Computer Simulation of the Exciton Transfer in the Coupled Ring

10892

J. Phys. Chem. B 1999, 103, 10892-10909

Computer Simulation of the Exciton Transfer in the Coupled Ring Antenna Subunits of Bacteria Photosynthetic Systems Pavel Herˇ man* Department of Physics, UniVersity of Education, V. Nejedle´ ho 573, 50003 Hradec Kra´ loVe´ , Czech Republic

Ivan Barvı´k Institute of Physics, Charles UniVersity, Ke KarloVu 5, 12116 Prague, Czech Republic ReceiVed: March 12, 1999; In Final Form: July 19, 1999

A common accepted feature of the purple bacteria antenna systems seems to be a ring structure of their subunits LH2 and LH1. We concentrate our investigation on a delivery time of the energy through the subunits LH2 and LH1 of the antenna system to the reaction center. We are dealing with a model system consisting of one ring LH2 and one ring LH1. We have used structure data from Rhodopseudomonas acidophila. We investigate the exciton transfer inside and between LH2 and LH1 rings in the presence of the interaction with a bath. One can expect the incoherent (hopping) regime of the exciton transfer among the LH2 and LH1 rings. The exciton transfer inside the rings LH2 and LH1 is treated in a quasicoherent regime. One is therefore forced to deal with the time development of the full exciton density matrix of the exciton to take into account phase relations given by off-diagonal elements, completing in such a way information given by the site occupation probabilities Pm(t), diagonal elements of the exciton density matrix. Time dependence of the exciton density matrix is governed by dynamic equations which form an extension of the stochastic Liouville equation method with a Haken-Strobl-Reineker parametrization. Some known shortcomings of the original HSRSLE treatment are removed in our model: (a) we replace a classical stochastic field by a quantum field and (b) we introduce a new parameter A to provide a correct imbalance among the extended states at finite temperatures for long times due to energy relaxation. We discuss the influence of a local energy and a transfer integral heterogeneity, distance, and orientation dependence of the transfer rates between rings, relaxation, etc., on the energy delivery time.

1. Introduction The light-harvesting complexes that are present in photosynthetic systems perform two major functions: harvest (absorb) the incident light and transport excited state energy in the form of Frenkel excitons (eventually to the reaction center (RC)). Many investigations, both experimental and theoretical ones, have been directed toward understanding of the exciton transfer in antenna systems (AS) of purple bacteria photosynthetic units (PSU) and are reviewed in refs 1-4, and references therein. Although considerable progress has been made during recent years, our knowledge about the mechanism of energy transfer is still far from complete [see refs 5-12, and references therein]. 1.1. Motivation. Much progress has been made during the past years in structural knowledge of purple bacteria photosynthetic complexes. The photosynthetic membranes of purple bacteria show well-ordered two-dimensional arrays of photosynthetic subunits LH1 and at some times LH2. These lightharvesting ASs (LH1 and LH2) with different numbers of bacteriochlorophylls (BChls) have been described and isolated from various species of purple bacteria. The LH1 complex is closely associated with the RC and occurs in all bacteria, whereas the LH2 is not present in all species and the relative amount of LH2 with respect to LH1 and RC is dependent on growth conditions. * Corresponding author. E-mail: [email protected].

In 1995 the first high-resolution three-dimensional X-ray structure of a bacterial antenna complex LH2 from Rhodopseudomonas (Rps.) acidophila was published by Mc Dermott et al.13,14 It consists (see Figure 2 in ref 14) of nine identical units, protomers, each of which is formed by two proteins (R and β) with bound BChls. The nine R subunits are packed in an inner ring to form a hollow cylinder of radius 1.8 nm. The nine β subunits are arranged radially outward with respect to the R subunits to form another ring with a radius of 6.8 nm. The protein serves as a scaffold for the BChls and furthermore specifically influences the spectroscopic properties of the BChls by supplying a special environment for them. A ring of 18 BChl molecules is sandwiched between the R and β subunits, and a further 9 BChl molecules are positioned between the outer β subunits. A common accepted feature of the LH1 and LH2 ASs seems to be a circular (ring) structure (ring plane parallel to the x-yplane and perpendicular to the z-axis of a Cartesian coordinate system). In general, it can be concluded now that isolated LH2 of purple bacteria can have a different number of Rβ units. This conclusion could also be valid for LH1 complexes surrounding the reaction center. In the latter case, however, the fixed shape and size of the reaction center in the membrane likely constrains the number of possibilities. However, if the LH1 becomes separated from the reaction center, it may be possible that artificial ringlike structures with lower or higher numbers of units appear.

10.1021/jp9908855 CCC: $18.00 © 1999 American Chemical Society Published on Web 11/16/1999

Exciton Transfer in Antenna Systems In the past knowledge of the energy transfer was mainly derived from steady state spectroscopic experiments leading to several absorption bands at different wavelengths. Pigment molecules, characterized by an absorption wavelength, in the LH1 (B875) and LH2 (B850,B800) subunits of the ASs of the purple bacteria are close enough to interact each with other. This interaction is described by Coulomb and exchange interaction matrix elements between their molecular orbitals. This interaction gives rise to a modification of the electronic excitation energies and to a transfer of the electronic excitation energy between the molecules of LH1 and LH2 and between LH2 and LH1.15-36 Despite intensive study, the precise nature of the excited states of the pigments involved and the role of the protein moiety in governing the dynamics of the excited states are still under debate.1-12 At room temperature the solvent and protein environment fluctuate with characteristic time scales ranging from femtoseconds to nanoseconds. The dynamic aspects of the system are reflected in the line shapes of electronic transitions. To fully characterize the line shape of a transition and thereby the dynamics of the system, one needs to know not only the fluctuation amplitude (coupling strength) but also the time scale of each process involved. The observed line width reflects the combined influence of static disorder and exciton coupling to intermolecular, intramolecular, and solvent nuclear motions. The simplest approach is to decompose the line profile into homogeneous and inhomogeneous contributions. In more sophisticated models, each process is defined with its characteristic time scale as well as a coupling strength. A sharp division into static and dynamic disorder is therefore not possible. The observed optical absorption bands, e.g., at 800, 850 nm in LH2 and 875 nm in LH1 antenna subsystems of the purple bacteria are determined by the Qy optical transition dipole of the BChls. They form a circular structure with BChl Qy transition moments in a single plane parallel to the membrane. The nature of a big red shift of the optical absorption spectrum in comparison with that of the isolated BChl in a solution, which is characterized by the peak at 770 nm, is still unknown. The position of the principal optical absorption bands in LH1 and LH2 shifts also with the temperature and could be also changed by single and double mutations. Hydrogen bonds (between protein and BChl) and the exciton splitting (due to strong interaction between BChls) also play a role here. The extent of the exciton delocalization, which could be reduced by dynamic and static disorders, has been discussed. The principal questions read: Are the states contributing to the optical and transort properties of the LH1 and LH2 rings localized or delocalized? What are consequences on the exciton transport regime? Both the exciton-phonon coupling and the static disorder lead to localization of excitons. Many different and mutually nonconsistent quantities which should characterize this effect have been defined up to now. In ref 8 an attempt has been made to set up a correlation between some of them: the localization length,37 the superradiance coherence size, the exciton coherence size, the Debye-Waller factor size and the polaron size. Recent investigations27 indicate a time dependence of the localization length on the time scale of 2 ps, this means on the time interval during which the exciton transfer, e.g., inside LH2 and from LH2 to LH1 takes place. We would like to point out very clearly that often mentioned so-called coherence length has nothing to do with the coherence time used in the GME and SLE theories (next section) of the exciton transfer.

