equation. The
electronic transition probability is calculated from the Landau-Zener expression.
The rate constant of the reaction is calculated as a function of
the strength of the electronic interaction between the reactants. For weak
interactions, the rate is the same as that obtained form first-order perturbation
theory. For strong interactions, solvent dynamics is rate-determining.
The calculations presented here bridge these two limits. |
| format |
Article |
| author |
Schmickler, W. |
| spellingShingle |
Schmickler, W. The rate of electron-transfer reactions in the diffusive limit Condensed Matter Physics |
| author_facet |
Schmickler, W. |
| author_sort |
Schmickler, W. |
| title |
The rate of electron-transfer reactions in the diffusive limit |
| title_short |
The rate of electron-transfer reactions in the diffusive limit |
| title_full |
The rate of electron-transfer reactions in the diffusive limit |
| title_fullStr |
The rate of electron-transfer reactions in the diffusive limit |
| title_full_unstemmed |
The rate of electron-transfer reactions in the diffusive limit |
| title_sort |
rate of electron-transfer reactions in the diffusive limit |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120528 |
| citation_txt |
The rate of electron-transfer reactions in the diffusive limit / W. Schmickler // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 773-778. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT schmicklerw therateofelectrontransferreactionsinthediffusivelimit AT schmicklerw rateofelectrontransferreactionsinthediffusivelimit |
| first_indexed |
2025-07-08T18:04:40Z |
| last_indexed |
2025-07-08T18:04:40Z |
| _version_ |
1837102930376785920 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 4(28), pp. 773–778
The rate of electron-transfer reactions
in the diffusive limit∗
W.Schmickler
Abteilung Elektrochemie, University of Ulm, D-89069 Ulm, Germany
Received August 8, 2001, in final form October 10, 2001
A simple electron-transfer reaction is treated in the diffusive limit, in which
the motion of the solvent is governed by the Smoluchowski equation. The
electronic transition probability is calculated from the Landau-Zener ex-
pression. The rate constant of the reaction is calculated as a function of
the strength of the electronic interaction between the reactants. For weak
interactions, the rate is the same as that obtained form first-order perturba-
tion theory. For strong interactions, solvent dynamics is rate-determining.
The calculations presented here bridge these two limits.
Key words: electron transfer reactions, diffusive limit
PACS: 82.30.F, 82.20.Gk
1. Introduction
Electron-transfer reactions play a major role in physics, chemistry, and biology
alike [1]. Much of our present understanding of the way these reactions proceed
in condensed phases is based on the works of Marcus [2] and Hush [3], which are
based on classical statistical mechanics, and of Levich and Dogonadze [4], who em-
ployed first-order perturbation theory. These two classes of theories refer to different
strengths of the electronic interaction between the reacting partners: Marcus and
Hush consider the adiabatic limit, in which the interaction is strong, while the Levich
and Dogonadze theory treats nonadiabatic reactions with weak electronic coupling.
Till the present day, despite many efforts (see e.g. [5–8]), the gap between the
adiabatic and the nonadiabatic limits has never been quite closed. For the special
case of electrochemical reactions Mohr and Schmickler [9] have recently derived an
expression for the rate constant that is valid for all interaction strengths. However,
solvent dynamics is not considered in this work, so that its validity is limited to
timescales shorter than the solvent relaxation time.
In this work, we treat the simple case in which the electron transfer is coupled
to classical solvent modes only, and in which solvent dynamics is overdamped. The
∗Dedicated to Jean-Pierre Badiali on the occasion of his 60th birthday.
c© W.Schmickler 773
W.Schmickler
motion of the system is then governed by the Smoluchowski equation, which we solve,
using the electronic transition probability as a boundary condition. In this way we
obtain an expression for the rate constant which is valid for arbitrary strengths of
the electronic interaction.
