Energy Transfer and Relaxation Dynamics in Light-Harvesting

3432

J. Phys. Chem. B 1997, 101, 3432-3440

Energy Transfer and Relaxation Dynamics in Light-Harvesting Antenna Complexes of Photosynthetic Bacteria O. Ku1 hn and V. Sundstro1 m* Department of Chemical Physics, Lund UniVersity, P.O. Box 124, S-22100 Lund, Sweden ReceiVed: October 31, 1996; In Final Form: February 12, 1997X

The dissipative dynamics of excitons in the outer antenna system of photosynthetic bacteria is investigated using an equation of motion approach. The coupling to environmental degrees of freedom is treated employing Redfield relaxation theory. It is shown that within the secular approximation the concept of essential excitonic states provides a convenient means for reducing the number of coupled differential equations to be solved. Further we derive the appropriate quenching matrix that accounts for the flow of excitation energy between weakly interacting pigment pools. Numerical simulations are presented to emphasize the influence of the various relaxation and dephasing mechanisms as well as the excitonic band structure on the energy transfer.

I. Introduction Ultrafast energy transfer (ET) in photosynthetic antennae has attracted considerable attention after high-resolution structural data for several pigment-protein complexes became available recently. The primary processes of bacterial photosynthesis are among the best characterized in this respect.1-3 Owing to the relatively simple structure of their light-harvesting antennae (LHA), these systems are well suited for exploring the relationship between microscopic structure and excited state dynamics. The core antenna (LH1) of bacteriochlorophyll a (BChl a) containing purple bacteria such as Rhodobacter (Rb.) sphaeroides consists of approximately 16 structural subunits in which two BChl a molecules are noncovalently attached to pairs of transmembrane polypeptides, i.e. (Rβ-BChl a2).4 These subunits are arranged in a ringlike structure suggested to surround the reaction center (see Figure 1). The outer antenna (LH2) exhibits the same ringlike organization of subunits. These, however, are made of 3 BChl a molecules per Rβ pair as found for Rhodopseudomonas (Rps.) acidophila in ref 5. Two BChl a molecules are part of the so-called inner ring, in which the distance between the centers of adjacent pigments is as close as 9 Å. The third BChl a contributes to an outer ring, where nearest neighbor distances are about 23 Å. The minimum separation between pigments belonging to the inner and outer ring is about 18 Å. The absorption spectrum between 750 and 900 nm is shaped by the Qy transitions of the pigments located in the different rings. According to the strongest absorption peaks, typical for a number of different bacteria, the rings are categorized as B875 (LH1), B850 (LH2, inner ring), and B800 (LH2, outer ring). A model of the whole photosynthetic unit is shown schematically in Figure 1. There is a rapidly growing number of experimental investigations of the ET dynamics in LHAs employing ultrafast coherent spectroscopic techniques (for a recent review see refs 1-3). Particular interest in this respect has been focused on the B800 intraband and the B800-B850 interband transfer. One-color pump-probe measurements at 77 K yielded intraband transfer times increasing from 0.3 to 0.6 ps when moving the excitation wavelength from the blue to the red of the main absorption peak of the B800 band.6 A similar wavelength dependence of the pump-probe signal decay time has been observed in ref 7. Twocolor measurements performed at 19 K yielded a 0.4 ps rise X

Abstract published in AdVance ACS Abstracts, April 1, 1997.

S1089-5647(96)03411-6 CCC: $14.00

Figure 1. Schematical view of the photosynthetic unit of purple bacteria as proposed in ref 3. The reaction center (RC) is surrounded by the core antenna (LH1). Both are embedded into a lattice consisting of peripheral antennae (LH2, only one shown). The LH2 is composed of an inner ring (B850) and an outer ring (B800).

time for the bleaching/stimulated emission signal recorded at 805 nm after excitation at 783 nm.8 The transfer times for B800-B850 ET observed in pump-probe experiments range from 0.7-0.8 ps at room temperature9,10 and 1.2 ps at 77 K7 to 1.6 ps at 19 K8 and 4 K.11 Fluorescence upconversion studies resulted in a 0.65 ps transfer time at room temperature.12 All mentioned room-temperature, 77 K, and 4 K results have been obtained for Rb. sphaeroides, whereas the 19 K measurement was performed on LH2 for Rps. acidophila. The most surprising result in this respect was the inability to describe two-color pump-probe and hole-burning data for Rps. acidophila simultaneously without invoking an additional transfer channel for B800 excitations.8 Small and co-workers8 proposed a model in which excitonic levels of the B850 band located at the upper band edge mediate the energy relaxation within the B800 band. This concept, but also the rather large number of vibrational Franck-Condon active BChl a modes which could serve as acceptor modes for the B800-B850 ET, has also been used to explain the flexibility of the interband transfer with respect to changes of the energy gap separating the B800 and B850 bands.13 As a matter of fact, excitonic calculations based on the available structural data recently suggested the excistence of an upper excitonic band belonging to the B850 pigment ring.14 In the present paper we address the issue of B800 intraband and B800-B850 interband ET from a theoretical point of view. The exciton motion in molecular aggregates is commonly described using the Frenkel Hamiltonian.15 The crucial point in exciton theory when applied to photosynthetic aggregates is the incorporation of the interaction between excitons and environmental degrees of freedom (DOF) such as phonons. The description of exciton dynamics using the stochastic Liouville © 1997 American Chemical Society

