Download On the adequacy of the Redfield equation and related approaches

On the adequacy of the Redfield equation and related approaches to the study of
quantum dynamics in electronic energy transfer
Akihito Ishizaki and Graham R. Fleming
Citation: The Journal of Chemical Physics 130, 234110 (2009); doi: 10.1063/1.3155214
View online: http://dx.doi.org/10.1063/1.3155214
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THE JOURNAL OF CHEMICAL PHYSICS 130, 234110 共2009兲
On the adequacy of the Redfield equation and related approaches to the
study of quantum dynamics in electronic energy transfer
Akihito Ishizaki and Graham R. Fleminga兲
Department of Chemistry, University of California, Berkeley, California 94720, USA and Physical
Bioscience Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
共Received 12 December 2008; accepted 11 May 2009; published online 18 June 2009兲
The observation of long-lived electronic coherence in photosynthetic excitation energy transfer
共EET兲 by Engel et al. 关Nature 共London兲 446, 782 共2007兲兴 raises questions about the role of the
protein environment in protecting this coherence and the significance of the quantum coherence in
light harvesting efficiency. In this paper we explore the applicability of the Redfield equation in its
full form, in the secular approximation and with neglect of the imaginary part of the relaxation terms
for the study of these phenomena. We find that none of the methods can give a reliable picture of
the role of the environment in photosynthetic EET. In particular the popular secular approximation
共or the corresponding Lindblad equation兲 produces anomalous behavior in the incoherent transfer
region leading to overestimation of the contribution of environment-assisted transfer. The full
Redfield expression on the other hand produces environment-independent dynamics in the large
reorganization energy region. A companion paper presents an improved approach, which corrects
these deficiencies 关A. Ishizaki and G. R. Fleming, J. Chem. Phys. 130, 234111 共2009兲兴. © 2009
American Institute of Physics. 关DOI: 10.1063/1.3155214兴
I. INTRODUCTION
The photosynthetic conversion of the physical energy of
sunlight into its chemical form suitable for cellular processes
involves a variety of physical and chemical mechanisms.1,2
Photosynthesis starts with the absorption of a photon of sunlight by one of the light-harvesting pigments, followed by
transfer of the excitation energy to the reaction center, where
charge separation is initiated. The transfer of this excitation
energy toward the reaction center occurs with a near unity
quantum yield.1
In green sulfur bacteria such as Prosthecochloris aestuarii and Chlorobium tepidum, the energy transfer between
the main chlorosome antenna and the reaction centers is mediated by a protein containing bacteriochlorophyll 共BChl兲
molecules, called the Fenna–Matthews–Olson 共FMO兲
protein.3,4 The FMO protein is a trimer made of identical
subunits, each of which contains seven BChl molecules and
no carotenoids. By virtue of its relatively small size, it represents an important model in photosynthetic energy transfer
and has been extensively studied experimentally and theoretically. Recently, Engel et al.5 investigated the FMO protein isolated from Chlorobium tepidum by means of twodimensional Fourier transform electronic spectroscopy6–8
and succeeded in observing long-lasting quantum beating
providing direct evidence for long-lived electronic coherence. The observed coherence clearly lasts for time scales
similar to the excitation energy transfer 共EET兲 time scales,
implying that electronic excitations move coherently through
the FMO protein rather than by incoherent hopping motion
as has usually been assumed.1,9 Furthermore Lee et al.10 revealed coherent dynamics in a bacterial reaction center by
a兲
Electronic mail: [email protected].
0021-9606/2009/130共23兲/234110/8/$25.00
applying a two-color electronic coherence photon echo experiment. However, the relevance and physical role of the
long-lived quantum coherence and its interplay with the protein environment in the EET process are to a large extent
unknown. In order to elucidate these issues, detailed theoretical investigations are required in addition to further experimental studies.11–15
Theoretically, photosynthetic EET processes are usually
described by two perturbative limits. When the electronic
coupling between chromophores is small in comparison with
the electron-environment coupling, the original localized
electronic state is an appropriate representation and the electronic coupling can be treated perturbatively. This treatment
yields the Förster theory,16–18 which describes only incoherent hopping between chromophores. In the opposite limit,
when the electron-environment coupling is small, it is possible to treat the electron-environment coupling perturbatively to obtain a quantum master equation. The most commonly used theory from this limit is the Redfield
equation.18–20 Innovated and established in the field of
nuclear magnetic resonance,21–23 the equation has been employed in wide areas of condensed phase chemical dynamics
such as electron transfer reactions,24–29 conical
intersections,30–32 and electronic energy transfer.9,33–37 Although the Redfield equation has been broadly applied as
mentioned above, its original form is based on the Markov
approximation and on the assumption that the systemenvironment interaction is sufficiently weak that a secondorder perturbative truncation should be valid. In the early
1990s, Walsh and Coalson38 showed numerically that the
Redfield equation actually has a wider scope of application
than previously thought. Recently, Ishizaki and Tanimura39
proved analytically that the Redfield equation can be derived
130, 234110-1
© 2009 American Institute of Physics
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J. Chem. Phys. 130, 234110 共2009兲
A. Ishizaki and G. R. Fleming
in a nonperturbative manner for any system linearly coupled
to a phonon bath. Their discussions are, however, applicable
to nontransfer processes such as spin relaxation and vibrational relaxation problems. In the EET process, each site of a
multichromophoric array is coupled to its local environmental phonons. In such systems, electronic de-excitation of a
donor chromophore and excitation of an acceptor occur via
nonequilibrium phonon states in accordance with the
Franck–Condon principle. The phonons coupled to each
chromophore then relax to the respective equilibrium states.