J. Phys. Chem. B, Vol. 103, No. 49, 1999 10893 TABLE 1: Transfer Integrals inside a Basic Dimer J12 and between the Dimers J23 together with a Parameter K ) J23/J12 Grondelle1 Pullerits2 Fleming6 Freiberg5 Novoderezhkin22,23 Sauer24 Schulten28 Leegwater32 Koolhaas36

J12

J23

κ

250 410 238 325 785 280 806 250 200

100 310 213 325 566 273 377 250 200

0.40 0.76 0.89 1 0.72 0.98 0.47 1 1

The extent of localization depends on relative values of the relevant parameters: the interaction between the BChls, the strength and the temperature dependence of the exciton-phonon coupling, and the strength and statistics of the static disorder. Several models of the interaction between the BChls have been applied. In the lowest approximation the interaction is described as the Coulombic one between molecular transition dipole moments (IDA, ideal dipole approximation).1,15 IDA has been used, e.g., by Pullerits,12 who found J12 ) 410 cm-1 and J23 ) 310 cm-1, and Novoderezhkin,22,23 who found J12 ) 785 cm-1 and J23 ) 566 cm-1. This approximation has been recently questioned by Fleming.6 Due to short distances between BChl molecules in and between dimers (9.1 A and 9.5 A, respectively), the Coulombic coupling within the intraprotomer BChl pair is, according to recent results in the framework of the TDC (transition density cubes) approximation, J12 ) 238 cm-1 and within the interprotomer BChl pair J23 ) 213 cm-1. It is seen that inclusion of higher orders of the multipole expansion diminishes the difference between J12 and J23, as follows from the comparison with the IDA J12 ) 367 cm-1 and J23 ) 284 cm-1 estimates. Monopole calculations by Sauer24 that are intermediate to TDC and IDA methods give J12 ) 280 cm-1 and J23 ) 273 cm-1. Fleming’s result6 does not support, in correspondence with Sauer’s former conclusions, the idea of the dimerization of the LH2 ring due to large difference between J12 and J23. Very often the average transfer integral J and the ring with whole 18-fold symmetry is therefore used in the investigation of the energy structure7,20,32 and exciton transfer. The role of a dielectric constant of the medium and possible exchange contributions should be examined very carefully in the future to approve the so-called “weak coupled dimer model” which is based on the large difference between J12 and J23 transfer integrals. Two excitonicaly interacting BChl a molecules should be responsible for the spectral properties of the B820 subunit, isolated from the LH1 of Rs. rubrum and Rb. sphaeroides, with little or no interaction between different dimers. By contrast, the recent investigation of the electronic excitations in Rs. molischianum by Schulten28-30 led to a big difference between the transfer integrals J12 and J23. Quantum chemical calculations of the ring reveal that the transfer integrals between more distant two neighboring places are larger than that between two closer sites in the effective model Hamiltonian. Transfer integrals inside a basic dimer J12 and between the dimers J23 together with a parameter κ ) J23/J12 are collected in Table 1. Without any disorder, the absorption spectrum, e.g., of the homogeneous ring of N molecules, with their transition dipole moments symmetrically oriented mostly in the ring plane with the nearest neighbor transfer integral J, would be characterized by a very weak peak at the energy E0 in the z-polarization and much stronger absorption at the double degenerate energy E1 in the x,y-polarization. The whole exciton energy structure

10894 J. Phys. Chem. B, Vol. 103, No. 49, 1999 multiplet En ) + 2J cos(2πn/N) is therefore not observable in the steady state optical experiments due to the symmetry. Novoderezhkin22,23 used the center of the energy spectrum of the LH2 subunit at 777 nm. He was able to explain the energetic difference between the observed optical transitions in the steady state absorption and in the hole-burning experiments 150-180 cm-1 by the theoretical energy difference E1 - E0. However, his conclusion requires very large transfer integrals. The whole red shift of the free BChla exciton optical absorption from 777 to 850 nm of the BChls in the LH2 should be explained as only due to the large transfer integrals J12 and J23. In other treatments12,28-30,36 a partial influence of the local environment on the site local energies 1,2 and their red shift from the free molecule value is admitted. Exciton dynamics has been probed by steady state optical experiments to determine the role of both the excitonic effects and the homogeneous and inhomogeneous broadening in the observed absorption and circular dichroism (CD) profiles and by various time domain optical measurements.4 These include fluorescence depolarization, hole burning, pump-probe, and photon echoes. The interpretation of these experiments requires a theory which incorporates exciton-phonon coupling and static disorder. Starting with the delocalized picture of the undisturbed exciton states (in which the influence of the exciton-phonon interaction is supposed not to be dominant), the influence of the exciton coupling to nuclei is incorporated via a relaxation calculated perturbatively in exciton-phonon coupling (not taking the detailed form of nuclear spectral densities into account). In the case of the stronger exciton-phonon interaction one should resort to the so-called dressed exciton picture, to reduce a strong-coupling system to a weak-coupling one. Two different conclusions have been drawn from the experimentally observed absorption and CD spectra about the influence of the static disorder [refs 22, 28-30, and 36, and references therein]. In the models by Novoderezhkin and Schulten relatively weak site energy inhomogeneity ∆ ∼ J/2 is required to obtain the correspondence with the optical experimental results. On the other hand, intepretation of stationary optical properties of the so-called B820 subunit, the strongly coupled dimer, led to the conclusion1 that a strong static disorder, besides the dynamic one, plays an important role. The molecules in the dimer B820 should have a head-tail arrangement of their dipole moments, putting most of the oscillator strength in the red-shifted transition. The high-energy component could only be observed by a small dip in the polarized excitation spectrum around 790 nm. The B820 absorption band is inhomogeneously broadened and in fact behaves like a single state system, as expected for a strongly coupled dimer. The strength of the static disorder ∆ is in this model larger in comparison with the transfer integral J12 ) 230 cm-1 inside the dimers and leads to the large monomer inhomogenous bandwidth fwhm ) 460 cm-1. In the model of the Grondelle group, the LH1 and LH2 rings are taken as a collection of the loosely coupled dimers. Due to the large difference between transfer integrals J12 and J23, the steady optical absorption of loosely coupled dimers should resemble the absorption of one dimer. Also, in the theory by Koolhaas, which should correlate the experimentally observed absorption and CD spectra with a model starting from the exciton extended states with a large J, the strong static disorder ∆ ≈ J has been required.36 Fluorescence, singlet-singlet annihilation, hole burning, and time-resolved absorption measurements, in which the signals were found to depend on the wavelength of the excitation within

