Article Information

• PDF (300 kb)
• Full Text with Large Figures

PubMed

• Articles by Vladimir Novoderezhkin
• Articles by Rienk van Grondelle

Related Articles

• Solvation Effect of Bacteriochlorophyll Excitons in Light-Ha...

  Solvation Effect of Bacteriochlorophyll Excitons in Light-Harvesting Complex LH2 Biophysical Journal, Volume 93, Issue 6, 15 September 2007, Pages 2188-2198 V. Urbonienė, O. Vrublevskaja, G. Trinkunas, A. Gall, B. Robert and L. Valkunas Abstract We have characterized the influence of the protein environment on the spectral properties of the bacteriochlorophyll (Bchl) molecules of the peripheral light-harvesting (or LH2) complex from . The spectral density functions of the pigments responsible for the 800 and 850nm electronic transitions were determined from the temperature dependence of the Bchl absorption spectra in different environments (detergent micelles and native membranes). The spectral density function is virtually independent of the hydrophobic support that the protein experiences. The reorganization energy for the B850 Bchls is 220cm, which is almost twice that of the B800 Bchls, and its Huang-Rhys factor reaches 8.4. Around the transition point temperature, and at higher temperatures, both the static spectral inhomogeneity and the resonance interactions become temperature-dependent. The inhomogeneous distribution function of the transitions exhibits less temperature dependence when LH2 is embedded in membranes, suggesting that the lipid phase protects the protein. However, the temperature dependence of the fluorescence spectra of LH2 cannot be fitted using the same parameters determined from the analysis of the absorption spectra. Correct fitting requires the lowest exciton states to be additionally shifted to the red, suggesting the reorganization of the exciton spectrum. Abstract | Full Text | PDF (201 kb)

• Pigment–pigment interactions and energy transfer in the ante...

  Pigment–pigment interactions and energy transfer in the antenna complex of the photosynthetic bacterium Rhodopseudomonasacidophila Structure, Volume 4, Issue 4, 1 April 1996, Pages 449-462 Andy Freer, Steve Prince, Ken Sauer, Miroslav Papiz, Anna Hawthornthwaite Lawless, Gerry McDermott, Richard Cogdell and Neil W Isaacs Summary The structure of the antenna complex not only shows Nature at its most aesthetic but also illustrates how clever and efficient the energy transfer mechanism has become, with singlet–singlet excitation being passed smoothly down the spectral gradient to the reaction centre. Summary | Full Text | PDF (983 kb)

• Comparative Study of Spectral Flexibilities of Bacterial Lig...

  Comparative Study of Spectral Flexibilities of Bacterial Light-Harvesting Complexes: Structural Implications Biophysical Journal, Volume 90, Issue 7, 1 April 2006, Pages 2463-2474 Danielis Rutkauskas, John Olsen, Andrew Gall, Richard J. Cogdell, C. Neil Hunter and Rienk van Grondelle Abstract This work presents a comparative study of the frequencies of spectral jumping of individual light-harvesting complexes of six different types: LH2 of , , and ; LH1 of ; and two “domain swap mutants” of LH2 of : PACLH1 and PACLH2mol, in which the -polypeptide C-terminus is exchanged with the corresponding sequence from LH1 of or LH2 of , respectively. The quasistable states of fluorescence peak wavelength that were previously observed for the LH2 of were confirmed for other species. We also observed occurrences of extremely blue-shifted spectra, which were associated with reversible bleaching of one of the chromophore rings. Different jumping behavior is observed for single complexes of different types investigated with the same equivalent excitation intensity. The differences in spectral diffusion are associated with subtle differences of the binding pocket of B850 pigments and the structural flexibility of the different types of complexes. Abstract | Full Text | PDF (276 kb)

• …more

Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 77, Issue 2, 666-681, 1 August 1999

doi:10.1016/S0006-3495(99)76922-5

Biophysical Theory and Modeling

Disordered Exciton Model for the Core Light-Harvesting Antenna of Rhodopseudomonas viridis

Vladimir Novoderezhkin^*, René Monshouwer^# and Rienk van Grondelle^#, ,  

^* A. N. Belozersky Institute of Physico-Chemical Biology, Moscow State University, Moscow 119899, Russia
^# Department of Biophysics, Faculty of Sciences, Vrije Universiteit, 1081 HV Amsterdam, the Netherlands

Address reprint requests to Dr. Rienk van Grondelle, Faculty of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, the Netherlands. Tel.: 31-20-444-7930; Fax: 31-20-444-7999.