These site-dependent reorganization processes cannot be described by the Redfield equation. These processes become
more significant when the reorganization energies are not
small in comparison with the electronic coupling—a typical
situation in photosynthetic EET. In the FMO protein, for example, the electronic coupling strengths span a wide range,
1 – 100 cm−1, while the suggested reorganization energies
span a similar range.8,40,41 Modified Redfield theory, first
suggested by Zhang et al.42 and later further examined by
Yang and Fleming,43 allows us to overcome this shortcoming
of the conventional Redfield equation. It was derived by employing a projection operator technique that projects only on
populations 共diagonal elements兲 of the density matrix in the
eigenstate basis. The theory reduces to the conventional Redfield theory and Förster theory in the respective limiting
cases.43 By construction, however, the modified Redfield
theory cannot take into account any processes related to the
quantum coherences 共the off-diagonal elements兲, e.g., coherent wavelike motion, coherence transfer, and so forth.
Consequently, the conventional Redfield equation is one
of the few viable methods to explore the relevance and
physical role of the quantum coherence and its interplay with
the protein environment in EET processes. In the original
paper of Redfield19,20 the zeroth-order Hamiltonian for the
perturbation expansion describes only the system and does
not provide any information about the temperature environment. In some of the literature,11,15,44–47 on the other hand, a
different quantum master equation is derived based on a different zeroth-order Hamiltonian, which is dressed by the environmental degrees of freedom and temperature due to the
small-polaron transformation. This quantum master equation
can be applied for cases of strong system-environment interaction and may well give better results compared to the Redfield equation in the original sense. Sometimes it is also
called the Redfield equation.43 In order to avoid confusion,
the discussion given in this paper is limited to the original
form of the Redfield equation, which assumes weak systemenvironment interaction, since this is the approach in most of
the current literature on photosynthetic EET.
In applications of the conventional Redfield equation,
the secular approximation18–20,48 has been widely employed.
This is based on mainly two lines of argument. The first is
that the approximation is attractive since numerical time
propagation is simplified. Under the approximation, the
populations are divorced from the coherences. Moreover,
each coherence generally follows its own time-evolution
equation irrespective of the rest. In this case, the Redfield
relaxation terms can be interpreted straightforwardly as
population relaxation and coherence dephasing rates. An-
other reason for its popularity is to guarantee that any diagonal element of the density matrix will be non-negative. The
Redfield equation sometimes violates this positivity requirement especially for low temperature systems.49,50 As shown
in Ref. 48, when the electron-environment coupling Hamiltonian is written as a sum of products of electronic and environmental operators, the secular approximation applied to
the Redfield equation leads to the form of the Lindblad
equation,51 whose relaxation operators are constructed based
on the assumption that the density matrix keeps positive definite.
In so far as simplicity and positivity are concerned, the
conventional Redfield equation with the secular approximation or the corresponding Lindblad equation has many advantages. However, the secular approximation, which eliminates some of the relaxation terms, corresponds to the partial
destruction of the interplay between the EET system and its
environment. Hence, there is the danger that the approximation distorts the true dynamics within the Redfield framework. As a result, the approximation might lead us to erroneous insights or incorrect conclusions regarding the
quantum coherence and its interplay with the protein environment. In this work, we re-examine the adequacy and limitations of the conventional Redfield equation to photosynthetic EET by employing Förster theory as a benchmark for
discussion of cases involving large reorganization energy.
Special attention is paid to an anomaly, which originates
from the secular approximation in the incoherent hopping
region. The paper is organized as follows: In Sec. II, we
describe the model and briefly review the Redfield equation
and the secular approximation. In Sec. III, we present numerical results of the Redfield equation with and without the
secular approximation. Here, we employ the simplest EET
system, a heterodimer, in order to make our discussion clear.
Finally, Sec. IV is devoted to concluding remarks.
II. MODEL
We employ the following Frenkel exciton Hamiltonian
to study EET dynamics,43
Htot ⬅ Hel + Hph + Hreorg + Hel-ph ,
共2.1兲
where
N
H = 兺 兩n典␧0n具n兩 +
el
n=1
兺 Jmn共兩m典具n兩 + 兩n典具m兩兲,
共2.2a兲
m⬍n
N
H = 兺 Hph
n ,
ph
共2.2b兲
n=1
N
Hreorg = 兺 兩n典␭n具n兩,
共2.2c兲
n=1
and
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N
N
H
el-ph
J. Chem. Phys. 130, 234110 共2009兲
Adequacy of the Redfield equation
=兺
Hel-ph
=
n
兺 V nu n .
共2.2d兲
n=1
n=1
In the above, 兩n典 represents the state where only the nth site
is in its excited electronic state 兩␸ne典 and all others are in
their ground electronic states 兩␸mg典, that is 兩n典
⬅ 兩␸ne典兿n⫽m兩␸mg典. Equation 共2.2兲 is the electronic Hamiltonian in which the Hamiltonian of the ground electronic
state is set to be zero by definition. ␧0n is the excited electronic energy of the nth site in the absence of phonons 共see
also Fig. 2兲. Jnm is the electronic coupling Hamiltonian between the nth and mth sites, which is responsible for EET
2
between the individual sites. In Eq. 共2.2兲, Hph
n ⬅ 兺␰ប␻␰共p␰
2
+ q␰ 兲 / 2 is the phonon Hamiltonian associated with the nth
sites, where q␰, p␰, and ␻␰ are the dimensionless coordinate,
conjugate momentum, and frequency of the ␰th phonon
mode, respectively. In Eq. 共2.2兲, ␭n ⬅ 兺␰ប␻␰dn2␰ / 2 is the reorganization energy of the nth site, where dn␰ is the dimensionless displacement of the equilibrium configuration of the
␰th phonon mode between the ground and excited electronic
is the coupling
states of the nth site. In Eq. 共2.2兲, Hel-ph
n
Hamiltonian between the nth site and the phonon modes and
we defined un ⬅ −兺␰ប␻␰dn␰q␰ and Vn = 兩n典具n兩.
