J. Phys. Chem. B 2001, 105, 6655-6665
6655
Characterization of the Static Disorder in the B850 Band of LH2† Seogjoo Jang, Sara E. Dempster, and Robert J. Silbey* Department of Chemistry and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ReceiVed: January 17, 2001
A systematic simulation study, guided by the expressions of the first-order perturbation theory, was performed for a model of the B850 band in the light harvesting complex 2 of purple bacteria. General types of Gaussian disorder, which include diagonal, off-diagonal, coexisting but independent diagonal and off-diagonal, and correlated diagonal and off-diagonal disorder, were considered. The effects of these types of disorder were studied by calculating four quantities that characterize the energy distribution of the three lowest levels, varying the magnitude of the disorder. For all the cases considered, except for one of the correlated disorder models, a common scaling relation was found among the four quantities, which is in reasonable agreement with available low temperature experimental data. On the basis of this fact and further physical reasoning, it is concluded that the disorder in the B850 is likely Gaussian and that comparable magnitudes of both diagonal and off-diagonal disorder coexist. Three distinct geometrical mechanisms were considered, and it was found that radial fluctuations in the transition dipole positions and fluctuations in the transition dipole angles are plausible sources of the off-diagonal Gaussian disorder.
I. Introduction The functional unit of photosynthesis, the photosynthetic unit (PSU),1 varies with organism, but its modular design, consisting of light harvesting (LH) complexes and a reaction center (RC), is common. One of the simplest examples of this can be seen in purple bacteria, which were subject to thorough structural and spectroscopic studies.1,2 In most of these species, two different types of LH complexes, denoted as 1 and 2, exist, and the PSU consists of a RC enclosed within an LH1 complex and many surrounding LH2 complexes. As this arrangement suggests, LH2 performs the role of a peripheral collector of solar energy, and LH1 mostly serves as an intermediate reserve of excitation energy, which is finally utilized for the charge-transfer reaction in the RC. According to the analysis of X-ray crystallography data,3,4 LH2 has a highly symmetric but intriguing structure with CN symmetry (N ) 8 or 9). Each unit in this structure, called a protomer, consists of two types of polypeptides (R and β), two carotenoids, and three bacteriochlorophylls (BChl). LH2 has two major absorption bands due to the BChls, B800 and B850, named after the wavelengths of their peak maxima at room temperature. One BChl in each protomer unit constitutes the former band while the remaining two BChls constitute the latter. Because of the larger radius and smaller number of chromophores, the nearest inter-BChl distance in the B800 ring is more than twice that in the B850 ring. As a result, resonance interactions between BChls in the former are much smaller than in the latter, and the general consensus is that excitation in B800 ring can be viewed as being localized at one BChl. However, the B850 band does not yield to such a simple description because resonance interactions between BChls are comparable to experimentally observed homogeneous and inhomogeneous broadenings.2 In this regime, the excitonic nature of the system †
Part of the special issue “Bruce Berne Festschrift”.
can be sensitive to the details of its coupling with the environment. As is typical in a biological system, in LH2 bath modes coupled to the BChls exhibit a broad spectral range, and physical sources for these are various.2,5 To name a few, they are intraBChl vibronic coupling, fluctuations in the van der Waals and the hydrogen bonding interactions between BChls and polypeptides, variations in the local dielectric response, and structural deformations. Some of these can be considered static while others should be treated as dynamical, although this classification depends on the temperature and the time scale of experimental observations. For a given set of conditions, however, they can be treated separately at the lowest level of approximation, where it is assumed that the inhomogeneous broadening is solely explained by static disorder and dynamic disorder is responsible for the homogeneous broadening only. As yet, a satisfactory understanding of the nature of the static disorder in light-harvesting systems has not been reached. In the site excitation basis of B850, there can be disorder in both diagonal and off-diagonal elements. It is not clear whether only the consideration of the former is enough or the latter should be included as well. If both are considered, then there remains a question about whether they are independent or correlated in some fashion. In addition, the question of the adequacy of the Gaussian statistics has not been fully resolved. Most experimental analyses6-14 have used a rather simple procedure of fitting the data in terms of only Gaussian diagonal disorder. Some accounts or considerations15-18 of the offdiagonal disorder exist, but no systematic study is available. Although each analysis was successful in explaining the given experimental results, the determined parameter sets varied with the type of the experiment. A strikingly different interpretation is the one based on a recent low temperature single molecule experiment,12 where uniform elliptic distortion was proposed in addition to Gaussian disorder in order to explain the large gap between the two brightest states. More recently, new
10.1021/jp010169e CCC: $20.00 © 2001 American Chemical Society Published on Web 05/10/2001
6656 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Jang et al.
theoretical18 analysis reinforcing this view has appeared. This explanation may indeed be correct, and might provide a basis for understanding some of recent intriguing results of 300 K SMS experiment19 and nonlinear spectroscopy experiments.20 However, no clear physical justification for the elliptic distortion is available yet. More importantly, we ask another question: Is there another possible explanation for the experimental results? The motivation of the present work is thus the lack of a consistent picture for the disorder in the B850 band, and the realization that a systematic study including the off-diagonal disorder might be important in resolving the issues. Physically, possible sources for off-diagonal disorder are dielectric fluctuations and/or structural variations. Because of the fact that the BChls are buried in hydrophobic environments,5 contributions of the former may be minor. Given that all the components in LH2 are held together through basket-like interlocking mechanisms,5 however, the structure of each LH2 in its natural condition may not exhibit perfect symmetry of the crystal state. This suggests the possibility that the off-diagonal disorder originates from the structural disorder. In the present paper, a systematic simulation study is performed on a model of B850, which examines the energy level structures for four cases of diagonal, off-diagonal, independent diagonal and off-diagonal, and correlated diagonal and off-diagonal disorder, within Gaussian statistics. As a guideline, the expressions of first-order perturbation theory are used which were recently found to be successful in understanding qualitative features of finite size circular aggregates even in the intermediate disorder regime.21,22 The main purpose of the present work is to investigate the general behavior of the model system with the variation of the strength and type of the disorder, thus providing a broad perspective from which a consistent understanding of experimental results may be possible. The sections are organized as follows. In section II, the exciton Hamiltonian of the B850 is introduced and a standard formal analysis is made. Section III describes the detailed aspects of the model of disorder and provides the first-order perturbation theory expressions of the distribution functions. Section IV introduces the parameters of the model Hamiltonian and describes the simulation results. In Section V, discussions of the physical implications of the simulations results in relation to known experimental facts are made and interpretations are made. The concluding remarks are provided in section VI.
