Article pubs.acs.org/JPCB

Assignment of Exciton Domain in Light Harvesting Systems Based on the Variational Polaron Approach Yuta Fujihashi† and Akihiro Kimura* Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan ABSTRACT: In large light harvesting systems, not all pigments are coupled strongly. This is evidenced by the formation of delocalized states in certain domains of strongly coupled pigments. The threshold value for assigning pigments to domains is usually defined, and the pigment pairs in which the electronic coupling is greater than this value are included in the same domain to describe the dynamic localization effect implicitly. However, domain assignment by a single threshold value may make it difficult to include the possible localization of exciton states by temperature and the difference in the electronic excitation energy between pigments. In this study, we use the variational polaron approach for domain assignment to include such possible localization. To demonstrate the validity of domain assignment by the variational approach, we applied it to pigments in photosystem II (PSII) and compared the domain model constructed by the single threshold value. We showed that domain assignment by the variational approach could be used to determine the valid domain model in PSII without using the empirical threshold value at least at 77 K.



INTRODUCTION Excitation energy transfer (EET) is an important process in photosynthetic light harvesting systems and organic materials.1−5 A photon of sunlight is absorbed by one of the pigments in photosynthetic light harvesting systems. This is followed by EET at the reaction center, where charge separation is initiated.6 The quantum efficiency of the transfer process is near unity despite significant environment-induced fluctuations; however, the precise mechanism responsible for the high efficiency remains unknown. With the recent observation of long-lived oscillations in the two-dimensional electronic spectra of light harvesting systems,7−10 studies have aimed to theoretically describe the quantum dynamics of the light harvesting system.11−29 In general, accurate quantum dynamical calculations of large light harvesting systems such as photosystem I (PSI) and photosystem II (PSII) require massive computational resources. At present, averaging over the static disorder in the transition energy of pigments for simulating their spectra is impractical. In large light harvesting systems, not all pigments are coupled strongly. This is evidenced by the formation of delocalized states in certain domains of strongly coupled pigments. The Redfield theory30 or modified Redfield theory31,32 can be used to describe exciton relaxation within the domains. Furthermore, the generalized Förster theory33−36 can be used to describe the EET between different exciton domains. This combined theoretical approach was successfully applied to study EET in LHCII,37,38 CP29,39 PSI,40,41 and PSII core complexes42,43 and PSII supercomplexes.44 The threshold value Vc is usually defined to assign pigments to domains, and pigment pairs in which the electronic coupling is greater than Vc are included in the same domain. The threshold value is often assumed to be in the order of the environmental reorganization energy to describe dynamic localization effects45 implicitly. However, domain assignment by a single threshold value may make it difficult to include the possible localization of © 2015 American Chemical Society

exciton states induced by temperature and the difference in electronic excitation energy (energy gap) between pigments. In this study, we employ the variational polaron approach as an alternative approach for domain assignment to include the possible localization of exciton states by the temperature and energy gap in addition to the reorganization energy. Silbey and Harris originally developed the variational polaron technique to study the dynamics of the spin-boson model.46 In this formalism, they transformed the total Hamiltonian into a frame such that the system Hamiltonian is dressed by the environment. The dressed system Hamiltonian is optimized by the variational principle for free energy minimization based on the Bogoliubov inequality. McCutcheon et al. developed a theory of time convolutionless quantum master equation based on the variational polaron approach.23,47−49 This formalism can interpolate between the strong and the weak coupling regimes and describe the correct quantum dynamics over a wide range of parameters. The authors have recently proposed an alternative theory50,51 for the variational master equation by using the second Bogoliubov inequality.52 The remainder of this paper is organized as follows. The first section introduces the formalism of the variational polaron approach. The second section suggests an alternative approach for domain assignment to include the possible localization of exciton states by the temperature and energy gap. We present the phase diagram of the exciton state based on our approach. We apply our domain assignment to pigments in PSII, and we compare the domain model determined by domain assignment based on the single threshold value. Finally, we consider the population dynamics of the strongly coupled pigment pairs, and Received: May 11, 2015 Revised: June 15, 2015 Published: June 15, 2015 8349

DOI: 10.1021/acs.jpcb.5b04503 J. Phys. Chem. B 2015, 119, 8349−8356

Article

The Journal of Physical Chemistry B we discuss whether the domain assignment by the variational polaron approach is valid.