The primary processes of photosynthesis include the absorption of solar photons by the pigments of the light-harvesting antenna, followed by ultrafast energy transfer until a reaction center is reached in which a charge separation can be initiated (van Grondelle, 1985,van Grondelle et al,Sundström et al). Recent advances in high-resolution structural studies of bacterial light-harvesting antenna systems (McDermott et al,Koepke et al,Hu et al) have strongly stimulated experimental and theoretical investigations to unravel the fundamental physical principles that are the basis for the very efficient energy transfer in these antenna complexes. From the structural and earlier biochemical (Zuber and Cogdell, 1995) studies it is now well established that antenna complexes of purple bacteria consist of α-β pigment-protein subunits arranged in high-symmetry ring-like structures. Each subunit consists of a pair of transmembrane polypeptides, α and β, binding two or three bacteriochlorophyll (BChl) molecules, and in some species a third polypeptide γ was identified (Zuber and Brunisholz, 1991). The spatial organization of the pigments in these rings is currently known with high precision in two peripheral LH2 complexes (McDermott et al,Koepke et al); all extensions to other complexes (in particular the core complex LH1) are educated guesses (Hu et al). Although the spatial organization is known, the nature of the spectra and dynamics of electronic excitations still remains unclear.

Using the structural parameters from the x-ray data, several groups have modeled the exciton spectra of these ring-like complexes, taking into account the site inhomogeneity of the pigments (Dracheva et al,Dracheva et al,Jimenez et al,Sauer et al,Alden et al,Monshouwer et al,Hu et al,Wu et al,Wu et al). Different estimations of the interaction energy and the site inhomogeneity values in these papers have resulted in different predictions for the degree of exciton delocalization in the light-harvesting antenna. A direct estimate of the delocalization degree is obtained from the shape and amplitude of difference absorption spectra (Novoderezhkin and Razjivin, 1993,Van Burgel et al,Pullerits et al). Analysis of the room-temperature pump-probe spectra for the LH2 antenna complexes of purple bacteria suggests exciton delocalization over four BChls (Pullerits et al,Kühn and Sundström, 1997). Relative difference absorption measurements of the LH2 antenna and the B820 dimeric subunit revealed delocalization over five BChls (Novoderezhkin et al., manuscript submitted for publication). Analysis of the shapes and relative amplitudes of difference absorption of the B866 band and the B808 monomeric band of the B808–866 complex from green bacteria (analogous to LH2) gave a delocalization value of about five or six BChls (Novoderezhkin and Fetisova, manuscript submitted for publication). From nonlinear absorption experiments an even larger delocalization degree was proposed (Leupold et al).

Probably the most special and mysterious among all of the species of purple bacteria is the BChl b–containing bacterium Rhodopseudomonas (Rps.) viridis. Electron microscopy studies have shown a regular circular organization of the LH1 antenna (Miller, 1982,Stark et al,Ikedayamasaki et al) around the reaction center. The main absorption band around 1000nm is very redshifted and was found to be heterogeneous, with at least three spectral bands contributing to the major LH1 peak (Monshouwer et al). The corresponding maxima of the second derivative of the membrane absorption spectrum at 4.2K are 1049, 1042, and 1030nm. This is in remarkable contrast to the LH1 absorption band of BChl a–containing species (van Mourik et al).

Low-temperature emission spectra of membranes of Rps. viridis were studied for different excitation wavelengths (Monshouwer et al). The shape of the red edge of the emission spectrum and the position of the maximum of the emission (1054nm) do not change when the excitation wavelength is tuned from the blue to the red edge of the absorption band (from 1018 to 1047nm). Polarization of the emission for these excitation wavelengths is constant (r=0.1) and only starts to increase upon excitation in the very red edge (from 1050nm). All of these data strongly suggest very efficient excitation transfer (or relaxation) to the redmost pigment pool (or exciton level), which was denoted as the B1045 spectral form (Monshouwer et al).