ET in Light-Harvesting Antenna Complexes equation with Haken-Strobl-Reineker parametrization has been widely applied in this connection.16 Here, the effect of the weak interaction with the phonon DOF is modeled as a Gaussian-Markovian process. This treatment is strictly valid only in the case of infinite temperatures but capable of interpolating between the coherent and the incoherent limit of exciton motion. C ˇ a´pek and co-workers generalized the HakenStrobl model to account for finite temperature effects using a convolutionless form of the stochastic Liouville equation for the reduced density operator.17 The application to the exciton population dynamics in photosynthetic units was recently presented in ref 18. On the other hand, if it comes to accounting for the influence of weak interactions with an environment having arbitrary temperature, multilevel Redfield theory19 seems to be most appropriate for treating the exciton dynamics in LHAs. However, it has been applied so far to the formal description of molecular dimers only.20 Much effort has also been invested in the study of nonlinear optical properties of linear chainlike aggregates of coupled twolevel molecules (for a recent review see ref 21). To investigate the nonlinear optical response of aggregates having arbitrary geometry, it is quite convenient to start with the Heisenberg equations of motion for relevant excitonic variables in real space.22 This leads to a Green’s function formulation that has been given recently for aggregates composed of coupled two23 and three-level24 molecules. A more general framework for incorporating exciton-phonon coupling based on equations of motion for generating functions has also been proposed.25 A number of theoretical contributions have been devoted to the description of situations where the weak exciton-phonon coupling limit does not apply. The Fo¨rster theory26 is the approach that is widely used in this respect. It has recently been combined with numerical simulations of the Pauli master equation to describe the energy migration in antenna complexes.6,27 The Fo¨rster description is based on two approximations: The transfer is incoherent and proceeds from vibrationally relaxed states. In a number of papers it has been shown how to remove the latter restriction and treat the so-called hot transfer.28,29 Approaches capable of incorporating coherent exciton-vibrational motion have been proposed quite recently.30-33 Numerical applications, however, were only presented for ET in heterodimers so far.30-32 Here we will focus attention on the weak coupling limit and apply Redfield theory to model dissipative ET dynamics and its relationship to the excitonic band structure in the LH2 system. In section II we outline the equations of motion approach and discuss the approximations to the Redfield relaxation matrix. Further, the contributions to the equations of motion due to excitation energy quenching are derived. Numerical simulations in section III are presented for the B800 band separately as well as for the whole LH2 system. The paper is finally summarized in section IV. II. Theory A. Hamiltonian and Equations of Motion. In the following we take the point of view that the electronic DOF are coupled to an environment that includes, for instance, nuclear DOF of the BChl a molecules and the protein, as well as DOF of the ambient solvent. This environment comprises dynamics taking place on a multitude of time scales. There are some very slow DOF giving rise to a distribution of monomer transition energies and dipole-dipole interactions. They will be taken into account by numerical averaging over an ensemble of LHAs. On the other hand, the interaction of each member of the ensemble with fast environmental DOF leads to rapid fluctuations of the respective system energies.

J. Phys. Chem. B, Vol. 101, No. 17, 1997 3433 The Hamiltonian is taken as

Htot(t) ) Hex + δHex + Henv + Hf(t)

(2.1)

where the excitonic part

Hex )

)

hmn|m〉〈n| ∑ mn (Emδmn + Jmn)|m〉〈n| ∑ mn

(2.2)

and the semiclassical coupling to the external field, b (t),

b(t) Hf(t) ) -

b µ m(|m〉〈g| + c.c.) ∑ m

(2.3)

are independent of the fast environmental DOF. Instead of the flip operators one could introduce exciton creation (annihilation) operators, B†m ) |m〉〈g| (Bm ) |g〉〈m|), which obey Bose commutation relations in the present context. Further, Em and Jmn denote the monomer transition energy and the dipole-dipole interaction energies, respectively. In eq 2.3 b µm is the dipole matrix element for S0 f S1 transition at site m. While Henv is not further specified, the interaction between system and environment is taken as

δHex )

δhmn|m〉〈n| ∑ mn

(2.4)

For convenience we assume 〈δHex〉env ) 0, where 〈...〉env ) Trenv(Fenv...) denotes the average with respect to the environmental DOF, with Fenv being their equilibrium density operator. In the following we will use the representation in terms of the single-exciton eigenstates defined by Hex|R〉 ) ER|R〉, with

∑n cn,R|n〉

|R〉 )

(2.5)

We are interested in the time evolution of the one-exciton density matrix averaged with respect to the environmental DOF, i.e. FRβ(t) ) 〈R|Fred(t)|β〉, with Fred being the reduced density operator for the electronic DOF. For this quantity equations of motion can be derived that read in second order with respect to the external field as well as the fluctuations

d FRβ ) -iωRβFRβ -

dt

RRβ,R′β′FR′β′ ∑ R′β′ i b µ βb µ Rb *(t)FRg + b (t)Fgβ (2.6) p p i

Here, the relaxation matrix, which is frequently termed the Redfield tensor, is given by19,20

RRβ,R′β′ ) -

1

∫∞dt[〈δhβ′βδhRR′(t)〉enve-iω

2 0

p

β′βt

+

∑γ 〈δhRγ(t)δhγR′〉enve-iω δR′R∑〈δhβ′γδhγβ(t)〉enve-iω t] γ

〈δhβ′β(t)δhRR′〉enve-iωRR′t - δβ′β

β′γ

γR′t

(2.7)

In eq 2.6 we further introduced the matrix elements of the transition dipole operator with respect to the eigenstates (eq 2.5),

b µR )

∑n bµncn,R

(2.8)

3434 J. Phys. Chem. B, Vol. 101, No. 17, 1997

Ku¨hn and Sundstro¨m

TABLE 1: Parameters for the Different Configurations Used in the Numerical Calculationsa scaling max(|Jmn|) λupper λ800 λ850 case factor [cm-1] [nm] [nm] [nm] A B C

1 0.75 0.5

400 300 200

760 780 800

800 800 800

L(790)

L(800)

800 0.48 (0.41) 0.28 (0.30) 811 0.34 (0.36) 0.33 (0.33) 825 0.86 (0.79) 0.40 (0.41)

numerical simulations of absorption difference spectra for different sizes of connected segments of the B850 ring with experimental results and obtained a coherence size of Ncoh ) 4 ( 2. Another approach is based on the inverse participation ratio defined as

L(E) )

a

The energy gap between B800 and B850 is kept constant at about 720 cm-1, which corresponds to the low temperature value for Rb. sphaeroides.13 This is done by adjusting the monomer transition energies of the B850 pigments. The resulting position of the upper band edge is denoted λupper. The participation ratios are given for the LH2 using σ800 ) 75 cm-1 and σ850 ) max(|Jmn|/2) (σ850 ) max(|Jmn|/ 1.5)).