of the static disorder. Its general expression in the site excitation basis is as follows:
N
H ˆ0 )
|s˜k〉 ≡
∑ ∑ Jss′(n - m)|sn〉〈s′m|
(2)
N
1
xN
e2πikn/N|sn〉 ∑ n)1
(4)
with k ) -3, ..., 4 for N ) 8 and -4, ..., 4 for N ) 9, matrix elements of H ˆ 0 can be simplified as
ˆ 0|s˜′k′〉 ) δkk′J˜ss′(k) 〈s˜k|H
(5)
with
J˜ss′(k) ≡
N
1
∑ e-2πik(n-m)/NJss′(n - m) Nn,m)1
(6)
N-1
∑ e-2πikn/NJss′(n)
)
(7)
n)0
where s,s′ ) R,β as above. Diagonalization of the two state Hamiltonian in each k subspace leads to the following expression for the eigenvalues of the zeroth-order Hamiltonian: (0)
E l ,k ) u
J˜RR(k) + J˜ββ(k)
{(
-
2
(8)
)
}
J˜RR(k) - J˜ββ(k) 2 + |J˜Rβ(k)|2 2
1/2
(9)
where l and u respectively imply lower and upper bands. The corresponding eigenstates are given by
(
( ) ( ) |R˜ k〉 |l, k〉 ) Gk |β˜ k〉 |u, k〉
() ()
)
() ( ) ()
χk -iηk/2 χk iηk/2 e cos e |R˜ k〉 2 2 ) χk -iηk/2 χk iηk/2 |β˜ k〉 cos e sin e 2 2 -sin
(1)
β
(3)
where all the matrix elements are assumed to be real and δJss′(n,m) ) δJs′s(m,n). One can find explicit eigensolutions of H ˆ 0 due to the CN symmetry and the two state nature within each symmetry unit. Transforming the site excitation states into the following k space states
A. General Expressions. The single excitation Hamiltonian of the B850 ring is given by
where the first term is the zeroth-order excitation Hamiltonian corresponding to an idealized B850 ring with CN symmetry and can be written as
∑ ∑ δJss′(n,m)|sn〉〈s′m|
n,m)1s,s′)R
II. Model Hamiltonian
H ˆ )H ˆ 0 + δH ˆ
β
N
δH ˆ )
where
χk ≡ tan-1
(
2|J˜Rβ(k)|
)
J˜RR(k) - J˜ββ(k)
ηk ≡ arg(J˜βR(k))
(10)
(11) (12)
n,m)1s,s′)R
where Rn and βn represent single Qy excitation of the BChl bound, respectively, to the nth R-polypeptide and to the nth β-polypeptide. The ring symmetry imposes the conditions of Jss′(n + N) ) Jss′(n) and Jss′(n) ) Js′s(N - n). The second term in eq 1 is a disorder Hamiltonian that includes all contributions
At this level, the allowed optical transitions are from the ground state to the k ) (1 and k ) 0 states, but the optical properties of LH2 are dominated by the k ) (1 states in the lower band. Equations 4 and 10 define the transformation from the site excitation basis into the basis of the eigenstates of H ˆ 0. In terms of this transformation, δH ˆ can now be expressed as
B850 Band of LH2
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6657 u
δH ˆ )
∑ ∑ δHbb′(k,k′)|b,k〉〈b′,k′| k,k′ b,b′)l
(13)
with
(
) (
)
δHll(k,k′) δHlu(k,k′) δJ˜ (k,k′) δJ˜Rβ(k,k′) T ) G/k RR G δHul(k,k′) δHuu(k,k′) δJ˜βR(k,k′) δJ˜ββ(k,k′) k′ (14)
where G/k and Gk′T represent the complex conjugate and transpose of Gk and Gk′, and
δJ˜ss′(k,k′) )
1
N
e-2πi(kn-k′m)/NδJss′(n,m) ∑ Nn,m)1
(15)
B. First-Order Degenerate Perturbation Expressions. For weak disorder such that the maximum of δHbb′(k,k′) is much smaller than the minimum level spacing of H ˆ 0, first-order perturbation theory serves as a good approximation. Since there are two degenerate states of |b,k〉 and |b,-k〉 for each of 0 < k < N/2 and for b ) l, u, degenerate perturbation theory needs to be used, which amounts to diagonalizing the subspaces where H ˆ 0 is degenerate. The projection of the total Hamiltonian into one of those subspaces reads
H ˆ b,k ) (E(0) b,k + δHbb(k,k))|b,k〉〈b,k| + (0) + δHbb(-k,-k))|b,-k〉〈b,-k| + (Eb,-k δHbb(k,-k)|b,k〉〈b,-k| + δHbb(-k,k)|b,-k〉〈b,k| (16) (0) where Eb,-k ) E(0) b,k and δHbb(k,k) ) δHbb(-k,-k). Diagonalization of eq 16 leads to the following expressions for the eigenvalues:
(1) ) E(0) Eb,k( b,k + δHbb(k,k) ( Db,k
(17)
where
Db,k ≡ |δHbb(k,-k)| ) |δHbb(-k,k)|
(18)
The zeroth-order eigenstates corresponding to above eigenvalues are given by (0) 〉) |ψb,k(
1 ((e-iφb,k/2|b,k〉 + eiφb,k/2|b,-k〉) x2
(19)
φb,k ≡ arg(δHbb(-k,k))
(20)
where
For k ) 0, which is nondegenerate, the corresponding expressions are as follows: (1) (0) Eb,0 ) Eb,0 + δHbb(0,0)
(21)
(0) |ψb,0 〉 ) |b,0〉
(22)
If N is even, there is an additional nondegenerate state with k ) N/2, for which the same expression as above, replacing 0 with N/2, can be used. Equations 17-22 provide expressions for nondegenerate energy levels correct to the first order of δH ˆ and the relevant eigenstates correct to the zeroth order. For further corrections, one can now apply standard nondegenerate perturbation theory starting from these expressions. However, we have found
recently,21,22 the simplest first-order expression of eq 17 provides exceptionally good approximation even for moderately large disorder. Therefore, analyses based on above expressions can provide simple qualitative guidelines in understanding the effects of various disorder as will be shown in the following section. III. Disorder A. Model for Off-Diagonal Disorder and Correlation. Simplifications can be made for off-diagonal matrix elements of the disorder Hamiltonian based on the nature of the B850 band. According to the dipole model estimates or ab initio calculations,23-25 nearest-neighbor interaction terms of H ˆ 0 are at least one order of magnitude larger than the other terms. Given that physical sources are common for all elements and that disorder is some fraction of the zeroth-order Hamiltonian, a similar disparity is likely to exist between the corresponding matrix elements of the disorder Hamiltonian. Therefore, without affecting the qualitative nature of the result, all the off-diagonal elements of δH ˆ may be approximated to be zero except for the nearest neighbor interaction terms. That is, the only nonzero off-diagonal elements of δH ˆ are assumed to be
δJRβ(n,n) ) V0(n)
(23)
δJRβ(n,n - 1) ) V1(n)
(24)
where V0(n) and V1(n) are two different sets of random variables. The diagonal elements of the disorder Hamiltonian are assumed to consist of two components. One is the disorder affecting only the site excitation energy, which has pure energetic origin. The other is the shift in the excitation energy induced by the change of the off-diagonal elements. Thus, we assume