2. Diffusion controlled electron transfer
U(q)
qqa
initial
well
final
well
0 qf
Figure 1. Potential energy curves for a
simple electron-transfer reaction
We consider electron exchange be-
tween two reactants in a solution, and
assume that this exchange is coupled to
a classical bath. In this case, the system
can be characterized by a single, effec-
tive bath coordinate q. Within the har-
monic approximation the diabatic po-
tential energy curves for the initial and
final states are parabolas, which inter-
sect at the saddle point qa (see figure 1),
and which have their equilibrium posi-
tions at q = 0 and q = qf . Within this
model, which originated with Marcus [2]
and Hush [3], electron transfer is an ac-
tivated process: A thermal fluctuation
takes the system from the bottom of the
initial well to the saddle point, where
electron exchange may occur. The orig-
inal papers by Marcus and Hush treated only the adiabatic limit, in which the
electronic coupling between the reactants is so strong that an electron is exchanged
every time the system passes the saddle point. Here we consider the general case, in
which the coupling has an arbitrary strength, and the system may pass the saddle
point without the occurrence of an electron transfer.
Before the reaction, the system undergoes a stochastic motion in the initial well.
In the overdamped case, which we consider here, the probability P (q, t) to find the
system at position q at the time t obeys a Smoluchowski equation of the form [10]:
∂P (q, t)
∂t
= − ∂
∂q
j(q, t) = D
∂
∂q
[
∂P (q, t)
∂q
+ βU ′
i(q)P (q, t)
]
. (1)
Here, j(q, t) is the corresponding probability current, D is the diffusion coefficient,
which we take as constant, β = 1/kBT , kB is Boltzmann’s constant, T the tempera-
ture, and Ui(q) =
1
2
mω2q2 the potential in the initial well. For numerical calculations
we will set the effective mass m and the frequency ω equal to unity.
At q = qa the system can exchange an electron and escape from the initial well.
This results in a radiative boundary condition of the form:
j(qa, t) = −κP (qa, t). (2)
774
Electron transfer
For the electronic transition probability t(v) we use the Landau-Zener expression
[11,12]:
t(v) =
1− exp(−2πν)
1− 1
2
exp(−2πν)
with ν =
|V |2
h̄mω2|qfv|
. (3)
Here, V is the matrix element for the electronic coupling, and v is the velocity of
the system near the saddle point. The denominator in the first part of equation (3)
takes account of multiple crossing of the saddle point [1].
The Landau-Zener equation results in the following expression for the escape
rate κ:
κ = (2πkBT/m)(1/2)
∫
∞
0
dv exp
(
− mv2
2kBT
)
t(v). (4)
The rate of electron transfer can be calculated as the inverse of the first mean
passage time τ , for which Szabo et al. [13] and Deutch [14] have obtained explicit
expressions. For the case in which the initial distribution equals the equilibrium
distribution, P (q, 0) = p0(q), the result is:
τ = [κp0(a)]
−1 + I2, with I2 =
1
D
∫
∞
qa
dq[p0]
−1
[
∫
∞
q
dxp0(x)
]2
. (5)
Note that the first term is just the inverse of the rate for the case of a small electronic
transition rate κ, while the second term is independent of the electronic coupling.
For the harmonic potential the integral I2 can be simplified:
I2 =
2γ
ω2
{
ln 2
2
+
√
π
4
∫ αqa
0
dy exp(y2) [1 + erf(y)]2
}
with α =
√
mω2
2kBT
, (6)
where we have introduced the friction coefficient γ = kBT/mD.
The remaining integration is easily performed numerically. The two limiting cases
of weak and strong electronic interaction can be calculated explicitly. For weak
coupling the first term in equation (5) dominates, and taking the appropriate limit
of equation (4) gives for the rate constant:
k = 1/τ =
|V |2
h̄
√
π
kBTλ
exp(−βEact), (7)
where λ = mω2x2
f/2 is the energy of reorganization, and Eact = mω2x2
a/2 the energy
of activation. This expression is identical to that obtained by first order perturbation
theory [1,4].
For strong coupling, the first term in equation (5) is negligible. For a sufficiently
high energy of activation, when βEact ≫ 1, the error function in equation (6) can
be replaced by unity, and the integral can be replaced by its asymptotic value. The
leading term is:
k =
ω2
2πγ
√
πβEact exp−βEact (8)
and is identical to the expression derived by Kramers [15,16] for the case of a cusp-
shaped barrier.
775
W.Schmickler
-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0
-20
-18
-16
-14
-12
ln σ
ln
k
Figure 2. Rate constant versus the
Landau-Zener parameter σ; full line:
γ/ω = 10, dotted line: γ/ω = 50; dashed
line γ/ω = 100.