The dynamics of FRβ is coupled to FRg, which is of first order in the external field and obeys the equation

i FRg ) - ERFRg dt p d

i

µ Rb RRg,R′gFR′g + b (t) ∑ p R′

(2.9)

We will focus our analysis of the ET on the population dynamics as obtained from FRR(t). To describe nonlinear optical experiments, e.g. pump-probe or photon echo measurements, the set of eqs 2.6 and 2.9 has to be supplemented by third-order variables that bring into play the two-exciton states.22 We will study a model system that resembles the LH2 of purple bacteria. The geometry is taken from the high-resolution structural data obtained for Rps. acidophila,5 which are in general believed to be representative for other bacterial LH2s, too. Given the knowledge of the mutal distances between the BChl a molecules as well as the orientations of the Qy transition dipoles, one can calculate the dipole-dipole interaction energies, Jmn, which enter eq 2.2. There are, however, several points leading to uncertainties in the numerical values of Jmn. First, the dielectric constant of the medium in which the pigments are embedded is not known. Thus, different values for the effective monomeric dipole strengths have been used in previous calculations. Second, the compact structure of the B850 in particular suggests that more elaborate methods going beyond the point-dipole approximation should be used for calculating the interaction energies. Consequently, the values reported for the strongest interaction within the B850 ring range from 450 cm-1 35 down to 290 cm-1.14 One of the most significant differences introduced by these ambiguities is in the position of the upper edge of the B850 band relative to the B800 absorption maximum. The predicted values range from 760 nm35 to about 785 nm.14 As discussed in the Introduction, this is likely to influence the B800-B850 transfer. In the following we will use the interaction energies of ref 35 but scale them by a common factor such that the upper band edge of the B850 band is gradually moved from 760 to 800 nm. The monomeric transition energies of the B850 pigments are adjusted to keep the energy gap between B800 and B850 main absorption bands constant. The chosen parameters are summarized in Table 1. One of the central questions raised in the context of exciton dynamics in LHAs concerns the size of the excitation coherence domain. The strong dipole-dipole coupling between the inner ring pigments suggests the possibility of rather delocalized exciton states; that is, the response becomes collective in nature and the ET is partly coherent. On the other hand, the interaction with the environmental DOF tends to localize the exciton. This localization is believed to be rather effective in the weakly coupled outer ring. There are several ways of approaching the determination of the coherence domain size. Pullerits et al.35 compared their

∑Rδ(E - ER)(∑nc4n,R)〉disorder

1〈 N

D(E)

(2.10)

with

∑Rδ(E - ER)〉disorder

1 D(E) ) 〈 N

(2.11)

being the aggregate density of states (DOS). Equation 2.10 has been used by Fleming and co-workers12 to estimate Ncoh ) 5 for the B850 pigments. These considerations do not involve the influence of interactions with fast environmental DOF; that is, the size of the coherence domain as obtained from L(E) has to be viewed as an upper boundary to the real value. A generalized participation ratio including the effect of excitonphonon interaction was proposed recently in the framework of a Green’s function approach.37 In any case, however, one should keep in mind that L(E) always resembles the symmetry of the system. For a linear aggregate, for instance, the maximum coherence domain size according to eq 2.10 would be L-1(E) ) Ncoh ) 2(N + 1)/3 and not Ncoh ) N. Therefore in section III we will use L(E) in a more qualitative way to explain the features of ET dynamics but not to draw conclusions about the real value of Ncoh. B. Excitation Energy Quenching. In photosynthetic systems one quite often encounters situations where two pigment pools are interacting rather weakly with each other due to, for example, large spatial separations. Examples are the LH1 and LH2 antenna systems or the LH1 and the pigments belonging to the reaction center (See Figure 1). If one is interested in the dynamics of a particular (main) pool only without neglecting, for example, the excitation energy quenching due to the presence of another (sink) pool, a consistent way of incoporating the mutual interaction in the equations of motion is needed.38 Below we will show how this goal can be achieved by making the following approximations: (i) The interaction between the pigment pools is weak enough to neglect frequency changes in the main pool due to the coupling. (ii) Only self-energy like contributions containing the interaction in second order are considered. (iii) There is no optical excitation of the sink pool. (iv) Energy relaxation in the sink pool is fast enough that there are always empty final states for scattering from the main to the sink pool; that is, no blocking occurs. The sink contribution to the equations of motion, eq 2.6, for the main pool reads

( )

i

d

FRβ

dt

S

)

∑Rˆ (hRˆ βFRRˆ - hRRˆ FRˆ β)

(2.12)

p

with hRˆ R ) ∑mmˆ c* m,Rhmm ˆ cm ˆ ,Rˆ , where indices with a caret denote states belonging to the sink pool. The equations of motion for the mixed RDM, which appears on the rhs of eq 2.12, are given by (neglecting damping terms for brevity)

i FRˆ R ) -iωRˆ RFRˆ R + ( dt p d

∑β hRˆ βFβR - ∑βˆ hβˆ RFRˆ βˆ )