HI =



+

N

∑ En|n⟩⟨n| + ∑ Vmn|m⟩⟨n| n=1

(2)

m≠n

where |n⟩ denotes the state in which only the nth pigment is in its electronic excited state and another is in its electronic ground state. En is the electronic excitation energy of the nth pigment (site) and Vmn the electronic coupling between |m⟩ and |n⟩. Hsb is the exciton−phonon coupling Hamiltonian

∑ |n⟩⟨n| ∑ gk ,n(bk†,n + bk ,n) n=1

k

where ρB is the thermal equilibrium state ρB = e / (TrB{e−βHB}) at inverse temperature β = 1/kBT. If the variational parameter is set to f k = gk for all k, the variational transformation corresponds to the full polaron transformation.11,15,21,22 On the other hand, f k = 0 for all k corresponds to no transformation. In this case, HI corresponds to the exciton−phonon coupling, that is, Redfield-theory-type perturbation term. To find the optimal variational parameters over a wider range of parameters, we use the Bogoliubov inequality as follows

(3)

where (bk,n) is defined as a creation (annihilation) operator with the kth phonon mode at the nth pigment and gk,n the coupling strength of the state |n⟩ to the kth phonon mode. We assume that gk,n has the same strength as that of the excited states |n⟩. Then, we have gk,n = gk. Hb is the phonon Hamiltonian as Hb = ∑Nn=1 ∑k ωkb†k,nbk,n, where we set ℏ = 1, and ωk is the quantum energy of the kth phonon mode. We introduce the spectral density to treat the exciton−phonon coupling. The spectral density is defined as J(ω) = ∑k |gk|2δ(ω − ωk). By using the above definitions of the spectral densities, the environmental reorganization energy is expressed as

∫0





J(ω) ω

A ≤ AB = −

(4)



−1 ⎡ ⎛ βω ⎞⎤ ⎛ βη ⎞ 2V 2 F(ωk) = ⎢1 + R tanh⎜ ⎟ coth⎜ k ⎟⎥ ⎝ 2 ⎠ ⎝ 2 ⎠⎦ ηωk ⎣

Hsb =

⎡ B = exp⎢ − ⎣

(5)

∑ n = 1,2

|n⟩⟨n| ∑

gk (bk†, n

(6)

∑ n = 1,2

(En + R )|n⟩⟨n| +





⎛ βω ⎞⎤ J(ω) ⎟ F(ω) coth⎜ 2 ⎝ 2 ⎠⎥⎦ ω

(12)

RESULTS Definition for Domain Assignment Based on the Variational Polaron Approach. In this subsection, we discuss the property of the renormalized factor B in the dimer model system to obtain a definition for domain assignment that includes the possible localization of exciton states by the temperature and energy gap in addition to the reorganization energy. Equation 12 indicates that B is directly dependent on T and the spectral density, that is, the reorganization energy. In Figure 1, we plot B as a function of

∑ VR|n⟩⟨m| n≠m

∫0



We apply a variational transformation 46 to the total Hamiltonian as HT = eGHe−G, where the operator G is defined † as G = ∑n=1,2 |n⟩⟨n|∑k f k,nω−1 k (bk,n − bk,n), and f k,n is the variational parameter of the nth site. We assume that the variational parameters for the two sites are equivalent. Then, we have f k,1 = f k,2 = f k. We divide the transformed Hamiltonian into a nonperturbation term H0 and perturbation term HI as HT = H0 + HI. The details of H0 and HI are written as H0 =

(11)

4VR2)1/2

As discussed in the following section, this factor is employed as the criterion for our domain assignment. As F(ω) is also a function of B, the above equation must be solved selfconsistently.

+ bk , n)

k

(10)

where η = (ϵ + and ϵ = E1 − E2 is the energy gap between pigments. Introducing the spectral density allows eq 9 to be written in the integral form as 2

En|n⟩⟨n| + V(|1⟩⟨2| + |2⟩⟨1|)

n = 1,2

1 ln⟨e−βH0⟩H0 − ⟨HI⟩H0 β

where A is the true free energy of the total Hamiltonian H. By construction, ⟨HI⟩H0 in eq 10 is equal to zero. We try to make this bound as small as possible by minimizing AB with respect to f k. The minimization condition leads to f k = F(ωk)gk, with

which is a good measure of the strength of the exciton−phonon coupling. Variational Transformation. In order to consider domain assignment by the variational polaron approach, we introduce the formalism of the variational polaron approach for a dimer system (N = 2). The exciton Hamiltonian and the exciton− phonon coupling Hamiltonian are as follows: Hs =

(9) −βHB

b†k,n

λ=

(8)