Owing to Wick’s theorem,52 all the fluctuationdissipation processes of phonon modes can be specified by
the two-point correlation function, Cmn共t兲 ⬅ 具um共t兲un共0兲典ph.
Introducing a spectral distribution function of the electronphonon coupling constants, Jmn共␻兲, one can express the correlation function as
Cmn共t兲 =
ប
␲
冕
⬁
d␻Jmn共␻兲关nBE共␻兲 + 1兴e−i␻t ,
共2.3兲
−⬁
where nBE共␻兲 ⬅ 1 / 共e␤ប␻ − 1兲 is the Bose–Einstein distribution
function and we postulated the antisymmetry, Jmn共−␻兲 =
−Jnm共␻兲. The absolute magnitude of the spectral distribution
function is related to the reorganization energy by ␭n
= 兰⬁0 d␻Jnn共␻兲 / 共␲␻兲.
Now, we identify the electronic excitations as the relevant system. The phonon degrees of freedom constitute the
heat bath responsible for electronic energy fluctuations and
dissipation of the coherent exciton motion. An adequate description of the EET dynamics is given by the reduced density operator ␳, that is, the partial trace of the total density
operator ␳tot over the irrelevant phonon degrees of freedom:
␳ = Trph兵␳tot其.
There are two possibilities to represent the reduced density operator ␳ for EET processes. First, one can choose a
site representation in terms of the 兵兩n典其 basis, which gives the
site
⬅ 具m兩␳兩n典. Since Förster theory
reduced density matrix ␳mn
describes a hopping motion between sites, the site representation is well suited for establishing the link between Förster
theory and the present reduced density approach. On the
other hand, the alternative to the site representation is provided by an eigenstate representation of the excitons 兵兩e␯典其.
This gives a reduced density matrix such as ␳␮exc␯ ⬅ 具e␮兩␳兩e␯典.
In this paper, Roman letters are employed for the sites while
Greek letters are used for the excitons. In order to obtain
exciton energies 兵E␯其 and states 兵兩e␯典其, the electronic Hamil-
tonian He ⬅ Hel + Hreorg is diagonalized by the orthogonal matrix U via U−1HeU = ⍀, where the ␯th diagonal element of ⍀
is identical to the ␯th eigenenergy E␯. An exciton ket state
can be expanded as
兩e␯典 = 兺 共U−1兲␯n兩n典 = 兺 Un␯兩n典.
n
共2.4兲
n
Hence, the two representations of the reduced density operator are transformed to each other as follows:
U−1␳siteU = ␳exc .
共2.5兲
This orthogonal transformation between the two representations does not affect the dynamics. In the exciton representation, the diagonal element of the coupling Hamiltonian between the nth site and the phonon modes, 具e␯兩Vnun兩e␯典
= Un␯Un␯un, induces energy fluctuations of the ␯th exciton,
whereas the off-diagonal matrix element, 具e␮兩Vnun兩e␯典
= Un␮Un␯un, contributes to the exciton relaxation process between the ␮th and ␯th excitons.
When the electron-phonon coupling Hel-ph is weak, it can
be treated perturbatively. In this regime, the EET dynamics
can be described by the Redfield equation in the exciton
basis,18–20
⳵ exc
exc
␳ 共t兲 = − i␻␮␯␳␮exc␯ 共t兲 + 兺 R␮␯,␮⬘␯⬘␳␮⬘␯⬘共t兲,
⳵ t ␮␯
␮ ,␯
共2.6兲
⬘ ⬘
where the Markov approximation has been employed. In Eq.
共2.6兲, the first term corresponds to the coherent evolution
governed by He and ប␻␮␯ ⬅ E␮ − E␯ is the energy gap between excitons. The second term describes the phononinduced relaxation dynamics. The tetradic relaxation matrix
R␮␯,␮⬘␯⬘ is the transfer rate from ␳␮exc␯ to ␳␮exc␯ . This is called
⬘ ⬘
the Redfield tensor and expressed as
R␮␯,␮⬘␯⬘ ⬅ ⌫␯⬘␯,␮␮⬘ + ⌫␮⬘␮,␯␯⬘ − ␦␯␯⬘ 兺 ⌫␮␬,␬␮⬘
ⴱ
␬
− ␦␮␮⬘ 兺 ⌫␯␬,␬␯⬘ ,
ⴱ
␬
共2.7兲
in terms of the damping matrix,
⌫ ␮␯,␮⬘␯⬘ ⬅
1
兺 具e␮兩Vm兩e␯典具e␮⬘兩Vn兩e␯⬘典Cmn关␻␯⬘␮⬘兴, 共2.8兲
ប2 m,n
where the complex quantity Cmn关␻兴 is given by the Fourier–
Laplace transform of the phonon correlation function Eq.