Recently, time-resolved pump-probe measurements of membranes of Rps. viridis were performed at different temperatures (Monshouwer et al). It was found that the major relaxation processes take place within the first picosecond after excitation. The ultrafast redshift of the difference absorption spectrum (with a time constant of ∼130fs) accompanied by an anisotropy decay (time constant of 150fs) were taken to reflect the electronic energy transfer and dephasing processes. The coupling of the electronic transitions with two vibrational modes (65 and 103cm⁻¹) gives rise to strong oscillations at all detection wavelengths and all temperatures. The surprisingly long decay of these oscillations (700–800fs) indicates that the ultrafast electronic energy transfer does not shorten the vibronic dephasing.

In this paper we have modeled the linear absorption and pump-probe spectra obtained by Monshouwer et al, using an exciton model for the LH1 antenna of Rps. viridis. The spectral heterogeneity of the LH1 antenna was explained in terms of the exciton splitting of the major electronic transition due to resonant interactions in a ring-like aggregate of BChls. Within the context of this model, we calculated the degree of exciton delocalization at different temperatures, using the site inhomogeneity value and the other parameters obtained from a simultaneous fit of absorption and pump-probe spectra.

The elementary subunit of the core antenna of Rps. viridis consists of three transmembrane polypeptides α, β, and γ (Miller, 1982,Stark et al,Zuber and Brunisholz, 1991). The α and β polypeptides, binding one BChl molecule each, are analogous to the LH1 proteins found in BChl a–containing bacteria. The γ-polypeptide probably does not bind BChl (Zuber and Brunisholz, 1991). The available structural information suggests that the photosynthetic unit of Rps. viridis consists of one reaction center surrounded by six antenna subunits (α₂β₂γ₂BChl₄) (each containing two α, two β, and two γ polypeptides and four BChls) arranged in a ring-like structure with sixfold symmetry. The spatial arrangement of the BChls should have at least the same (sixfold) or even higher (12-fold) symmetry (the exact nature of the BChl organization in this antenna complex is not known). Alternatively, the LH1 may be a 16-fold symmetrical ring of 16 (αβγBChl₂) subunits, as was shown to be the case for the BChl a–containing species (Karrasch et al,Walz et al). In this case the spatial arrangement of the BChls should also exhibit 16-fold symmetry.

We suppose that the pigment arrangement in the antenna of Rps. viridis is analogous to that of the BChl a–containing bacteria. As a model for the antenna of Rps. viridis we consider a circular aggregate of 24–32 BChl b molecules with either C₁₂ or C₁₆ symmetry (the elementary unit cell contains two BChl b molecules, bound to the α- and β-polypeptides). The Q_y transition dipole moments of the two BChls in a dimeric unit cell form angles ψ₁, ψ₂ with the plane of the circle and angles φ₁, φ₂ with the tangent to the circle (ψ₁ and ψ₂ take values from −90° to 90°; φ₁ and φ₂ from 0° to 360°). The unperturbed Q_y electronic transition energies of these two BChls are E₁ and E₂. The Mg-Mg distance between BChls in a dimeric unit is r₁₂ and between nearest BChls from different units is r₂₃. We further assume that ψ₁=10°, ψ₂=5°, φ₁=20°, φ₂=200°, r₁₂=0.87nm, r₂₃=0.97nm. These parameters are approximately the same as those for the strongly coupled B850 ring of BChl a's in the LH2 antenna from Rhodopseudomonas (Rps.) acidophila (McDermott et al). The difference between E₁ and E₂ was varied from 0 to 600cm⁻¹. The ratio of the transition dipoles for the S₁-S₂ and S₀-S₁ transitions in the BChl monomer, χ, was varied from 0 to 1.5. In our simulations of the long-wavelength absorption band, we have taken into account the interactions between the Q_y transitions of BChls, neglecting their mixing with Q_x, B_y, B_x transitions as well as with charge transfer states (Alden et al).

We have assumed that the interaction energies between BChl b molecules are M₁₂=400cm⁻¹, M₂₃=290cm⁻¹, and M₁₃=−52cm⁻¹, where M₁₂ corresponds to the intradimer interaction, M₂₃ to the interdimer nearest-neighboring interaction, and M₁₃ to the next nearest-neighbor interaction, respectively. Furthermore, we tested two alternative sets of interaction energies: “high” energies, M₁₂=600cm⁻¹, M₂₃=440cm⁻¹, M₁₃=−78cm⁻¹, and “low” energies, M₁₂=260cm⁻¹, M₂₃=190cm⁻¹, M₁₃=−34cm⁻¹. Note that microscopic calculations using the point charge approximation gave M₁₂=806cm⁻¹, M₂₃=377cm⁻¹, M₁₃=−152cm⁻¹ for the LH2 complex of Rhodospirillum molischianum (Hu et al), and M₁₂=197–545cm⁻¹, M₂₃=158–461cm⁻¹, with various treatments of the dielectric screening for the LH2 complex of Rps. acidophila (Alden et al). For the LH2 complex of Rhodobacter (Rb.) sphaeroides, an analysis of the experimental CD spectrum yielded M₁₂=300cm⁻¹ and M₂₃=233cm⁻¹ (Koolhaas et al).