Herˇman and Barvı´k the band, should further help to obtain reliable values of the strength of the static disorder.1-4 Pump-probe spectroscopy, e.g., provides a direct view into excitonic motions through the differential absorption of a probe pulse as a function of its frequency and the time delay with respect to a pump pulse. The frequency-dependent differential absorption typically contains a negative peak related to bleaching and to stimulated emission from the one-exciton band to the ground state (BL) and a positive peak which reflects excited state absorption from oneexciton to two-exciton states (ESA). Taking into account only the influence of the static disorder, to obtain the shift ∆Ω between these two features in correspondence with the experiment (≈200 cm-1), it had to be assumed that only a fraction of the monomers (2-4) contributes.2 It was not possible to reproduce the experimentally observed shifts by calculations which include the entire ring. Pronounced exciton localization due to the static disorder should take place. This modeling of the pump-probe spectra included disorder but neglected exciton phonon coupling. In ref 7 influence of the exciton-phonon interaction has also been taken into account. In the absence of specific information the simplest model, in which the entire exciton-phonon coupling is represented by a single parameter Γ, has been applied. The homogeneous contribution to the linear absorption line width results in fwhm ) 2Γ. Three models (I, Γ ) 180 cm-1, ∆ ) 0 cm-1; II, Γ ) 100 cm-1; ∆ ) 377 cm-1; III, Γ ) 50 cm-1, ∆ ) 527 cm-1) have been used for J12 ) 291 cm-1, J23 ) 273 cm-1 in calculation of the pump-probe and the linear absorption spectra at 4.2 K. The linear absorption line width is fwhm ) 360 cm-1 for model I, and fwhm ) 310 cm-1 for models II and III, which fall within the range of experimentally reported values. All three models have yielded about the same ratio of BL and ESA and the same ∆Ω. This demonstrates that the pump-probe experiment alone can be interpreted using very different models without static disorder and with strong static disorder. Estimates of the static local energy fluctuations ∆ between 150 and 500 cm-1 can be found in literature. Not only different estimates of the static disorder strength have been presented. Differences can be also found for correlation length of the static disorder. Schulten,28-30 Novoderezhkin,22,23 Barvı´k,41,42 and Knoester43,44 have supposed no correlation between the static disorder on different places. Grondelle with his collaborators40 and Small’s group45,46 accepted that BChls inside one dimer feel the same disorder. Freiberg, on the other hand, assumed5 that one should work with an ensemble of spectrally disordered excitons, each representing a separate B850 ring. This picture corresponds to former conclusions by Small,33,34 who evaluated his hole-burning data of a Rb. sphaeroides LH2 in terms of weakly disordered aggregate. The lowest exciton levels, characterized by narrow holes with full width at half-maximum (fwhm) equal to 3.2 cm-1 at 4.2 K are distributed over a band (denoted as B870) with fwhm at about 60 cm-1. The peak was found to be roughly 260 cm-1 below the B850 absorption maximum. According to more recent data,45,46 the B870 bandwidth is twice as large, and closer to the B850 band maximum (185 cm-1 below the maximum). Models using strong localization due to the strong static disorder as well as strong exciton-phonon interaction (polaron picture) have been recently applied8-10 in three-pulse femtosecond spectroscopy. At the end we can pick up main conclusions. There is at present a general understanding that the electronic excitations in the LH2 and LH1 rings are at least to some extent delocalized,5 even at room temperature. The important role of

Exciton Transfer in Antenna Systems the excitonic effect due to the rather large transfer integrals in J12 and between dimers J23 is now without question. A wide range of estimates for the transfer integrals J between 100 and 806 cm-1 exists. The resulting excitonic states are properties of a molecular group, not of a single molecule. The possibility of a (partial) coherent exciton transfer (strong-coupling limit in J) in the ring bacteria antenna subunits has been discussed in several papers.15-36 A consensus is lacking, however, when the exciton coherence size, the localization length, the energy level structure, the exciton transfer regime, and relaxation properties in the LH2 and LH1 rings are considered. The exciton properties are, generally, influenced by the interaction of the exciton with vibrations and by the static disorder. Both theoretical and experimental arguments have been raised, supporting a wide variety of possible exciton coherence sizes in the B850 aggregate at room temperature covering 2, 2-3, 3-4, 4 ( 2, 12, 16 ( 4, and the whole ring of 18 molecules [refs 3 and 5, and references therein]. We shall be dealing with the exciton transport properties. In the limit of the localized excited electronic states, the so-called weak-coupling limit, referring to the relative strength of the transfer integral J compared with vibrational interactions broadening the electronic transitions, is based on papers by Fo¨rster and Dexter.38,39 This picture for the description of the exciton transfer uses the Pauli master equation (PME) for the time development of the site occupation probabilities Pm(t) and in addition random walk computer modeling of the exciton transfer on homogeneous AS with a trap. An argument against the use of the pure incoherent Fo¨rster mechanism lies in the recent observation of coherent nuclear motions in the antenna complexes of the photosynthetic bacteria.2,47 The vibrational coherence is preserved up to 1 ps, the time of many single transfer steps. This means one of the main Fo¨rster assumptions that the vibronic relaxation occurs much faster than the excitation transfer is apparently not fulfilled. Pullerits has also not succeded in explaining the time-dependent optical experiments with the transfer rates determined according to the Fo¨rster prescription.2,12 In the limit of the extended excited electronic states, one is forced to use in this case more advanced methods, to obtain, e.g., the time dependence of the site occupation probabilities Pm(t). Applying the generalized master equations method (GME)48 to a system of N molecules, one has to work with a set of coupled N integrodifferential equations in which the kernel is formed from the memory functions. The memory functions are determined in a very complicated manner by the Hamiltonian with the exciton interaction with a bath and so on. In practice only simple models could be solved. To avoid complications connected with calculations of such memory functions, various types of convolution or convolutionless dynamic equations for the exciton density matrix Fmn(t) has been often used to describe the coherence effects in the exciton transfer. A disadvantage of these methods is a necessity to work with a very high number of coupled diferential equations (N × N). To keep the numerical effort tractable, various approximations have been used for generation of the dynamic equations. The simplest treatment is based on the stochastic model for the interaction of the exciton with a bath.49 Only several phenomenological parameters (γ0, j 1) are needed to build up the set of equations in the γ1, and γ Haken-Strobl-Reineker parametrization (HSR-SLE). To overcome the most important disadvantage of the HSR-SLE method, namely, the intrinsic high temperature limit, several attempts to incorporate, e.g., the energy relaxation have been made.50-52

J. Phys. Chem. B, Vol. 103, No. 49, 1999 10895 1.2. Goal of This Paper. We would like to follow in our simulations the intrinsic problem of photosynthesis:53 Recently experiments have yield estimates of ≈0.7 ps at 296 K and ≈1.2 ps at 77 K for the transfer time from the B800 ring to the B850 ring of the LH2 and of ≈3 ps at 296 K and ≈5 ps at 77 K for the transfer time from the B850 ring of the LH2 to the B875 ring of the LH1 in Rb. sphareoides.2,54 Subsequent transfer from the LH1 to the RC takes place. The exciton disappears due to the trapping on the RC within 35 ps [refs 1-3, and references therein]. This means that the energy is transferred extremely rapidly in the B850 and B875 rings, on account of their favorable spacing and orientation. Where the rings touch in the close-packed membrane, the energy can easily jump the short distance to an adjacent complex where it again spreads in the ring. In this way, the energy contained in a single photon is conducted in a very short time, and with minimal loss, from the point where it is absorbed to where it is needed. During the past years it has become clear that a single energy transfer step in the ring antenna subunits occurs on a time scale less than 0.01 fs. To model the energy transfer from the LH2 subunit through the LH1 subunit to the reaction center, we use the simplest system which consists only from one LH2 ring subunit with 18 B850 BChls, one LH1 subunit with 32 B875 BChls, and a RC with 2 B896 BChls. We have shown in the previous subsection that short distances between BChl molecules in the ring structure of different subunits13,14 of the purple bacteria antenna systems lead to a strong mutual interaction J. We accept therefore the model of extended states for the LH2 and the LH1, in which the exciton properties could be explained by the combined influence of the local energy heterogeneity, the transfer integral heterogeneity, the weak interaction with a bath, and the relatively weak static disorder. We shall use the delocalized picture of the exciton states disturbed by the influence of the dynamic and static disorders to investigate the time development of the occupation probabilities in the LH2 and LH1 antenna complexes and describe the energy transfer through the antenna subsystems LH2 and LH1 to the reaction center. Use of the extended exciton state picture does not exclude the exciton transfer inside the LH2 and LH1 rings. We apply the model of the exciton extended states to the loosely coupled ring antenna subunits LH2 and LH1 of Rps. acidophila. The exciton Hamiltonian He of the LH2 and LH1 subunits without interaction with the bath takes into account possible heterogeneity both the BChl local energies (δE ) 1 - 2) in the dimer and the transfer integrals inside and between the dimers (J12 * J23). We suppose41,42 that the strength of the possible local static disorder ∆ is relatively weak. This allows us to treat the exciton transfer inside the rings LH2 and LH1 in the quasicoherent regime.56 To describe the influence of the exciton-bath interaction, one is forced to deal with very complicated equations for the exciton density matrix to take into account phase relations given by off-diagonal density matrix elements, completing in such a way information given by diagonal elements, the site occupation probabilities Pm(t). The off-diagonal density matrix elements (after averaging over the bath) are dependent on the representation used in the course of the calculations to include properly the influence of the exciton interaction with the bath (phonons).50 Our starting point will be the dynamic equations of motion for the density matrix of the exciton under the influence of the bath50 which are proper (in the framework of the used approximation) for weak interaction (bare exciton picture) as for strong