共2.3兲 as
Cmn关␻兴 ⬅
冕
⬁
dtei␻tCmn共t兲,
共2.9兲
0
whose real and imaginary parts are expressed as
Re Cmn关␻兴 = បJmn共␻兲关nBE共␻兲 + 1兴,
共2.10a兲
冕
共2.10b兲
Im Cmn关␻兴 =
1
P
␲
⬁
−⬁
d␻⬘
Re Cmn关␻⬘兴
,
␻ − ␻⬘
respectively. The symbol “P” denotes the principal value of
the integral. The real part affects the relaxation of the system,
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J. Chem. Phys. 130, 234110 共2009兲
A. Ishizaki and G. R. Fleming
whereas the imaginary part introduces terms, which can be
interpreted as a modification of the transition frequencies.
Sometimes, it is suggested that the frequency shifts give no
qualitatively new contribution to the dynamics of the reduced density matrix and, consequently, the imaginary parts
are disregarded. However, this statement is not correct. Actually the frequency shifts correspond to the reorganization
energies and the imaginary parts are responsible for the phonon equilibrium, as will be discussed in Sec. III. In addition,
when we discuss the Redfield equation without the secular
approximation,18–20 the disregard of the imaginary contribution results in noticeable changes to the dynamics. The popular Haken–Strobl type model,12,53 which uses a stochastic
Liouville equation to describe the coherent-incoherent transition of excitonic motion due to site-energy fluctuations, is
in this category.
The secular approximation has been widely employed in
applications of the Redfield equation. In the interaction picture, Eq. 共2.6兲 takes the form,
⳵ exc
˜␳ 共t兲 = 兺 R␮␯,␮⬘␯⬘ei共␻␮␯−␻␮⬘␯⬘兲t˜␳␮exc␯ 共t兲,
⬘ ⬘
⳵ t ␮␯
␮ ,␯
共2.11兲
⬘ ⬘
where ˜␳exc indicates the density matrix in the interaction picture. The secular approximation consists of neglecting the
tensor elements R␮␯,␮⬘␯⬘ satisfying 兩␻␮␯ − ␻␮⬘␯⬘兩 ⫽ 0. The origin and applicability of the approximation can be described
as follows. Equation 共2.11兲 can be solved formally as
˜␳␮exc␯ 共t兲 = ˜␳␮exc␯ 共0兲eR␮␯,␮␯t +
⫻
冕
t
0
兺
共␮⬘,␯⬘兲⫽共␮,␯兲
R ␮␯,␮⬘␯⬘
d␶eR␮␯,␮␯共t−␶兲ei共␻␮␯−␻␮⬘␯⬘兲␶˜␳␮⬘␯⬘共␶兲,
exc
共2.12兲
which clearly shows the mixture of the different kinds of
density matrices, e.g., a coherence transfer process 共␳␮exc⬘␯⬘
→ ␳␮exc␯ 兲,54,55 a conversion of coherence into population proexc
cess 共␳␮exc␯ → ␳␮␮
兲, and vice versa 共␳␮exc⬘␮⬘ → ␳␮exc␯ 兲.56 In Eq.
⬘ ⬘
共2.12兲, when the condition 兩␻␮␯ − ␻␮⬘␯⬘兩 Ⰷ 兩R␮␯,␮␯兩 holds and
the time variation of ˜␳␮exc␯ 共␶兲 is slower than 兩␻␮␯ − ␻␮⬘␯⬘兩−1,
⬘ ⬘
the integrand in the second term rapidly oscillates, and then
is averaged to zero in the time integration. Under the secular
approximation, the above mixed processes are omitted and
the populations and the coherences are decoupled from each
other. However, if more than one energy gap coincides in a
multilevel system, coherence transfer processes are still possible.
U=
冋
cos ␪ − sin ␪
sin ␪ cos ␪
册
,
共3.1兲
with the mixing angle,
1
2J12
.
␪ = arctan 0
2
共␧1 + ␭1兲 − 共␧02 + ␭2兲
共3.2兲
In the dimer system, the secular approximation leads to the
following equation of motion involving no coherence transfer:
⳵ exc
␳ = − i␻␮␯␳␮exc␯ + 共1 − ␦␮␯兲R␮␯,␮␯␳␮exc␯
⳵ t ␮␯
exc
+ ␦␮␯ 兺 R␮␮,␬␬␳␬␬
.
␬
共3.3兲
This is the so-called Bloch model, which derives its popularity from its straightforward interpretation and implementation; R␮␯,␮␯ and R␮␮,␬␬ are exciton coherence dephasing and
exciton population relaxation rates, respectively. The Bloch
expression can be derived directly from the Lindblad equation.
For simplicity, we assume that the phonon modes associated with one site are uncorrelated with those of another
site. Moreover, we assume that the phonon spectral distribution functions for the two monomers are equivalent. Then,
we have
Cmn共t兲 ⬅ ␦mnC共t兲
Jmn共␻兲 ⬅ ␦mnJ共␻兲.
共3.4兲
Several forms of J共␻兲 are employed in the literature, either
based on model assumptions or analyses of molecular simulations. In this paper, we employ the Drude–Lorentz density
共the overdamped Brownian oscillator model兲 for our model
calculations,57,58
J共␻兲 = 2␭
␻␥
,
␻ + ␥2
2
共3.5兲
where we set ␭1 = ␭2 ⬅ ␭.
We employ the site representation via the transformation, Eq. 共2.5兲. This representation is useful for considering
the link between the present Redfield framework and Förster
theory. From the secular-Redfield equation in Eq. 共3.3兲, we
can obtain the equations of motion for the site population
dynamics as
⳵ site
exc
共t兲 − ␭ · Rsec关␳exc共t兲兴,
␳ 共t兲 = − 2J12 Im ␳12
⳵ t 11
共3.6a兲
⳵ site
exc
␳ 共t兲 = + 2J12 Im ␳12
共t兲 + ␭ · Rsec关␳exc共t兲兴,
⳵ t 22
共3.6b兲
III. DISCUSSION: DIMER SYSTEM
In order to examine the validity of the secular approximation to EET processes, we discuss the simplest transfer
system, a dimer consisting of heterogeneous molecules
whose excited electronic state are denoted by 兩1典 and 兩2典,
respectively. For the system, the transformation matrix in Eq.