(2.13)

The last term will be neglected according to our assumptions. In passing we note that this term would give the appropriate

ET in Light-Harvesting Antenna Complexes

J. Phys. Chem. B, Vol. 101, No. 17, 1997 3435

source term if one wants to consider the dynamics in some main pool after excitation of a source pool. The formal solution to eq 2.13 can be obtained straightforwardly, after invoking the Markov approximation for FβR(t), as

∑β hRˆ β δ(ERˆ - Eβ) FβR(t)

FRˆ R(t) ) -iπ

(2.14)

Introducing the interaction-weighted spectral density for the sink pool as

SRβ(E) )

π p

∑Rˆ hRRˆ hRˆ βδ(E - ERˆ )

(2.15)

the sink contribution that follows from eq 2.12 reads

( ) d

FRβ

dt

)-

S

QRβ,R′β′FR′β′ ∑ R′β′

(2.16)

with QRβ,R′β′ ) SRR′(ER′)δββ′ + δRR′Sββ′(Eβ′) being the quenching matrix. In a similar way we obtain

( ) d

FRg dt

S

)-

SRR′(ER′)FR′g ∑ R′

(2.17)

It is important to note that usually excitation energy quenching is introduced by adding an imaginary part to the transition energy of a particular sink molecule.18,37 In the present approach, however, it is the spectral overlap between main and sink pool that determines the quenching. C. Approximations. The explicit calculation of the Redfield relaxation matrix, eq 2.7, requires some approximations: First, one has to specify the relevant correlations. We will assume that

〈δhkl(t)δhmn〉env ≈ δklδmnδkm〈δhkk(t)δhkk〉env + (1- δkl)[δlmδkn〈δhkl(t)δhlk〉env + δkmδln〈δhkl(t)δhkl〉env] (2.18) That is, there are no correlations between fluctuations of different site energies and dipole-dipole interaction matrix elements belonging to different pairs of monomers. In view of the complexity of LHAs, where many quantities such as positions and orientations of the pigments and the charge distribution over the porphyrin planes are likely to fluctuate independently, eq 2.18 appears rather reasonable. Second, the question of the time dependence of the correlation functions has to be answered. In general, it is to be expected that correlations decay rather rapidly in biological systems. Numerical simulations for the reaction center of Rb. sphaeroides, for instance, showed that the correlation function for the energy gap coordinate decays appreciably within less than 100 fs.41 Similar conclusions could be drawn from the behavior of the peak shift in a three-pulse photon-echo experiment on B800.10 Thus, we will restrict ourselves to correlation functions that decay exponentially with a time constant τc. Further we will invoke a Markov approximation neglecting the frequency dependence in eq 2.7 and use the properly symmetrized correlation functions to account for detailed balance.30,42 It should be noted here that the “coarse graining” introduced by the Markov approximation limits the range of validity of the present theory to time scales for the system dynamics that are larger than τc.19,39,40 Fortunately, typical time scales for the ET in the LH2 at room temperature are about 0.4-0.7 ps (see the Introduction), i.e. much larger than τc, which is likely to be on the order of some tens of femtoseconds.

Since it is this ET dynamics that is of primary interest here, we can also invoke the secular approximation to the Redfield relaxation matrix, i.e. neglect those elements of RRβ,R′β′ for which |ωRβ - ωR′β′| * 0. Their contribution to the system dynamics averages out for times greater than |ωRβ - ωR′β′|-1.19,34,39 Within this approximation we have to consider terms responsible for population relaxation and coherence dephasing as well as matrix elements of the type RRβ,R′β′ leading to the so-called coherence transfer if ωRβ ) ωR′β′.40 The energy level structure in the disordered LHAs, however, is rather anharmonic; that is, the condition ωRβ ≈ ωR′β′ may only be accidentally fulfilled. Furthermore, our numerical simulations will be focused on the population dynamics which is not influenced by the coherence transfer. Thus we will neglect the latter and use

RRβ,R′β′FR′β′ ≈ (1 - δRβ)RRβ,RβFRβ + δRβ∑RRR,γγFγγ ∑ R′β′ γ

(2.19)

which is analogous to the Bloch model.40 The same type of approximation is invoked in the treatment of the quenching matrix, QRβ,R′β′ (eq 2.16). The secular approximation provides the basis for a considerable reduction of numerical effort in the solution of eqs 2.6 and 2.9. From the calculation of the linear absorption spectrum of the homogeneous LH2, for instance, it is known that only a few exciton states are optically active. Inclusion of static disorder, of course, distributes oscillator strength over more eigenstates. However, if one introduces a certain cutoff, osc, and takes the point of view that all transitions with |µR|2 < osc can be neglected for the problem at hand, one arrives at the picture of essential exciton states24 governing the system dynamics. In terms of eqs 2.6 and 2.9 this means that only the matrix elements of FRg for the important exciton eigenstates have to be considered. Consequently, the off-diagonal elements of FRβ are restricted to combinations of these eigenstates too, whereas all populations, FRR, have to be taken into account since they can be reached via relaxation (eq 2.19). In the numerical calculations presented below we chose osc ) 0.01, which led to a reduction of CPU time by about 25%. With these approximations the phase relaxation part of the Redfield matrix becomes

RRβ,Rβ ) Γˆ Rβ +

∑ ΓRγ + γ*β ∑ Γβγ (R * β)

(2.20)

γ*R

and the population relaxation is governed by

RRR,ββ ) -2ΓβR + 2δRβ

∑γ ΓRγ

(2.21)

Here we introduced the energy relaxation rates for transitions from |R〉 to |β〉 as

ΓRβ )

1 1 + exp(-pωRβ

(〈|δhmn|2〉env|cm,R|2|cn,β|2 + ∑ /k T) mn B

(1 - δmn)〈δh2mn〉envc* m,Rcn,βc* m,βcn,R) (2.22) and the pure dephasing rate

Γˆ Rβ )