δJRR(n,n) ) uR(n) + A{V0(n) + V1(n)}
(25)
δJββ(n,n) ) uβ(n) + A{V0(n) + V1(n + 1)}
(26)
where uR(n) and uβ(n) are independent random variables accounting for the energetic disorder only. Possible mechanisms for this can be variations in the van der Waals and hydrogen bonding interactions between BChls and polypeptides. The coefficient A reflects the size of the site excitation energy shift induced by the physical changes responsible for the disorder in the nearest neighbor interactions. If A is negative, increase of the nearest neighbor interaction induces a decrease of the site excitation energy and vice versa for a positive A. Sources for these correlations may be variations in the local dielectric constant, inter-BChl distances, and their orientations, which can affect both the off-diagonal matrix elements and the electronic wave function of an individual BChl. The linear model of eqs 25 and 26 is the simplest approximation that can mimic these situations to some degree, and A is considered as an unknown phenomenological parameter. Inserting above expressions into eqs 14 and 15, one can obtain the corresponding matrix elements of the disorder Hamiltonian expressed in terms of
u˜ R(k) )
1
xN
N
e2πikn/NuR(n) ∑ n)1
(27)
and u˜ β(k), V˜ 0(k), and V˜ 1(k) defined in the same way.15 For example, the matrix elements appearing in the expressions of the degenerate perturbation theory, for the case of b ) l, are
6658 J. Phys. Chem. B, Vol. 105, No. 28, 2001
{ ()
Jang et al.
()
χk χk 1 sin2 u˜ R(0) + cos2 u˜ (0) +[A 2 2 β xN 2πk sin(χk) cos(ηk)]V˜ 0(0) + A - sin(χk) cos ηk V˜ 1(0) N (28)
δHll(k,k) )
[
)] }
(
in the weak disorder limit. On the other hand, the quantity determining the splitting between k( states within the firstorder perturbation theory, Dl,k of eq 18 with b ) l, can be shown to have the following form:
P(Dl,k) )
and
{ ()
()
χk -iηk χk iηk 1 δHll(-k,k) ) sin2 e u˜ R(2k) + cos2 e u˜ β(2k) 2 2 xN + [A(cos(ηk) + i sin(ηk) cos(ηk)) - sin(χk)]V˜ 0(2k) + 2πk 2πk + i sin ηk cos(χk) e-2πik/N A cos ηk N N
[( (
)
(
)
)
]
sin(χk) V˜ 1(2k)
}
(29)
Similar expressions can be found for the case of upper band, b ) u. However, experimentally, this upper band is of minor concern compared to the lower band, and only the latter will be considered hereafter. The expressions of eqs 28 and 29 show that the average shift and the splitting of originally degenerate (k states are affected by both the diagonal and the off-diagonal disorder of the corresponding symmetry with their relative contributions determined by χk and ηk, which vary with the detailed nature of H ˆ 0. For instance, when the R and β site excitation energies are equal, χk ≈ π/2 and ηk ≈ 0 for small k. Expressions of eqs 28 and 29 indicate that, for these cases, the contributions of the off-diagonal disorder matrix elements, V0 and V1, are about twice that of the diagonal. On the other hand, in the limit where there is a large difference in the R and β site excitation energies, one can show that the relative importance of the off-diagonal disorder diminishes by examining the changes in the phase factors. Regarding the effect of correlation, the negative value of A amplifies the effect of off-diagonal disorder if 0 < χk < π, -π/2 < ηk < π/2, and -π/2 < ηk - 2πk/N < π/2. For model parameters that will be considered later, these conditions are met for the states considered. Thus, one can expect that the effect of disorder will be larger for negative A according to the prediction of the first-order perturbation theory. B. Gaussian Disorder. If all the random variables entering the disorder Hamiltonian obey Gaussian statistics, analytic expressions can be found for the moments of the matrix elements of the Hamiltonian. We assume the distribution function of the random variables are given by
{ (
P(uR,uβ,V0,V1) ∝ exp -
N
uR(n)2
n)1
2σ2R
∑
+
uβ(n)2 2σ2β
+
V0(n)2 2σ20 V1(n)2 2σ21
+
)}
()
(
{
()
(30)
)) } (31)
This determines the statistics of the mean energy of k( states
(32)
()
1 2 4 χk 4 χk σ sin + σ2β cos + N R 2 2 σ20(A2(cos2(ηk) + sin2(ηk) cos2(χk)) - 2Asin(χk) cos(ηk) +
〈|δHll(-k,k)|2〉 )
sin2(χk)] + σ21[A2(cos2(ηk - 2πk/N) + sin2(ηk - 2πk/N) cos2(χk)) -2Asin(ηk) cos(ηk - 2πk/N) + sin2(χk)]
}
(33)
The explicit forms of eqs 32 and 33 provide valuable insights into how different parameters affect the splitting patterns. IV. Simulation A. Model Parameters. The diagonal elements of H ˆ 0 in the site excitation basis are
Jss(0) ) s
(34)
where s with s ) R or β is the Qy excitation energy of the R or β BChl. The off-diagonal elements are calculated under the transition dipole approximation and using a simplified structure of the B850 ring. We assume the transition dipoles are placed on the same x-y plane and the angles of their directions relative to the z axis are the same. As a result, the positions of the transition dipoles can be specified by the following twodimensional vectors in the x-y plane:
r R n ) RR β
β
(
cos(2πn/N - ν) sin(2πn/N - ν)
)
(35)
with 2ν being the relative angle between rRn and rβn. The unit dipole moments of BChls are specified by
(
sin θ cos(2πn/N - ν + φR) β
µRn ) sin θ sin (2πn/N - ν + φR) β
β
cos θ
)
(36)
with φR and φβ being the angles of the dipole moments of R and β BChls relative to the above position vectors in the x-y plane. Then, defining the relative distance vector between sn and s′m BChls as ′
χk χk 1 + σ2βcos4 + σ20(A 〈δHll(k,k)2〉 ) σ2Rsin4 N 2 2 2πk 2 sin(χk) cos(ηk))2 + σ21 A - sin(χk) cos ηk N
(
where
}
lss nm ≡ rsn - rs′m
Then, by use of eq 28, one can show that the probability distribution of δHll(k,k) is also Gaussian with zero mean and with the following variance:
{ ()
{
D2l,k exp 〈|δHll(-k,k)|2〉 〈|δHll(-k,k)|2〉 2Dl,k
(37)
the off-diagonal elements of H ˆ 0 are given by ′
Jss′(n - m) )
Cκss nm
(38)
′
3 |lss nm|
where ′
′
′
′