共2.4兲 can be given analytically as
where the second term of the right hand side in each equation
is the relaxation part originating only from the secular terms.
Although the explicit expression is not shown here, it is obvious that the relaxation part is proportional to the reorganization energy ␭ from Eqs. 共2.7兲–共2.9兲 and 共3.5兲. The important point to notice is that Eq. 共3.6兲 shows that the intersite
transfer rate increases without bound as the reorganization
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J. Chem. Phys. 130, 234110 共2009兲
Adequacy of the Redfield equation
FIG. 1. 共Color online兲 Intersite energy transfer rates from 兩1典 to 兩2典, k2←1, as
a function of the reorganization energy, ␭, predicted by the full-Redfield
equation 共open circles兲, the secular-Redfield equation 共open squares兲, and
the nonsecular-Redfield equation without the imaginary parts of the relaxation tensors 共open triangles兲. As a benchmark for a strong reorganization
energy region, the rate calculated from Förster theory is also shown 共solid
line兲. The other parameters are ␧01 − ␧02 = 100 cm−1, J12 = 20 cm−1, ␥
= 53 cm−1共␥−1 = 100 fs兲, and T = 300 K. For these parameters, the intersite
dynamics is dominantly incoherent for the entire region depicted and the
positivity problem does not occur.
energy increases. On the other hand, the full-Redfield equation for the dimer leads to the equations,
⳵ site
exc
␳ 共t兲 = − 2J12 Im ␳12
共t兲,
⳵ t 11
⳵ site
exc
␳ 共t兲 = + 2J12 Im ␳12
共t兲.
⳵ t 22
共3.7a兲
⳵ site
site
site
␳ 共t兲 = − k2←1␳11
共t兲 + k1←2␳22
共t兲,
⳵ t 11
共3.8a兲
⳵ site
site
site
␳ 共t兲 = + k2←1␳11
共t兲 − k1←2␳22
共t兲.
⳵ t 22
共3.8b兲
The transition rates k2←1 and k1←2 are determined as follows.
site
共0兲 = 1, we calculate the time
For the initial condition ␳11
site
evolution of ␳mm共t兲 by using the full- or secular-Redfield
equation. We used the fourth-order Runge–Kutta method as
the numerical propagation scheme. The time evolution of
site
␳mm
共t兲 does not strictly follow exponential decay kinetics
exc
共t兲 as can be
since it is affected by the exciton coherence ␳12
seen in Eqs. 共3.6兲 and 共3.7兲. Therefore, we determined the
intersite transfer rates k2←1 and k1←2 with the least-squares
routine. As a benchmark for discussion of strong reorganization energy cases, we also show the rate predicted by Förster
theory for the same parameters. The Förster rate is given by43
F
k2←1
=
2
J12
ប2
冕
⬁
−⬁
d␻
A2关␻兴F1关␻兴,
2␲
共3.9兲
where An关␻兴 and Fn关␻兴 are the absorption and fluorescence
spectra of site n, respectively, and are expressed as
A n关 ␻ 兴 =
冕
⬁
0
dtei␻te−i共␧n+␭n兲t/ប−gn共t兲 ,
共3.10a兲
0
F n关 ␻ 兴 =
冕
⬁
0
ⴱ
dtei␻te−i共␧n−␭n兲t/ប−gn共t兲 ,
共3.10b兲
0
with the line-shape function57 gn共t兲 defined by
gn共t兲 =
共3.7b兲
In Eq. 共3.7兲, the contributions of the secular terms and the
nonsecular terms cancel each other unlike in the case of the
secular approximation. For the Hamiltonians, Eqs. 共2.1兲, the
phonon-induced relaxation processes enter through coherence transfer or the conversion of coherence into population
and vice versa.
In Fig. 1, we show the intersite energy transfer rates
from site 1 to site 2, k2←1, as a function of the reorganization
energy, ␭, predicted by the full-Redfield equation and the
secular-Redfield equation. The other parameters are fixed to
be ␧01 − ␧02 = 100 cm−1, J12 = 20 cm−1, ␥ = 53 cm−1共␥−1
= 100 fs兲, and T = 300 K, which are typical for photosynthetic EET. These parameters do not cause the positivity
problem. The intersite dynamics calculated by using these
parameters is dominantly incoherent for the entire region depicted in Fig. 1. For ␭ ⬍ 5 cm−1, a weak coherent oscillation
is observed at a short time; however, the oscillation does not
affect the overall exponential-decay behavior. Therefore, we
can assume the overall dynamics can be adequately analyzed
by the following master equations:59
1
ប2
冕 冕
t
s
ds
0
ds⬘Cnn共s⬘兲.
共3.11兲
0
The rate predicted by the secular-Redfield equation 共the
open squares兲 increases monotonically over the whole range
of ␭. This behavior is easily understandable from Eq. 共3.6兲,
in which the relaxation part linearly depends on the reorganization energy ␭. However, it is completely unphysical.
When the reorganization energy is large compared to the
electronic coupling J12共=20 cm−1兲, the excitation will be
trapped in a deep well of the site potential associated with
environmental phonons. Thus, the intersite transfer rate
should decrease with increasing reorganization energy, as
shown by the solid line of Föster theory. Clearly the secular
approximation
overestimates
the
contribution
of
environment-assisted incoherent EET.
