JOURNAL OF CHEMICAL PHYSICS
VOLUME 120, NUMBER 9
1 MARCH 2004
Bridge mediated two-electron transfer reactions: Analysis of stepwise
and concerted pathways
E. G. Petrov
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,
UA-03143 Kiev, Ukraine
V. May
Institüt für Physik, Humboldt Universität zu Berlin, D-12489, Berlin, Germany
共Received 26 September 2003; accepted 4 December 2003兲
A theory of nonadiabatic donor (D) –acceptor (A) two-electron transfer 共TET兲 mediated by a single
regular bridge (B) is developed. The presence of different intermediate two-electron states
connecting the reactant state D ⫺⫺ BA with the product state DBA ⫺⫺ results in complex
multiexponential kinetics. The conditions are discussed at which a reduction to two-exponential as
well as single-exponential kinetics becomes possible. For the latter case the rate K TET is calculated,
which describes the bridge-mediated reaction as an effective two-electron D – A transfer. In the limit
of small populations of the intermediate TET states D ⫺ B ⫺ A, DB ⫺⫺ A, D ⫺ BA ⫺ , and DB ⫺ A ⫺ ,
(step)
(sup)
and K TET
. The first rate describes stepwise TET
K TET is obtained as a sum of the rates K TET
originated by transitions of a single electron. It starts at D ⫺⫺ BA and reaches DBA ⫺⫺ via the
intermediate state D ⫺ BA ⫺ . These transitions cover contributions from sequential as well as
superexchange reactions all including reduced bridge states. In contrast, a specific two-electron
(sup)
. An analytic dependence of
superexchange mechanism from D ⫺⫺ BA to DBA ⫺⫺ defines K TET
(step)
(sup)
K TET and K TET on the number of bridging units is presented and different regimes of D – A TET
are studied. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1644535兴
I. INTRODUCTION
which the overall SET rate follows by additive contributions
from the sequential and the superexchange SET
mechanism.20
Turning back to TET reactions, it has to be considered as
a continuous problem to correctly characterize the different
mechanisms contributing to the observed overall processes.
At present, there are some basic results on the description of
TET in polar liquids where the transfer directly proceeds.21,22
Current theoretical studies on TET reactions in biological
systems concentrate on electronic structure calculations of,
e.g., the active centers of enzymes and their dependency on
the position of the transferred electrons4,7 or on possible TET
pathways.14,15 To estimate the rate constants characterizing
the TET process, so far phenomenological versions of the
Marcus theory16 have been used. In particular, the matrix
element responsible for the concerted two-electron hopping
transition has been introduced as a phenomenological
parameter.17 Such treatments of TET reactions indicates the
need for a theory, which is founded on nonequilibrium quantum statistics and, in this way, gives a correct description of
the concerted as well as stepwise TET pathways mediated by
the molecular bridge.
In the following we will present such a theoretical description of nonadiabatic long-range TET reactions. The reaction to be investigated starts with two electrons located at
the donor (D). Both are transferred through a chain of bridging molecules (B), and thereafter they are captured at the
acceptor (A). The whole course of the reaction will be covered by a description which generalizes earlier approaches
derived for the study of nonadiabatic D – A SET 共for an over-
Two-electron transfer 共TET兲 processes and processes
which incorporate even more electrons are common for enzyme regulated reactions in biological systems. For example,
multielectron transfer could be clearly attributed to different
enzyme complex mediated oxidation-reduction reactions.
TET reduction process have been reported for different types
of reductases,1,2 oxidases,3–5 and hydrogenases.6 – 8 All these
reactions involve numerous intermediate states1,3,9–11 and
proceed in a much more complex way as single-electron
transfer 共SET兲 processes. Protein film voltametry represents
an experimental technique which enables direct observation
of such electron transfer reactions.12,13
In biological systems, SET, TET, and higher multielectron reactions incorporate charge motion along specific structures like peptide chains and DNA strands as well as complete redox chains. In the various types of fumarate
reductases, for example, the redox chains are given either by
the chain of hemes or by Fe–S clusters.2 Such structures
form molecular bridges and mediate the electron transfer
across large spatial distances. To account for the influence of
molecular bridges on electron transfer reactions represents a
long-standing task of theoretical chemical physics. We consider it as a particular challenge to formulate a theory of
bridge-mediated multielectron transfer reactions.14 –17
In recent studies on nonadiabatic bridge-mediated
SET18 –20 the derivation of kinetic equations could be demonstrated as well as rate constants which account for different transfer mechanisms 共sequential and superexchange
SET兲. As a particular result conditions could be specified for
0021-9606/2004/120(9)/4441/16/$22.00
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© 2004 American Institute of Physics
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4442
J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
E. G. Petrov and V. May
view see, e.g., Refs. 23–29 and the textbooks30–33兲. It is just
the nonadiabatic character of the transfer process which enables us to utilize a coarse-grained approach and to simplify
the necessary rate equations considerably.18,20,34
The paper is organized as follows: In the next section we
set up a model for the TET process covering all two-electron
configurations of the DBA system. This is continued by the
description of a particular density matrix approach which
enables us to derive rate equations and respective rate constants for the nonadiabatic TET. The latter include expressions originated by the sequential and the superexchange
electron transfer mechanism. A reduction of the TET kinetics
to two-exponential and single-exponential ones is carried out
in Sec. III. The conditions to be fulfilled to have this simplified behavior are specified. Section IV is devoted to an
analysis of the bridge-length dependence of the stepwise and
the concerted part of the overall D – A TET rate. A detailed
discussion of those mechanisms originating the bridgeassisted D – A TET is given in the concluding Sec. V. Computational details are displaced to the Appendix.
II. COARSE-GRAINED DESCRIPTION OF TET
As it is well-known nonadiabatic SET takes place across
different sites of electron localization. Due to the weak intersite electronic coupling the transfer proceeds against the
background of considerably faster intrasite relaxation processes, and the whole mechanism is known as the sequential
SET. If the SET occurs between a D and A center connected
by a molecular bridge long-range superexchange D – A SET
may represent an alternative charge transfer mechanism.18 –20
Since for both mentioned mechanisms the characteristic time
scale ⌬t of the overall electron motion exceeds the characteristic time ␶ rel of intrasite relaxation the SET kinetics can
be described by a simplified set of coarse-grained rate equations 共see Refs. 18 and 20兲. The sequential and superexchange mechanisms enter via respective rate expressions. We
must expect similar conditions for nonadiabatic TET reactions if the characteristic transfer time, ␶ TET , substantially
exceeds ␶ rel . Therefore, the inequality
␶ relⰆ ␶ TET
共1兲
represent the precondition to employ a coarse-grained description of TET processes on a time scale ⌬tⰇ ␶ rel .
A. Basic model
Theoretical studies on TET processes in a DA complex
dissolved in a polar liquid are based on the use of three
distinct two-electron configurations. The TET starts at the
initial configuration, 兩 D ⫺⫺ A 典 , with the two electrons at the
D site. It is followed by the intermediate configuration,
兩 D ⫺ A ⫺ 典 , and the TET ends at the final configuration,
兩 DA ⫺⫺ 典 , with the two electrons located at the A site.21,22
This course of TET directly corresponds to the stepwise
mechanism D ⫺⫺ AD ⫺ A ⫺ DA ⫺⫺ , which has to be confronted with the concerted TET, D ⫺⫺ ADA ⫺⫺ . Thus, the
intermediate electronic configuration D ⫺ A ⫺ is of basic importance for the TET process between the contacting redox
centers.
FIG. 1. Pathway scheme of bridge-assisted TET reactions. Thick dashed
lines indicate the sequential single-electron pathways. Sole and double solid
lines display single-electron and two-electron superexchange pathways, respectively. The thin dashed lines correspond to the pathways via twofold
reduced bridging states.
In the present paper, we consider a generalization of this
TET model to the case of long-range TET mediated by a
bridge of N units connecting the D and the A. According to
a possible large number of bridge units the number of accessible two-electron configurations become also large 共cf. Fig.
1兲. Let us start with those states in which the bridge has not
been reduced. To these types of states belongs the initial state
of the TET reaction
兩 D 典 ⬅ 兩 D ⫺⫺ B 1 B 2 ¯B m ¯B N A 典 ,
共2兲
the intermediate state
兩 I 典 ⬅ 兩 D ⫺ B 1 B 2 ¯B m ¯B N A ⫺ 典 ,
共3兲
and the final state of the reaction
兩 A 典 ⬅ 兩 DB 1 B 2 ¯B m ¯B N A ⫺⫺ 典 .
共4兲
Furthermore there exist two categories of states including a
singly reduced bridge,
⫺
兩 B m 典 ⬅ 兩 D ⫺ B 1 B 2 ¯B m
¯B N A 典 ,
共5兲
⫺
兩 B̃ n 典 ⬅ 兩 DB 1 B 2 ¯B ⫺
n ¯B N A 典 .
共6兲
and
Every category has to be subdivided into N different states.
If both transferred electrons in the bridge, we arrive at the
manifold of N 2 doubly reduced bridge states 共note that a
double occupation of a single bridge unit is not forbidden兲,
⫺
兩 B mn 典 ⬅ 兩 D ⫺ B 1 B 2 ¯B m
¯B ⫺
n ¯B N A 典 .
共7兲
The possible TET routes are depicted in Fig. 1. Below, however, we will exclusively restrict our studies to TET pro-
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J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
Two-electron transfer reactions
cesses where doubly reduced bridge states do not take part.
In particular, this would be the case for TET through short
bridges where the Coulomb repulsion between the transferred electrons strongly increases the energetic position of
the states 兩 B nm 典 relative to all other TET states 共see also the
discussion in Ref. 35兲.
Resulting from this restriction, the mechanisms of TET
named stepwise TET from now on can be characterized as
follows 共see Fig. 1兲. It consists of two sequential SET routes
each related to a single-electron hopping transitions between
nearest neighbor bridge units. The first type of a sequential
SET
route,
i.e.,
the
sequence
of
reactions:
⫺
⫺
¯
B
AD
B
B
¯B
D ⫺⫺ B 1 ¯ B N AD ⫺ B ⫺
N
1 2
N A ¯
1
D ⫺ B 1 ¯B N⫺ AD ⫺ B 1 ¯B N A ⫺ , represents an electron
transfer from the initial state 兩 D 典 to the intermediate state
兩 I 典 . Then, the second type of sequential SET let move the
second
electron
from
the
D
to
the
A:
⫺
⫺
⫺
D ⫺ B 1 ¯ B N A ⫺ DB ⫺
¯
B
A
DB
B
¯
B
A
¯
N
1 2
N
1
DB 1 ¯B N⫺ A ⫺ DBA ⫺⫺ . In the scheme of state Eqs. 共2兲–
共6兲 the overall TET is completed by the transition from the
intermediate state 兩 I 典 to the final state 兩 A 典 .