ss ss ss 2 κss nm ) µsn ‚ µs′m - 3(µsn ‚ lnm)(µs′m ‚ lnm)/|lnm|
(39)
and C is a constant factor accounting for the local dielectric constant and the magnitude of transition dipole moment. The structural and energetic data for the B850 system of Rhodopseudomonas acidophila (Rps. acidophila), optimized by
B850 Band of LH2
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6659 TABLE 3: Eigenvalues of the Zeroth-Order Hamiltonian and the Phase Factors Calculated for Dipole Model (Dp) Parameters and the KSF Parameters of Rps. acidophila model
k
-1 E(0) l,k (cm )
χk/π
ηk/π
Dp
0 1 2 3 4 0 1 2 3 4
-96.11 0 195.75 420.30 629.77 -84.07 0 161.46 335.43 494.52
0.5061 0.5041 0.5003 0.4942 0.4778 0.5074 0.5048 0.5002 0.4924 0.4670
0 -0.09556 -0.1862 -0.2593 -0.2335 0 -0.1035 -0.2055 -0.3000 -0.3403
KSF
Figure 1. Comparison of the energy levels between the KSF model and the corresponding dipole approximation.
TABLE 1: Parameters for the Dipole Interaction Model of Rps. acidophila RR (Å)
Rβ
θ
ν
φR
φβ
C(cm-1 Å3)
26.04
27.18
84.91°
10.26°
-112.47°
63.20°
1.895 × 105
TABLE 2: Structural Data and Matrix Elements of the Zeroth-Order Hamiltonian for Rps. acidophila s,s′
n-m
lss′ nm(Å)
κss′ nm
Jss′(n - m) (cm-1)
R,R
1 2 3 4 1 2 3 4 0 1 2 3 4 5 6 7 8
17.812 33.476 45.103 51.289 18.592 34.942 47.077 53.534 9.542 26.834 40.928 50.093 53.219 49.930 40.622 26.422 9.077
-1.426 -1.132 -0.797 -0.579 -1.255 -0.961 -0.627 -0.409 1.692 1.335 0.965 0.638 0.484 0.560 0.819 1.111 1.126
-47.786 -5.715 -1.647 -0.814 -37.000 -4.269 -1.139 -0.505 368.932 13.090 2.667 0.962 0.608 0.853 2.314 11.405 285.183
β,β
R,β
(KSF),25
Kruger, Scholes, and Fleming were used to fit the present model. This system corresponds to N ) 9, and Table 1 lists the corresponding set of parameters of our model. The choice of C was made such that Jββ(1) of the present model agrees with that of KSF data. Table 2 provides the corresponding structural and energetic data. Comparison of these with the original KSF data show that the simplified structure of the present model can reproduce the main features of the original structure. As has been emphasized by KSF, the dipole approximation results in overestimation of the nearest neighbor interaction by about 25%. To examine the effects of this error, energy levels were calculated first using the data in Table 2 and second using the same set but with the three largest interactions replaced with those of KSF. It was assumed that R ) β, and the value of this in each case was adjusted such that the energy of the degenerate k ) (1 states is zero. Figure 1 compares the energy levels calculated by the two sets of parameters. The main differences of the two results are in the upper band and in the high lying states of the lower band, but very small differences are found in the low lying states of the lower band. Therefore, if the focus is on the spectroscopic properties of the B850 system as is the case here and as long
as the site excitation energy is adjusted by a proper reference value, use of dipole approximation seems valid because this does not affect the relative energetics of optically active states significantly. Table 3 provides the values of the lower band energy levels shown in Figure 1 and the phase factors defined by eqs 11 and 12. Again, the differences in the phase factors are small for the low lying states. This similarity of phase factors in the two parameter sets is important because they determine how the disorder affects the zeroth-order Hamiltonian. As a result, for moderately large disorder, where dominant mixing comes from neighboring eigenstates, the effects of the disorder for the low lying states will be similar for both the present transition dipole model and the KSF parameters. B. Results Simulations were performed using the parameters of the zeroth-order Hamiltonian provided in the preceding subsection and assuming the Gaussian distribution of eq 30 for the disorder Hamiltonian matrix elements. The goal is to examine the pattern of the distributions and correlations of the three lowest levels. These patterns are characterized in terms of the following two standard deviations: 2 w0 ≡ x〈El,0 〉 - 〈El,0〉2
(40)
2 〉 - 〈El,1m〉2 w1 ≡ x〈El,1m
(41)
with El,1m ) (El,1- + El,1+)/2, and in terms of the average gaps between levels given by
∆1,0 ≡ 〈El,1m - El,0〉
(42)
∆1,-1 ≡ 〈El,1+ - El,1-〉
(43)
First, the behavior due to diagonal and off-diagonal disorder is examined separately, where σ0 ) σ1 ) 0 in the former and σR ) σβ ) A ) 0 in the latter. This permits a straightforward comparison of the effects of diagonal vs off-diagonal disorder on energy levels. In each case, the two nonzero values of the standard deviations were kept the same, and varied from 0 to 2JRβ(0). For a given value of the standard deviation, sampling was made over 10 000 realizations of the Gaussian random variables. Figure 2 shows w0 and w1 of eqs 40 and 41, and Figure 3 shows ∆1,0 and ∆1,-1 of eqs 42 and 43. Also shown are the first-order perturbation theory results based on the expressions of section IIIB. The results of the simulation agree well with the predictions of the first-order degenerate perturbation theory in the weak disorder limit. The degree of agreement is better for w1 than for w0, which corresponds to the edge state. The qualitative fact
6660 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Figure 2. Values of w0 and w1 calculated with the variation of the degree of disorder. Filled circles represent the situation of only diagonal disorder (σ0 ) σ1 ) 0), while the open squares represent that of only off-diagonal disorder without any correlation (σR ) σβ ) A ) 0). Dashed lines are the first-order perturbation theory results for the former and the dot-dashed lines are those for the latter. The x-axis represent either σR ) σβ or σ0 ) σ1. All the results are shown in the unit of JRβ(0) ) 368.932 cm-1.