⎡ f2 ⎛ βω ⎞⎤ B = TrB{BnmρB } = exp⎢ −∑ k 2 coth⎜ k ⎟⎥ ⎝ 2 ⎠⎥⎦ ⎢⎣ k ωk

N

Hsb =

∑ V (Bnm − B)|n⟩⟨m|

The excitation energies after transformation are shifted in comparison to the original frame by a factor R, defined as R = ∑k ω−1 k f k(f k − 2gk). The first term of HI is the exciton−phonon coupling with the renormalized coupling strength. The second term of HI describes the fluctuations of the phonon-dressed electronic coupling. Bnm is a phonon displacement operator † −1 † defined by Bnm = exp[∑k ω−1 k f k(bk,n − bk,n) − ∑k ωk f k(bk,m − bk,m)]. The phonon-dressed electronic coupling strength VR is defined as VR = VB, and the renormalized factor B is expressed as

(1)

N

k

n≠m

Hs is the exciton Hamiltonian, Hs =

|n⟩⟨n| ∑ (gk − fk )(bk†, n + bk , n)

n = 1,2

THE MODEL Hamiltonian. To study EET in light harvesting systems, we employ the following Frenkel exciton Hamiltonian including the singly excited state of N pigments: H = Hs + Hsb + Hb



(7) 8350

DOI: 10.1021/acs.jpcb.5b04503 J. Phys. Chem. B 2015, 119, 8349−8356

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Figure 2. Renormalized factor B as a function of V and ϵ at (a) T = 77 K and (b) T = 300 K. The Huang−Rhys factor is set to S = 0.5, that is, λ = 39.1 cm−1.

Figure 1. Renormalized factor B as a function of T and λ. The energy gap between pigments is set to ϵ = 100 cm−1. The constants of the electronic coupling are set to (a) V = 20 cm−1 and (b) V = 100 cm−1.

T and λ. The energy gap between the pigments is set to ϵ = E2 − E1 = 100 cm−1, and the constants of the electronic coupling are set to (a) V = 20 cm−1 and (b) V = 100 cm−1, which are typical parameters for pigments in a light harvesting system. We take the functional form of the spectral density, which was extracted from the fluorescence line-narrowing spectra of the B777 complex,53 as follows S J(ω) = s1 + s2

∑ i = 1,2

siω5 7! 2ωi4

e

same domain. Vc(ϵpair) is determined by solving the following condition ∂ 2B(V , ϵpair) ∂V 2

=0 V = Vc(ϵpair)

(14)

For a discontinuous transition, Vc(ϵpair) is def ined as the discontinuity of a f unction B(V, ϵpair). Phase Diagram in Exciton States. To demonstrate the validity of domain assignment based on the variational polaron approach, we consider the PSII core complex from cyanobacteria. The core complex is a dimer made of identical subunits, each containing 35 chlorophyll a and 2 pheophytin a. We consider the comparison between the criterion for domain assignment by our approach and that defined by Raszewski and Renger.42 Raszewski and Renger’s calculations of the electronic coupling between the pigments and the fittings of the site energies were based on the structure of the PSII core complex from Thermosynechococcus elongatus as determined by Loll et al.57 It was assumed that the spectral density form for all pigments is equivalent and has the form of eq 13. Therefore, the pigment pairs are only characterized by the electronic coupling and energy gap. A pigment pair assigned to a domain is represented by drawing the phase diagram of the exciton state as a function of V and ϵ. Figure 3 shows the phase diagram of the exciton states in the pigment pair. The red and green curves are the criteria given by the variational polaron approach at T = 77 K and T = 300 K, respectively. The dashed line indicates the criterion defined by Raszewski and Renger, in which the threshold value was set to Vc = 36 cm−1. The circles represent the absolute value of V and ϵ for all pigment pairs in PSII. The V value obtained by Raszewski and Renger for some pigment pairs in PSII is negative. As shown in eqs 11 and 12, f k and B depend only on V2 and are independent of the sign of V. f k and B also depend only on ϵ2 and are independent of the sign of ϵ. Therefore, we draw the phase diagram as a function of the absolute value of V and ϵ. Figure 3 shows that the criterion based on the variational polaron approach captures the localization of the exciton state induced by the energy gap and temperature. At T = 77 K, the curve of the domain criterion based on the variational polaron approach is similar to the criterion line obtained by Raszewski and Renger in the region of the small energy gap ϵ, and consequently, the number of delocalized pigment pairs judged by the domain criterion based on the variational polaron