Apparently, the transitions 兩 D 典 → 兩 I 典 and 兩 I 典 → 兩 A 典 can
also proceed directly via the 共single-electron兲 superexchange
mechanisms 共full lines in Fig. 1兲. It will be the particular aim
of the following derivations to present an approach which
accounts for stepwise TET as well as includes all possible
superexchange mechanism. Beside the described superexchange we have also to expect the so-called concerted TET
leading directly from state 兩 D 典 to state 兩 A 典 共double solid line
in Fig. 1兲.
In order to derive rate equations which incorporate the
different types of TET pathways, in a first step, we have to
set up a respective Hamiltonian. We abbreviate all twoelectron states, Eqs. 共2兲–共6兲, by 兩 M 典 (M ⫽D,B m ,I,B̃ n ,A)
and get TET Hamiltonian as
H TET⫽H 0 ⫹V,
共8兲
4443
are zero we will denote the respective energy by E M . If an
energetic bias is present we assume ⌬⫽E m ⫺E m⫹1 ⫽Ẽ n
⫺Ẽ n⫹1 and may write 共for a regular bridge兲
E m ⫽E B ⫺ 共 m⫺1 兲 ⌬,
Ẽ n ⫽Ẽ B ⫺ 共 n⫺1 兲 ⌬.
共11兲
共There is a variety of reasons for ⌬. Its presence may result
from externally applied as well as internal intermembrane
electric fields acting along the bridge. However, the bias can
also be introduced by slightly changing the chemical structure of the bridge units or of the groups surrounding the
bridge.兲 Moreover, non-Condon effects are neglected and the
coupling matrix elements are approximated by
V M ␯ M N ␯ ⬘ ⫽MM N 具 ␯ M 兩 ␯ N⬘ 典 ,
N
共12兲
where 具 ␯ M 兩 ␯ N⬘ 典 denotes the vibrational overlap integral. The
transfer matrix elements MM N ⫽ 具 M 兩 V̂ tr兩 N 典 characterize the
transitions between the electronic states 兩 M 典 and 兩 N 典 , Eqs.
共2兲–共6兲 of the whole DBA system. We shall specify these
matrix elements utilizing the tight binding model where the
transfer operator V̂ tr describes the single-electron transitions.
It allows to express MM N via the couplings V ␭␭ ⬘ between
the molecular orbitals related to the neighboring bridge sites.
For instance, if 兩 M 典 ⫽ 兩 D 典 and 兩 N 典 ⫽ 兩 B 1 典 , then MDB 1
⬘ .
⫽V D1 , and if 兩 M 典 ⫽ 兩 I 典 and 兩 N 典 ⫽ 兩 B̃ 1 典 , then MIB˜ 1 ⫽V D1
⬘ . In
Analogously, one obtains MIB N ⫽V AN and MAB˜ N ⫽V AN
the case of a regular bridge we set for all sites in the bridge
MB m B m⫾1 ⫽M˜B n ˜B n⫾1 ⫽V mm⫾1 ⬅V B .
All bridge-assisted electron transfer processes caused by
the single-electron couplings, are depicted in Fig. 2. The
presence of two single-electron stepwise pathways can be
clearly identified. The direct couplings T DI and T IA are the
result of a superexchange mechanism 关cf. the Appendix C,
Eqs. 共C1兲 and 共C2兲兴. The energetic position of all considered
TET states are represented in Fig. 3.
with the zero-order part
B. Derivation of kinetic equations
H 0⫽
H M 兩 M 典具 M 兩 ,
兺
M
共9兲
where the H M denote the vibrational Hamiltonian which belong to the states 兩 M 典 . The couplings 共transfer integrals兲
H M N between the different two-electron states are included
in the interaction Hamiltonian,
V⫽
兺
M ,N
共 1⫺ ␦ M ,N 兲 H M N 兩 M 典具 N 兩 .
共10兲
However, only those H M N are included for which the state
兩 M 典 and 兩 N 典 can be translated into one another by a singleelectron exchange.
Let us introduce the eigenstates and energies of the
Hamiltonian H M as 兩 M ␯ M 典 and E(M , ␯ M ), respectively,
where ␯ M denotes the quantum number of the vibrational
state belonging to the electronic state 兩 M 典 . In the case that
all vibrational quantum numbers of the electronic state 兩 M 典
Our considerations on TET reactions will be based on
the fact that inequality 共1兲 is fulfilled. The related fast vibrational relaxation within every electronic state of the TET
results in electron vibrational level broadening18 and
dephasing.36 –38 Here, we denote the broadening of the energy level E(M , ␯ M ) by ⌫(M , ␯ M ).
It has been discussed at length in Ref. 18 how to compute the electron vibrational dynamics in a case characterized
by Eq. 共1兲. In a first step one has to set up equations of
motion for the electron vibrational density matrix,
␳ M ␯ M ,N ␯ ⬘ 共 t 兲 ⫽ 具 M ␯ M 兩 ␳ 共 t 兲 兩 N ␯ N⬘ 典 ,
N
共13兲
where ␳ (t) is the related reduced density operator. The influence of an additional heat bath causing fast vibrational
relaxation has been taken into consideration by energy relaxation and dephasing rates. Next, one changes to a time region
which is large compared to the fast vibrational relaxation
processes 共coarse-graining approximation兲. This change is
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4444
J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
E. G. Petrov and V. May
FIG. 2. Energetic scheme of the DBA states participating in the TET. The single-electron transitions along the
⫺ ⫺
first,
(D ⫺⫺ BA)(D ⫺ B ⫺
1 A)¯ (D B N A)
⫺
⫺
⫺
(D BA ),
and
the
second,
(D BA ⫺ )
⫺ ⫺
⫺
⫺⫺
A
)¯
(DB
A
)(DBA
),
stepwise
(DB ⫺
1
N
pathways are shown 关panels 共a兲 and 共b兲, respectively兴.
Wavy lines indicate the fast relaxation within the various electronic state.
achieved in reducing the exact density matrix equations to
the following approximate versions valid for the diagonal
density matrix elements:
⳵
i
␳ M ␯ M ,M ␯ M 共 t 兲 ⫽ 具 M ␯ M 兩 关 V, ␳ 共 t 兲兴 ⫺ 兩 M ␯ M 典 ,
⳵t
ប
共14兲
and the off-diagonal density matrix elements 共note in particular M ⫽N)
␳ M ␯ M ,N ␯ ⬘ 共 t 兲 ⫽
N
具 M ␯ M 兩 关 V, ␳ 共 t 兲兴 ⫺ 兩 N ␯ N⬘ 典
.
⌬␧ 共 M ␯ M ,N ␯ N⬘ 兲
共15兲
The reduced set of equations of motion assumes that the time
dependence of the diagonal density matrix elements is exclusively determined by the coupling to other TET states
whereas relaxation processes do not contribute 共they are just
finished at every time step兲. In contrast the time derivative
has been removed in the equation of motion for the offdiagonal density matrix elements. This approximation results
in a determination of the time dependence mainly by the
dephasing rates and is justified by the rapidness of the
dephasing. The related dephasing rates enter the equation via
⌬␧(M ␯ M ,N ␯ N⬘ )⫽E(M , ␯ M )⫺E(N, ␯ N⬘ )⫺i(⌫(M , ␯ M )
⫹⌫(N, ␯ N⬘ )).
Equations 共14兲 and 共15兲 will be used to derive rate equations for the electronic state populations
P M共 t 兲⫽
PM␯
兺
␯
m
M
共 t 兲⫽
␳M␯
兺
␯
m
M ,M ␯ M
共 t 兲,
共16兲
where the transition rates are obtained as expansions with
respect to the transfer integrals V M N . In order to derive the
related perturbation theory we first introduce the projector
PA⫽
AM␯
兺
␯
m
M ,M ␯ M
兩 M ␯ M 典具 M ␯ M 兩 ,
共17兲
and the orthogonal complement Q⫽1⫺P. The latter
projects any operator A defined in the electron vibrational
state space on the part which has only off-diagonal matrix
elements. A projection on the part of A including only diagonal matrix elements with respect to the states 兩 M ␯ M 典 is
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J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
Two-electron transfer reactions
4445
but in any case they only contain an even number of interactions.
Taking the matrix elements of Eq. 共21兲 we obtain for the
electron vibrational state populations
⳵
P
共 t 兲 ⫽⫺
具 M , ␯ M 兩 共 R兩 N ␯ N⬘ 典具 N ␯ N⬘ 兩 兲 兩 M ␯ M 典
⳵t M␯M
N, ␯ ⬘
兺
N
⫻ P N␯⬘ 共 t 兲.
N
共24兲
According to the coarse graining approximation there is vibrational equilibrium established at every electronic state and
at every time tⰇ ␶ rel . Therefore, we can assume the following thermal equilibrium relation to be fulfilled:
FIG. 3. Energetic positions E M ⬅E(M ,0M ) of the states involved in the TET
together with respective energy gaps.
achieved by P. Furthermore, we change from the matrix notation, Eqs. 共14兲 and 共15兲, to the operator notation
⳵
P␳ 共 t 兲 ⫽⫺iPL V Q␳ 共 t 兲
⳵t
共18兲
and
Q␳ 共 t 兲共 t 兲 ⫽⫺iQRL V 共 P␳ 共 t 兲 ⫹Q␳ 共 t 兲兲 .
共19兲
Note the introduction of L V ⫽(1/ប) 关 V, . . . 兴 ⫺ . It allows to
replace L V ␳ (t) by L V Q␳ (t) in the first equation. The resolvent superoperator R⫽ 兰 ⬁0 d ␶ S 0 ( ␶ ) could be introduced in
the second equation since it generates an expression as in Eq.
共15兲 when changing to matrix elements. The newly defined
time-propagation superoperator S 0 is determined by the zeroorder Liouvillian L 0 ⫽(1/ប) 关 H 0 , . . . 兴 ⫺ as well as by a part
describing dissipation.
To get the required rate equations, in a next step, we
have to derive a master equation for the diagonal part P␳ of
the density operator. This is easily achieved by formally
solving Eq. 共19兲 as
Q␳ 共 t 兲 ⫽⫺i 共 1⫹iQRL V 兲 P␳ 共 t 兲
共20兲
and inserting the result into Eq. 共18兲. It follows
⳵
P␳ 共 t 兲 ⫽⫺RP␳ 共 t 兲 .
