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. Čápek 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 Förster 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 Förster 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
Kühn and Sundström
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
)
∑R̂ (hR̂βFRR̂ - hRR̂FR̂β)
(2.12)
p
with hR̂R ) ∑mm̂c*
m,Rhmm̂cm̂,R̂, 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
FR̂R ) -iωR̂RFR̂R + (
dt
p
d
∑β hR̂βFβR - ∑β̂ hβ̂RFR̂β̂)
(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 hR̂β δ(ER̂ - Eβ) FβR(t)
FR̂R(t) ) -iπ
(2.14)
Introducing the interaction-weighted spectral density for the sink
pool as
SRβ(E) )
π
p
∑R̂ hRR̂hR̂βδ(E - ER̂)
(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 ) Γ̂Rβ +
∑ 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
Γ̂Rβ )
(〈|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
3436 J. Phys. Chem. B, Vol. 101, No. 17, 1997
Kühn and Sundström
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)
ET in Light-Harvesting Antenna Complexes
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
3438 J. Phys. Chem. B, Vol. 101, No. 17, 1997
Kühn and Sundström
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
ET in Light-Harvesting Antenna Complexes
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.
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