On the other hand, the rate predicted by the full-Redfield
equation 共the open circles兲 shows quite different behavior
from that of the secular-Redfield equation. For the small reorganization energy region, ␭ ⬍ J12, the rate increases in a
similar way to the secular-Redfield result. However, in the
intermediate and strong reorganization regions, ␭ ⲏ J12, the
rate becomes independent of ␭. This is because the Redfield
equation cannot describe the reorganization dynamics of the
phonon modes. The Markov approximation requires the phonon modes in the Redfield framework to relax to their equilibrium states instantaneously, that is, the phonons are always
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234110-6
J. Chem. Phys. 130, 234110 共2009兲
A. Ishizaki and G. R. Fleming
Arb. Unit
λ
ε0
-300 -200 -100
0
100 200 300 400 500
ω − ε 0 (cm−1 )
FIG. 2. 共Color online兲 Linear absorption spectrum for a monomer. In the left
figure, the lower parabola is an electronic ground state while the upper
parabola is an electronic excited state. The gray packets illustrate the phonon
states in each electronic state. In the right graph, the solid line is obtained
from the full-Redfield equation while the dashed line is calculated by disregarding the imaginary parts of the Redfield tensor. The parameters are set to
be ␭ = 100 cm−1, ␥ = 100 cm−1, and T = 77 K. The frequency shift between
the two spectra is identical to the reorganization energy, ␭ = 100 cm−1.
in equilibrium under the electron-phonon interaction. The
imaginary parts of the Redfield tensors are responsible for
this equilibrium. In order to clarify this, we show the linear
absorption spectrum for a monomer in Fig. 2. The solid line
was calculated from the full-Redfield equation, whereas the
dashed line was calculated by neglecting the imaginary part
of the Redfield tensor. The parameters were set to ␭
= 100 cm−1, ␥ = 100 cm−1, and T = 77 K. The frequency
modification induced by the imaginary part of the Redfield
tensor corresponds to the reorganization energy and the
phonons are in the potential minimum. Thus, in a strong
reorganization region of the Redfield framework, EET occurs
from the potential minimum of the donor state 兩1典 to that of
the acceptor state 兩2典, as shown in Fig. 3. Figure 3 illustrates
the EET mechanism of the full-Redfield equation and of
(a) Full-Redfield Equation
ε10
ε 20
(b) Förster Theory
ε10 − λ
ε 20 + λ
FIG. 3. Schematic of the EET mechanism of the full-Redfield equation 共a兲
and Förster theory 共b兲 in a region of large reorganization energy. In the
full-Redfield equation 共a兲, the phonons are always in equilibrium under the
electron-phonon interaction due to the Markov approximation. Thus, the
electronic de-excitation 共down-pointing arrow兲 of site 1 and excitation 共uppointing arrow兲 of site 2 occur involving only the equilibrium phonon state.
This process is independent of reorganization energy. On the other hand, in
the Förster theory 共b兲, the de-excitation and excitation occur from the equilibrium phonons of the initial state, 兩1典 = 兩␸1e典兩␸2g典, to the nonequilibrium
phonons of the final state, 兩2典 = 兩␸2e典兩␸1g典, in accordance to the Franck–
Condon principle. The nonequilibrium states are dependent on the magnitude of reorganization energy.
Förster theory in a region with large reorganization energy.
In the full-Redfield equation 关Fig. 3共a兲兴, electronic deexcitation 共down-pointing arrow兲 of site 1 and excitation
共up-pointing arrow兲 of site 2 involve only the equilibrium
phonon states. This process is independent of reorganization
energy. Consequently, the rate predicted by the full-Redfield
equation shows a ␭-independent region in Fig. 1. On the
other hand, in Förster theory 关Fig. 3共b兲兴, the electronic deexcitation and excitation occur from the equilibrium phonons
of the initial state, 兩1典 = 兩␸1e典兩␸2g典, to the nonequilibrium
phonons of the final state, 兩2典 = 兩␸2e典兩␸1g典, in accordance with
the Franck–Condon principle. After that, the phonons associated with each site relax to the respective equilibrium states
共site-dependent reorganization兲. The nonequilibrium states
are dependent on the magnitude of reorganization energy.
For a large reorganization region, the difference between
␧01 − ␭ and ␧02 + ␭ becomes large with increasing ␭ and then
the overlap of the emission spectrum of site 1 and absorption
spectrum of site 2 become very small in Eq. 共3.9兲. As the
result, the Föster rate in Fig. 1 decreases with increasing ␭.
This is the reason why the rate predicted by the full-Redfield
equation is qualitatively different from the rate given by
Förster theory.
Finally, we discuss the open triangles in Fig. 1. These
were calculated by disregarding the imaginary parts of the
Redfield tensor, but the secular approximation was not employed. This equation produces a turnover of the intersite
transfer rate similar to Förster theory. The imaginary parts of
relaxation terms describe dissipation processes of the phonon
bath, whereas the real parts describe the fluctuations.57,58
Then, as can be seen in Fig. 2, the phonons are frozen in the
nonequilibrium states impulsively generated through the
Franck–Condon transition when the imaginary parts are disregarded. The lack of the bath dissipative effect corresponds
to phenomenological stochastic approaches.57,58 For example, the Haken–Strobl model,53 which uses a stochastic
Liouville equation to describe the coherent-incoherent transition of excitonic motion due to site-energy fluctuations, can
produce a decreasing trend with increasing ␭ similar to the
open triangles.60 However, the form of the open triangles
arises as a result of an inadequate treatment of the phonon
relaxation: The phonon-induced fluctuations are overestimated, whereas the dissipation 共the reorganization process兲 is
neglected. Clearly this treatment will not give us realistic
information on the role of the protein environment in the
EET. This is obvious from Fig. 1, where the open triangles
greatly underestimate the transfer rate compared with the
Förster rate.