⳵t
共21兲
Here R denotes the superoperator which describes any type
of TET. Its expansion with respect to the transfer coupling
共10兲,
R⫽R2 ⫹R4 ⫹R6 ⫹¯
共22兲
enables one to arrange the different TET processes in a
proper order. The various terms of the expansion can be written as
R2k ⫽⫺ 共 ⫺i 兲 2k
冕
⬁
0
d ␶ 2k⫺1
冕
⬁
0
d ␶ 2k⫺2 ¯
冕
⬁
0
d␶1
Changing to the total TET state population, Eq. 共16兲, we may
set 共see, e.g., Refs. 18 and 34兲
P M ␯ M 共 t 兲 ⫽W 共 E 共 M , ␯ M 兲兲 P M 共 t 兲 ,
共26兲
where
W 共 E 共 M , ␯ M 兲兲 ⫽Z ⫺1
M exp关 ⫺E 共 M , ␯ M 兲 /k B T 兴 ,
Z ⫺1
M ⫽
兺
␯
exp关 ⫺E 共 M , ␯ M 兲 /k B T 兴 ,
共27兲
M
is the statistical weight of the electron-vibrational state
兩 M ␯ M 典 . Noting Eq. 共26兲 we obtain from Eq. 共24兲 the set of
balancelike equations,
Ṗ M 共 t 兲 ⫽⫺
兺N KM N P N共 t 兲 ,
共28兲
with
KM N ⫽
兺
兺 W 共 E 共 N, ␯ N⬘ 兲兲 ⫻ 具 M ␯ M 兩 共 R兩 N ␯ N⬘ 典
␯
M
␯N
⬘
⫻具 N ␯ N⬘ 兩 兲 兩 M ␯ M 典 .
共29兲
Now, noting the fact that the symbols M and N indicate
electronic states 共2兲–共6兲 related to the whole DBA system
we obtain 共see Appendix A兲 the set of coupled rate equations
共A8兲–共A16兲 with respective sequential and superexchange
rate constants.
III. OVERALL TET RATES
The concrete form of the balancelike equations 共28兲, as
given by the set of rate equations, Eqs. 共A8兲–共A16兲, reflects
the rather complicated kinetics of the TET process, also
shown in scheme 共a兲 of Fig. 4. Generally, such kinetics has a
multiexponential character for each of the 2N⫹3 state populations P M (t) (M ⫽D,I,A,B 1 ,...B N ,B̃ 1 , . . . B̃ N ),
2N⫹2
P M共 t 兲⫽ P M共 ⬁ 兲⫹
⫻PL V QS 0 共 ␶ 2k⫺1 兲 L V QS 0 共 ␶ 2k⫺2 兲
⫻L V ¯QS 0 共 ␶ 1 兲 L V ,
P M ␯ M 共 t 兲 / P M ␯ ⬘ 共 t 兲 ⫽exp兵 ⫺ 关 E 共 M , ␯ M 兲 ⫺E 共 M ␯ ⬘M 兲兴 /k B T 其 .
M
共25兲
共23兲
兺
r⫽1
B (r)
M exp共 ⫺K r t 兲 .
共30兲
The steady-state populations, P M (⬁)⫽lims→0 (sF M (s)), as
well as the overall transfer rates K r can be deduced from the
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4446
J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
E. G. Petrov and V. May
If the total bridge state population P B (t) remains small
during the TET reaction the solution 共31兲 describes TET only
between the electronic states 兩 D 典 , 兩 I 典 , and 兩 A 典 . Just the inequality
N
P B共 t 兲 ⫽
兺
m⫽1
共 P m 共 t 兲 ⫹ P m̃ 共 t 兲兲 Ⰶ1
共32兲
represents a necessary and sufficient condition for the reduction of the multiexponential TET kinetics to a twoexponential process. If t→⬁ the inequality 共32兲 can be rewritten as
N
lim sF B 共 s 兲 ⫽
s→0
FIG. 4. Complete kinetic scheme of the bridge-assisted TET process 关part
共a兲兴. For a small bridge-state population the kinetics is reduced to the transitions between the three electronic states, 兩 D 典 , 兩 I 典 , and 兩 A 典 关the related
transfer rates are also shown, part 共b兲兴. If the population of the intermediate
state 兩 I 典 becomes also small the three-state kinetics reduce to D – A TET
kinetics with an effective single forward and single backward rate only 关part
共c兲兴.
exact solution of the set 共A8兲–共A16兲 for all Laplacetransformed populations F M (s) 共the solution is given in Appendix B兲. In most cases a numerical computation of the
overall transfer rates becomes necessary. However, further
analytical considerations are possible whenever the intermediate states become less populated. 共This has been discussed
at length for SET which is reduced to a single-exponential
D – A SET kinetics.20兲 Therefore, we will concentrate on the
description of such a simple overall time behavior. A twoexponential TET process will be described in the following
section whereas single-exponential behavior is discussed in
the section afterwards.
共 lim sF m 共 s 兲 ⫹ lim sF m̃ 共 s 兲兲 Ⰶ1.
共31兲
which, of course, is only valid in a time domain t
⫺1
⫺1
ⰇK ⫺1
3 ,K 4 , . . . ,K 2N⫹2 .
共33兲
s→0
(1)
Ṗ D 共 t 兲 ⫽⫺ 共 k (1)
f ⫹k DA 兲 P D 共 t 兲 ⫹k b P I 共 t 兲 ⫹k AD P A 共 t 兲 ,
(1)
(1)
(2)
Ṗ I 共 t 兲 ⫽⫺ 共 k (2)
f ⫹k b 兲 P I 共 t 兲 ⫹k f P D 共 t 兲 ⫹k b P A 共 t 兲 ,
共34兲
(2)
Ṗ A 共 t 兲 ⫽⫺ 共 k (2)
b ⫹k AD 兲 P A 共 t 兲 ⫹k f P I 共 t 兲 ⫹k DA P 2 共 t 兲 .
The rates introduced here have to be identified with Laplacetransformed expressions as introduced in the Appendix B,
Eq. 共B10兲,
k (1)
f ⫹k DA ⫽R DD 共 0 兲 ,
(1)
k (2)
f ⫹k b ⫽R II 共 0 兲 ,
k (2)
b ⫹k AD ⫽R AA 共 0 兲 ,
k (1)
f ⫽R ID 共 0 兲 ,
k (1)
b ⫽R DI 共 0 兲 ,
k (2)
f ⫽R AI 共 0 兲 ,
共35兲
k (2)
b ⫽R IA 共 0 兲 .
Using the Eqs. 共B3兲, 共B4兲–共B6兲, and 共B11兲 as well as the
recursion formula 共B7兲 we may identify the rates of the Eqs.
共34兲 with basic rate expressions of sequential and superexchange transitions
(seq)
(sup)
k (1)
f ⫽k DI ⫹k DI ,
According to the general solution, Eq. 共30兲, of the rate
equations 共A8兲–共A16兲, two-exponential TET kinetics requires the existence of two rates which are much smaller
than all other ones. Let these rates be K 1 and K 2 . Then, the
exact solution 共30兲 reduces to
s→0
This relation has to be considered as a necessary condition
for the appearance of two-exponential kinetics 共see also Ref.
18兲. Since all the Laplace-transformed populations F M (s)
have a rather complicated form this is also valid for the inequality 共33兲.
To get analytical expressions for the two smallest transfer rates K 1 and K 2 we shall start from the exact Eqs. 共B10兲
which, however, only determine the Laplace-transformed
populations of the DBA states 兩 D 典 , 兩 I 典 , and 兩 A 典 关the bridge
populations are given by Eqs. 共B1兲 and 共B2兲兴. If this reduced
set of rate equations is transformed back into the time domain we get the following three kinetic equations 关see the
scheme 共b兲 of Fig. 4兴:
A. Two-exponential TET kinetics
⫺K 1 t
⫺K 2 t
⫹B (2)
,
P M 共 t 兲 ⯝ P M 共 ⬁ 兲 ⫹B (1)
M e
M e
兺
m⫽1
(seq)
(sup)
k (1)
b ⫽k ID ⫹k ID ,
(seq)
⫽k D1 k NI ␣ N⫺1 /D 1 ,
k DI
(sup)
k DI
⫽k DI ,
(seq)
⫽k 1D k IN ␤ N⫺1 /D 1 ,
k ID
(sup)
k ID
⫽k ID
共36兲
and
(seq)
(sup)
k (2)
f ⫽k IA ⫹k IA ,
(seq)
(sup)
k (2)
b ⫽k AI ⫹k AI ,
(seq)
⫽k I1 k NA ␣ N⫺1 /D 2 ,
k IA
(sup)
k IA
⫽k IA ,
(seq)
⫽k 1I k AN ␤ N⫺1 /D 2 ,
k AI
(sup)
k AI
⫽k AI ,
共37兲
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J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
Two-electron transfer reactions
where
D 1 ⬅k NI ␣ N⫺1 ⫹k 1D ␤ N⫺1 ⫹k 1D k NI D 共 N⫺2 兲 ,
共38兲
D 2 ⬅k NA ␣ N⫺1 ⫹k 1I ␤ N⫺1 ⫹k 1I k NA D 共 N⫺2 兲 .
The main bridge length dependence of rates of transfer is
originated by the function
D 共 M 兲 ⬅D 共 s⫽0;M 兲 ⫽ ␣ M
1⫺ ␥ M ⫹1
,
1⫺ ␥
共39兲
where
␥ ⬅ ␤ / ␣ ⫽exp共 ⫺⌬/k B T 兲 ⭐1
共40兲
denotes the ratio of the backward and the forward site-to-site
rate constants. In Eq. 共40兲 ⌬ is the intersite energy bias 关introduced in Eq. 共11兲兴. The transfer rates 共36兲 and 共37兲 define
two overall TET rates,
K 1 ⫽ 21 关 a 1 ⫹d 1 ⫹ 冑共 a 1 ⫺d 1 兲 2 ⫹4a 2 d 2 兴 ,
共41兲
K 2 ⫽ 21 关 a 1 ⫹d 1 ⫺ 冑共 a 1 ⫺d 1 兲 2 ⫹4a 2 d 2 兴 ,
where we have introduced the following abbreviations:
(2)
a 1 ⬅k (2)
f ⫹k b ⫹k AD ,
a 2 ⬅k DA ⫺k (2)
f ,
(1)
d 1 ⬅k (1)
f ⫹k b ⫹k DA ,
d 2 ⬅k AD ⫺k (1)
b .
共42兲
Having determined the two overall TET rates we may
present the other quantities entering the Eqs. 共31兲 for the
time-dependent populations. They read
P D共 ⬁ 兲 ⫽
(2)
a 1 k (1)
b ⫹d 2 k f
K 1K 2
,
(2)
K 1 K 2 ⫺ 共 a 1 ⫹a 2 兲 k (1)
b ⫺ 共 d 1 ⫹d 2 兲 k f
P I共 ⬁ 兲 ⫽
K 1K 2
P A共 ⬁ 兲 ⫽
,
共43兲
(1)
d 1 k (2)
f ⫹a 2 k b
K 1K 2
as well as ( j⫽1,2)
( j)
⫽ 共 ⫺1 兲 j⫹1
BD
B I( j) ⫽ 共 ⫺1 兲 j
⫹
冋
(2)
共 a 1 ⫺K j 兲共 k (1)
b ⫺K j 兲 ⫹d 2 k f
K j 共 K 1 ⫺K 2 兲
共 a 1 ⫺K j 兲共 k (1)
b ⫺K j 兲
K j 共 K 1 ⫺K 2 兲
(2)
a 2 共 k (1)
b ⫺K j 兲 ⫹k f 共 d 1 ⫹d 2 ⫺K j 兲
B A( j) ⫽ 共 ⫺1 兲
,
K j 共 K 1 ⫺K 2 兲
册
k (2) 共 d 1 ⫺K j 兲 ⫹a 2 共 k (1)
b ⫺K j 兲
j⫹1 f
K j 共 K 1 ⫺K 2 兲
and s⫽⫺K 2 关 M ,N⫽D,I,A; R ⬘M N (0)⬅(dR M N (s)/ds) s⫽0 ].