Figure 3. Values of ∆1,0 and ∆1,-1 with the variation of the degree of disorder. Filled circles represent the situation of only diagonal disorder (σ0 ) σ1 ) 0), while the open squares represent that of only off-diagonal disorder without any correlation (σR ) σβ ) A ) 0). Dashed lines are the first-order perturbation theory results for the former and the dotdashed lines are those for the latter. The x-axis represent either σR ) σβ or σ0 ) σ1. All the results are shown in the unit of JRβ(0) ) 368.932 cm-1.
that the off-diagonal disorder has a larger effect than the diagonal remains valid even up to the moderately large value of disorder tested. According to the predictions of the first-order perturbation theory, the value of ∆1,0 should remain constant, but it is seen that they deviate from the initial constant even from the very early stage. For the gap average, ∆1,-1, simulation results agree well with the value of the first-order perturbation theory results up to about ∆1,-1 ≈ JRβ(0)/4, beyond which the actual values tend to increase much more slowly than the estimates of the perturbation theory. Figures 2 and 3 demonstrate the differences of the diagonal and the off-diagonal disorder. However, there are some similarities between the two cases in relative variations of the four different quantities defined by eqs 40-43. On the basis of this observation, another plot was made from the same set of data using the value of w1 as the x-axis. Figure 4 shows the results,
Jang et al.
Figure 4. Values of w0, ∆1,0, and ∆1,-1 relative to those of w1 for the two situations of only diagonal disorder (filled circles) and of only off-diagonal disorder without correlation (open squares). All the results are shown in the unit of JRβ(0) ) 368.932 cm-1.
which exhibit striking similarity between two effects of diagonal and off-diagonal disorder. This indicates that the difference between the diagonal and the off-diagonal disorder, within the quantities considered and the moderate degree of disorder, is rather a quantitative one that can be measured in terms of the value of w1 and no major qualitative difference exists between the two. The fact that the off-diagonal disorder has larger effect than the diagonal can be inferred from the expressions of eqs 3133 and the values of the phase factors provided in Table 3. That is, due to the facts that χ0 ≈ χ1 ≈ 0 and η0 ≈ η1 ≈ 0, δHll(0,0), δHll(1,1), and δHll(1,-1) are affected about twice more by the off-diagonal disorder than by the diagonal in the weak disorder limit, although such a discrepancy diminishes as the degree of disorder increases. It should be noted, however, that this quantitative difference between the diagonal and the off-diagonal disorder holds true only when R ≈ β. If they differ by a large amount, the values of the phase factors appearing in Table 3 change, which in turn alter relative contributions of the diagonal and off-diagonal disorder. Physically, it is not likely that the standard deviation of the off-diagonal disorder can be comparable or larger than JRβ(0), but we considered these cases for a general understanding of the model. As the second case, the effects of the correlation between the diagonal and the off-diagonal disorder were examined without independent sources of the diagonal disorder (σR ) σβ ) 0). The case where A ) 1 and A ) -1 were considered. The previous case of only off-diagonal disorder corresponds to A ) 0. Simulations were performed in the same way as before, and the values of σ0 ) σ1 were varied from 0 to 2JRβ(0). Figure 5 shows w1 with varying σ0 ) σ1. The case of A ) 0 considered before is shown as a reference. The large difference between the cases of A ) 1 and A ) -1, predicted by the first-order perturbation theory, can be seen up to the value of σ0 ) σ1 ) JRβ(0)/2, but for stronger disorder the results of A ) 1 approach those of A ) -1. In the whole regime, it is seen that the negative correlation amplifies the effects of the off-diagonal disorder more than the positive correlation. One interesting fact is that, for the case of A ) -1, the simulation results agree fairly well with the first-order perturbation theory result even for a very large value of disorder, while it is not the case for A ) 1.
B850 Band of LH2
Figure 5. Values of w1 calculated with the variation of σ0 ) σ1. Uptriangles represent the case with A ) 1, while the down-triangles represent that with A ) -1. The dashed line is the first-order perturbation theory result for the former and the dot-dashed line is that for the latter. The results with A ) 0 were shown as squares for a reference. All the results are given in the unit of JRβ(0) ) 368.932 cm-1.
Figure 6. Values of w0, ∆1,0 and ∆1,-1 relative to those of w1. No independent source of diagonal disorder exists (σR ) σβ ) 0). The squares represent the case with A ) 0, the up-triangles A ) 1, and the down-triangles A ) -1. All the results are shown in the unit of JRβ(0) ) 368.932 cm-1.
To examine whether the common scaling behavior observed in Figure 4 exists, a similar plot was made, and the results are shown in Figure 6. The case of A ) 0 was shown again for a reference. The values of w1 were shown only up to JRβ(0), beyond which the same linear trend persists. The cases with A ) 0 and A ) -1 converge into almost one line while the case with A ) 1 deviates, the degree of which is largest for w0 while it is the smallest for ∆1,0. However, even for A ) 1, the slope of the variation seems to be approximately the same as those for other cases. Finally, the effects of the correlation between the diagonal and the off-diagonal disorder were considered in the presence of an independent source of diagonal disorder. For a fixed value of σR ) σβ ) 100 cm-1, which has some physical relevance as will be clear in the discussion, the value of σ0 ) σ1 was varied from 0 to JRβ(0), and the three different possibilities of the correlation, A ) 0, 1, and -1, were considered. Figure 7 shows the values of w1 with the variation of σ0 ) σ1. The patterns are similar to those of Figure 5 except that w1 starts from a finite value, due to the presence of the additional independent diagonal
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6661
Figure 7. Values of w1 calculated with the variation of σ0 ) σ1. There is an independent source of diagonal disorder such that σR ) σβ ) 100 cm-1. Squares represent the case with A ) 0, up-triangles A ) 1, and the down-triangles A ) -1. All the results are given in the unit of JRβ(0) ) 368.932 cm-1.