The site inhomogeneity of the LH1 antenna was described by uncorrelated perturbations δE of the electronic energies of the BChl pigments (uncorrelated diagonal disorder). The δE values were randomly taken from a Gaussian distribution W(δE)=π^−1/2Δ⁻¹exp(−δE²/Δ²). The width (full width at half-maximum, FWHM) of this distribution, σ=2Δ(ln 2)^1/2, was varied from 0 to 1000cm⁻¹. The Monte Carlo calculations of the linear and nonlinear (pump-probe) absorption spectra included:

We start with a study of the structure of one-exciton states of the antenna that are responsible for the linear absorption line shape. In Table 1 the parameters of the one-exciton states for the 12-fold symmetrical circular aggregate are shown for the “normal” set of interaction energies (M₁₂=400cm⁻¹, M₂₃=290cm⁻¹, and M₁₃=−52cm⁻¹), in the homogeneous limit (σ=0) and in the presence of disorder (σ=440cm⁻¹). For both cases we have calculated the exciton structure with either equal or nonequal transition energies of BChls in an α-β unit, i.e., taking E₁−E₂=0, and E₁−E₂=600cm⁻¹. The zero of energy is taken to be (E₁+E₂)/2. The dipole strengths D_x, D_y, D_z, D=D_x+D_y+D_z of the exciton components are normalized to the dipole strength of the monomeric S₀-S₁ transition (x and y axes are in the plane of the ring, the z axis is perpendicular to the plane). The dipole strengths are averaged over disorder, for example, D_x for any particular exciton state actually means 〈D_x〉, where brackets indicate averaging over realizations of diagonal energies. Energies of one-exciton transitions E were calculated as 〈ED〉/〈D〉, i.e., they correspond to the center of the spectral line (which can be slightly different from the line maximum because of asymmetry of the inhomogeneous line shape).

For a homogeneous aggregate with a dimeric unit cell, the lowest exciton level is the out-of-plane (z-) polarized k=0 level of the lower Davydov component, where k is the exciton wavenumber. The next two are the in-plane polarized twofold degenerate levels, k=±1. The higher levels (in increasing order of energy) are the k=±2, … k=±5, k=6 levels of the lower Davydov component, and k=6, k=±5, … ±1, k=0 levels of the higher Davydov component. In the homogeneous model only the k=0 and k=±1 levels are dipole allowed. If the angles ψ₁, ψ₂ are small, and the φ₁−φ₂ value is close to 0° or 180°, then the largest part of the dipole strength of the circular aggregate will be concentrated in the k=±1 levels of the lower Davydov component. In this case the k=0 levels of both Davydov components are very weak, and their intensities are proportional to (ψ₁−ψ₂)² for the lower and (ψ₁+ψ₂)² for the higher Davydov component (see Table 1). If we increase the asymmetry of the dimeric unit cell by increasing the energy difference, E₁−E₂, this will give rise to a larger Davydov splitting and to some redistribution of dipole strength between the Davydov components (without any changes of the exciton structure within the Davydov components; see Table 1). The exciton structure within the Davydov components can be changed only by a perturbation that breaks the symmetry of the ring, but not the symmetry within a dimeric unit cell. Such a situation is in fact realized in a spectrally disordered ring. In this case the k=0, k=±2, … k=±5, k=6 levels become dipole allowed, borrowing some fraction of the dipole strength from the k=±1 levels (see Table 1). The situation is best described as a mixing of the wavefunctions of the homogeneous aggregate, induced by the site inhomogeneity. Note that the mixture of the k=±1 and k=0 wavefunctions results in a change of polarization of the lowest exciton level from out-of-plane to in-plane, together with an increase of its intensity. Another important result concerns the splitting between the k=±1 levels as well as the increase in the splitting between these and the lowest k=0 level. For example, in the homogeneous limit the gap between the k =±1 and k=0 levels is 38cm⁻¹, whereas for σ=440cm⁻¹ the k=−1 and k=1 levels are shifted by 71cm⁻¹ and 126cm⁻¹ from the k=0 level (Table 1). All of these effects of inhomogeneity become more pronounced in the presence of intradimer asymmetry, E₁−E₂ (Table 1).