Then, one can see that the multistate DBA system shows a
two-exponential TET kinetics, Eq. 共31兲 only if
兩 R ⬘M N 共 0 兲 兩 Ⰶ 兩 ␦ M ,N ⫺K ⫺1
j R M N共 0 兲兩
共 j⫽1,2兲 .
共44兲
B. Single-exponential D – A TET kinetics
So far we arrived at a description of bridge-assisted TET
(2)
where the two forward (k (1)
f ,k f ) and two backward
(1)
(2)
(k b ,k b ) transfer rates contain additive contributions from
the single-electron sequential and the single-electron superexchange mechanisms. Remember that this additivity is a
direct consequence of the small population of all bridging
states. This two-exponential TET corresponds to an effective
three-state system. If the intermediate state 兩 I 典 carries a pronounced population the overall transfer rates K 1 and K 2
共note that K 1 ⬎K 2 ) contain a mixture of rate expressions
caused by the single-electron stepwise and the two-electron
superexchange 共concerted兲 mechanism. The situation
changes substantially when
K 1 ⰇK 2 ,
共46兲
which indicates that the TET process, Eq. 共31兲, is determined
by two strongly separated kinetic phases, a fast and a slow
⫺1
one with the characteristic times ␶ 1 ⫽K ⫺1
1 and ␶ 2 ⫽K 2 , respectively. If tⰆ ␶ 2 we have exp(⫺K2t)⯝1, and, thus, the fast
phase is defined by the time-dependent factor
exp(⫺K1t) for all three populations, P D (t), P I (t), and
P A (t). The first phase of the TET is finished for tⰇ ␶ 1 , but
tⰆ ␶ 2 . As a result, a redistribution of the initial populations
P M (0)⫽ ␦ M ,D took place, and for ␶ 2 ⰇtⰇ ␶ 1 the transient
populations, P M (tⰇ ␶ 1 )⬅ P M (⫹0), have been formed. They
are easily derived from Eq. 共31兲 at exp(⫺K1t)⫽0 and
exp(⫺K2t)⫽1.
Below we present concrete expressions for the overall
transfer rates K 1 and K 2 as well as for the 共intermediate兲
populations P M (⫹0), bearing in mind that the energy E I
exceeds the energies E D and E A 共cf. Fig. 3兲. For such a
condition, the following inequality is satisfied:
At this point of our discussion let us formulate the conditions
for which the exact solution, Eqs. 共B10兲 becomes identical
with that of the kinetic equations 共34兲. Therefore, we have to
consider the quantities R M N (s) which have been introduced
when considering the Laplace-transformed rate equations 共cf.
Appendix B兲. We expand the R M N (s) with respect to s and
assume that the approximations s⫹R M M (s)⬇s⫹R M M (0)
⫹sR ⬘M M (0)⬇s⫹R M M (0) as well as R M N (s)⬇R M N (0)
⫹sR ⬘M N (0)⬇R M N (0) are fulfilled for both roots s⫽⫺K 1
共47兲
Note that in line with Eq. 共41兲 the inequality 共46兲 is reduced
to
共 a 1 ⫹d 1 兲 2 Ⰷ4 共 a 1 d 1 ⫺a 2 d 2 兲 .
.
共45兲
A comparison of this inequality with the condition, Eq. 共33兲,
for a small steady bridge population indicates their complete
correspondence.
(2)
(1)
(2)
k (1)
b ,k f Ⰷk f ,k b ,k DA ,k AD .
,
4447
共48兲
Therefore, the following approximate expression for the
overall transfer rates can be presented
K 1 ⬇a 1 ⫹d 1 ,
K 2⬇
a 1 d 1 ⫺a 2 d 2
.
a 1 ⫹d 1
共49兲
According to the inequalities 共47兲 this can be rewritten as
(2)
K 1 ⬇k (1)
b ⫹k f ,
K 2⬇
(2)
(2) (1)
k (1)
f k f ⫹k b k b
(2)
k (1)
b ⫹k f
共50兲
⫹k DA ⫹k AD ,
Downloaded 10 Mar 2005 to 141.20.41.167. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
4448
J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
E. G. Petrov and V. May
states of a whole DBA system. Using the definitions 共38兲 and
共39兲 one can rewrite the sequential contribution as
and the transient populations take the following form:
(1)
(2)
P D 共 ⫹0 兲 ⫽1⫺ 共 k (1)
f ⫹k DA 兲 / 共 k b ⫹k f 兲 ,
(1)
(2)
P I 共 ⫹0 兲 ⫽k (1)
f / 共 k b ⫹k f 兲 ,
(seq)
k DI(IA)
⫽
共51兲
(1)
(2)
P A 共 ⫹0 兲 ⫽k (2)
b / 共 k b ⫹k f 兲 .
From these expressions it is seen that the ratio
P I (⫹0)/ P A (⫹0) is determined by the ratio of the stepwise
and the concerted rate constants, k (1)
f /k DA . Note that
P I (⫹0) gives the maximal value possible for the population
of the intermediate state I 共cf. Fig. 4兲. But inequality 共47兲
tells us also that P I (⫹0)Ⰶ1, i.e., the population of state I
remains very small at any time t,
P I 共 t 兲 Ⰶ1.
共52兲
P A 共 t 兲 ⯝ P A 共 ⬁ 兲共 1⫺e
兲.
共55兲
(conc)
(f )
(b)
⫽K conc
⫹K conc
,
K TET
(b)
K conc
⫽k AD .
seq)
k (1
⫽
f0
k D1 k NI
,
k NI ⫹k 1D
(1 seq)
k b0
⫽
k IN k 1D
,
k NI ⫹k 1D
共59兲
seq)
⫽
k (2
f0
k I1 k NA
,
k NA ⫹k 1I
(2 seq)
k b0
⫽
k AN k 1I
k NA ⫹k 1I
共60兲
define rates of sequential transfer through a bridge with a
single unit. The related decay parameters read
共54兲
and
(f )
K conc
⫽k DA ,
共58兲
provided that ␥ ⫽0 and that the sequential decay parameters
␰ 1 and ␰ 2 are not too small.
The quantities
共56兲
IV. DISCUSSION OF RESULTS
We consider the derivation of the overall TET rates and
the description of the conditions at which a multiexponential
TET is reduced to a two-exponential as well as a singleexponential TET as the main result of the present studies.
Additionally, it could be shown in which way the overall
transfer rates are expressed by the sequential and the superexchange rate constants.
When considering distant electron transfer it is of interest to describe the dependence of rate expressions on the
number of bridge units. Concentrating on the 共singleexponential兲 D – A TET this dependence is contained in the
stepwise and concerted components of the overall transfer
rate 共54兲. First let us consider the stepwise channel of the
D – A TET. The stepwise transfer rates 共36兲 and 共37兲 contain
contributions from the single-electron sequential and singleelectron superexchange transfer rates between the 兩 D 典 and
the 兩 I 典 as well as between the 兩 I 典 and the 兩 A 典 electronic
␰ 1⫽
k 1D 共 k NI ⫺ ␣ 共 1⫺ ␥ 兲兲
␣ 共 k NI ⫹k 1D 兲
共61兲
␰ 2⫽
k 1I 共 k NA ⫺ ␣ 共 1⫺ ␥ 兲兲
.
␣ 共 k NA ⫹k 1I 兲
共62兲
and
(step)
(f )
(b)
⫽K step
⫹K step
,
K TET
(1)
k (2)
b kb
(b)
K step
⫽ (1) (2) ,
k b ⫹k f
.
and
Since K TET⫽k f ⫹k b contains a forward and a backward part
(step)
(conc)
and K TET
separate corre关cf. scheme 共c兲 in Fig. 4兴, K TET
spondingly. We have
(2)
k (1)
f kf
(f )
⫽ (1) (2) ,
K step
k b ⫹k f
共57兲
R 共 N 兲 ⫽ 共 1⫺ ␥ N⫺1 兲 / 共 1⫺ ␥ 兲 ,
共53兲
This kinetics describes a D – A TET with an overall transfer
rate K 2 ⫽K TET . The latter can be represented as the sum of a
stepwise and a concerted contribution,
⫺1
(step)
(conc)
⫽K TET
⫹K TET
.
K TET⫽ ␶ TET
,
It follows from Eq. 共57兲 that the N dependence of the sequential mechanism of SET is concentrated in the factor
This inequality indicates that at
the two-exponential
kinetics 共31兲 is reduced to a single-exponential one,
⫺K TETt
1⫹ ␰ 1(2) R 共 N 兲
(1(2)seq)
k b0
关 1⫺ 共 1⫺ ␥ 兲 R 共 N 兲兴
(seq)
k ID(AI) ⫽
1⫹ ␰ 1(2) R 共 N 兲
tⰇK ⫺1
1
P D 共 t 兲 ⯝ P D 共 ⬁ 兲 ⫹ 共 1⫺ P D 共 ⬁ 兲兲 e ⫺K TETt ,
k (1(2)seq)
f0
Note that due to a single-electron character of sequential
pathway the rate constants in Eqs. 共59兲–共62兲 are identical to
the hopping rate constants between the nearest sites of electron localization. Therefore, e.g., the rate k D1 (k I1 ) characterizes the hopping of an electron from the state with two electrons 共single electron兲 at the D to the adjacent wire unit
while the rate k AN (k IN ) characterizes the similar process
with respect to the A.
