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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 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/130/23?ver=pdfcov Published by the AIP Publishing 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 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 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 234110-2 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 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 234110-3 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បdn2 / 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បdnq 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 ⬁ dJmn共兲关nBE共兲 + 1兴e−it , 共2.3兲 −⬁ where nBE共兲 ⬅ 1 / 共eប − 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 dJnn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−1siteU = 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 = UnUnun, induces energy fluctuations of the th exciton, whereas the off-diagonal matrix element, 具e兩Vnun兩e典 = UnUnun, 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 = − iexc 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 ⬁ dteitCmn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, 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 234110-4 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 ,⬘⬘ deR,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 = − iexc + 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 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 234110-5 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←111 共t兲 + k1←222 共t兲, t 11 共3.8a兲 site site site 共t兲 = + k2←111 共t兲 − k1←222 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 dteite−i共n+n兲t/ប−gn共t兲 , 共3.10a兲 0 F n关 兴 = 冕 ⬁ 0 ⴱ dteite−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 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 234110-6 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 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 234110-7 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. 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