The shape of the difference absorption spectra as measured in pump-probe spectroscopy is determined by photobleaching (PB) and stimulated emission (SE) of the one-exciton levels and by excited state absorption (ESA) due to transitions from the one- to the two-exciton states. In Fig. 1 the calculated steady-state difference absorption spectrum and its PB, SE, and ESA components are shown (by “steady state” we mean with respect to excitonic and vibrational relaxation in the exited state of the aggregate). In general, the PB and SE spectra consist of N exciton components, whereas the ESA spectrum is a sum of about N³/2 transitions from N one-exciton levels to N(N+1)/2 two-exciton levels. The most intense ESA lines, corresponding to transitions from a few low one-exciton levels, are blue-shifted with respect to the PB/SE lines, giving rise to a specific sigmoid spectrum, which is characteristic for a circular aggregate (Novoderezhkin and Razjivin, 1993,Novoderezhkin and Razjivin, 1995a), as well as for a quasilinear aggregate (Van Burgel et al,Pullerits et al).

Display large version of this figure

Figure 1

The calculated spectral shapes of the photobleaching (PB), stimulated emission (SE), excited state absorption (ESA), and the resulting difference absorption spectrum at 77K. Parameters correspond to fit no. 3 (see Table 2), i.e., N=24, M₁₂=400cm⁻¹, γ_1L=40cm⁻¹, γ_1H=182cm⁻¹, γ₂=230cm⁻¹, σ=440cm⁻¹, E₁−E₂=0, and the Stokes shift is 110cm⁻¹.

In Figure 2 and Figure 3 and Figure 4 and Figure 5 the experimental absorption and pump-probe spectra are shown together with the calculated spectra. For this fit we used the steady-state pump-probe spectra measured at 1.6ps after excitation. Parameters γ_1L, γ_1H, γ₂, σ, and the value of Stokes shift, which gave the best fit for different N, M₁₂, M₂₃, M₁₃, and E₁−E₂ values are listed in Table 2. Typically, excitonic interactions explain only part of the red shift of the absorption maximum of LH1 with respect to the absorption peak of the BChl b monomer. To obtain the correct position of the experimental absorption maximum, we have assumed that the in situ electronic transition energies of both BChl b's in the dimeric unit are E₁+ΔE and E₂+ΔE, where ΔE is a free parameter, different for the fit nos. 1–9 shown in Table 2, but the same for each pair of absorption and pump-probe spectra. The ratio of the transition dipoles for the S₁-S₂ and S₀-S₁ transitions in the BChl monomer, χ, was taken to be 0.5 (for larger χ it is more difficult to reproduce the shape of the pump-probe spectra).

Display large version of this figure

Figure 2

Simultaneous fit of linear absorption (top) and pump-probe spectra (bottom). Circles show experimental data; solid lines show calculated spectra. The absorption profiles for individual exciton components are also shown by solid lines. T=77K, N=24, M₁₂=600cm⁻¹, γ_1L=40cm⁻¹, γ_1H=191cm⁻¹, γ₂=230cm⁻¹, σ=495cm⁻¹, E₁−E₂=0, and the Stokes shift is 105cm⁻¹ (fit no. 1). The ΔA values are in arbitrary units.

Display large version of this figure

Figure 3

The same as in Fig. 2, but parameters are from fit no. 3: T=77K, N=24, M₁₂=400cm⁻¹, γ_1L=40cm⁻¹, γ_1H=182cm⁻¹, γ₂=230cm⁻¹, σ=440cm⁻¹, and the Stokes shift is 110cm⁻¹.

Display large version of this figure

Figure 4

The same as in Fig. 2, but parameters are from fit no. 6: T=77K, N=24, M₁₂=260cm⁻¹, γ_1L=40cm⁻¹, γ_1H=166cm⁻¹, γ₂=230cm⁻¹, σ=418cm⁻¹, and the Stokes shift is 110cm⁻¹.