10896 J. Phys. Chem. B, Vol. 103, No. 49, 1999 interaction (exciton polaron picture). We have accepted Small’s conclusion33,34,45,46 that the interaction of the exciton with vibrations in purple bacteria is not very strong. Therefore we should start with the bare exciton representation.50 In this case one can work with a simpler form of these dynamic equations, resembling in form the equations used in the HSR-SLE.50 Influence of the weak exciton-bath interaction on the exciton transfer inside the LH2 and LH1 rings is included by a phenomenological local parameter γ0 (known from standard HSR parametrization). The arrangement of the LH2 and LH1 rings leads55 to the 2.5 times longer distance between nearest neighbors of such two rings. One can expect, in contrast to the exciton transfer inside the rings, the weak coupling regime of the exciton transfer among such distant sites leading to the incoherent hopping transfer regime. Additional generalizations are included to remove known shortcomings of the original SLE treatment.51,57,58 The detailed balance condition between nonlocal parameters γmn and γnm which are connected with the incoherent (hopping) transfer rates allows us to work only with nonzero transition rates 2γLH1-LH2 accepting in such a way a low-temperature approximation in which the back transfer from LH1 to LH2 is negligible. Results of our investigation of the exciton transfer in the antenna complexes of the purple bacteria presented in coming sections below extend our former studies of the ring antenna structures16-19,35,59,60 as well as conclusions by Leegwater,32 Schulten,28-30 Novoderezhkin,22,23,61 and Ku¨hn27 obtained in the framework of the extended exciton state model. We have investigated the exciton transfer inside the ring antenna structures, also in the presence of a trap,16-19 and later inside35 and between incoherently coupled rings.59,60 We have investigated the influence of the transfer regimestransition from the pure incoherent to the pure coherentson the site occupation probabilities of the ring structures and of the RC. Here we use a more realistic Hamiltonian for the LH1 and LH2 subunits together with the incoherent transfer rates given by the geometry of the AS of the purple bacteria Rps. acidophila (presented by Papiz55). Leegwater investigated32 in the framework of the stochastic Liouville equation model the transition from the exciton coherent transfer regime to the incoherent one for one ring with one trap. We extend the Leegwater investigation in two directions. We use a more sophisticated Liouvillian which takes into account also the exciton energy relaxation, and we suppose that the exciton hops from the LH2 to the LH1 ring. Schulten with his collaborators developed28-30 on the basis of the model structure of the Rhodospirillum molischianum a quantum mechanical description of the entire light-harvesting process. On the basis of the quantum chemical calculations, an effective Hamiltonian has been established and employed to describe the LH2 f LH1 f RC cascade of the excitation transfer. Results suggested that the excitons described in the extended state model could be the key carriers. We extend the investigation by Schulten taking into account the modern quantum statistical description of the exciton transfer in the framework of the dynamic equations for the exciton density matrix. Novoderezhkin has been61 also dealing with the exciton transfer between loosely coupled ring antenna subunits. We extend his investigations of the exciton transfer in the incoherent transfer regime between the rings taking into account the possible partially coherent character of the exciton transfer inside the rings.

Herˇman and Barvı´k Ku¨hn investigated27 the dissipative dynamics of the excitons in the ring antenna system. The coupling to environmental degrees of freedom has been treated by employing Redfield relaxation theory.62 In a separate paper we give63 the theory of the exciton transfer and relaxation which shows shortcomings of the Redfield model. The remainder of this article is organized as follows. In section 2 we review basic theories of the exciton transfer we have used. In section 3 we calculate the exciton energy spectrum of the rings and the time development of the site occupation probabilities in the ring subunits LH2 and LH1. In section 4 we draw some conclusions. 2. Model 2.1. The Hamiltonian H, Eigenenergies, and Eigenstates. The most simple model Hamiltonian describing all important features of the energy transfer inside and between the LH1 and LH2 rings has three parts:

H ) He + Hph + He-ph

(1)

The pure coherent transfer of the bare exciton is described by

He )

Jrsa+ ∑r ra+r ar + ∑ r as r*s

(2)

where Jrs is a transfer integral between places r and s, r are the local energies, and a+ r , ar are exciton creation and annihilation operators. In the framework of the extended exciton states it is supposed both intradimer J12 and interdimer J23 transfer integrals are strong enough that the exciton states spanning over the whole rings are built. Inside one ring the pure exciton Hamiltonian He (2) could be diagonalized using B k representation with corresponding delocalized “Bloch” states and energies. Considering, e.g., only nearest neighbor transfer matrix elements (Jmn ) -J(δm,n+1 + δm,n-1)), the same local energies n and using Fourier transformed excitonic operators (Bloch representation)

ak )

∑n an eikn,

k)

2π

l, l ) 0, (1, ..., (N/2 (3)

N+1

the simplest exciton Hamiltonian in B k representation reads

He )

∑k Ek a+k ak

(4)

with the dispersion of the excitonic energies

Ek ) -2J cos k

(5)

A splitting of the local degenerate exciton energies in one ring, to a band of energies Ek, corresponding to the exciton eigenstates, is produced. The static disorder, inhomogeneity in the local energies r, could lead at the end to the exciton localization.37 The dynamic disorder represented by the interaction with a bath could be modeled48-50 as interaction of the exciton He-ph with vibrations (phonons) described by the Hamiltonian

Hph )

(

1

∑Bk pωBk 2 + b+Bk bBk

)

(6)

Exciton Transfer in Antenna Systems

J. Phys. Chem. B, Vol. 103, No. 49, 1999 10897

Here ωk are phonon frequencies; bBk+, and bBk are creation and annihilation operators for phonons. It is impossible to obtain eigenenergies and eigenfunctions for the Hamiltonian (1), even in the case of the simplest model unit, namely dimer. One should resort in any case to some approximations. Without any approximation, it would be not of such importance from which basis one starts his calculations of the physical properties. Only the necessity to accept some approximations makes the choice of the proper initial basis crucial. Historically, there have been used two opposite limit modelss bare excitons and exciton polarons. Neither bare exciton nor exciton polaron representation could generally serve as a proper description of the moving (after the initial relaxation) excitation in the molecular aggregate. One should take into account the exciton and phonon dispersion relations, strength of the mutual interaction, temperature, and so on, to try to describe, e.g., in an “optimal way” the stationary regime of the moving excitation. The so-called partial dressed approximation has been used in the past50 for a better description of the real particle which is moving, after the initial stage of the relaxation into its stationary state. However, let us point out very clearly. The initial state prepared in any experiment we should be dealing with is a highly nonequilibrium state. Such excitation should simultaneously polarize a lattice and move. Only after such an initial stage, in which the memory of the initial condition is lost, does the excitation reach the stationary partially dressed regime of its transfer. 2.2. Exciton Transfer Theories. Having defined the Hamiltonian of the problem, one should, generally, solve the equations of motion for the whole density matrix F, namely the Liouville equation

∂ ip F(t) ) LF ∂t

(7)

It is not a simple task, because one should find the time development of all matrix elements of the density matrixs diagonal and off-diagonalsin any representation which takes into account the exciton and phonon states. Nevertheless, this treatment is in many cases not necessary. Information which is used in further investigation is in many cases limited. For example, the site occupation probabilities Pm(t) of the exciton (the diagonal matrix elements in the site indices after the averaging over the bath (phonon) variables) are in many cases the most interesting quantities in theoretical investigations of the exciton transfer. 2.2.1. Pauli Master Equations. In the incoherent exciton transfer regime (assumed very often in the past1-3 for the exciton transfer in the antenna systems of the photosynthetic units) the site occupation probabilities Pm(t) fulfill the master equation:

∂

Pm(t) )

∂t

∑ [FmnPn(t) - FnmPm(t)],

m, n ) 1, 2, ... (8)

n(*m)

In the original formulation the transfer rate Fnm is given by the absorption and emission spectrum of the two fully relaxed molecules. As pointed out by Pullerits,12 the original Fo¨rster formula and formulas derived from it contain a number of uncertain parameters. The problem with the Fo¨rster radius and the natural lifetime of the pigments have been discussed. The refractive index occurs in the fourth power, and even the applicability of this macroscopic constant on a microscopic level is not obvious. Thermal relaxation on the molecules is supposed before the transfer of the exciton takes place. Application of

the PME method to describe the exciton transfer in the purple bacteria LH2 antenna systems with the Fo¨rster transition rates failed (e.g., ref 12). Much further experimental evidence has been recently collected [ref 5, and references therein], that the exciton states are in LH2 and LH1 partially delocalized. The correct treatment of the exciton transfer requires dealing with the time development of the full exciton density matrix under the influence of the dynamic and static disorder. 2.2.2. Time ConVolution GME Theories with Diagonalizing Projector. Many of the transport properties of a system can be determined only from a knowledge of the site occupation probabilities Pn(t). Thus, it is not always necessary to know all density matrix elements. In cases where the probabilities suffice, generalized master equations (GME) for the relevant probabilities may be obtained. The time development of the site occupation probabilities Pm(t) is given by

∂Pm(t)/∂t )

∑ ∫0 [wmn(t - τ) Pn(τ) - wnm(t - τ) Pm(τ)] dτ n*m t

(9)

The convolutional GME method48 is a natural framework for explaining a transition from the pure incoherent exciton transfer regime (limit of the small J) to the pure coherent one. The main problem within the GME formalism consists in determining a proper form of the memory functions (MFs) under the influence of the dynamic65 and static disorders. The transition from the incoherent regime (which allows use of the PME) to the quasicoherent one is well pronounced in the time dependence of the MFs, which reflect the interaction of the exciton with the bath and the static disorder. The MFs wmn(t) are very complicated functions of Hamiltonian matrix elements of the whole system with the static and dynamic disorders, temperature, etc. In most cases they cannot be found explicitly. They reflect the character of the exciton motionscoherent or incoherents mainly in their decay for long times. For time intervals much shorter than a typical reciprocal vibrational frequency, the molecular aggregate is practically rigid; i.e., no effect of vibrations suppressing the coherent character of the exciton motion is to be expected. For longer times, in the presence of the exciton-bath interaction, the time development of the site occupation probabilities, which should be independent of the basis used in our calculation,50 is determined by equations in which the MFs, which are dependent on the basis we have used (due to approximations made in reality in their determination), appear. As usual, perturbational expansions are the most common way of deducing physical information from general formulas with the aim of determining time decay of the noncoherent MFs wmn(t). Two schemes of expansion are at hand:65 expansion in terms of coupling to phonons or one in terms of resonance (hopping) integrals J. Another important question, namely, the choice of the Hamiltonian H0, also enters the perturbational expansions for the MFs. The exciton-vibration interaction in the LH2 should not be very strong.33,34,45,46 This estimate indicates that the most appropriate description of the exciton transfer regime in the LH2 should be based on the partial dressed exciton picture.50 The microscopical prescription for γ0, which mainly contributes to the damping of the MFs wqcoh mn (t) and in such a way determines the transfer rates, is given in ref 50. The general form of the MFs, mentioned above, reveals another important feature of the coherent transfer regime, namely

10898 J. Phys. Chem. B, Vol. 103, No. 49, 1999

Herˇman and Barvı´k

the nonlocal character of MFs.48 The nonzero MFs connect also the places m and n not directly connected by the nonzero transfer integrals Jmn. We have mentioned that it is not easy to calculate the MFs from the microscopic Hamiltonian. The so-called Kenkre-Knox prescription48 determines, as was also done in the Fo¨rster theory for transition rates, the MFs from the optical observables. The coherence time of the exciton has been very often determined in the past in such a way. However, automatic application of this prescription could be misleading.48 Nedbal proved66 that Kenkre-Knox’s relation holds only in the so-called “weak coupling” (in J) regime. The coherence time entering the quasicoherent part wqcoh mn (t) of the exciton polaron MF has nothing to do with the time given by Kenkre-Knox’s prescription. The time derived from Kenkre-Knox’s prescription gives the decay only of the incoherent part wincoh mn (t) of the polaron MFs. Transition to the incoherent regime of the exciton transfer (decreasing of the J) is accompanied by the rising role of the Fo¨rster incoherent channel wincoh mn (t). In the weak coupling limit Fmn ∞ incoh ) ∫0 wmn (t). Barvı´k and C ˇ a´pek65 guessed on the basis of the microscopic model that the exciton coherence time entering the quasicoherent part of the MF could be in the ASs of the PSU much longer than that one given by Kenkre and Knox.48 Only a few attempts to touch the problem of the influence of the static disorder in the framework of the GME can be found in the literature. One of the rare examples is the investigation by Kasner and Reineker67 of the influence of a colored noise on the diffusion of a quantum particle in infinite chain. We have recently succeeded in an analytical calculation of the MFs for the exciton transfer in dimer under a dichotomic colored noise.68,69 The investigation of the time development of the occupation probabilities in the GME method is a very complicated problem. To avoid the projection diagonalizing to the site representation, the dynamic equations for the diagonal and nondiagonal elements of the exciton density matrix are very often used. 2.2.3. Stochastic Theories. In the stochastic treatment of the exciton interaction with the dynamic and static disorders the influence of the disorder is described by a stochastic process with prescribed properties. In this approach the interaction part of the microscopic Hamiltonian is replaced by a stochastically time dependent model Hamiltonian. In the original version49 of the SLE method the bare exciton was influenced by a stochastic field

H1(t) )

∑ m,n

hmn(t)a+ m an

(10)

Grover and Silbey suggested64 to understand by the excitation in a molecular aggregates also a full dressed exciton (polaron) obtained from the microscopic Hamiltonian using a canonical transformation. The stochastic field acting on the dressed excitation has a different microscopic meaning in comparison with that one acting on the bare exciton.50 The stochastic field obtained from the microscopic Hamiltonian for the local and linear exciton-phonon interaction leads50 in the bare exciton representation mainly to the local energy fluctuations while the corresponding stochastic field acting on the exiton polaron leads mainly to the fluctuations of the renormalized transfer integrals. Because the exciton-phonon interaction in the ring subunits LH1 and LH2 of the bacteria should not be very strong,33,34,45,46 the Hamiltonian H1(t) would model the influence of the dynamic disorder mainly via fluctuations of the local exciton energies hm(t). Mean values and correlation functions are modeled by

〈hn(t)〉 ) 0 〈hn(t) hm(t)〉 ) δmn∆2 exp(-λ(t - τ))

(11)

Multitime correlation functions entering the equations of motion can be calculated from two time correlation functions (11) according to different rules. Dichotomic, Gaussian, and white noise statistics have been mostly applied in the past. The simplest description of the exciton dynamics in the framework of the stochastic theories is obtained in the so-called white noise limit ∆2/λ f 2γ0 (∆ f ∞, λ f ∞) in which the time correlation function obtains the δ-function shape. The stochastic Liouville equations for the bare exciton density matrix in the Haken-Strobl-Reineker parametrization (HSR-SLE) (for a review see ref 49) have been very often used in the past to describe the coupled coherent and incoherent regime of the exciton transfer in molecular aggregates using a few phenomenological parameters γ:

∂ ∂t

Fmn(t) ) -

i

([H0, F(t)])mn + p [2γmpFpp(t) - 2γpmFmm(t)] + δmn

∑p

- (1 - δmn)[2ΓmnFmn(t) -2γ j mnFnm] (12) 2Γmn )