To derive the single-electron superexchange contribution
to the stepwise transfer rates 共36兲 and 共37兲 one has to specify
the superexchange rate constants k DI(ID) and k IA(AI) . These
rate constants are defined in Appendix A, Eq. 共A7兲, and are
proportional to the square of the superexchange couplings
兩 T DI 兩 2 and 兩 T IA 兩 2 , respectively. Here, we restrict ourself to
the consideration of a small intersite bridge bias ⌬, where the
effective transfer couplings are given by Eqs. 共C21兲 and
共C23兲, Appendix C. A small change in the notation leads to
the following expressions:
(sup)
sup) ⫺ ␨ 1 (N⫺1)
⫽k (1(2)
e
k DI(ID)
f0
共63兲
(sup)
(1(2) sup) ⫺ ␨ 2 (N⫺1)
⫽k b0
e
.
k IA(AI)
共64兲
and
(sup) (sup)
(k AI ) characterizes the distant coherent
Here, the rate k DI
transfer of an electron from the state with two electrons at
the D(A) to the state with one electron at the A(D). The rate
(sup) (sup)
(k IA ) describes the reverse process. 关Compare the
k ID
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J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
Two-electron transfer reactions
definition of DBA electronic states 共2兲–共4兲 and the scheme
共a兲 of Fig. 4.兴 The superexchange decay parameters
冋
␨ 1 ⫽⫺2 ln
and
冋
␨ 2 ⫽⫺2 ln
兩 V B兩
共 ⌬E D ⌬E I 兲 1/2
册
兩 V B兩
共 ⌬Ẽ A ⌬Ẽ I 兲 1/2
共65兲
册
are mainly responsible for the distant dependence of the superexchange rate. Note that in the presence of an energetic
bias in the bridge an additional N dependence also appears in
the quantities
sup)
⫽
k (1(2)
f0
2 ␲ 兩 V D1 V NA 兩 2
共 FC 兲 DI(ID)
ប ⌬E D ⌬E I
共67兲
and
(1(2) sup)
⫽
k b0
⬘ V NA
⬘ 兩2
2 ␲ 兩 V D1
ប
⌬Ẽ A ⌬Ẽ I
共 FC 兲 IA(AI) .
共68兲
The N dependence is located in the Franck–Condon factors
through the driving forces ⌬E ID and ⌬E IA 关cf. Eqs. 共C10兲
and 共C11兲兴 characterizing the corresponding single-electron
reaction.
Two couples of decay parameters, ( ␨ 1 , ␰ 1 ) and ( ␨ 2 , ␰ 2 ),
characterize the efficiency of the first and the second singleelectron bridge-mediated stepwise pathway, respectively. At
the same time, the efficiency of the concerted pathway is
defined by the two-electron superexchange rate constants
k DA and k AD 关compare Eq. 共56兲兴. Introducing the twoelectron superexchange decay parameter as
冋
␨ ⫽⫺2 ln
兩 V B兩 2
共 ⌬E D ⌬E A ⌬Ẽ D ⌬Ẽ A 兲 1/2
册
共69兲
and utilizing Eq. 共C25兲, a respective simplified notation of
each concerted rate can be given,
(0)
e ⫺ ␨ (N⫺1) .
k DA(AD) ⫽k DA(AD)
共70兲
Here, the rate k DA (k AD ) characterizes the distant coherent
transfer of two electrons from the state with two electrons at
the D(A) to the state without extra electrons at the A(D).
Such a process is possible owing to a virtual electronic configuration D ⫺ BA ⫺ corresponding to the state with one electron at the D center and one electron at the A center.
In the case of a bridge with an energy bias the factor
(0)
⫽
k DA(AD)
2␲
⬘ V NA
⬘ 兩2
兩 V D1 V NA V D1
ប ⌬E D ⌬E A ⌬Ẽ D ⌬Ẽ A ⌬E ID ⌬E IA
⫻ 共 FC兲 DA(AD)
␻ M N be the single characteristic frequency accompanying
the M →N electronic transition and ␭ M N be the corresponding reorganization energy. Then Jortner’s form of the rate
constant 共A7兲 can be written as 共see also Refs. 18, 25, 30, 31,
and 33兲
k M N⫽
共66兲
共71兲
includes an additional N dependence originated by the energy gaps ⌬E ID and ⌬E IA as well as by the Franck–Condon
factor 关through the driving force ⌬E DA of the TET reaction,
cf. Eq. 共C12兲兴.
In order to demonstrate the bridge-length dependence of
the rate we take Jortner’s form39 for all rate constants. Let
4449
2 ␲ 兩 V M N兩 2
⌽
,
ប ប␻ MN ␯MN
where we have introduced
冋
⌽ ␯ M N ⫽exp ⫺S M N coth
ប␻MN
k BT
共72兲
册冉
1⫹n 共 ␻ M N 兲
n共 ␻ MN兲
冊
␯ M N /2
⫻I 兩 ␯ M N 兩 共 2S M N 关 n 共 ␻ M N 兲共 1⫹n 共 ␻ M N 兲兲兴 1/2兲 . 共73兲
The expression contains the modified Bessel function I ␯ (z),
the Bose distribution function n( ␻ )⫽ 关 exp(ប␻/kBT)⫺1兴⫺1,
and we have set S M N ⬅␭ M N /ប ␻ M N , ␯ M N ⬅⌬E M N /ប ␻ M N .
First let us discuss the reduction of the multiexponential
TET kinetics to the two-exponential one. In line with the
theory proposed, such a reduction is only possible at a small
integral bridge population, which is completely justified by
the results of Fig. 5. Moreover, the derived analytic results
on the two-exponential kinetics 关cf. Eqs. 共31兲, 共41兲–共44兲兴
coincide completely 关cf. Fig. 5共a兲兴 with those obtained by a
numerical solution of the basic set of Eqs. 共A8兲–共A16兲 provided that 共32兲 is fulfilled during the TET process. Note that
the bridge population evolve in time as well 关cf. the numerical results depicted in Fig. 5共b兲兴. But due to the small population of each bridging state this kinetics is well separated
from the kinetics of the populations P D (t), P I (t), and
P A (t).
If in the course of the TET process the bridging states
兩 B m 典 , Eq. 共5兲, and 兩 B̃ n 典 , Eq. 共6兲, as well as the intermediate
state 兩 I 典 , Eq. 共3兲, have a small population, then the TET is
reduced to a single-exponential D – A TET. This statement is
supported by the results shown in Fig. 6. Again, despite the
fact that both types of bridge populations, P m (t) and P ñ (t)
关denoted in Fig. 6共b兲 by the numerals 1 and 2, respectively兴,
as well as the intermediate state population P I (t) show a
time dependence according to their respective transfer rates
the basic populations P D (t) and P A (t) display a singleexponential kinetics 共53兲 with overall D – A TET rate 共54兲.
Figures 7–10 show the D – A TET rate in dependency on
the number N of bridging units. Special attention is put on
the temperature dependence since it determines the contribution by the thermally activated stepwise transfer mechanism.
Figure 7共a兲 demonstrates that the stepwise contribution to the
overall transfer rate K TET , Eq. 共54兲 exceeds the concerted
(step)
. Indeed, the analysis of the interone, so that K TET⬇K TET
state transfer rates 共36兲 and 共37兲 indicates that at a given set
of parameters 关specifying the elementary rate constants 共72兲兴
(1)
the inequality, k (2)
f Ⰷk b 关cf. Fig. 7共b兲兴, is fulfilled, and thus
the stepwise transfer rate, Eq. 共55兲 follows as
(step)
K TET
⬇k (1)
b exp共 ⫺⌬E ID /k B T 兲 .
共74兲
This expression indicates that the limiting step of the stepwise TET process is related to a single-electron 兩 D 典 → 兩 I 典
transition. Note that a noticeable contribution in this thermally activated transition is given by the superexchange 共at
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4450
J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
FIG. 5. TET kinetics in a bridge with five units, without energy bias and for
a small population P m (t) and P m̃ (t) of the two types of bridge states. Analytic results for the populations P D (t), P I (t), and P A (t) according to Eqs.
共31兲, 共41兲–共44兲 and the numerical solution of the set of Eqs. 共A8兲–共A16兲
completely agree 关part 共a兲兴. The bridge populations related to the first 共1兲
and the second type of bridge states 共2兲 show their own kinetics 关panel 共b兲兴.
The calculations are based on the Eqs. 共54兲–共73兲 and by choosing the following parameters: ⌬E D ⫽0.29 eV, ⌬E I ⫽0.27 eV, ⌬Ẽ I ⫽0.21 eV, ⌬Ẽ A
⫽0.26 eV, ⌬E⫽0.03 eV; ␭ 1D ⫽␭ NI ⫽␭ DI ⫽0.6 eV, ␭ I1 ⫽␭ NA ⫽␭ IA
⫽0.4 eV, ␭ DA ⫽0.8 eV, ␭ m m⫾1 ⬅␭ B ⫽0.8 eV; ␻ M N ⫽ ␻ 0 ⫽100 cm⫺ ; V D1
⬘ ⫽V ⬘NA ⫽0.03 eV, V B ⫽0.04 eV.
⫽V NA ⫽V D1
N⫽1 – 4) and the sequential 共at N⬎4) single-electron pathway 关compare the N dependence of both contributions plotted in Fig. 7共b兲兴. At lower temperatures the contribution of
the concerted mechanism increases 关compare Fig. 7共a兲 with
Fig. 8共a兲 and Fig. 8共b兲兴. For instance, at T⫽150 K 关Fig. 8共a兲兴
the concerted mechanism dominates the TET at N⫽1 – 2
while at T⫽100 K the same mechanism works effectively up
to N⫽4 关Fig. 8共b兲兴. In contrast to the stepwise mechanism
the concerted mechanism works also at small temperatures
down to T⫽0. Note that the stepwise pathway also includes
a superexchange 共single-electron兲 contribution. But due to
the thermally activated origin of the stepwise rate, Eq. 共74兲,
the superexchange single-electron contribution can only
work at the presence of thermal activation. This fact reflects
the specificity of the D – A TET process. 共However, in the
case of single-electron D – A transfer the superexchange
channel contributes down to T⫽0.) Therefore, in the case of
D – A TET the contribution of the concerted mechanism increases at decreasing temperature.
It follows from the comparison of Fig. 7共b兲 with Fig. 9
that the contribution of the single-electron superexchange
E. G. Petrov and V. May
FIG. 6. D – A kinetics of a TET reaction in a bridge with five units, without
energy bias and for a small bridge state population and a small population of
the intermediate state. Analytic results 关cf. Eqs. 共53兲 and 共54兲–共56兲兴 and the
numerical solution of the set of Eqs. 共A8兲–共A16兲 completely agree 关part
共a兲兴. The bridge populations related to the first 共1兲 and the second type of
bridge states 共2兲 as well as the population of the intermediate state I show
their own kinetics 关panel 共b兲兴. The calculations are based on the ⌬E D
⫽0.29 eV, ⌬E I ⫽0.09 eV, ⌬Ẽ I ⫽0.03 eV, ⌬Ẽ A ⫽0.3 eV, ⌬E⫽0.07 eV.
All other parameters are identical to those used for Fig. 5.
channel to the interstate transfer rates 共characterizing the
stepwise pathway兲 strongly increases at decreasing temperature. This means that at low temperatures even the stepwise
pathway is used since it covers single-electron superexchange channels 共which however have to be thermally activated in contrast with the two-electron superexchange
channel兲.