Figure 8. Values of w0, ∆1,0 and ∆1,-1 relative to those of w1. There is an independent source of diagonal disorder such that σR ) σβ ) 100 cm-1. The squares represent the case with A ) 0, the up-triangles A ) 1, and the down-triangles A ) -1. All the results are shown in the unit of JRβ(0) ) 368.932 cm-1.
disorder. To examine whether a similar scaling behavior is observed, the variations of other quantities were plotted against that of w1, in Figure 8. Almost the same patterns of variations as those of Figure 6 can be observed. The above analyses of eigenvalues suggest that w1 can be considered as the effective measure of disorder, and can be used to determine how different types of disorder affect the system. Although the case of A ) 1 showed some deviations in this respect, other cases cover a broad class of Gaussian disorder. Thus, the generality found here has important practical implications. However, examination of eigenvalues is not enough to fully understand the effect of disorder. Understanding the characteristics of wave functions in the presence of disorder is also important. One important quantity in this respect is the delocalization length, which provides crucial information on the excitonic nature of the system. Although this quantity26-29 is frequently discussed in disordered systems, the mathematical definition of the term itself is not unique. The most commonly used one is the inverse participation ratio.26 Here, we consider the inverse participation ratio of the lowest excited-state only, for which there is less ambiguity than other states due to the absence of nodes. The corresponding mathematical definition
6662 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Jang et al. broadening of the mean energy of the second and the third excited states can serve as an approximate measure for the effective strength of disorder, based on the study of eigenvalue structures and the delocalization length. V. Discussion
Figure 9. Delocalization lengths for different values of correlation in the presence of independent diagonal disorder such that σR ) σβ ) 100 cm-1. The x-axis is shown in the unit of JRβ(0) ) 368.932 cm-1.
is given by
1 1 ≡ P 〈Σ |c |4〉 j
(44)
j
where cj is the probability amplitude of the lowest excited state wave function at the jth site. Although this inverse participation ratio has a well-defined dynamical meaning and approaches correct values in both limits of completely delocalized and localized states, it tends to overestimate the degree of localization in the intermediate regime due to the quadratic account of each probability density. For this reason, we also considered an alternative measure given by
LD ≡ 〈
∑j min{1,2N|cj|2}〉
(45)
Unlike the inverse participation ratio, this does not give a weight more than unity per site and is expected to give a better quantification of the delocalization in the intermediate regime of the disorder strength. The above two measures of delocalization lengths were calculated only for the last model case, corresponding to Figures 7 and 8. The results are shown in Figure 9 as a function of w1. The inverse participation ratio shows a similar scaling behavior as the eigenvalues. That is, for the cases of A ) 0 and A ) -1, the same value of w1 results in almost the same value of 1/P. On the other hand, for A ) 1, it is somewhat smaller, which is consistent with larger values of w0 in Figure 8. The delocalization lengths defined by eq 45 are shown in the lower part of Figure 9, and they exhibit different patterns. First of all, for w1 g 0.2JRβ(0), this newly defined delocalization length is larger than 1/P by about 2 - 3. Second, LD is more sensitive to different models than 1/P, as can be seen from differences between the two cases of A ) 0 and A ) -1. However, even for this quantity, w1 seems to serve as an approximate measure for the delocalization length within the error of (1. Several conclusions can be drawn based on the results of the present simulation study. First, within the Gaussian statistics, the off-diagonal disorder has similar qualitative effects as the diagonal, and can be more effective in disrupting the symmetry of the system. Second, it was shown that possible existence of correlation between the diagonal and the off-diagonal disorder can either amplify or suppress the effect of disorder in a nontrivial way. Third, it was found that the inhomogeneous
According to the simulation results presented in the preceding section, common scaling relations exist among w0, w1, ∆1,0, and ∆1,-1 for a broad class of Gaussian disorder. Since these quantities are experimental observables, one can compare them for the specific case of the B850 band in the LH2 ring of Rps. acidophila and then examine whether the assumption of Gaussian disorder provides a consistent picture. To this end, considerations of only low-temperature experimental results are suitable because the role of dynamic disorder is minimized. We use the low-temperature hole-burning experimental data as the starting point. Wu et al.8 report that, at 4.2 K, the inhomogeneous broadening of the lowest excited state is 120 cm-1 and the gap between the maximum of the B850 peak and the lowest excited state is 200 cm-1. If the former is identified as the full width at halfmaximum of the lowest excited state and the latter as the average gap between the mean of the second and the third excited states and the lowest one, these data imply that w0 ≈ 100 cm-1 (w0/ JRβ(0) ≈ 0.27) and ∆1,0 ≈ 200 cm-1 (∆1,0/JRβ(0) ≈ 0.54). Inspection of Figures 4, 6, and 8 shows that these two values correspond to a common value of w1 ≈ 74 cm-1 (w1/JRβ(0) ≈ 0.2) for all the types of Gaussian disorder, except for the positively correlated case of A ) 1. Thus, the four situations: only diagonal disorder, only off-diagonal disorder, independent diagonal and off-diagonal disorder, and negatively correlated diagonal and off-diagonal disorder, are all consistent with the hole-burning experiment. From Figures 4, 6, an 8, we find the common corresponding value of ∆1,-1 is about 100 cm-1 (∆1,-1/ JRβ(0) ≈ 0.27). Interestingly, this value is quite close to 110 cm-1, the average gap reported by van Oijen et al.12 from their single molecular spectroscopy (SMS) experiment at 1.2 K. The value of w1 estimated above is also comparable to the width of the distribution of the mean energy of k ) 1( states obtained from the same SMS experiment, but reported more recently.18 In the original analysis12 of the SMS data and in a quite recent theoretical analysis,18 an elliptic distortion in the B850 ring was considered as the most plausible explanation of the large gap between k ) 1( states. However, the analyses provided above suggest that the large gap may be explained in a way fully consistent with the hole-burning experiment based on the Gaussian disorder only. Even though the two explanations give the same value of ∆1,-1, they can be distinguished in terms of other quantities, w1, w0, and ∆10. In addition, the distribution functions of ∆1,-1 should be different. According to the analysis of Mostovoy and Knoester,18 however, the large value of w1 can be reconciled with the elliptic distortion if additional interLH2 disorder is assumed. However, the values of ∆10 are still different. While assumption of only Gaussian disorder predicts it to be about 200 cm-1 in favor of the hole-burning data, the value based on the elliptic distortion is about 66-100 cm-1.18 Thus, a careful determination of ∆10 is critical in clarifying which assumption holds true. Alternatively, considering the fact that the distribution functions of ∆1,-1 are different, SMS data with enough statistics can also clarify this issue. At present, however, no definite statement seems possible. With this inconclusive issue left behind, in the following, we further present more detailed aspects of our analysis based on simple Gaussian disorder.