−(ω / ωi)1/2

(13)

where s1 = 0.8, s2 = 0.5, ω1 = 0.069 meV, and ω2 = 0.24 meV. S is the Huang−Rhys factor. In Figure 1a, as λ and T increase, B decays to zero rapidly and monotonically. This indicates that B can capture the impact of the environmentally induced fluctuation on exciton delocalization. This behavior is also captured by the renormalized factor based on the full polaron transformation.25 In Figure 1b, at a fixed temperature, B changes abruptly from B ≫ 0 to B ≈ 0 as the reorganization energy increases. On the other hand, at a fixed reorganization energy, B decays to zero abruptly as the temperature increases. This discontinuity is an artifact caused by the variational transformation rather than the physical phase transition, and it has been predicted by Silbey and Harris.46 This discontinuity is caused by the existence of multiple solutions to the self-consistent equation in eq 12 over a certain range of the reorganization energy,54 and may indicate the breakdown of the ansatz due to insufficient degrees of freedom in the variational transformation.55,56 Additionally, eq 12 indicates that B is indirectly dependent on ϵ and V via the variational parameter f k. In Figure 2, we plot B as a function of V and ϵ at (a) T = 77 K and (b) T = 300 K. The Huang−Rhys factor is set to S = 0.5, that is, λ = 39.1 cm−1. In Figure 2a, as V decreases, B decays to zero rapidly and monotonically. B also decays to zero rapidly and monotonically as ϵ increases. This indicates that B can capture the localization of the exciton state induced by the energy gap between the pigments. Therefore, the renormalized factor B can potentially be a criterion for domain assignment that includes the possible localization of exciton states by the temperature, energy gap, and reorganization energy. As in Figure 1b, there is a discontinuity in B at T = 300 K in Figure 2b. From the above behavior of B, we define the criterion for domain assignment as follows: The pigment pairs with V = Vpair and ϵ = ϵpair for which the renormalized factor B(Vpair, ϵpair) is greater than a threshold value B(Vc(ϵpair), ϵpair) are included in the 8351

DOI: 10.1021/acs.jpcb.5b04503 J. Phys. Chem. B 2015, 119, 8349−8356

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Figure 3. Phase diagram of exciton states in a pigment pair. The dashed line is the criterion defined by Raszewski and Renger, for which the threshold value is set to Vc = 36 cm−1. The red and green curves are the criteria given by the variational polaron approach at T = 77 K and T = 300 K. The circles represent the absolute value of V and ϵ for all pigment pairs in PSII. The pigment pairs for which the electronic coupling is larger than 70 cm−1 and is assigned to the localized pigments by the domain assignment of the variational polaron approach at T = 300 K are marked by the numbers of pigments according to the nomenclature proposed by Loll et al.57

Figure 4. Domain models in the PSII core complex from Thermosynechococcus elongatus. All pigments with the same color are included in the same domain; localized pigments are shown in gray. (a) The domain model defined by Raszewski and Renger was redrawn for reference; in this model, the threshold value is set to Vc = 36 cm−1. Raszewski and Renger’s domain assignment is independent of the temperature, as the threshold value does not have temperature dependence. Parts b and c represent the domain model determined from the variational polaron approach at T = 77 K and T = 300 K, respectively.

approach is similar to that judged by Raszewski and Renger’s criterion. Raszewski and Renger’s criterion line does not change with increasing temperature because the threshold value Vc defined by them is independent of the temperature. On the other hand, in the criterion obtained by the variational polaron approach at T = 300 K, only three pigment pairs in PSII form the delocalized exciton. Assignment of Exciton Domain in PSII. On the basis of the phase diagram in Figure 3, we assign the pigments in PSII to the domains (Figure 4). We consider the comparison between the domain model by the variational polaron approach and Raszewski and Renger’s approach. The simulation of the time-resolved pump−probe spectra of subunits CP43 and CP47 of PSII based on the Raszewski and Renger’s domain model was in good agreement with the experimental data at T = 77 K.42 Therefore, we consider their domain model to be the optimized domain model at T = 77 K. In Raszewski and Renger’s domain model, there are 13 exciton domains in which six domains contain at least two pigments and the remaining domains are single-pigment domains, as shown in Table 1. Raszewski and Renger’s domain assignment is independent of the temperature, as the threshold value does not have temperature dependence. In domain assignment by the variational polaron approach, the pigments were assigned to 12 exciton domains in which five domains contain at least two pigments and the remaining domains are single-pigment domains at T = 77 K. The domain model constructed by the variational polaron approach at T = 77 K is in agreement with that by Raszewski and Renger except for one domain. However, the domain model obtained by the variational polaron approach differs significantly from that obtained by Raszewski and Renger at T = 300 K. At T = 300 K, the pigments were assigned to 34 exciton domains in which three domains contain at least two pigments and the remaining domains are single-pigment domains according to the domain assignment by the variational polaron approach.