∑r [γrm + γrn] ) 2Γnm

In application of the HSR-SLE mostly nearest neighbor parameters γ0 ) γnm for n ) m, γ1 ) γnm for n ) m ( 1, and γ j1 ) γ j nm for n ) m ( 1 have been used. The structure of the SLE corresponds to two-channel exciton transfer: (a) quasicoherent channel given by the first row in (12); (b) incoherent (hopping) channel given by the δ-function in the second row in (12), which describes the incoherent hops between different places with transfer rates 2γ1. Lost of coherence in the exciton transfer is determined as by the pure decay of the nondiagonal elements of the density matrix given by 2γ0 as by the hopping transfer integrals 2γ1. In the original HSR-SLE treatment of the weak local exciton-phonon interaction, the inverse of the dominant parameter 2γ0 determines the coherence time τc entering, e.g., the quasicoherent part of the MF (see the following subsection)

τc )

1 2γ0

(13)

The corresponding exciton optical line shape in the HSRSLE treatment

I(ω) )

d 1 π d2 + ω2

(14)

reveals a broadening given by fwhm parameter 2d ) 2γ0. 2.2.4. Generalizations of the HSR-SLE Method. The original HSR-SLE method49 represents the simplest stochastic treatment of the microscopic interaction Hamiltonian. Resorting to the stochastic treatment of the microscopic Hamiltonian, e.g., the usual application of the white noise is very restrictive. The original HSR-SLE method leads to improper long-time asymptotics of the occupation probabilities of the exciton extended states Pk(∞). There have been many attempts to determine more general equations using different representations of the exciton interacting with the bath and different ways of obtaining the



dynamic equations for the exciton density matrix from the whole Hamiltonian (1). The equations obtained in such a way could be after some further approximations and manipulations rewritten51,57,58 in the form which resembles the nonconvolutional SLE but brings several important improvements: (a) Results of the original HSR-SLE approximation are near the proper one only for very high temperatures (kT > W, where W represents the exciton bandwidth). Only in such a case can one accept that in a long time limit the local states are occupied with the same site occupation probabilities even in the case of the local energy heterogeneity. Taking into account the quantum character of the bath, the asymptotic form of the site occupation probabilities is in accordance with the equilibrium statistical mechanics. (b) Replacing the classical field V(t) leading to just induced transitions by a quantum field yielding both induced and spontaneous transitions breaks the symmetry relations γmn ) γnm which are being (in the lowest order in Jmn) replaced by the detailed balance conditions. (c) Later on, however, it was shown that still keeping the HSR-like parametrization (now, owing to finite-temperature effects, already with the possible left-right bath-induced hopping rate asymmetry 2γmn * 2γnm!) leads (irrespective of the success with the site-diagonal elements Fmm(t)) to persistent problems with the asymptotic form of diagonal matrix elements of Fkk(t) in the basis of the eigenstates of the exciton Hamiltonian. Thus it became clear that the famous HSR parametrization must be generalized by inclusion of further parameters. During a long-time investigation51 several different ways of obtaining the convolutional and convolutionless dynamic equations for the exciton density matrix have been applied by Cˇ a´pek under diverse conditions of the exciton properties and excitonbath interaction. The equations of motion obtained by C ˇ a´pek read

d dt

Fmn(t) )

iωmn,pq(t) Fpq(t) ∑ pq

(15)

Figure 1. Energy structure of LH2 subunit with J12 inside dimers and J23 between dimers for κ ) 0.89 (a) and κ ) 0.4 (b) with the local energy heterogeneity δE ) 0, 0.5, 1, 2 in units J12 ) 1.

other Apmn coefficients being omitted. Let us notice that Ammn plays the same role as the usual γ0 ≡ γmm coefficient of the HSR parametrization. As for the Anmn coefficient, it provides the bath-induced coupling of the site-off-diagonal to sitediagonal elements of Fmn(t). In a separate paper63 we touch more thoroughly on a relation between the C ˇ a´pek and other theories. We know50 that in the case of the weak local linear exciton-bath interaction the microscopically derived parameters entering the HSR-SLE γ1 ) 0 and γ0 * 0. This conclusion has been also approved by a direct microscopical derivation of the MFs,65 which have only the quasicoherent channel in this case

where

wHSR-SLE (t) ) 2 12

iωmn,pq(t) ) iΩmn,pq + iδΩmn,pq(t) iΩmn,pq )

A

p mn

)

i [δ δ ( - m) + Jqnδmp - Jmpδnq] p mp nq n iδΩmn,pq(t) ftωD.1 - δmpA qnm - δnqA pmn* (16)

ip N

∑k

2

m ωk (G-k

-

n G-k )

∑r

{

〈ν2|r〉 〈r|ν1〉 〈ν1|m〉 〈p|ν2〉

Grk

∑

Eν1 - Eν2 + pωk + iε

+

nB(pωk) Eν1 - Eν2 - pωk + iε

}

Here |ν〉 and Eν designate the corresponding eigenvalues and eigenvectors of He. The parametrization suggested by C ˇ a´pek is then as follows:

Apmn ) Re A pmn, Bpmn ) Im A pmn

2

(18)

0

Extension of the HSR-SLE method to include the energy relaxation performed by C ˇ a´pek does not modify the MF of the dimer ˇ a´pek (t) ) 2 wC12

(pJ) exp(-2γ t) 2

(19)

0

The Lindenberg52 and Redfield62 theories open a fictitious incoherent channel of the exciton transfer in the dimer

ν1,ν2

1 + nB(pωk)

(pJ) exp(-2γ t)

(17)

In the fast carrier regime Bpmn coefficients become negligible (no renormalization of Jmn appears) and can be omitted. As for the Apmn coefficients, only Ammn and Anmn should be kept, with

(pJ) exp(-Γt) J )δ(t) + 2 ( ) exp[-(γ + γ p 2

wLind 12 (t) ) Γ/2δ(t) + 2 wRedf 12 (t) ) (γ+- + γ-+

2

+-

-+)t]

(20)

Results of the investigations (e.g., ref 27) of the exciton transfer in LH2, which have used the Redfield approximation, could be questioned. We have therefore used in our investigation of the exciton transfer in the coupled antenna ring subunits the method suggested by C ˇ a´pek.51 3. Results and Discussion In our calculations and presentation on Figures 1-11, energy is taken in cm-1 (or is scaled by J12), p ) 1, time t is scaled to



Figure 2. Time dependences of the occupation probability P10(τ) for δE ) 0, γ19,10 ) 0.001, ..., 0.05, and (a) γ0 ) 0.1 (κ ) 0.89), (b) γ0 ) 0.5 (κ ) 0.89), (c) γ0 ) 0.1 (κ ) 0.4), and (d) γ0 ) 0.5 (κ ) 0.4).