B850 Band of LH2
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If one assumes that the disorder is solely diagonal, Figure 2 indicates that the required value of its standard deviation producing w1 ≈ 74 cm-1, is about 280 cm-1 (σR/JRβ(0) ) σβ/ JRβ(0) ≈ 0.76). However, this is not a likely situation considering that the standard deviation of the disorder in the site excitation energy of B800 is about 100 cm-1.12 There is no specific reason the BChl in the B850 band should have much larger site energy disorder. By a similar reason, assuming only off-diagonal disorder is not so plausible. Therefore, it is most likely that there are both diagonal and off-diagonal disorder in the B850 band. In fact, this corresponds to the last model case considered in the preceding section, where we have deliberately chosen the standard deviation of the diagonal disorder as 100 cm-1. Both the independent disorder model and the negatively correlated disorder model are consistent with all the physical facts and reasoning. According to Figure 7, the required values of the standard deviations so as to produce w1 ≈ 74 cm-1, is about 130 cm-1 (σ0/JRβ(0) ) σ1/JRβ(0) ≈ 0.35) for the independent case and about 70 cm-1 (σ0/JRβ(0) ) σ1/JRβ(0) ≈ 0.19) for the negatively correlated case of A ) -1. The remaining question is whether there can be actual physical mechanisms producing this amount of off-diagonal Gaussian disorder. Some qualitative answers for this can be found by considering the actual physical mechanisms of the off-diagonal disorder. Instead of assuming Gaussian statistics for the off-diagonal elements of the disorder Hamiltonian, we have performed simulations assuming Gaussian fluctuations of the structural parameters of the B850 ring. Three different realizations of geometric fluctuations were considered independently. The first type is called R-Gaussian disorder, which assumes radial fluctuations in the position vector of each dipole moment as follows:
rRn ) (RR + δRRn) β
β
β
(
cos(2πn/N - ν) sin(2πn/N - ν)
)
(46)
where δRRn and δRβn are Gaussian random variables with the standard deviation of σR. The second type is called ν-Gaussian, which assumes angular fluctuations in the position vector of each dipole moment as follows:
r R n ) RR β
β
(
cos(2πn/N - ν + δνRn) β
sin(2πn/N - ν + δνRn) β
)
(47)
where δνRn and δνβn are Gaussian random variables with the standard deviation of σν. In both cases of R-Gaussian and ν-Gaussian disorder, the dipole moment vectors maintain their original symmetry and are given commonly by the expression of eq 36. The third type, however, corresponds to the opposite situation where only the orientation of dipole moment vectors fluctuate. This is called φ-Gaussian, and the dipole moment vectors have angular fluctuations in the plane of B850 ring as follows:
(
sin θ cos(2πn/N - ν + φR + δφRn) β
β
β
β
µRn ) sin θ sin (2πn/N - ν + φR + δφRn) β
cos θ
)
(48)
where δφRn and δφβn are Gaussian random variables with the standard deviation of σφ. All these models do not conform exactly to the one proposed in section IIIA because the nonnearest neighbor terms do not vanish, although small. In this sense, the results provided here can also serve as tests for the validity of the simplification made in section IIIA.
Figure 10. Results for three types of off-diagonal disorder. Upper part shows w1 with the variation of the scaled standard deviation, σs, which is equal to σR/(10 Å), 10σν, and σφ. There are independent diagonal disorder such that σR ) σβ ) 100 cm-1. The lower part shows w0 vs w1 for the three types and the result for Gaussian off-diagonal disorder in the previous section are shown as dashed line.
Simulation was performed as described in section IIIB. For all of the three types of off-diagonal disorder, it was assumed that σR ) σβ ) 100 cm-1 and A ) 0. The ranges of the standard deviations were chosen as follows: 0 e σR e 4 Å, 0 e σν e 0.04, and 0 e σφ e 0.4, and the results are shown in Figure 10. The upper part provides the values of w1 with the variation of the scaled standard deviation σs ) σR/(10 Å) ) 10σν ) σφ, and the lower part shows the variation of w0 vs w1. The dashed line corresponds to the Gaussian off-diagonal disorder with only nearest neighbor terms and with A ) 0. Interestingly, the behavior of R-Gaussian and φ-Gaussian remain quite close to that of the Gaussian off-diagonal disorder even up to the value of w1 ≈ 74 cm-1 (w1/JRβ(0) ≈ 0.2), while the results of ν-Gaussian show deviations starting from the region of w1 ≈ 37 cm-1 (w1/JRβ(0) ≈ 0.1). This indeed indicates that the R-Gaussian and φ-Gaussian disorder can be possible sources of the off-diagonal disorder that might be found in the B850 band, even without assuming negative correlation. However, ν-Gaussian shows substantial deviation from the Gaussian offdiagonal before it can produce the required amount of inhomogeneous broadening. Thus, unless there is a mechanism which brings negative correlation of the diagonal disorder to the φ-Gaussian disorder, it may not play important roles compared to other types of disorder. For the detailed geometric model of the off-diagonal disorder, one can examine spectroscopic properties such as the intensity of each excitonic state. Figure 11 provides the average intensities of the five lowest states, for the cases of R-Gaussian and φ-Gaussian, in the unit of the single BChl intensity. For the R-Gaussian, the intensities of k ) 1- and k ) 1+ states remain quite close, while for the φ-Gaussian, the intensity of k ) 1state becomes larger as the magnitude of disorder increases. We have performed similar calculations for Gaussian diagonal disorder, Gaussian off-diagonal disorder, and ν-Gaussian disorder. Interestingly, the former two cases show similar behavior as R-Gaussian, while the latter shows a behavior opposite to that of φ-Gaussian. These results imply that different geometric mechanisms of the off-diagonal disorder may be distinguished in terms of the intensity ratio of the two brightest states, which can be confirmed by SMS experiment. For both models of offdiagonal disorder, the average intensity of k ) 0 state when w1 ) 0.2JRβ(0) is about 2.3. This is 13% of the total intensity. While
6664 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Figure 11. Intensities of the lowest five states for the two types of R-Gaussian and φ-Gaussian disorder with the variation of w1.