Table 1. Number Ndomain of All Domains in PSII from T. elongatus and Thermosynechococcus vulcanus (Numbers in Parentheses) for the Domain Models Obtained by the Two Approachesa variational approach Ndomain Ndelocalized Nlocalized

single threshold approach

77 K

300 K

13 (15) 6 (6) 7 (9)

12 (15) 5 (5) 7 (10)

34 (36) 3 (1) 31 (35)

a

Ndelocalized is the number of domains that contains at least two pigments. Nlocalized is the number of domains that contains only one pigment.

Second, we consider the comparison between the domain model obtained by the variational polaron approach and by Shibata et al. Shibata et al.’s calculations of the electronic coupling and fittings of the site energies were based on the structure of the PSII core complex from T. vulcanus determined by Umena et al.58 For reference, Figure 5a redraws Shibata et al.’s domain model in which the threshold value is set to Vc = 30 cm−1. In Shibata et al.’s domain model, there are 15 exciton domains in which six domains contain at least two pigments and the remaining domains are single-pigment domains. Figure 5b and c show the domain model constructed by the variational polaron approach at T = 77 K and T = 300 K, respectively. In the domain assignment by the variational polaron approach, the pigments were assigned to 15 exciton domains in which five domains contain at least two pigments and the remaining domains are single-pigment domains at T = 77 K. At T = 300 K, the pigments were assigned to 36 exciton domains in which 8352

DOI: 10.1021/acs.jpcb.5b04503 J. Phys. Chem. B 2015, 119, 8349−8356

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The Journal of Physical Chemistry B

according to Loll et al.’s nomenclature in Figure 3. These pigment pairs are assigned to the localized pigments by domain assignment by the variational polaron approach at 300 K even if eq 15 is applied to the calculation of B instead of eq 13. Figure 6 shows the time evolution of the donor population in the four pigment pairs at T = 300 K. We selected the pigment

Figure 6. Time evolution of donor population in pigment pairs calculated by the 2CTNL equation. We selected the pigment with higher site energy as the donor. The calculated pigment pairs are as follows: Chls 12 and 13, Chls 21 and 23, Chls 41 and 43, and Chls 45 and 47, which are marked by the numbers of the pigments according to Loll et al.’s nomenclature in Figure 3.

Figure 5. Domain models in the PSII core complex from T. vulcanus. All pigments with the same color are included in the same domain; localized pigments are shown in gray. (a) The domain model defined by Shibata et al. was redrawn for reference, in which the threshold value is set to Vc = 30 cm−1. Shibata et al.’s domain assignment is independent of the temperature, as the threshold value does not have temperature dependence. Parts b and c represent the domain model determined from the variational polaron approach at T = 77 K and T = 300 K, respectively.

with the higher site energy as the donor. The values of the site energy and electronic coupling of the pigment pair are taken from Raszewski and Renger’s study. The donor populations of the pigment pairs calculated by the 2CTNL equation except Chls 41 and 43 clearly show the oscillation. This result suggests that these pigment pairs are not localized, and therefore, domain assignment by the variational polaron approach should overestimate the localization of the exciton states by the temperature at T = 300 K. Figure 2b shows the discontinuity in B at T = 300 K. As mentioned in the subsection of definition for domain assignment, the discontinuity in B may indicate the breakdown of the ansatz, and therefore, the artifact transition might not provide an accurate position of the sign that the exciton states are localized. This may be a cause that overestimates the localization of the exciton states at 300 K. To conclude whether domain assignment by the variational polaron approach can be applied to the high-temperature case, we need to calculate the time-resolved spectra of our domain model and compare them with the experimental results for PSII. In this study, we used the parameter sets determined by Raszewski and Renger and by Shibata et al. for constructing the domain model based on the variational polaron approach. The fittings of the site energies are based on the calculation of the linear spectra of the domain model constructed by their domain criterion. If we apply their parameter sets to the calculation of the linear spectra of the domain model by the variational polaron approach, the calculated linear spectra may deviate somewhat from the experimental data. We have to reperform the fitting of the site energies in PSII based on the calculation of the linear spectra of the domain model by the variational polaron approach. However, this fitting procedure may require searching for a larger parameter space than in Raszewski and Renger’s study because the domain model based on the variational polaron approach is dependent on the energy gap. The theory of the variational master equation can be used to describe the quantum dynamics of the multiple site system. As

only one domain contains at least two pigments and the remaining domains are single-pigment domains. The domain model constructed by the variational polaron approach at T = 77 K is in agreement with that obtained by Shibata et al., except for three pigments. This result indicates that domain assignment based on the variational polaron approach can be used to determine the adequate domain model regardless of the particular parameter set of PSII at T = 77 K. However, the domain model obtained by the variational polaron approach differs significantly from that obtained by Shibata et al. at T = 300 K.