(〈|δhmn|2〉env|c*m,Rcn,R - c*m,βcn,β|2 + ∑ mn 2 (1 - δmn)〈δh2mn〉env(c*m,Rcn,R - c* m,βcn,β) ) (2.23)

Note that we scaled the fluctuation matrix such that δhmm f δhmmxτc/p. In the numerical simulations we will use the



Figure 2. Inverse participation ratio (upper panel) and DOS (lower panel) as obtained for the B800-only system for the cases A-C. The variance for the diagonal static disorder is taken as σ800 ) 75 cm-1, and the averaging is performed using 2000 realizations. The DOS is calculated using a Lorentzian having a width of 1 cm-1 instead of the delta function in eq 2.11.

parameter γn ) 〈|hnn|2〉env, characterizing the strength of onsite fluctuations; that is, the linear absorption of each monomer has a Lorentzian shape and width pγn. For the off-diagonal fluctuations we assume for simplicity that 〈|hmn|2〉env ) Jfl|Jmn|/ p. In general the effect of the off-diagonal fluctuations is to open additional channels for population relaxation. This can be seen as follows: The total rate for scattering out of a certain eigenstate |R〉 is proportional to ∑βΓRβ, while the total rate for scattering into this state is approximately ∑βΓβR. If Jfl ) 0 and γn ) γ ) const, the summations can be carried out to give both total rates approximately proportional to ∑m|cm,R|2 ≈ |µR|2. This estimate thus shows that scattering takes place between states that carry oscillator strength. This is no longer the case for Jfl * 0; that is, optically dark states become involved into the relaxation process thus providing additional channels. In addition to the rapid fluctuations we will take into account static energetic disorder by numerical averaging over Gaussian ensembles of LHAs. In particular we assume an uncorrelated distribution of site energies having variance σn. Static variations of the dipole-dipole interaction energies are likely to exist as well but are assumed to be neglegible compared to the rather strong diagonal disorder.

the inverse participation ratio L(E) (eq 2.10) for the cases A-C (see Table 1). The variance of the distribution of monomer energies is σ800 ) 75 cm-1, i.e. larger than the maximum interaction energy between adjacent pigments (∼30 cm-1 for case A). While in case A the excitonic wave function extends over about 2.5 monomers at the absorption maximum, this number decreases to about 1.6 in case C. At the band egdes the excitation will be localized almost to a single monomer in all cases. As mentioned in section II.A, these numbers present only upper limits for Ncoh, suggesting that the B800 system behaves at least in case C rather monomeric. Also shown in Figure 2 (lower panel) is the DOS for the same configurations. As expected the one-exciton bandwidth decreases upon decreasing the dipole-dipole interaction energies from A to C. To address the question to what extent the observed subpicosecond B800 intraband dynamics is mediated through excitonic states of the B850 band, we solved the equations of motion, eqs 2.6 and 2.9, using the standard fourth-order Runge-Kutta method to obtain the wavelength- and time-dependent population,

III. Numerical Results

The initial decay of P(t) after the pulse is over can be expected to reflect the behavior of a one-color pump-probe signal approximately. For negative and short delay times the overlapping pump and probe pulses will modify the signal, for instance, due to direct two-photon transitions to the two-exciton manifold. For delay times on the order of the B800-B850 transfer time excited state absorption of the B850 band starts to dominate the signal.6,8 Further it should be mentioned that in general the one-exciton contribution to the third-order pump-probe signal depends on the dynamics of FRR(t) and FRβ(t). The coherences, FRβ(t), are excited by the finite width Gaussian pulse -1 and decay on a time scale ∝ RRβ,Rβ (see eq 2.6). In other words, they could be of some importance for an accurate determination of the short time evolution of the spectrum yielding, for instance, excitonic quantum beats. In view of the

A. B800 Dynamics. We start our discussion of the ET dynamics by considering the B800 subsystem only. The presence of the B850 pigment pool is accounted for by adding a quenching term to the equations of motion. For simplicity we choose SRβ(E) as a Lorentzian centered at the B850 absorption maximum with state-independent coupling, kquench. For kquench ) 0.012 cm-1 the overall population decay of the B800 pool was about 1.5 ps, which corresponds to the experimental value at 77 K.9 The rather large spatial separation of the outer ring pigments causes the dipole-dipole interaction to be small compared, for example, with the static disorder. In other words, the excitonic wave functions are expected to be more localized than in the strongly coupled B850 pigment pool. This can be seen from Figure 2 (upper panel), where we plotted

P(t) ) 〈

∑Rδ(E - ER)FRR(t)〉disorder

(3.1)


Figure 3. Population dynamics 10 nm to the blue of the B800 absorption maximum for case A according to eq 3.1 (B800-only system). The Gaussian pulse (fwhm 80 fs) was centered at the same wavelength. The parameters are σ800 ) 75 cm-1, pγ800 ) 50 cm-1, Jfl ) 0 (solid line); σ800 ) 60 cm-1, pγ800 ) 75 cm-1, Jfl ) 0 (dashed line); σ800 ) cm-1, pγ800 ) 25 cm-1, Jfl ) 0.3 (dotted line). The strength of the quenching is taken as kquench ) 0.012 cm-1. The B850 band is represented by a Lorentzian centerd at 850 nm and having a width of 220 cm-1. The inset shows the corresponding dynamics at the absorption maximum for the parameters of the dotted curve in the main figure. Experimental data for the one-color pump-probe signal at 77 K are included as circles.