Display large version of this figure

Figure 5

The same as in Fig. 2, but parameters are from fit no. 8: T=300K, N=24, M₁₂=400cm⁻¹, γ_1L=280cm⁻¹, γ_1H=510cm⁻¹, γ₂=600cm⁻¹, σ=550cm⁻¹, and the Stokes shift is 140cm⁻¹.

It is remarkable that the fitting parameters in Table 2 vary only slightly when we change the interaction energies or the value of E₁−E₂. For example, for the low-temperature (77K) absorption and pump-probe spectra, we obtain γ_1L=40cm⁻¹, γ_1H=166–199cm⁻¹, γ₂=230cm⁻¹, σ=370–495cm⁻¹, and the Stokes shift of 100–110cm⁻¹, upon variation of the interaction energy from 260 to 600cm⁻¹ with N=24. For N=32 all fitting parameters are approximately the same as for N=24, but the disorder value is larger (increasing from 440 to 545cm⁻¹ for the interaction energy of 400cm⁻¹). For these parameters the shape of the absorption spectrum is dominated by the five lowest exciton components (k=0, k=±1, k=±2), which have maxima at 1046–1049, 1039–1042, 1032–1036, 1020–1031, and 1012–1025nm. The lowest component (k=0) has the width of 15–18nm, and its dipole strength is 3.4–4.1 (N=24) or 4.7–5.5 (N=32), i.e., 14–17% of the total dipole strength of the aggregate in this range of N values. The two higher levels (k=±1) are more intense and broader. The next two (k=±2) are also broad but much less intense. The additional broadening of the higher levels is explained by additional homogeneous broadening due to relaxation (see the difference between γ_1L and γ_1H). Notice that within the limits of our model we are not able to explain the blue wing of absorption profile as well as the wings of the pump-probe spectra (blue wing of ESA and red wing of SE), which are most probably determined by the vibronic structure associated with each of the exciton levels.

It is important to note that the exciton model, used here to calculate the spectral features of LH1, is in good agreement with the earlier observation of spectral heterogeneity of LH1 of Rps. viridis (Monshouwer et al). The calculated maxima of the three lowest exciton levels at 77K are very close to the 1049-, 1042-, and 1030-nm maxima observed in the second derivative of the absorption spectrum at low temperature (Monshouwer et al). Position, spectral width, and relative intensity of the lowest exciton level show good correlation with the same parameters of the B1045 spectral form (Monshouwer et al). The width of the B1045 band was determined as 12.4nm at 4K (Monshouwer et al), whereas the calculated width of the lowest exciton component is 15–18nm at 77K. The difference may easily be explained by the additional homogeneous broadening of the lowest level at 77K (γ_1L=40cm⁻¹, or 4nm in our model).

Notice that the spectral heterogeneity obtained here will be a common property of a spectrally disordered circular aggregate. The same (or, at least, similar) features may also be expected for the LH1 antenna of the BChl a–containing bacteria (recall that in our model the pigment arrangement is analogous to that of the BChl a–containing bacteria). However, no spectral heterogeneity was observed for the BChl a–containing species. For example, it was concluded that the long-wavelength side of the absorption spectrum for isolated LH1 complex of Rb. sphaeroides is dominated by inhomogeneous broadening (van Mourik et al). In principle, the effect of heterogeneity may be masked by overlapping of spectral components. For example, the absorption peak of the lowest exciton level of a ring-like aggregate may be hidden under more intense absorption of higher levels. The possibility of resolving a fine structure of the overall spectrum may be strongly dependent on the positions, spectral widths, and shapes of individual exciton components. These parameters, determined by the static disorder and exciton-phonon coupling, are generally different for different species.

Let us now consider the problem of delocalization of the exciton states of the spectrally disordered antenna. For a particular realization of the disorder, the kth one-exciton state is characterized by its energy E_k and wave function |k〉:

(1)

(2)

The inverse participation ratio, (L_k)⁻¹, determines the delocalization length of kth exciton state. For example, for a localized state c_n^k=δ(n−n₀), and (L_k)⁻¹=1, whereas for a completely delocalized wave function c_n^k=N^−1/2, and (L_k)⁻¹=N. Typically, the (L_k)⁻¹ values are different for different eigenstates. In this case an effective delocalization length can be defined as the thermally averaged inverse participation ratio (Meier et al):

(3)

(4)

Display large version of this figure

Figure 6

Localization function for the LH1 antenna with N=24 at 77K. (Top) M₁₂=600cm⁻¹. (Middle) M₁₂=400cm⁻¹. (Bottom) M₁₂=260cm⁻¹. E₁−E₂=0 (lower curves), 600cm⁻¹ (upper curves). The σ values are taken from Table 2.