IV. CONCLUDING REMARKS
For the study of photosynthetic energy transfer, the use
of the Redfield equation is appropriate only when the reorganization energy of each chromophore is small compared to
the electronic coupling between the chromophores. In many
photosynthetic complexes, the reorganization energy and the
electronic coupling are similar in magnitude. Nonetheless,
the Redfield equation is one of the few viable methods of
exploring the role of the quantum coherence observed in
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234110-7
J. Chem. Phys. 130, 234110 共2009兲
Adequacy of the Redfield equation
light harvesting proteins, since neither Förster theory nor the
modified Redfield theory can treat quantum coherence.
In this paper, we discussed the adequacy and limitations
of the Redfield approach to photosynthetic EET processes by
considering a heterodimer system. The Redfield equation
without any additional approximations 共the full-Redfield
equation兲 predicts that the intersite transfer rate becomes independent of the reorganization energy when the reorganization energy is not small compared to the electronic coupling
strength. On the other hand, the Redfield equation with the
secular approximation 共the secular-Redfield equation兲, or the
corresponding Lindblad equation, predicts that the rate can
increase without limit with increasing reorganization energy.
This feature leads to the conclusion that a larger reorganization energy is advantageous for rapid EET, which implies
that incoherent hopping associated with a large reorganization energy is preferable to quantum coherent wavelike motion in photosynthetic EET. However, the monotonically increasing behavior is an anomaly caused by the secular
approximation. By comparing with the full-Redfield equation
or with Förster theory, it is obvious that the secular approximation overestimates the contribution of incoherent EET in
the intermediate and strong reorganization energy cases. We
also examined the Redfield equation without the imaginary
parts of the relaxation tensors. Disregarding the imaginary
parts of the relaxation terms corresponds to the neglect of
bath dissipation or reorganization. In this treatment, the phonon modes induce fluctuations in the EET system but they
are frozen in nonequilibrium states. Because the balance between the bath dissipation and fluctuations is disrupted, the
influence of fluctuations is overestimated, and this treatment
cannot give reliable information on the role of the protein
environment in energy transfer.
In summary none of the approaches we discussed is adequate to explore the interplay of bath dynamics and system
coherence in photosynthetic light harvesting. Indeed some of
the methods currently being used produce pathological behavior. On the ultrashort time scale relevant to photosynthetic EET, it seems unreasonable to expect a time scale
separation to exist between phonon relaxation and electronic
dynamics. For this we need a new theoretical framework that
allows the treatment of the quantum coherence and the sitedependent reorganization dynamics in a unified manner. The
theory suggested by Jang et al.15 is one such approach. An
approach based on a quantum master equation incorporating
the relaxation dynamics of the environment has been developed by the authors and is described in the companion
paper.61
ACKNOWLEDGMENTS
We thank Dr. Yuan-Chung Cheng for critical reading of
the manuscript and valuable comments. This work was supported by the Director, Office of Science, Office of Basic
Energy Sciences, U.S. Department of Energy under Contract
No. DE-AC02-05CH11231 and by the Chemical Sciences,
Geosciences and Biosciences Division, Office of Basic Energy Sciences, U.S. Department of Energy under Contract
No. DE-AC03-76SF000098. A.I. appreciates the support of
the JSPS Postdoctoral Fellowship for Research Abroad.
1
H. van Amerongen, L. Valkunas, and R. van Grondelle, Photosynthetic
Excitons 共World Scientific, Singapore, 2000兲.
R. E. Blankenship, Molecular Mechanisms of Photosynthesis 共World Scientific, London, 2002兲.
3
R. E. Fenna and B. W. Matthews, Nature 共London兲 258, 573 共1975兲.
4
Y.-F. Li, W. Zhou, R. E. Blankenship, and J. P. Allen, J. Mol. Biol. 271,
456 共1997兲.
5
G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mancǎl, Y.-C.
Cheng, R. E. Blankenship, and G. R. Fleming, Nature 共London兲 446, 782
共2007兲.
6
D. M. Jonas, Annu. Rev. Phys. Chem. 54, 425 共2003兲.
7
T. Brixner, T. Mancǎl, I. V. Stiopkin, and G. R. Fleming, J. Chem. Phys.
121, 4221 共2004兲.
8
T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G.
R. Fleming, Nature 共London兲 434, 625 共2005兲.
9
S. I. E. Vulto, M. A. de Baat, S. Neerken, F. R. Nowak, H. van Amerongen, J. Amesz, and T. J. Aartsma, J. Phys. Chem. B 103, 8153 共1999兲.
10
H. Lee, Y.-C. Cheng, and G. R. Fleming, Science 316, 1462 共2007兲.
11
S. Rackovsky and R. Silbey, Mol. Phys. 25, 61 共1973兲.
12
R. Silbey, Annu. Rev. Phys. Chem. 27, 203 共1976兲.
13
S. Jang, M. D. Newton, and R. J. Silbey, Phys. Rev. Lett. 92, 218301
共2004兲.
14
Y.-C. Cheng and R. J. Silbey, Phys. Rev. Lett. 96, 028103 共2006兲.
15
S. Jang, Y.-C. Cheng, D. R. Reichman, and J. D. Eaves, J. Chem. Phys.
129, 101104 共2008兲.