DISCUSSION To investigate whether domain assignment by the variational polaron approach is valid at T = 300 K, we consider the population dynamics of the pigment pairs in which the electronic coupling is larger than 70 cm−1 and is assigned to the localized pigments by domain assignment by the variational polaron approach at T = 300 K. The time evolution of the population in the pigment pairs is calculated by the secondorder cumulant time-nonlocal (2CTNL) quantum master equation approach,13,59−61 which is limited to the specific form of the spectral density. Therefore, we take the overdamped Brownian oscillator form J(ω) =

2 λγω π ω2 + γ 2

(15)

The environmental reorganization energy and environmental relaxation time are set to λ = 39.1 cm−1 and γ−1 = 100 fs (γ = 53.0 cm−1), respectively. The calculated pigment pairs are as follows: Chls 12 and 13, Chls 21 and 23, Chls 41 and 43, and Chls 45 and 47, which are marked by the numbers of pigments 8353

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explained in the Results section, there are multiple solutions to the self-consistent equation in B around the discontinuity. For a multiple site system, more local minima of the free energy may emerge in the parameter space, and therefore, the variational parameters need to be carefully determined. Therefore, we considered the dimer system for obtaining the domain criterion by the variational polaron approach. There are two local minima of the free energy minima around the discontinuity of B for the dimer system. We can easily determine the global minima by comparing the values of the free energy of the two local minima. In a future study, it would be interesting to compare the quantum dynamics of the multiple sites calculated by the variational master equation and by the combined generalized Förster−Redfield approach with domain assignment by the variational polaron approach.



(Y.F.) Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585, Japan. Notes

The authors declare no competing financial interest.



ACKNOWLEDGMENTS This work was supported by the Program for Leading Graduate Schools “Integrative Graduate Education and Research in Green Natural Sciences” of MEXT, Japan.



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CONCLUSIONS In large light harvesting systems, not all pigments are coupled strongly, and therefore, the situation is well-described by the formation of delocalized states in certain domains of strongly coupled pigments. The excitation energy transfer between the different exciton domains is described by the generalized Förster theory. The threshold value for assigning pigments to domains is usually defined, and pigment pairs in which the electronic coupling is greater than a threshold value are included in the same domain to describe the dynamic localization effect implicitly. However, domain assignment by a single threshold value may make it difficult to include the possible localization of exciton states induced by the temperature and energy gap between pigments. In this study, we performed domain assignment by the variational polaron approach to include the possible localization of exciton states by the temperature and energy gap in addition to the reorganization energy. We explored the behavior of the renormalized factor B introduced by the variational transformation over a wide range of parameter regimes. We demonstrated that B can potentially be a criterion for domain assignment, which includes the possible localization of exciton states by the temperature, energy gap, and reorganization energy. We defined the criterion for domain assignment based on the property of B. To demonstrate the validity of domain assignment by the variational polaron approach, we applied our domain assignment approach to pigments in the photosystem II (PSII) core complex and compared the domain model determined by domain assignment based on a single threshold value. We demonstrated that domain assignment by the variational polaron approach could be used to determine the valid domain model in PSII at T = 77 K without using the empirical threshold value; however, it may overestimate the localization of the exciton states by the temperature at T = 300 K. The combined generalized Förster−Redfield approach was successfully applied to study large light harvesting systems, although the best threshold value for domain assignment needs to be determined empirically. We believe that domain assignment by the variational polaron approach is the first step in systematically determining the best threshold value, and it needs to be improved for determining the valid domain model at high temperature.



REFERENCES

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DOI: 10.1021/acs.jpcb.5b04503 J. Phys. Chem. B 2015, 119, 8349−8356

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Assignment of Exciton Domain in Light Harvesting Systems Based on the Variational Polaron Approach.

In large light harvesting systems, not all pigments are coupled strongly. This is evidenced by the formation of delocalized states in certain domains ...
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