Display large version of this figure

Figure 7

Localization function for the LH1 antenna at room temperature. N=24, M₁₂=400cm⁻¹, σ=550cm⁻¹, E₁−E₂=0.

From the data shown in Figure 6 and Figure 7 we conclude that the individual exciton states are highly delocalized in the middle of the exciton band, i.e., in the blue edge of the absorption profile of LH1, but more localized near the absorption maximum (1040 and 1015nm for low and room temperatures, respectively) and even more localized in the red wing. At low temperature (77K) and for N=24, the thermally averaged inverse participation ratio is equal to N_eff=9.1, 7.4, and 5.4 for “high,” “normal,” and “low” interaction energies, respectively, with E₁−E₂=0 or N_eff=8.7, 6.5, and 3.8 with E₁−E₂=600cm⁻¹ (see Table 3). At room temperature and for N=24 we obtained N_eff=8.1 for “normal” interaction energies and E₁−E₂=0. (Notice that the thermally averaged inverse participation ratio increases with temperature because higher exciton states that are more delocalized start to contribute.) For N=32 the N_eff value is slightly less than for N=24 (Table 3) because of the higher disorder values required for the N=32 fit. Using a fixed σ value, we will have approximately the same N_eff value for N=24 and N=32. It means that in our case the delocalization length is controlled mostly by the disorder (the exciton wave function does not “feel” the aggregate size). Notice that in the homogeneous limit the delocalization length is proportional to N (the inverse participation ratio for a homogeneous circular aggregate is equal to N for the lowest level and 2/3N for the higher ones). In general, the delocalization length increases with N and decreases with σ. The N values determine the delocalization length of the zero-order (homogeneous) wave functions, whereas the value of σ is responsible for mixing of these zero-order wave functions due to inhomogeneity, giving rise to more localized states.

Notice that the inverse participation ratio corresponds to a delocalization length for individual exciton states only. In reality one deals with some kind of superposition of exciton levels. For zero time delay (immediately after excitation) such a superposition may have been created because of the simultaneous excitation of several exciton levels. In the steady-state limit (for time delays longer than exciton relaxation) we will have a superposition of exciton states that are populated at thermal equilibrium. Evolution of the initially formed exciton wave packet (or selectively excited single exciton state) to the steady-state wave packet can be described by the density matrix in the site representation, ρ_m,n(t), where n and m are molecular numbers (Meier et al,Kühn and Mukamel, 1997,Kühn and Sundström, 1997). In the case of the disordered aggregate one should use the density matrix 〈ρ_m,n(t)〉 averaged over realizations of the disorder (everywhere below we omit these angular brackets). In the steady-state limit (with respect to exciton relaxation within the one-exciton manifold) the density matrix is given by (Meier et al)

(5)

(6)

(7)

Display large version of this figure

Figure 8

Three-dimensional view of the steady-state density matrix, ρ_m,n, at 77K (top) and room temperature (bottom). Parameters from fit nos. 3 and 8, i.e., N=24, M₁₂=400cm⁻¹, E₁−E₂=0, σ=440, and 550cm⁻¹, for 77K and 300K, respectively. x and y coordinates correspond to the molecular numbers, n and m.

The steady-state coherence functions, C(n), calculated with the parameters taken from the low-temperature and the room-temperature fits, are shown in Figure 9 and Figure 10, respectively. At low temperature the coherence length, N_coh, corresponds to 10, 8, and 6 molecules for “high,” “normal,” and “low” interaction energies, respectively, with E₁−E₂=0 (Figure 9top) and to 10, 8, and 5 molecules, respectively, with E₁−E₂=600cm⁻¹ (Figure 9bottom). These values are close to the inverse participation ratio, N_eff, so that the exciton wave packet length is approximately the same as the delocalization length for a single exciton level. Notice that in thermal equilibrium at 77K only the lowest exciton level is populated (population of the second level for our parameters is ∼0.17–0.2), and consequently there is no significant superposition of levels.

Display large version of this figure

Figure 9

The steady-state low-te