Figure 10 displays the influence of the energy bias ⌬ on
the concerted and the stepwise bridge-assisted superexchange contribution to the overall D – A TET. At low temperatures under consideration just the single-electron and the
two-electron superexchange channels determine the thermally activated stepwise and the concerted pathway of the
electron transfer, respectively. It is seen that at N⫽1 – 4 the
concerted contribution exceeds the stepwise one, while at
N⬎4 the distance dependence of the K TET is determined by
the stepwise mechanism of the D – A TET. This means that at
low temperatures the main contribution to the D – A TET
stems from the single-electron and the two-electron superexchange mechanisms 共stepwise and concerted pathways, respectively兲. Figure 10 also shows that a change of ⌬ generally affects the stepwise transfer rate. This follows from the
fact that at N⬎4 for which K TET is given by Eq. 共74兲, the
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J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
FIG. 7. Bridge length dependence of the overall D – A TET transfer rate
K TET 共at room temperature兲 as well as its stepwise (K (step)
TET ) and concerted
(K (conc)
TET ) components 关part 共a兲兴. The different contributions to the stepwise
transfer rate K (step)
共single-electron transfer channels, superexchange
TET
channel—up to N⬃3 – 4 bridge units—and the sequential transfer channel
at N⬎4) are shown in part 共b兲. The calculations are based on the Eqs.
共54兲–共73兲 and by choosing the following parameters: ⌬E D ⫽0.20 eV, ⌬E I
⫽0.10 eV, ⌬Ẽ I ⫽0.05 eV, ⌬E⫽0.15 eV; ␭ 1D ⫽␭ NI ⫽␭ 1I ⫽1.2 eV, ␭ DI
⫽0.8 eV, ␭ NA ⫽1.4 eV, ␭ IA ⫽0.5 eV, ␭ DA ⫽0.2 eV, ␭ m m⫾1 ⬅␭ B ⫽0.5 eV;
⬘ ⫽V NA
⬘ ⫽0.02 eV, V B ⫽0.07 eV.
␻ M N ⫽ ␻ 0 ⫽800 cm⫺ ; V D1 ⫽V NA ⫽V D1
Two-electron transfer reactions
4451
FIG. 8. Bridge length dependence of the overall D – A TET transfer rate
(conc)
K TET as well as its stepwise (K (step)
TET ) and concerted (K TET ) components at
different temperatures 共other parameters like those of Fig. 7兲. Part 共a兲 T
⫽150 K, part 共b兲 T⫽100 K.
kinetics of the TET process, Eq. 共30兲, is reduced to a twoexponential kinetics, Eq. 共31兲. The respective two overall
transfer rates K 1 and K 2 are given in Eq. 共41兲. It has been
demonstrated that such a reduction becomes possible if the
main dependence of K TET on ⌬ is concentrated at ⌬E ID , Eq.
共C10兲 rather than in the rate constant k (1)
b , Eq. 共36兲. This
circumstance indicates directly that the stepwise D – A TET
process is originated by a thermally activated mechanism.
V. CONCLUSION
The present paper has been devoted to the study of nonadiabatic two-electron transfer 共TET兲 in a donator (D) acceptor (A) complex mediated by a molecular bridge. Focusing
on nonadiabatic reactions the intersite electron transitions
take place on a time scale ⌬tⰇ ␶ rel , where ␶ rel characterizes
the time of intrasite vibrational relaxation. Therefore, the
whole analysis can be based on the coarse-grained kinetic
equations 共A8兲–共A16兲, valid for tⰇ ␶ rel .
An analysis of the exact solution based on the Laplacetransformed state populations allowed to formulate the conditions, Eqs. 共33兲 and 共45兲, for which the multiexponential
FIG. 9. Bridge length dependence of the fastest interstate transfer rates k (1)
b
and k (2)
characterizing the transitions from the intermediate state I in the
f
three-state DBA system 关see scheme 共b兲 in Fig. 4兴. The calculations have
been done with the same parameters as in Fig. 7.
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4452
J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
FIG. 10. Low temperature bridge length dependence of the overall D – A
TET rate K TET at different values of the bridge internal energy bias ⌬. The
calculations have been done with the same parameters as in Fig. 7.
integral population of the bridging states remains small 共less
then 10⫺2 ) during the whole TET process.
If, additionally, the population of the intermediate state
兩 I 典 , Eq. 共3兲 共with one electron at the D and one at the A),
remains also small, then the two-exponential kinetics turns to
a single-exponential one 关cf. Eq. 共53兲兴 characterized by a
single overall D – A TET transfer rate K TET . Certain conditions have been derived 关cf. Eqs. 共45兲 and 共46兲兴 at which the
transfer rate K TET can be given by a sum of the stepwise and
the concerted transfer rates, Eq. 共54兲.
Within the model of a regular bridge, we have presented
analytic expressions for both mentioned transfer rates, Eqs.
共55兲 and 共56兲 as well as Eqs. 共57兲–共70兲. The stepwise transfer rate characterizes the single-electron pathways of D – A
TET. It contains two contributions related to the sequential
and superexchange mechanism of bridge-assisted singleelectron transfer between the state 兩 D 典 共both electrons at the
D) and the state 兩 I 典 共one electron at the D and another electron at the A) as well as single-electron transfer between the
state 兩 I 典 and the state 兩 A 典 共both electrons at the A).
Since the single-electron pathways follow from thermal
activation the stepwise contributions to K TET are strongly
suppressed with decreasing temperature. In contrast, the concerted transfer rate, which is originated by a specific twoelectron superexchange coupling 共between the state with two
electrons at the D and the state with two electrons at the A),
can act at low temperatures down to T⫽0. Therefore, the
contribution of the concerted mechanism to the D – A TET
increases with decreasing temperature.
The model of a regular bridge has been used to derive
analytical results on the N dependence of the stepwise as
well as concerted contributions to the overall D – A TET rate.
The Coulomb interaction between the transferred electrons
located within the bridge as well as at the D and A induces a
shift of the site energies. This may also happen for the
bridge-site reorganization energies ␭ m m⫾1 which can differ
from ␭ B . 共In the case of a single-electron D – A transfer this
problem has been already discussed in, e.g., Ref. 40.兲 If,
however, among all possible bridging states only those with
E. G. Petrov and V. May
a single electron in the bridge are involved in the TET process the general form of the nonadiabatic rate constants 共72兲
关see also the more general expression 共A7兲兴 does not change.
关In particular, the square of the superexchange couplings are
further defined by Eqs. 共C1兲–共C3兲.兴 Only the energy gaps,
Eqs. 共C4兲–共C12兲 as well as the reorganization energies have
to be modified. In the present paper, however, we assumed a
strong screening of the D and A redox centers by surrounding polar groups 共as it is often the case in enzymes兲. Therefore, any Coulomb interaction of the transferred electrons
with these centers could be neglected. Studying TET in a
bridge with an energy bias a deviation from the standard
exponential law ⬃exp关⫺␨(N⫺1)兴 characterizing the decrease of the superexchange rate with the number of bridge
units N has been obtained. This deviation could be explained
by the change of the energy gap for every bridging unit.
The key result of the present paper is the specification of
the stepwise and concerted contributions to the overall transfer rate which characterizes the nonadiabatic D – A TET process mediated by a molecular bridge. This has been done in
the framework of a tight binding model where the electronic
energies and intersite couplings enter as phenomenological
parameters. Such a model allowed us to derive the analytic
dependence of both contributions on the bridge length 共number of bridging units N). Note that the examination of the
bridge-length dependence of the overall transfer rate is the
most direct way to specify the stepwise and concerted electron transfer routes in a concrete system. However, even for
a SET process such an experimental examination has been
done only recently 共see discussion in Refs. 19 and 20兲.
Therefore, a detailed test of bridge-length effects on nonadiabatic TET reactions by comparing with experimental data
has to be postponed to the future. Another way to distinguish
the stepwise and concerted routes is an examination of the
pH dependence of the overall transfer rate characterizing the
TET reactions in enzymes. Recently, we used this way to
analyze the reduction of micothione reductase by NADPH.41
It has been shown that such a reduction takes place along the
concerted route. Probably, further progress in the theoretical
description of bridge-assisted TET could be achieved with
the combination of semiphenomenological analytic expressions 共containing the precise bridge-length dependence of
transfer rates兲 and quantum-chemical estimations of
electronic energies and of the various 共site-to-site and superexchange兲 electronic couplings. 共In the case of SET,
just such methods allowed to specify single-electron
pathways between redox-centers of number protein
macromolecules.26 –29兲
ACKNOWLEDGMENTS
This work is supported in part by the Fund of Fundamental Research of NASU and the Volkswagen-Stiftung,
Germany 共priority area Intra- and Intermolecular Electron
Transfer兲.
APPENDIX A: DERIVATION OF KINETIC EQUATIONS
The concrete form of the kinetic equations follows from
Eq. 共28兲 if the expansion 共22兲 is introduced into Eq. 共29兲.
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J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
Two-electron transfer reactions
The first term R2 of this expansion corresponds to the standard 共second-order兲 Born approximation. It leads to rate constants which describe the sequential pathway of TET.
Higher-order terms result in superexchange single-electron
and two-electron rate constants. A similar result has been
presented earlier in Ref. 18 for the derivation of SET rate
equations. Therefore, we do not repeat details of the derivation. We only present the superexchange couplings for the
transitions between the two-electron states 兩 D 典 and 兩 I 典 , 兩 I 典
and 兩 A 典 , as well as 兩 D 典 and 兩 A 典 . Moreover, we will only
consider pure electronic energy gaps ⌬E M N 共in Fig. 3 these
gaps are shown for a bridge without energy bias兲.
Let 兩 M 典 be the donor state 兩 D 典 . Then, the Born approximation of the quantities KDN , Eq. 共29兲 follows from the
general expression 共29兲 as
(Born)
KDN ⫽K DN
⫽ ␦ N,D k D1 ⫺ ␦ N,1k 1D .
共A1兲
Here, k D1 ⫽k 1D exp关⫺(E1⫺ED)/kBT兴 and
k 1D ⫽
2␲
ប
兩V 1␯
兺
兺
␯
␯
D
1
1 D␯D
兩 2 ⫻W 共 E 共 1,␯ 1 兲兲 L 共 E 共 1,␯ 1 兲
⫺E 共 D, ␯ D 兲兲 .
共A2兲
The quantity L(x) denotes the Lorentzian-type broadening of
the ␦ function. Employing the Condon approximation, Eq.
共12兲, and following the approach given in Ref. 18 we can
write the rate constant k 1D in the standard form:
k 1D ⫽
2␲
兩 V 1D 兩 2 共 FC兲 1D .
ប
共A3兲
Here, (FC) 1D is the Franck–Condon factor for the 1→D
electron transition. In the considered case of nonadiabatic
electron transfer any rate constant characterizing electron
hopping between adjacent sites has a similar form.
Higher-order expansion terms in Eq. 共22兲 result in two
effects, the renormalization of those rate constants derived in
the Born approximation and the appearance of superexchange rate constants. In the case of nonadiabatic electron
transfer the rate renormalization is of less importance and
will be neglected. The superexchange contributions, if related to the D, are of the single-electron type 共rate constants
k DI and k ID ) and of the two-electron 共concerted兲 type 共rate
constants k DA and k AD ). Therefore we get
(sup 1)
(sup 2)
(Born)
⫹K DN
⫹K DN
,
KDN ⫽K DN
共A4兲
where the first term is given by Eq. 共A1兲 while
(sup 1)
K DN
⫽ ␦ N,D k DI ⫺ ␦ N,I k ID
共A5兲
and
(sup 2)
⫽ ␦ N,D k DA ⫺ ␦ N,I k AD .