τ ) tJ12/p, and parameters γ0 and A entering our dynamic equations for the exciton density matrix are scaled by J12/p. 3.1. Exciton Energy Structure. The position of the local site energies in LH2, LH1, and RC of the purple bacteria PSU is unclear. Several attempts have been performed to correlate the experimental absorption and CD spectra and the exciton energy structure in the framework of the extended exciton states model (e.g., refs 28-30, and 36). From the high-resolution picture of the LH2 complex of Rps. acidophila, it is seen that the β-bound pigment BChl 850 is physically bent due to interactions with the surrounding protein. The energy of the excited state should be lower than that of the unperturbed molecule. It is then natural, due to a bending of one of the BChls in the dimer, to admit energetic heterogeneity of the local energies 0 on positions R and β.35,36 We have calculated the energy structure (Figure 1) and the wave functions of the LH2 with 18 molecules of the B850 for two values of κ ) J23/J12 representing most of the estimates from Table 1. The ratio κ ) 0.4 represents estimates of Grondelle’s group1 while κ ) 0.89 14 was obtained from the structure data by application of the dipole-dipole approximation. We have calculated exciton energy structures for the local energy heterogeneity δE ) 1 - 2 ) 0, ..., 2. We have

continued in such a way our former investigation35 of the energy structure and the corresponding wave functions of dimer, tetramer, nonamer, and octadecamer. Small transfer integral heterogeneity (κ ) 0.89) does not change drastically the energy structure (Figure 1a) in comparison with the homogeneous case J12 ) J23. On the other hand, the large transfer integral heterogeneity (κ ) 0.4) leads (Figure 1b) to a pronounced splitting of the LH2 energy structure into two well-separated bands even in the case δE ) 0.28-30,70 We have used the simplest nearest neighbor approximation in the calculation of the ring energy spectrum. Inclusion of non nearest neighbor transfer integrals would modify the ring exciton energy spectrum.28-30,36 The width of the upper energy subband would be smaller than that of the lower energy one. We have learned from the literature that there are many uncertainties in input parameters of the exciton Hamiltonian H0, e.g., inaccuracy in the position of the local energies of the LH2 and LH1 subunits, the broad range of the transfer integrals J (Table 1), probable heterogeneity of both the local energies and transfer integrals in the basic dimer, and so on. Therefore it is hard at present to draw finite conclusions about the influence of dynamic and static disorders on the exciton transfer regime because they are determined by relations between such input



Figure 3. Time dependence of the occupation probability P10(τ) for δE ) 2, γ19,10 ) 0.001, ..., 0.05, and (a) γ0 ) 0.1 (κ ) 0.89), (b) γ0 ) 0.5 (κ ) 0.89), (c) γ0 ) 0.1 (κ ) 0.4), and (d) γ0 ) 0.5 (κ ) 0.4).

parameters. Computer simulations of the exciton transfer and relaxation for a broad range of input parameters are therefore desirable. 3.2. The GSLE Treatment of the Exciton Transfer. Results presented in this section for the time development of the site occupation probabilities Pm(t), i.e., the exciton transfer inside LH2 and between LH2 and LH1 represent a prolongation of our former investigations of the exciton transfer inside the ring structures (also in the presence of a trap)16-19 and between incoherently coupled rings.59,60 They form also an extension of the calculations by Leegwater32 of the exciton transfer inside one ring with a trap in the framework of the SLE model. Instead of being trapped,32 the exciton follows its transfer inside the second ring. We investigate (a) the influence of the local energy and transfer integral heterogeneity, (b) the bath-induced loss of the coherence in the exciton transfer, (c) the distance and orientation dependence of the transfer rates between molecules from different rings, etc., on the time development of the exciton site occupation probabilities Pn(t). Our primary aim is the investigation of the exciton delivery time inside the LH2, from the LH2 to the LH1, and in a subsequent paper from the antenna system to the RC. We investigate therefore the time development of the exciton

occupation probabilities Pm(t) in the LH2, LH1, and RC. We decided on the grounds of arguments presented in the previous section to work in the framework of the important generalization of the stochastic Liouville equation (GSLE) model.50,51,58 The real photosynthetic system consists of many LH1 ring subunits with the RCs inside in the lake of the LH2 ring subunits. To avoid a time-consuming work with very large density matrices in the framework of the GSLE method, we decided to treat the exciton transfer in the simplest model which consists of one LH2 and one LH1 only. We have used the arrangement of the BCHls in the LH2 and LH1 from the Rps. acidophila (presented by Papiz55). The distance between the nearest (touching) sites of LH2 and LH1 rings is 2.5 times longer in comparison with the nearest neighbor distance of two chromophores inside one such ring. One can therefore expect the incoherent (weak coupling in corresponding transfer integral given in the dipoledipole approximation) regime of the exciton transfer between such places. This enables us to (a) take into account the influence of the weak interaction with a bath on the bare exciton inside the rings LH1 and LH2 only through the (phenomenological) parameter γ0 and (b) to characterize the incoherent transfer between the chromophores from the LH2 and LH1 by the transition rates Frs given through the distance and orientation



Figure 4. Time dependences of the occupation probabilities P1(τ), P2(τ), and P18(τ) for δE ) 0 and (a) γ0 ) 0.1 (κ ) 0.89), (b) γ0 ) 0.5 (κ ) 0.89), (c) γ0 ) 0.1 (κ ) 0.4), and (d) γ0 ) 0.5 (κ ) 0.4).

dependent (phenomenological) parameters γrs (where r ∈ LH1 and s ∈ LH2). Transfer rates Frs between the molecules from the different rings are given by the distance and orientations of the chromophores (dipole-dipole approximation). Therefore γrs between LH2 and LH1 run with Rrs-6. We have admitted the heterogeneity of the local energies 1 and 2 and of the transfer integrals J12 and J23 in LH2 and LH1 ring subunits. We have chosen two different initial conditions: (a) the (hypothetical) site local excitation P1(0) ) 1 in the LH2 ring subunit on the place n ) 1 (the most distant one from the LH1 ring) to follow the exciton transfer through the LH2 and LH1 subunits to the RC and (b) the excitation of the state B k ) 1 to follow the exciton relaxation. 3.2.1. Exciton Transfer inside the LH2 Unit. We have investigated at first the influence of the local energy and the transfer integral heterogeneity on the exciton transfer inside one ring. The exciton transfer starts from the hypothetical initial condition P1(0) ) 1. The time development of the site occupation probability P10(t) in the ring LH2 is displayed in Figures 2 and 3 for the local energy heterogeneity δE ) 1 2 ) 0, 2 and for the transfer integral heterogeneity κ ) 0.89, 0.4 in the range of γ0 ) 0.1, 0.5. The transfer integral heterogeneity has a pronounced influence on the delivery time in the LH2 ring subunit from the most distant site m ) 1 (from LH1 subunit) to the closest site m ) 10. In Figure 4 short-time developments of the occupation probabilities P1(t), P2(t), and P18(t) show the influence of the

transfer integral heterogeneity (κ ) 0.89, 0.4) and the incoherence (γ0 ) 0.1, 0.5). 3.2.2. Exciton Transfer between the LH2 and LH1 Subunits. We extended our investigation of the exciton transfer between two coupled model rings59,60 to the investigation of the transfer between the LH2 and the LH1 subunits. The coupling between two rings LH2 and LH1 is taken in the weak coupling limit in the transfer integrals between sites from those two rings. The corresponding transfer rates are taken in the dipole-dipole approximation.55 Due to the different local energies in the LH2 and LH1 subunits, we shall suppose at low temperatures only one directional transfer from the LH2 to the LH1 subunit.57,58 We parametrize the transfer rates Frs(r ∈ LH1, s ∈ LH2) between molecules of both rings by the transfer rate 2γ19,10 between touching sites of two rings. The transfer rates Frs are modified due to the different positions and orientations of the corresponding dipoles. To follow possible coherence effects in the exciton transfer, we have obtained numerically the time development of diagonal elements of the exciton density matrix (12) and nondiagonal ones. We have solved numerically the set of coupled differential equations (12). We have used two initial conditions. The first one is the hypothetical localized excitation of the most distant LH2 molecule at t ) 0. The second one corresponds to the instantaneous “optical excitation” of the k ) 1 exciton state. The results for the “optical excitation” do not differ from that for the localized excitation with an exception of short times



Figure 5. Time dependences of the occupation probabilities PLH2(τ) and PLH1(τ) (κ ) 0.89, δE ) 0 and local initial excitation) for (a), (c), (e) γ0 ) 0.1 and (b), (d), (f) γ0 ) 0.5.

(till τ ≈ 50). We present therefore only results for the localized excitation. The diagonal matrix elements of the density matrix F(t) (the site occupation probabilities Pm(t) of all places in the LH2 and LH1) generally oscillate with time.