Figure 4. Same as Figure 3 but for case C.

experimental data, however, it seems reasonable that an analysis of the population dynamics alone can capture essential features of the ET in the LH2. In Figures 3 and 4 we show P(t) at a wavelength that is 10 nm to the blue of the B800 absorption maximum for cases A and C, respectively (circles in Figures 3 and 4 are experimental data). The 80 fs Gaussian pulse is centered at the same wavelength. In Figure 3 we display P(t) for a mainly inhomogeneous (solid line) and a mainly homogeneous (dashed line) case, neglecting off-diagonal fluctuations (Jfl ) 0). Further a situation that includes off-diagonal fluctuations is shown (dotted line). Comparing the three curves we note that for Jfl ) 0 increasing the ratio σ850/pγ850 leads to a slower decay of P(t) due to a temporary localization of excitation energy at the upper band edge. On the other hand, the fastest decay is observed if off-diagonal fluctuations are taken into account. The origin of this behavior has been discussed in section II.C: For Jfl * 0 the relaxation is no longer restricted to those states that carry appreciable oscillator strength, leading to an overall increase of the energy relaxation rates. This behavior is also observed in case C, as shown in Figure 4. Here, the decay is further slowed down due to stronger localization (compare Figure 2). Even though the agreement between P(t) and the experimental

J. Phys. Chem. B, Vol. 101, No. 17, 1997 3437 pump-probe signal is quite reasonable for certain parameters, simulation of a one-color pump-probe signal for excitation at the absorption maximum using the same parameters shows a far too slow decay (see insets in Figures 3 and 4). This leads us to the conclusion that within the present theoretical model it is difficult to explain the fast intraband dynamics in the B800 band with a single parameter set by considering the B800 pigments alone and including the B850 pool perturbatively via the quenching matrix. B. LH2 Dynamics with Mixing of B800 and B850 Exciton States and LH1 Quenching. Since the perturbative inclusion of the B850 band cannot account for the observed B800 intraband dynamics in the present model, the full LH2 will be considered next. Here, the weak coupling to the LH1 is treated within the framework of the quenching matrix approach derived in section II.B. Choosing a Lorentzian centered at 875 nm (fwhm 280 cm-1) and a coupling strength of kquench ) 4 × 10-4 cm-1 results in an overall LH2-LH1 transfer with a time constant of about 5 ps at 77 K.9 Furthermore, the static disorder and the strength of diagonal fluctuations for the B800 pigments are kept constant at σ800 ) 75 cm-1 and pγ800 ) 50 cm-1. Note, however, that all dipole-dipole interactions are scaled; that is, the ratio between these parameters and the Jmn belonging to the B800 molecules increases from A to C. We start with the examination of the static properties, i.e. L(E) and D(E), which are shown in Figure 5 for σ850 ) max(|Jmn|/2). Decreasing the dipole-dipole interactions leads to an overall reduction of the bandwidth of the one-exciton manifold, as can be seen from the DOS plotted in the lower panel of Figure 5. This effect is most pronounced for the B850 band. To the red of the B800 absorption maximum the B850 exciton states are becoming more closely spaced. More important for the efficiency of the ET, however, is that the upper band edge of the B850 band moves from about 760 nm (A) to about 800 nm (C), yielding a high DOS around 800 nm. The inverse participation ratio shown in the upper panel of Figure 5 indicates that on average the size of the coherence domain around 800 nm is larger than in the B800-only system. This is a manifestation of the mixing between B800 and B850 exciton states. (the B800-B850 coupling is slightly higher than the intra-B800 coupling.) Figure 5 further reveals that the value of Ncoh for a given energy within the B800 band strongly depends on the B850 bandwidth, i.e. on the position of the upper band edge of the B850 band (see Table 1). In the previous section we learned that the time constant for population relaxation is quite sensitive to the degree of localization of the excitonic wave functions. To connect L(E) and the band structure shown in Figure 5 with the dynamics in the LH2, we first investigate the ET for excitation and detection 10 nm to the blue of the B800 absorption maximum. In Figures 6 and 7 we plotted P(t) for case A and C. In both figures solid lines correspond to σ850 ) max(|Jmn|)/2 and dashed lines to σ850 ) max(|Jmn|)/1.5. They are grouped together for the configurations: (I) pγ850 ) 200 cm-1, Jfl ) 0; (II) pγ850 ) 100 cm-1, Jfl ) 0; and (III) pγ850 ) 50 cm-1, Jfl ) 0.05. First, we note from Figures 6 and 7 that increasing the disorder causes a slight acceleration of the decay. This somewhat counterintuitive result derives from the fact that increasing the disorder not only leads to a broadening of the DOS but also causes a modification of L(E) in a way that states at 790 nm are slightly more delocalized for the higher value of σ850 (compare Table 1). Even though this effect is rather minute and strongly dependent on the considered wavelength, it points to the complicated band structure encountered in LHAs. Comparing curves corresponding to the same strength of the disorder in Figures 6 and 7, we further conclude that for a fixed



Figure 5. Inverse participation ratio (upper panel) and DOS (lower panel) as obtained for the LH2 system for the cases A-C. The variance for the diagonal static disorder is taken as σ800 ) 75 cm-1 and σ850 ) max(|Jmn|)/2, respectively. The DOS is calculated using a Lorentzian having a width of 1 cm-1 instead of the delta function in eq 2.11.