K DN
共A6兲
In Eqs. 共A5兲 and 共A6兲, the superexchange rate constants take
the form
k M N⫽
2␲
兩 T M N 兩 2 共 FC兲 M N ,
ប
共A7兲
where expressions of the square of single-electron and twoelectron superexchange couplings defining the D – A TET are
given in Appendix C.
4453
Substituting Eqs. 共A4兲–共A6兲 in Eq. 共28兲 we get
Ṗ D 共 t 兲 ⫽⫺ 共 q D ⫹k DA 兲 P D 共 t 兲 ⫹k 1D P 1 共 t 兲 ⫹k ID P I 共 t 兲
⫹k AD P A 共 t 兲 ,
共A8兲
Ṗ 1 共 t 兲 ⫽⫺q 1 P 1 共 t 兲 ⫹k D1 P D 共 t 兲 ⫹ ␤ P 2 共 t 兲 ,
共A9兲
Ṗ m 共 t 兲 ⫽⫺q P m 共 t 兲 ⫹ ␣ P m⫺1 共 t 兲 ⫹ ␤ P m⫹1 共 t 兲 ,
共A10兲
Ṗ N 共 t 兲 ⫽⫺q N P N 共 t 兲 ⫹k IN P I 共 t 兲 ⫹ ␣ P N⫺1 共 t 兲 ,
共A11兲
Ṗ I 共 t 兲 ⫽⫺ 共 q I(1) ⫹q I(2) 兲 P I 共 t 兲 ⫹k NI P N 共 t 兲 ⫹k DI P D 共 t 兲
⫹k 1I P̃ 1 共 t 兲 ⫹k AI P A 共 t 兲 ,
共A12兲
Ṗ ˜1 共 t 兲 ⫽⫺q̃ 1 P ˜1 共 t 兲 ⫹k I1 P I 共 t 兲 ⫹ ␤ P ˜2 共 t 兲 ,
共A13兲
Ṗ ñ 共 t 兲 ⫽⫺q P ñ 共 t 兲 ⫹ ␣ P ñ⫺1˜ 共 t 兲 ⫹ ␤ P ñ⫹1˜ 共 t 兲 ,
共A14兲
Ṗ N˜ 共 t 兲 ⫽⫺q̃ N P N˜ 共 t 兲 ⫹k AN P A 共 t 兲 ⫹ ␣ P N˜ ⫺1̃ 共 t 兲 ,
共A15兲
Ṗ A 共 t 兲 ⫽⫺ 共 q A ⫹k AD 兲 P A 共 t 兲 ⫹k NA P N˜ 共 t 兲 ⫹k IA P I 共 t 兲
⫹k DA P D 共 t 兲 .
共A16兲
Note that in the Eqs. 共A10兲 and 共A14兲, m,n run over
2,3, . . . ,N⫺1. Furthermore, we used the abbreviation m
⬅B m , and ñ⬅B̃ n so that we can identify P m (t) with P B m (t)
and P ñ (t) with P ˜B n (t).
The Eqs. 共A8兲–共A16兲 have to be complemented by the
following abbreviations for the different rate expressions:
q D ⬅k D1 ⫹k DI ,
q̃ 1 ⬅k 1I ⫹ ␣ ,
q 1 ⬅k 1D ⫹ ␣ ,
q̃ N ⬅k NA ⫹ ␤ ,
q I(1) ⬅k IN ⫹k ID ,
q N ⬅k NI ⫹ ␤ ,
q A ⬅k AN ⫹k AI ,
q I(2) ⬅k I1 ⫹k IA ,
共A17兲
q⬅ ␣ ⫹ ␤ .
Here ␣ ⬅k mm⫹1 and ␤ ⬅k m⫹1m (m⫽1,2, . . . ,N⫺1), are
the rates characterizing the forward and backward jumps of a
single-electron between neighboring bridge units.
APPENDIX B: EXACT SOLUTION OF THE LAPLACE
TRANSFORMED SET OF RATE EQUATIONS
The solution of the Eqs. 共A8兲–共A16兲 can be constructed by changing to the Laplace transform F M (s)
⫽ 兰 ⬁0 exp(⫺st)PM(t) dt of all electronic populations.
Noting the initial condition P M (0)⫽ ␦ M ,D the solution of
the Laplace-transformed rate equations 共A9兲–共A11兲 and
共A13兲–共A15兲 governing the bridge populations P m and P ñ
are obtained as
F m共 s 兲 ⫽
1
关 k ␣ m⫺1 C N 共 m 兲 F D 共 s 兲
D 1 共 s 兲 D1
⫹k IN ␤ N⫺m C 1 共 m 兲 F I 共 s 兲兴
共B1兲
and as
F ñ 共 s 兲 ⫽
1
关 k ␣ n⫺1 C̃ N 共 n 兲 F I 共 s 兲
D 2 共 s 兲 I1
⫹k AN ␤ N⫺n C̃ 1 共 n 兲 F A 共 s 兲兴 .
共B2兲
The following abbreviations have been introduced:
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4454
J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
E. G. Petrov and V. May
D 1 共 s 兲 ⫽sD 共 s;N⫺1 兲 ⫹ 关 s 共 k 1D ⫹k NI 兲 ⫹ ␤ k 1D ⫹ ␣ k NI
⫹k 1D k NI 兴 D 共 s;N⫺2 兲 ⫺ ␣␤ 共 k 1D ⫹k NI 兲
⫻D 共 s;N⫺3 兲 ,
1
关共 s⫺R II 共 s 兲兲共 s⫺R AA 共 s 兲兲
Det共 s 兲
F D共 s 兲 ⫽
⫺R IA 共 s 兲 R AI 共 s 兲兴 ,
共B3兲
D 2 共 s 兲 ⫽sD 共 s;N⫺1 兲 ⫹ 关 s 共 k 1I ⫹k NA 兲 ⫹ ␤ k 1I ⫹ ␣ k NA
F I共 s 兲 ⫽
⫹k 1I k NA 兴 D 共 s;N⫺2 兲 ⫺ ␣␤ 共 k 1I ⫹k NA 兲
⫻D 共 s;N⫺3 兲
共B4兲
1
关 R 共 s 兲共 s⫺R AA 共 s 兲兲 ⫹R IA 共 s 兲 k DA 兴 ,
Det共 s 兲 ID
1
关 k 共 s⫺R II 共 s 兲兲 ⫹R ID 共 s 兲 R AI 共 s 兲兴 ,
Det共 s 兲 DA
F A共 s 兲 ⫽
as well as
where we introduced
C N 共 m 兲 ⫽ 共 k NI ⫺ ␣ 兲 D 共 s;N⫺m⫺1 兲 ⫹D 共 s;N⫺m 兲 ,
Det共 s 兲 ⫽ 共 s⫺R DD 共 s 兲兲共 s⫺R II 共 s 兲兲共 s⫺R AA 共 s 兲兲
共B5兲
C 1 共 m 兲 ⫽ 共 k 1D ⫺ ␤ 兲 D 共 s;m⫺2 兲 ⫹D 共 s;m⫺1 兲 ,
⫺ 关 R IA 共 s 兲 R AI 共 s 兲共 s⫺R DD 共 s 兲兲 ⫹R DI 共 s 兲 R ID 共 s 兲
and
⫻ 共 s⫺R AA 共 s 兲兲 ⫹k AD k DA 共 s⫺R II 共 s 兲兲兴
C̃ N 共 n 兲 ⫽ 共 k NA ⫺ ␣ 兲 D 共 s;N⫺n⫺1 兲 ⫹D 共 s;N⫺n 兲 ,
⫺R DI 共 s 兲 R IA 共 s 兲 k DA ⫺R AI 共 s 兲 R ID 共 s 兲 k AD ].
共B6兲
C̃ 1 共 n 兲 ⫽ 共 k 1I ⫺ ␤ 兲 D 共 s;n⫺2 兲 ⫹D 共 s;n⫺1 兲 .
In the Eqs. 共B3兲–共B6兲, we introduced a new function
D(s;M ), which obeys the recursion relation
D 共 s;M 兲 ⫽ 共 s⫹ ␣ ⫹ ␤ 兲 D 共 s;M ⫺1 兲 ⫺ ␣␤ D 共 s;M 兲 .
共B7兲
It follows
D 共 s;M 兲 ⫽ 共 ␣␤ 兲 M /2
sinh关 ⌳ 共 s 兲共 M ⫹1 兲兴
,
sinh ⌳ 共 s 兲
共B8兲
where
exp ⌳ 共 s 兲 ⫽ 共 s⫹ ␣ ⫹ ␤ 兲 / 共 2 ␣␤ 兲 1/2.
共B9兲
Introducing the expressions 共B1兲 and 共B2兲 into the Eqs. 共A8兲,
共A12兲, and 共A16兲 results in three coupled equations for the
Laplace-transformed population of the states 兩 D 典 , 兩 I 典 , and
兩 A 典 . The equations read
共 s⫺R DD 共 s 兲兲 F D 共 s 兲 ⫺R DI 共 s 兲 F I 共 s 兲 ⫺k AD F A 共 s 兲 ⫽1,
R ID 共 s 兲 F D 共 s 兲 ⫺ 共 s⫺R II 共 s 兲兲 F I 共 s 兲 ⫹R IA 共 s 兲 F A 共 s 兲 ⫽0,
共B10兲
k DA F D 共 s 兲 ⫹R AI 共 s 兲 F I 共 s 兲 ⫺ 共 s⫺R AA 共 s 兲兲 F A 共 s 兲 ⫽0,
共B13兲
Laplace-transformed bridge populations, F m (s) and F ñ (s),
are expressed by those of the donor, acceptor, and intermediate state population 关 F D (s), F I (s), and F A (s), respectively兴.
The exact solution for all Laplace-transformed populations F M (s) 关 M ⫽D,I,A,1, . . . ,N,1̃, . . . ,Ñ 兴 , Eqs. 共B1兲,
共B2兲, and 共B12兲 permits a description of the TET kinetics
within all time domains 共provided that tⰇ ␶ rel). Solving
Det(s)⫽0 共at least numerically兲 one obtains 2N⫹2 nonzero
roots s 1 ,s 2 , . . . , s 2N⫹2 which determine the transfer rates
according to K 1 ⫽⫺s 1 , K 2 ⫽⫺s 2 , . . . , K 2N⫹2 ⫽⫺s 2N⫹2 .
Therefore, state populations follow as Eq. 共30兲.