The general case has to be treat-
ed numerically. Figure 2 shows a plot
of the rate constant versus the Landau-
Zener parameter σ = |V |2/h̄mω2qf and
for three different values of the fric-
tion coefficient γ. For small interac-
tions, the rate is proportional to the
strengths |V |2, while for higher inter-
action strengths, solvent dynamics be-
comes rate-determining, and the rate at-
tains a constant value. The higher the
coefficient of friction, the lower is the
limiting rate for high σ. For very strong
electronic interactions, the energy of ac-
tivation is lowered, and the rate should
rise again [17]. However, this effect is not
considered here since we calculate the
rate from the diabatic potential energy
curves.
3. Conclusion
To the best of our knowledge, this is the first model that applies for all strengths
of the electronic interaction between the reactants. It encompasses both the nonadia-
batic and the adiabatic regions, and nicely illustrates the interplay between electron
transfer and solvent dynamics. From equation (5) it can be seen that the rate con-
stant k obeys a relation of the form:
1
k
=
1
kel
+
1
kdyn
, (9)
where kel depends on the electronic interaction, and kdyn on the solvent dynamics.
An equation of this form is sometimes assumed as an interpolation formula between
the adiabatic and the nonadiabatic limits [18,19]. Here this relation arises naturally
from the solution of the diffusion equation. Note, however, that kel goes beyond
first-order perturbation theory.
Our model applies only to the simplest possible case, in which quantum modes
play no role, and solvent dynamics are overdamped. It should be extendable to the
case where the dynamics are not overdamped and obey a Fokker-Plank involving
both the position q and the velocity v. However, in this case the solution is likely to
be wholly numerical.
Acknowledgement
Financial support by the Volkswagenstiftung is gratefully acknowledged.
776
Electron transfer
References
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Breach, 1995.
2. Marcus R.A. // J. Chem. Phys., 1956, vol. 24, p. 966.
3. Hush N.S. // J. Chem. Phys., 1958, vol. 28, p. 962.
4. Levich V.G. Kinetics of Reactions with Charge Transfer. Physical Chemistry, an Ad-
vanced Treatise, vol. Xb, ed. by Eyring H., Henderson D., Jost W. New York, Academic
Press, 1970.
5. Grote R.F., Hynes J.T. // J. Chem. Phys., 1980, vol. 73, p. 2715.
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11. Landau L.D., Lifshitz E.M. Quantum Mechanics. Oxford, Pergamon, 1965.
12. Zener C. // Proc. Roy. Soc., 1932, vol. 137, p. 696.
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14. Deutch J.M. // J. Chem. Phys., 1980, vol. 73, p. 4700.
15. Kramers H.A. // Physica, 1940, vol. 7, p. 84.
16. Hänggi P., Talkner P., Borkovec M. // Rev. Mod. Phys., 1990, vol. 62, p. 251.
17. Schmickler W. // J. Electroanal. Chem., 1986, vol. 204 p. 31.
18. Kuznetsov A. M. Stochastic and Dynamic Views of Chemical Reactions in Solutions.
Lausanne, Presses Polytechniques et Universitaires Romandes, 1999, p. 68ff.
19. Zusman L.D. // Chem. Phys., 1980, vol. 49, p. 295.
777
W.Schmickler
Ступінь реакцій електронного переносу в
дифузійній границі
В.Шміклер
Університет Ульму, D-89069, Ульм, Німеччина
Отримано 8 серпня 2001 р., в остаточному вигляді – 10 жовтня
2001 р.
Проста реакція електронного переносу розглядається в дифузій-
ній границі, в якій рух розчинника описується рівнянням Смолухов-
ського. Імовірність електронного переходу розраховується з вира-
зу Ландау-Зенера. Константа реакції розрахована як функція сили
електронної взаємодії між реактантами. Для слабких взаємодій сту-
пінь реакції співпадає з виразом, отриманим в першому наближенні
теорії збурень. Для сильних взаємодій динаміка розчинника визна-
чається ступенем реакції. Представлені розрахунки поєднують оби-
дві границі.
Ключові слова: реакції електронного переносу, дифузійна границя
PACS: 82.30.F, 82.20.Gk
778
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