Figure 6. Population dynamics according to eq 3.1 for the full LH2 system (case A) for excitation and detection at λ800 ) 10 nm. The parameters are (I) pγ850 ) 200 cm-1, Jfl ) 0; (II) pγ850 ) 100 cm-1, Jfl ) 0; and (III) pγ850 ) 50 cm-1, Jfl ) 0.05. Solid lines correspond to σ850 ) max(|Jmn|)/2; dashed lines to σ850 ) max(|Jmn|)/1.5. The dashed line of II shows the best agreement with the experimental one-color pump-probe signal at 77 K (circles).

pγ850 the decay is considerably faster for the more delocalized case (A) (see Table 1). In both figures the inclusion of offdiagonal fluctuations leads as expected to the fastest decay (see section II.C). While the dashed curve of set II in Figure 6 (pγ850 ) 100 cm-1, Jfl ) 0) shows a reasonable agreement with the experimental one-color signal; a similar behavior for case C could only be obtained for pγ850 ) 50 cm-1, Jfl ) 0.075 (dashdotted curve in Figure 7) and for case B with pγ850 ) 60 cm-1, Jfl ) 0.0 (not shown). (We chose σ850 ) max(|Jmn|)/1.5 in all optimized configurations.43) Using these optimized parameters, we simulated a one-color pump-probe signal for excitation at the absorption maximum of the B800 band. The results are plotted in Figure 8. The best agreement is obtained in case C, while the decay for cases A and B is too slow. Comparing the participation ratios given in Table 1, we are led to the conclusion that this behavior is caused not by temporary localization as in the blue part of the B800 absorption band but rather by competition between B800

Figure 7. Same as Figure 6 but for case C. In addition the dash-dotted line shows the best fit obtained for σ850 ) max(|Jmn|)/1.5, pγ850 ) 50 cm-1, Jfl ) 0.075.

Figure 8. Population dynamics for the LH2 system for excitation and detection at the B800 absorption maximum using the best fit paramters from Figures 6 and 7 as well as σ850 ) max(|Jmn|)/1.5, pγ850 ) 60 cm-1, Jfl ) 0.0 for case B (solid (A), dashed (B), dotted (C)).

intraband and B800-B850 interband ET. This intraband transfer, however, is most favored in case C, where the upper band edge of the B850 exciton band, which carries appreciable oscillator strength, is in resonance with the B800 absorption


Figure 9. Time- and wavelength-dependent population dynamics for the LH2 system using the optimized paramter set (panels correspond to cases A-C).

maximum, resulting in a high DOS at this wavelength (see Figure 5). To stress this point further, we have plotted P(t) over the relevant wavelength range for the optimized parameters and an excitation at 790 nm. Even though the one-color signal at 790 nm is reasonably fitted using these parameters (see Figures 6 and 7), the B800-B850 transfer is markedly different for cases A to C. Only in case C does the population around the 850 absorption maximum rise with a time constant of about 1-1.5 ps, as observed in the experiment.9 The B800-B850 ET in the cases A and B appears to be too slow for the present parameters. IV. Summary The energy transfer dynamics in the outer antenna complex of photosynthetic bacteria has been investigated using an equation of motion approach. The effects of the coupling between the exciton motion and environmental degrees of freedom were described using Redfield theory. Furthermore, a microscopic based quenching matrix was derived, which accounts for the weak coupling between different pigment pools such as the LH2 and LH1 in the photosynthetic unit. Comparing the time- and wavelength-resolved population dynamics initiated by an ultrashort laser pulse for different

J. Phys. Chem. B, Vol. 101, No. 17, 1997 3439 configurations, the following conclusions could be drawn: First, the fast intraband dynamics within the B800 band may be mediated through the coupling to B850 exciton states, as suggested in ref 8. This derives basically from the strong modification of the density of states as well as the size of the coherence domain due to the mixing of B800 and B850 states (compare Figures 2 and 5). Second, the upper band edge of the B850 band has to be close to the B800 absorption maximum to give B800 intraband and B800-B850 interband transfer times comparable to the experimentally obtained values.9 This is also in accord with the the estimate given by Wu et al., whose experiments on Rps. acidophila suggested the upper band edge to be about 50 cm-1 to the blue of the B800 absorption maximum.8 In view of the excitonic calculations presented in ref 35 this means that either the dielectric constant of the ambient medium is larger than ) 1 or the dipole strength for the Qy transition in the LH2 is smaller than the 41 D2 measured for BChl a in acetone. Since these parameters are not easily accessible in an experiment, and in fact the application of the concept of a macroscopic dielectric constant on atomic length scales is questionable at all, as pointed out in ref 35, in the future more detailed dynamical and excitonic calculations have to be combined with specific experiments to settle the question of the excitonic bandwidth in the LH2 system. As very recently proposed, the situation might be further complicated by the presence of the carotenoids, which could also promote the B800-B850 excitation energy transfer.11 One of the major shortcomings of the presented theoretical framework is the neglect of vibrational modes which can promote the B800-B850 transfer13 and cause quantum beats in the pump-probe signal due to coherent vibrational motion.44 Explicit incorporation of coupled exciton-vibrational motion on the level of the theory presented here appears to be rather unrealistic for systems as complex as the LHAs. Numerical simulations of the dissipative energy transfer in molecular heterodimers including one vibrational mode per monomer have been shown to approach the limits of present day computers.31 Clearly, appropriate approximation schemes have to be developed to obtain a deeper understanding of the role of vibrational coherences in the process of photosynthetic light harvesting. Acknowledgment. The authors thank Dr. T. Pullerits (Lund University) and Dr. V. M. Axt (University of Rochester) for many stimulating discussions. O.K. gratefully acknowledges a postdoctoral fellowship from the German Academic Exchange Service (DAAD) and Lund University. This work was supported by the Swedish Natural Science Research Council and EC Grant No. ERBCH-BGCT 930361. References and Notes (1) Van Grondelle, R.; Dekker, J. P.; Gillbro, T.; Sundstro¨m, V. Biochim. Biophys. Acta 1994, 1187, 1. (2) See related articles in: Blankenship, R. E.; Madigan, M. T.; Bauer, C. E., Eds.; Anoxygenic Photosynthetic Bacteria; Kluwer Academic: Dordrecht, 1995. (3) Pullerits, T.; Sundstro¨m, V. Acc. Chem. Res. 1996, 29, 381. (4) Karrasch, S.; Bullough, P. A.; Gosh, R. EMBO J. 1995, 14, 631. (5) McDermott, G.; Prince, S. M.; Freer, A. A.; HawthornthwaiteLawless, A. M.; Papiz, M. Z.; Cogdell, R. J.; Isaacs, N. W. Nature 1995, 374, 517. (6) Hess, S.; A° kesson, E.; Cogdell, R. J.; Pullerits, T.; Sundstro¨m, V. Biophys. J. 1995, 69, 2211. (7) Monshouwer, R.; De Zarate, I. O.; van Mourik, F.; Van Grondelle, R. Chem. Phys. Lett. 1995, 246, 341. (8) Wu, H. M.; Savikhin, S.; Reddy, N. R. S.; Jankowiak, R.; Cogdell, R. J.; Struve, W. S.; Small, G. J. J. Phys. Chem. 1996, 100, 12022. (9) Hess, S.; Chachisvilis, M.; Timpmann, K.; Jones, M. R.; Fowler, G. J. S.; Hunter, C. N.; Sundstro¨m, V. Proc. Natl. Acad. Sci. U.S.A. 1995, 92, 12333.