APPENDIX C: SUPEREXCHANGE COUPLINGS
AT D – A TET
Next we present the square of the superexchange couplings characterizing the distant single-electron ( 兩 D 典 兩 I 典
and 兩 I 典 兩 A 典 ) and the distant two-electron ( 兩 D 典 兩 A 典 )
transfer. Their precise form follows from the specification of
the transfer matrix elements KM N , Eq. 共29兲. It yields
with the abbreviations
k NI k D1 ␣ N⫺1
,
R ID 共 s 兲 ⫽
D 1共 s 兲
k NA k I1 ␣
D 2共 s 兲
k 1D k IN ␤ N⫺1
R DI 共 s 兲 ⫽
,
D 1共 s 兲
k NA k AN C̃ 1 共 N 兲
R AA 共 s 兲 ⫽
,
D 2共 s 兲
N⫺1
R AI 共 s 兲 ⫽
R II 共 s 兲 ⫽ 共 k IN ⫺k ID 兲 ⫺
⫺
兩 T DI 兩 ⫽
2
k 1D k D1 C N 共 1 兲
,
R DD 共 s 兲 ⫽
D 1共 s 兲
k 1I k AN ␤
D 2共 s 兲
兩 T IA 兩 2 ⫽
兩 V D1 V BN⫺1 V NA 兩 2
N
兿 m⫽1
⌬E mD ⌬E mI
⬘ V BN⫺1 V NA
⬘ 兩2
兩 V D1
N
兿 m⫽1
⌬Ẽ mI ⌬Ẽ mA
R IA 共 s 兲 ⫽
,
兩 T DA 兩 2 ⫽
k NI k IN C 1 共 N 兲
⫹ 共 k I1 ⫺k IA 兲
D 1共 s 兲
k 1I k I1 C̃ N 共 1 兲
.
D 2共 s 兲
Then, by solving the Eqs. 共B10兲, expressions for the Laplace
transform of P D , P I , and P D are obtained as
共C1兲
,
共C2兲
兩 V D1 V BN⫺1 V NA 兩 2
N
兿 m⫽1
⌬E mD ⌬E mA ⌬E ID
⫻
共B11兲
,
and
N⫺1
,
共B12兲
⬘ V BN⫺1 V NA
⬘ 兩2
兩 V D1
N
兿 m⫽1
⌬Ẽ mD ⌬Ẽ mA ⌬E IA
.
共C3兲
The gaps in the Eqs. 共C1兲–共C3兲 are defined as the energy
differences between the corresponding electron vibrational
states but taken at the vibrational ground state. If an energy
bias in the bridge is absent, a part of the energy gaps is
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J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
Two-electron transfer reactions
represented in Fig. 3. From this we may conclude that
⌬E mD ⫽⌬E D , ⌬E mI ⫽⌬E I , ⌬E mA ⫽⌬E A , ⌬Ẽ mD ⫽⌬Ẽ D ,
⌬Ẽ mI ⫽⌬Ẽ I , ⌬Ẽ mA ⫽⌬Ẽ A .
If a bridge with an energy bias mediates the D – A TET,
the energy gaps take a more complicated form. For instance,
if the energy gaps ⌬E 1D ⫽⌬E D , ⌬E NI ⫽⌬E I , ⌬Ẽ 1I
⫽⌬Ẽ I , and ⌬Ẽ NA ⫽⌬Ẽ A are fixed while the electronic energies of the interior bridge units change in line with Eq.
共11兲, one has to take
⌬E mD ⫽⌬E D ⫺ 共 m⫺1 兲 ⌬,
共C4兲
⌬Ẽ mD ⫽⌬Ẽ D ⫺ 共 m⫺1 兲 ⌬⫺ 共 N⫺1 兲 ⌬,
共C5兲
⌬E mI ⫽⌬E I ⫹ 共 N⫺m 兲 ⌬,
共C6兲
⌬Ẽ nI ⫽⌬Ẽ I ⫺ 共 n⫺1 兲 ⌬,
共C7兲
⌬E nA ⫽⌬E A ⫹ 共 N⫺n 兲 ⌬⫹ 共 N⫺1 兲 ⌬,
共C8兲
⌬Ẽ nA ⫽⌬Ẽ A ⫹ 共 N⫺n 兲 ⌬,
共C9兲
and
兩 T IA 兩 2 ⫽
共C10兲
⌬E IA ⫽ 共 ⌬E A ⫺⌬E I 兲 ⫹ 共 N⫺1 兲 ⌬,
共C11兲
⌬E DA ⫽⌬E⫹2 共 N⫺1 兲 ⌬.
共C12兲
The relations 共C4兲–共C11兲 allow to derive an analytical dependence of the superexchange rate constants on the number
of bridging units N 共see below兲.
Using Eq. 共C1兲 and Eq. 共C2兲 one can represent the
square of the two-electron coupling 共C3兲 as
兩 T DA 兩 2 ⫽
兩 T̃ DI T̃ IA 兩 2
.
⌬E ID ⌬E IA
共C13兲
兿
共 ⌬E mI /⌬E mA 兲 ,
m⫽1
共C14兲
N
兩 T̃ IA 兩 2 ⫽ 兩 T IA 兩 2
兿
m⫽1
共 ⌬Ẽ mI /⌬Ẽ mD 兲 .
If an energetic bias is not present in the bridge the latter
quantities simplify to give
2
N
共C15兲
兩 T̃ IA 兩 2 ⫽ 兩 T IA 兩 2 共 ⌬Ẽ I /⌬Ẽ D 兲 N .
It follows from the definition of the electronic states 共see also
Fig. 3兲 that ⌬Ẽ D ⫽⌬E ID ⫹⌬Ẽ I , ⌬E A ⫽⌬E IA ⫹⌬E I , and
thus if ⌬E ID Ⰶ⌬Ẽ I and ⌬E IA Ⰶ⌬E I , we may identify
兿
⌬E mD ⫽ 共 ⌬E D 兲
兩 T DI T IA 兩
兩 T DA 兩 2 ⫽
,
⌬E ID ⌬E IA
兩 T DI 兩 2 ⫽
冉
兩 V D1 V NA 兩 2
VB
⌬E D ⌬E I ⌬E D ⌬E I
共C16兲
冊
N
兿
m⫽1
.
共C18兲
关 1⫺ 共 m⫺1 兲共 ⌬/⌬E D 兲兴
再兺
冎
N
⫽ 共 ⌬E D 兲 N exp
m⫽1
ln关 1⫺ 共 m⫺1 兲
⫻共 ⌬/⌬E D 兲兴
共C19兲
and by noting the limit of a small intersite energy bias (N
⫺1)⌬Ⰶ⌬E D , one can expand ln关1⫺(m⫺1)(⌬/⌬ED)兴 with
respect to (m⫺1)(⌬/⌬E D ). The lowest order contribution
gives
N
兿
⌬E mD ⫽ 共 ⌬E D 兲 N e ⫺(1/2)(⌬/⌬E D )N(N⫺1) .
兩 T DI 兩 2 ⬇
共C20兲
N⫺1
,
共C17兲
兩 V D1 V NA 兩 2 ⫺ ␨ (N⫺1) ⫺ ␨ N(N⫺1)
e 1
e DI
.
⌬E D ⌬E I
共C21兲
Here the superexchange decay parameter ␨ 1 is defined by Eq.
共65兲. We have also introduced an additional parameter
冉
冊
1 ⌬
⌬
⫺
⬎0
2 ⌬E I ⌬E D
共C22兲
which modifies the N dependence of the superexchange coupling. Analogously, by using the approximate form of the
N
⌬Ẽ mI ⌬Ẽ mA one can represent Eq. 共C2兲 as
product 兿 m⫽1
兩 T IA 兩 ⬇
2
2
where now
冊
N
␨ DI ⫽
兩 T̃ DI 兩 ⫽ 兩 T DI 兩 共 ⌬E I /⌬E A 兲 ,
2
⌬Ẽ I ⌬Ẽ A
N⫺1
In the same way one can specify the approximate form of the
N
⌬E mI . Therefore, in accordance with Eq. 共C1兲
product 兿 m⫽1
one may derive
N
兩 T̃ DI 兩 2 ⫽ 兩 T DI 兩 2
冉
N
m⫽1
Here we have introduced
⌬Ẽ I ⌬Ẽ A
VB
关Note that the same expressions follow from the general
form 共C14兲 if the energy difference between the electronic
states 兩 D 典 , 兩 I 典 and 兩 A 典 is small, so that ⌬E mI ⬇⌬E mA and
⌬Ẽ mI ⬇⌬Ẽ mD .] Equation 共C16兲 reflects the fact that the
two-electron superexchange through a bridge of N units results from two single-electron D – A superexchange pathways associated with the couplings T DI and T IA . In other
words, the two-electron superexchange is originated by repeated single-electron superexchange transitions.
In the case of a regular bridge and at a small intersite
energy bias ⌬ the basic expressions 共C1兲–共C3兲 can be transformed into a more appealing form. Therefore, we rewrite
the denominators of the expressions 共C1兲–共C3兲 by utilizing
the relations 共C4兲–共C11兲. Let us start with the specification
N
⌬E mD . According to the identity
of the product 兿 m⫽1
m⫽1
⌬E ID ⫽ 共 ⌬E D ⫺⌬E I 兲 ⫺ 共 N⫺1 兲 ⌬,
⬘ V NA
⬘ 兩2
兩 V D1
4455
⬘ V NA
⬘ 兩2
兩 V D1
⌬Ẽ I ⌬Ẽ A
e ⫺ ␨ 2 (N⫺1) e ␨ IA N(N⫺1) .
共C23兲
Here, the superexchange decay parameter ␨ 2 is defined by
Eq. 共66兲 while
␨ IA ⫽
1
冉
⌬
2 ⌬Ẽ I
⫺
⌬
⌬Ẽ A
冊
⬎0.
共C24兲
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4456
J. Chem. Phys., Vol. 120, No. 9, 1 March 2004
E. G. Petrov and V. May
Note that due to the inequalities ⌬E I ⬍⌬E D and ⌬Ẽ I
⬍⌬Ẽ A 共cf. Fig. 3兲 the correction parameters ␨ DI and ␨ IA
become positive. Therefore, a different influence of the energy bias ⌬ exist on the factors 兩 T DI 兩 2 and 兩 T IA 兩 2 .
Taking the same approximations the two-electron coupling reads
兩 T DA 兩 2 ⬇
⬘ V NA
⬘ 兩2
兩 V D1 V NA V D1
⌬E D ⌬E A ⌬Ẽ D ⌬Ẽ A ⌬E ID ⌬E IA
⫻e ⫺ ␨ (N⫺1) e ␨ DA N(N⫺1) ,
共C25兲
where the two-electron superexchange decay parameter ␨ is
given by Eq. 共69兲, and
␨ DA ⫽
1
2
冋冉
⌬
⌬E D
⫹3
⌬
⌬Ẽ D
冊冉
⫺
⌬
⌬Ẽ A
⫹3
⌬
⌬E A
冊册
. 共C26兲
The Eqs. 共C21兲, 共C23兲, and 共C25兲 allow to analyze the
corrections to the distance dependence of the superexchange
transfer rates when a small intrabridge energy bias ⌬ is
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