Results are displayed for two values of the transfer integral heterogeneity with the same site local energies 1 ) 2 (Figure 5 (κ ) 0.89) and Figure 6 (κ ) 0.4)) and with the heterogeneity of the site local energies δE ) 1 - 2 ) 2 (Figure 7 (κ )




0.89) and Figure 8 (κ ) 0.4)). We have considered the exciton transfer in the nearly coherent (γ0 ) 0.1J12) and quasicoherent (γ0 ) 0.5J12) transfer regimes.

However, reducing the relevant information to only the occupation probabilities over the rings, the time dependences of PLH1(t) and PLH2(t) become nonoscillating. Decay of the




occupation probability of the first ring PLH2(t) and rise of the occupation probability of the second ring PLH1(t) run with an effective transfer rate FLH1-LH2 between the two ring subunits.

The quasicoherent exciton transfer inside the ring subunits is hidden in the “incoherent” hopping transfer regime between two rings.




The effective transfer rate FLH1-LH2 generally decreases with the decreasing of the coherent regime; this means going from γ0 ) 0.1J12 to γ0 ) 0.5J12. The dependence of the effective transfer rate FLH1-LH2 on the transfer rate F19,10 ) 2γ19,10 is

displayed for the same site local energies 1 ) 2 (Figure 9a) and for the heterogeneity of the site local energies δE ) 1 2 ) 2 (Figure 9b). We characterize the exciton transfer from the LH2 to the LH1


Figure 9. Time dependences of the effective transfer rate FLH1-LH2 on the transfer rate F19,10 ) 2γ19,10 for (a) δE ) 0 and (b) δE ) 2.

by time τ(PLH1(t) ) 0.9) at which PLH1(t) reaches the value 0.9. Time τ(PLH1(t) ) 0.9) is displayed in Figure 10 as a function of the transfer rate 2γ19,10. 4. Conclusions During the past years experimental evidence has been collected that the exciton transfer in the ring structures of the AS from the PSU is partially coherent. Present knowledge of the input parameters (local energies, nearest neighbor and other transfer integrals, ...) which determine the energy structure (the width and the energy position of the exciton energy multiplets) is unsufficient. The influence of the static and dynamic disorders in the ring subunits of the ASs of the purple bacteria is still far from being completely understood.1-4 We have been interested in the quasicoherent regime of the exciton transfer. Similar words, e.g., coherence, coherence length, and so on, have been used in connection with different physical properties, and have different meanings. We have been dealing with the nonequilibrium transport and relaxation properties of the exciton. We understand the coherence in the exciton transfer in the sence of GME method as the inverse of the decay time of the corresponding MFs. By contrast, many timedependent properties of the ASs with the static disorder have been interpreted in terms of the exciton hopping transfer regime. The excitation of the system rapidly relaxes to a localized state. Excitation hops subsequently by random walk to neighboring sites, and eventually to the special pair of the RC. The localization is thought to be the result of energy disorder and (or) exciton-phonon coupling. Estimates of the size of this localized exciton state vary between a dimer and at least half of the LH2 ring. The size of this localized exciton state cannot be interpereted as the size of a spectroscopic unit. Koolhaas, e.g., has shown36 that a correct simulation of the CD spectrum requires a spectroscopic unit of at least half the size of the ring.


Figure 10. Dependence of time τ(PLH1 ) 0.9) in which the occupation probability PLH1 ) 0.9 on parameter γ19,10 for (a) local initial excitation and (b) optical initial excitation.

Our paper should contribute to a better understanding of the transport properties of the exciton in the energy transfer.We have been interested in the delivery time of the energy inside the LH2 and from the LH2 to the LH1. Time dependence of the occupation probabilities of the ring subunits LH2 and LH1, obtained from the occupation probabilities of involved chromophores, have been calculated in the framework of the dynamic equations for the exciton density matrix, which represent important generalization of the original phenomenological SLE method. Influence of the static and dynamic disorder has been modeled by the large local site energy heterogeneity35,70 and by including several parameters, which characterize the loss of the coherence in the exciton transfer, exciton incoherent transfer between the rings LH2 and LH1, and the relaxation from the exciton extended states with higher energy to the lower energy ones. Our numerical results presented in Figures 2-11 allow the determination of required input phenomenological parameters (γ0, γ19,10, ...) for the given Hamiltonian. In Figure 11 the ratio of the coherence time τc and the transition time τ12 from site 1 to site 2 for J12 ) 1 is displayed. Comparison with Figure 4 shows that the oscillatory character of the exciton transfer is preserved for times longer then τc. We have scaled all parameters in the used dynamic equations by the transfer integral J12. We have supposed J12 to be between 100 and 800 cm-1. This transfer integral J12 determines also the excitation transfer time between two BChls in the basic dimer in LH2 and LH1. The time interval used for presentation of our results in Figures 2, 3, and 5-8 reaches 14 ps for J12 ) 100 cm-1 and 3 ps for J12 ) 800 cm-1. Most of the estimates of the transfer integral J12 concentrate (see Table 1) into the interval J12 ) 300-450 cm-1. Experimental investigation revealed the following: The excitation of the LH2 should be transported within 3-5 ps from


Herˇman and Barvı´k remain, according to the SLE method, in the quasicoherent (oscillatory) regime only a few steps after the exciton creation at the place m ) 1 at t ) 0. The transfer integral heterogeneity influences strongly the departure of the exciton from the ring LH2. Occupation PLH2(t) of the ring LH2 in the case γ0 ) 0.5 decreases 2.5 times faster for κ ) 0.89 than for κ ) 0.4. In the near-coherent regime γ0 ) 0.1 the influence of the transfer integral heterogeneity is weaker (Figures 9 and 10). Similarly, the incoherence of the exciton transfer and the transfer integral heterogeneity influence the shift of PLH1(t) occupation probability maximum to higher times t (Figures 5-8) with decreasing γ19,10. The experimentally verified time of the exciton transfer from the LH2 to the LH1 subunit is about 3-5 ps. As shown in Figure 11a for the transfer integral J12 ) 300 cm-1, time interval 3-5 ps corresponds to times τ ≈ 150-300 and for the transfer integral J12 ) 450 cm-1 time interval 3-5 ps corresponds to times τ ≈ 250-450. In Figure 10 the necessary transfer rates 2γ19,10 could be determined: 2γ19,10 ≈ 0.01-0.04. A more precise (microscopic) description of the dynamic and static disorders in the ring subunits is desirable to reach a better understanding of the exciton transfer regime. Calculations in this direction will be presented in our forthcoming paper. Acknowledgment. This work was supported by Contract Nos. GAC ˇ R 202/98/0499 and GA UK 345/1998. While preparing this work I.B. experienced the kind hospitality of the Humboldt University Berlin (DAAD). Discussions with V. May and T. Renger are gratefully acknowledged. References and Notes

Figure 11. (a) Dependence of time t (ps) on the transfer integral J12 for the constant τ ) 100, 200, 300, 400, (b) dependence of the fwhm on the transfer integral J12, and (c) dependence of the coherence time τc (reduced to single transfer step time τ12) on the parameter of incoherence γ0.

the LH2 subunit to the LH1 subunit.2,54 Subsequent exciton transfer from the LH1 subunit to the RC takes place. Excitation disappears due to the trapping on the RC within 35 ps.1,2,54 Figure 11 could help us to specify the transfer integral J12 and the coherence time from the comparison of our simulations (Figures 2-8) with the experimental results in the steady and time-dependent optical measurements. In the HSR-SLE model the experimentally verified homogeneous broadening with the fwhm ) 200 cm-1 corresponds to the values of the parameter γ0 between 0.5J12 for J12 ) 200 cm-1 and 0.15J12 for J12 > 800 cm-1. We have therefore presented all our simulations for γ0 ) 0.1 and 0.5. The exciton transfer time between the two places in the dimer (Figure 4) is used for scaling of the coherence time for different transfer integral heterogeneity and various values of the parameter γ (Figure 11). We see that the exciton transfer would

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Computer Simulation of the Exciton Transfer in the Coupled Ring

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