16
T. Förster, Ann. Phys. 437, 55 共1948兲.
17
T. Förster, in Modern Quantum Chemistry Part III, edited by O. Sinanoğlu 共Academic, New York, 1965兲, pp. 93–137.
18
V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems 共Wiley-VCH, New York, 2004兲.
19
A. G. Redfield, IBM J. Res. Dev. 1, 19 共1957兲.
20
A. G. Redfield, Adv. Magn. Reson. 1, 1 共1965兲.
21
P. N. Argyres and P. L. Kelley, Phys. Rev. 134, A98 共1964兲.
22
C. P. Slichter, Principles of Magnetic Resonance 共Springer-Verlag, Berlin, 1990兲.
23
J. Jeener, A. Vlassenbroek, and P. Broekaert, J. Chem. Phys. 103, 1309
共1995兲.
24
R. Friesner and R. Wertheimer, Proc. Natl. Acad. Sci. U.S.A. 79, 2138
共1982兲.
25
J. M. Jean, R. A. Friesner, and G. R. Fleming, J. Chem. Phys. 96, 5827
共1992兲.
26
W. T. Pollard and R. A. Friesner, J. Chem. Phys. 100, 5054 共1994兲.
27
O. Kühn, V. May, and M. Schreiber, J. Chem. Phys. 101, 10404 共1994兲.
28
D. Egorova, A. Kühl, and W. Domcke, Chem. Phys. 268, 105 共2001兲.
29
V. I. Novoderezhkin, A. G. Yakovlev, R. van Grondelle, and V. A. Shuvalov, J. Phys. Chem. B 108, 7445 共2004兲.
30
A. Kühl and W. Domcke, Chem. Phys. 259, 227 共2000兲.
31
A. Kühl and W. Domcke, J. Chem. Phys. 116, 263 共2002兲.
32
B. Balzer, S. Hahn, and G. Stock, Chem. Phys. Lett. 379, 351 共2003兲.
33
J. A. Leegwater, J. R. Durrant, and D. R. Klug, J. Phys. Chem. B 101,
7205 共1997兲.
34
O. Kühn and V. Sundström, J. Phys. Chem. B 101, 3432 共1997兲.
35
O. Kühn and V. Sundström, J. Chem. Phys. 107, 4154 共1997兲.
36
T. Renger, V. May, and O. Kühn, Phys. Rep. 343, 137 共2001兲.
37
R. van Grondelle and V. I. Novoderezhkin, Phys. Chem. Chem. Phys. 8,
793 共2006兲.
38
A. M. Walsh and R. D. Coalson, Chem. Phys. Lett. 198, 293 共1992兲.
39
A. Ishizaki and Y. Tanimura, Chem. Phys. 347, 185 共2008兲.
40
S. I. E. Vulto, M. A. de Baat, R. J. W. Louwe, H. P. Permentier, T. Neef,
M. Miller, H. van Amerongen, and T. J. Aartsma, J. Phys. Chem. B 102,
9577 共1998兲.
41
M. Cho, H. M. Vaswani, T. Brixner, J. Stenger, and G. R. Fleming, J.
Phys. Chem. B 109, 10542 共2005兲.
42
W. M. Zhang, T. Meier, V. Chernyak, and S. Mukamel, J. Chem. Phys.
108, 7763 共1998兲.
43
M. Yang and G. R. Fleming, Chem. Phys. 275, 355 共2002兲.
44
I. I. Abram and R. Silbey, J. Chem. Phys. 63, 2317 共1975兲.
45
R. Silbey and R. A. Harris, J. Chem. Phys. 80, 2615 共1984兲.
46
R. A. Harris and R. Silbey, J. Chem. Phys. 83, 1069 共1985兲.
47
D. R. Reichman and R. J. Silbey, J. Chem. Phys. 104, 1506 共1996兲.
2
This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.32.208.2 On: Wed, 06 Nov 2013 21:04:37
234110-8
48
J. Chem. Phys. 130, 234110 共2009兲
A. Ishizaki and G. R. Fleming
H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems
共Oxford University Press, New York, 2002兲.
49
A. Suárez, R. Silbey, and I. Oppenheim, J. Chem. Phys. 97, 5101 共1992兲.
50
Y.-C. Cheng and R. J. Silbey, J. Phys. Chem. B 109, 21399 共2005兲.
51
G. Lindblad, Commun. Math. Phys. 48, 119 共1976兲.
52
G. C. Wick, Phys. Rev. 80, 268 共1950兲.
53
H. Haken and G. Strobl, Z. Phys. 262, 135 共1973兲.
54
Y. Ohtsuki and Y. Fujimura, J. Chem. Phys. 91, 3903 共1989兲.
55
J. M. Jean and G. R. Fleming, J. Chem. Phys. 103, 2092 共1995兲.
O. Kühn, V. Sundstrom, and T. Pullerits, Chem. Phys. 275, 15 共2002兲.
S. Mukamel, Principles of Nonlinear Optical Spectroscopy 共Oxford University Press, New York, 1995兲.
58
Y. Tanimura, J. Phys. Soc. Jpn. 75, 082001 共2006兲.
59
R. Zwanzig, Nonequilibrium Statistical Mechanics 共Oxford University
Press, New York, 2001兲.
60
P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and A. Aspuru-Guzik,
New J. Phys. 11, 033003 共2009兲.
61
A. Ishizaki and G. R. Fleming, J. Chem. Phys. 130, 234111 共2009兲.
56
57
This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.32.208.2 On: Wed, 06 Nov 2013 21:04:37