this value is comparable to the emission strength of B850 in the zero temperature limit,2 it is much larger than the estimate of 3-5% obtained from the hole-burning experiment,8 and is somewhat larger than the estimates of the SMS experiment.18 On the other hand, the intensities of k ) 2( states are about 1, respectively. Although not shown here, the higher states have negligibly small intensities. Thus, the higher states observed in the SMS experiment12 may be k ) 2( states according to our interpretation based on Gaussian disorder only, but it is not clear whether this view is consistent with the energies and the polarizations observed in the experiment. VI. Concluding Remarks On the basis of the simulations of a general model of Gaussian disorder and the comparison of the common scaling relation observed with major experimental data at low temperature, we have found that the energy distribution pattern in the B850 band of Rps. acidophila is roughly consistent with Gaussian disorder. The estimates of the four quantities defined by eqs 40-43, for this system, are as follows: w0 ≈ 100 cm-1, w1 ≈ 74 cm-1, ∆1,0 ≈ 200 cm-1, and ∆1,-1 ≈ 100 cm-1. One may ask how sensitive these results are to the specific parameter set we used. Because of the near linearity of the scaling curves among the four quantities in the regime of interest and the insensitivity of the phase factors of the low lying states to changes in the parameter set, it is expected that the final numbers do not change much even if one uses other sets of available parameters. The major effect will be to rescale all the figures with the ratio of JRβ(0)’s. Since we use the figures to relate experimental data, the estimated values of broadenings and the disorder, reported in cm-1, the delocalization length, and the intensities are relatively insensitive to the choice of parameters. To confirm this, we performed similar calculations using the KSF parameters. The resulting scaling curves among the four were almost identical to those in the present paper. The maximum deviation was seen for the curve of ∆10/JRβ(0) vs w1/JRβ(0), and this difference was at most 5%, leading to differences in the four quantities on the order of 10 cm-1. Assuming that the site energy disorder in the B850 ring is the same as that in the B800 ring, 100 cm-1, we concluded that the desired amount of total Gaussian disorder can be achieved with the participation of off-diagonal disorder. Detailed considerations of the geometric mechanisms of the off-diagonal disorder and test simulations demonstrated that fluctuations in
Jang et al. the radius of the transition dipole positions and/or the transition dipole angles are plausible sources. The estimated values of the standard deviations of these two mechanisms, according to Figure 10, are, respectively, σR ≈ 2.6 Å and σφ ≈ 0.3 ≈ 17°, which are not unreasonable. If both of these mechanisms coexist, other sources of disorder participate, or there is negative correlation between diagonal and the off-diagonal, the necessary values of the standard deviations of off-diagonal disorder decrease, making the possibility of the off-diagonal disorder even more likely. The conclusion drawn here may provide an alternative to the elliptic distortion explanation of the SMS data. However, our intention is not to exclude the possibility of elliptic distortion. Rather, we tried to make a clear distinction of the two different interpretations, by providing analyses based on pure Gaussian disorder. For the clarification of this, further SMS experiments and hole-burning experiments under the same condition are necessary. In the present work, we did not consider correlated Gaussian disorder for which the standard deviations are different depending on the symmetry of the disorder. This possibility cannot be totally excluded especially in the low-temperature limit and for off-diagonal disorder because some low energy phonon modes may still play a role. In addition, there is evidence that the actual site excitation energy of R and β BChls can be different.6,13,14 This aspect can alter some of the results provided in the present work, but the conclusion that the off-diagonal disorder is important is likely to remain true. Acknowledgment. S.J. would like to thank Profs. G. R. Fleming and R. van Grondelle for helpful discussions. This research was supported by an NSF grant to MIT. References and Notes (1) Hu, X.; Schulten, K. Phys. Today 1997 August, 28. (2) Sundstro¨m, V.; Pullerits, T.; van Grondelle, R. J. Phys. Chem. B 1999, 103, 2327. (3) McDermott, G.; Prince, S. M.; Freer, A. A.; HawthornthwaiteLawless, A. M.; Paplz, M. Z.; Cogdell, R. J.; Issacs, N. W. Nature 1995, 374, 517. (4) Koepke, J.; Hu, X.; Muenke, C.; Schulten, K.; Michel, H. Structure 1996, 4, 581. (5) Freer, A.; Prince, S.; Sauer, K.; Papiz, M.; HawthornthwaiteLawless, A.; McDermott, G.; Cogdell, R.; Isaacs, N. W. Structure 1996, 4, 449. (6) Koolhaas, M. H. C.; van der Zwan, G.; Frese, R. N.; van Grondelle, R. J. Phys. Chem. B 1997, 101, 7262. (7) Chachisvilis, M.; Ku¨hn, O.; Pullerits, T.; Sundstrom, V. J. Phys. Chem. B 1997, 101, 7275. (8) Wu, H.-M.; Reddy, N. R. S.; Small, G. J. J. Phys. Chem. B 1997, 101, 651. (9) Wu, H.-M.; Ratsep, M.; Jankowiak, R.; Cogdell, R. J.; Small, G. J. J. Phys. Chem. B 1997, 101, 7641. (10) Freiberg, A.; Jackson, J. A.; Lin, S.; Woodbury, N. W. J. Phys. Chem. A 1998, 102, 4372. (11) Freiberg, A.; Timpmann, K.; Ruus, R.; Woodbury, N. W. J. Phys. Chem. B 1999, 103, 10032. (12) van Ojien, A. M.; Ketelaars, M.; Ko¨hler, J.; Aartsma, T. J.; Schmidt, J. Science 1999, 285, 400. (13) Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 2000, 104, 1854. (14) Koolhaas, M. H. C.; van der Zwan, G.; van Grondelle, R. J. Phys. Chem. B 2000, 104, 4489. (15) Wu, H.-M.; Small, G. J. J. Phys. Chem. B 1998, 102, 888. (16) Linnanto, J.; Korppi-Tommola, J. E. I.; Helenius, V. M. J. Phys. Chem. B 1999, 103, 8739. (17) Mukai, K.; Abe, S.; Sumi, H. J. Lumin. 2000, 87-89, 818. (18) Mostovoy, M. V.; Knoester, J. J. Phys. Chem. B 2000, 104, 12355. (19) Bopp, M. A.; Sytnik, A.; Howard, T. D.; Cogdell, R. J.; Hochstrasser, R. M. Proc. Nat. Acad. Sci. U.S.A. 1999, 96, 11271. (20) Book, L. D.; Ostafin, A. E.; Ponomarenko, N.; Norris, J. R.; Scherer, N. F. J. Phys. Chem. B 2000, 104, 8295. (21) Dempster, S. E.; Jang, S.; Silbey, R. J. J. Chem. Phys. In press. (22) Dempster, S. E.; Jang, S.; Silbey, R. J. In preparation.
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