3440 J. Phys. Chem. B, Vol. 101, No. 17, 1997 (10) Joo, T.; Jia, Y.; Yu, J.-Y.; Jonas, D. M.; Fleming., G. R. J. Phys. Chem. 1996 100, 2399. (11) Pullerits, T.; Hess, S.; Sundstro¨m, V. J. Phys. Chem., submitted. (12) Jimenez, R.; Dikshit, S. N.; Bradforth, S. E.; Fleming, G. R. J. Phys. Chem. 1996, 100, 6825. (13) Reddy, N. R. S.; Wu, H. M.; Jankowiak, R.; Picorel, R.; Cogdell, R. J.; Small, G. J. Photosynth. Res. 1996, 48, 277. (14) Sauer, K.; Cogdell, R. J.; Prince, S. M.; Freer, A. A.; Isaacs, N. W.; Scheer, H. Photochem. Photobiol. 1996, 64, 564. (15) Davydov, A. S. Theory of Molecular Excitons; Plenum: New York, 1971. (16) Reineker, P. Springer Tracts Mod. Phys. 1982, 94, 111. (17) C ˇ a´pek, V.; Szo¨cs, V. Phys. Status Solidi B 1985, 131, 667. (18) Szo¨cs, V.; Banacky´, P. Chem Phys. 1994, 186, 153. (19) Redfield, A. G. AdV. Magn. Reson. 1965, 1, 1. (20) Wertheimer, R.; Silbey, R. Chem. Phys. Lett. 1980, 75, 243. (21) Spano, F. C.; Knoester, J. AdV. Magn. Opt. Res. 1994, 18, 117. (22) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford: New York, 1995. (23) Chernyak, V.; Wang, N.; Mukamel, S. Phys. Rep. 1995, 263, 213. (24) Ku¨hn, O.; Chernyak, V.; Mukamel, S. J. Chem. Phys. 1996, 105, 8586. (25) Axt, V. M.; Mukamel, S. In IMA Volumes in Mathematics and Its Applications; Maloney, J., Sipe, J., Eds.; submitted. (26) Fo¨rster, T. Ann. Phys. (Leipzig) 1948, 2, 55. (27) Pullerits, T.; Freiberg, A. Biophys. J. 1992, 63, 879. (28) Tekhver, I. Y.; Khizhnyakov, V. V. SoViet Phys. JETP 1976, 42, 305. (29) Ku¨hn, O.; Rupasov, V.; Mukamel, S. J. Chem. Phys. 1996, 104, 5821.

Ku¨hn and Sundstro¨m (30) Matro, A.; Cina, J. A. J. Phys. Chem. 1995, 99, 2568. (31) Ku¨hn, O.; Renger, T.; May, V. Chem. Phys. 1996, 204, 99. (32) Renger, T.; Voigt, J.; May, V.; Ku¨hn, O. J. Phys. Chem. 1996, 100, 15654. (33) Chernyak, V.; Mukamel, S. J. Chem. Phys. 1996, 105, 4565. (34) Jean, J. M.; Friesner, R. A.; Fleming, G. R. J. Chem. Phys. 1992, 96, 5827. (35) Pullerits, T.; Chachisvilis, M.; Sundstro¨m, V. J. Phys. Chem. 1996, 100, 10787. (36) Note that it is quite advantageous to split off the high oscillatory part of the exciton variables, oscillating with the frequency of the respective driving field, and propagate the slowly varying envelopes only. (37) Leegwater, J. A. J. Phys. Chem. 1996, 100, 14403. (38) May, V. Physica D 1989, 40, 173. (39) Su¨sse, K. E.; Welsch, D. G. Relaxation Processes in Molecular Systems; Teubner: Leipzig, 1984. (40) Jean, J. M.; Fleming, G. R. J. Chem. Phys. 1995, 103, 2092. (41) Warshel, A.; Parson, W. W. Annu. ReV. Phys. Chem. 1991, 42, 279. (42) Oxtoby, D. W. AdV. Chem. Phys. 1981, 47, 487. (43) It should be noted that, of course, the used sets of parameters describing the influence of environmental DOF are not unique. In view of the lack of information and the various necessary approximations made, our intention is to give a more qualitative picture of the dynamics rather than to pinpoint parameters. (44) Chachisvilis, M.; Pullerits, T.; Jones, M. R.; Hunter, C. N.; Sundstro¨m, V. Chem. Phys. Lett. 1994, 224, 345.

Energy Transfer and Relaxation Dynamics in Light-Harvesting

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