##
1Exciton dynamics in ring-like photosynthetic light-harvesting complexes: a hopping model
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Exciton dynamics in ring-like photosynthetic light-harvesting complexes: a hopping model

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Excitation localization and dynamics in circular molecular aggregates is considered.

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It is shown that the Anderson localization of the excitons is taking place even in the finite size of the ring-type systems containing tens of pigments in the case of comparable values of the spectral inhomogeneity and of the intermolecular resonance interaction.

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The second type of localization comes from the dynamical disorder caused by exciton interactions with environment fluctuations.

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Because of these two reasons the hopping type migration of the small-size excitons is postulated to be responsible for the excitation dynamics in this kind of systems.

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6

This process is considered for the ensemble of independent rings and for the array of the interacting rings by means of Monte Carlo simulations.

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The intra-ring and inter-ring energy disorder with possible correlations is accepted in simulations performed for the cases of high and low temperatures.

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8

It is shown that for the typical parameters of the peripheral light-harvesting pigment-protein complexes LH2 of photosynthetic bacteria the excitation population reaches equilibrium within 1 ps in the case of the disconnected rings at nonselective excitation conditions, while equilibration on longer time scale is taking place in the system of connected rings.

Type: Observation |
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9

This nonexponential relaxation kinetics is observed at room temperature and is more pronounced by lowering the temperature.

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10

In the case of selective excitation the equilibration process is wavelength dependent for the disconnected rings at room temperature and becomes more pronounced by lowering the temperature.

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11

The wavelength dependence is resulted from the interplay between exciton population redistribution among pigments and the population, which stucks in the most red pigments.

Type: Conclusion |
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## Introduction

12

In natural photosynthesis two ultra-fast processes are at the basis of the high quantum efficiency: excitation energy transfer among the pigments of the light-harvesting antenna and energy transfer to a reaction center, where a charge separation is initiated.

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13

In the membrane of photosynthetic bacteria a complex system of pigment-proteins is responsible for this process to occur.

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14

Typically, a reaction center is surrounded by a core antenna, LH1, which in turn is associated to a more peripheral antenna, LH.2

^{1}
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15

From the high resolution structures of LH2 of

*Rhodopseudomonas*(*Rps.*)*acidophila*^{2}and*Rhodopsirillum*(*Rs.*)*molischianum*^{3}it follows that the bacteriochlorophyll (Bchl) molecules in peripheral light-harvesting complexes of photosynthetic bacteria (LH2 or B800-850 complexes) are arranged in a symmetric ring-like structure consisting of two concentric rings of polypeptides (α,β) and two rings of pigments.^{2,3}
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16

The first ring consists of 18, in the case of

*Rhodopseudomonas*(*Rps.*)*acidophila*(16 in the case of*Rhodopsirillum*(*Rs.*)*molischianum*), Bchl arranged as a tightly coupled ring of Bchl dimers and absorbs at about 850 nm (B850), while the second ring of nine, in the case of*Rps. acidophila*(eight in the case of*Rs. molischianum*), monomeric Bchl has its major absorption band at 800 nm (B800).
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17

These differences in the position of the absorption bands are (at least partly) attributed to differences in coherent exciton effects arising from the interaction between the molecules in the ring.

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18

A similar but larger ring of 15 or 16 Bchl dimers is expected for core complex (LH1) yielding the absorption band at 870–880 nm.

^{4}
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19

In LH1 the second ring of monomeric Bchl is absent.

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20

Due to their

*C*_{n}symmetry these ring-like structures seem to be very attractive molecular aggregates for studies of the collective excited states (excitons) by analysing the steady-state and transient spectra.
Type: Motivation |
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21

The optical transitions into the exciton states reflect the symmetry of the structural arrangement of the aggregate.

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22

For instance, a ring-like aggregate of identical molecules will obey different selection rules in comparison with those of linear aggregates.

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23

For a ring with the optical transition dipole moments of the molecules in plane of the ring (almost the actual arrangement of the B850 in LH2), the lowest exciton state (enumerated in the

*k*-space as*k*= 0) is optically forbidden and a pair of degenerate states just above it contains almost all the dipole strength.^{1,5,6}
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24

A difference in the orientation of the transition dipole moments, a probable difference in the transition frequencies of two molecules in a dimer, as well as slight alternating differences in the intermolecular distances between the adjacent molecules allow to transform the B850 ring of 18 (or 16) monomers into a ring of dimers.

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25

In this case the exciton energy band, due to the presence of two molecules per unit cell, splits into two Davydov subbands separated by the energy gap between them.

^{1,6,7}
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26

The optical transitions into both Davydov subbands obey the same selection rule already mentioned,

*i.e.*with the transition dipole moments in the plane of the ring, the states describing the edges of both Davydov components (corresponding to*k*= 0) are optically forbidden, while almost all dipole strength is concentrated in the transition to pairs of aggregate exciton states next to them.
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27

The distribution of the dipole strength between Davydov components is determined by the orientations and energies of the two Bchl within the unit cell.

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28

Multiple experiments have been carried out in order to understand the origin of the spectrum of LH2 and to define the exciton localization radius, and various approaches were used for the description of the exciton dynamics (see

^{ref. 1,8}for review).
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29

Since the interpigment interaction in the B800 ring is weak (less than the site inhomogeneity and the exciton–photon coupling), coherence effects play only a minor role in the description of spectroscopic properties and exciton dynamics.

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30

The hopping time between pigment molecules is of the order of 1 ps within the B800 ring.

^{1}
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31

The excitation energy transfer time from B800 to B850 is also estimated to be of the order of 1 ps and this process can be understood in terms of the incoherent energy transfer based on modified Förster theory, because of the spatial size of the coherently coupled pigments in B.850

^{1,9–11}
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32

Carotenoid molecules can play an additional role in this type of the energy transfer by modulating the interpigment interaction.

^{11}
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The Bchl molecules in the B850 ring are strongly coupled, therefore, coherent excitons are expected in this structure.

^{1}
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However, due to significant spectral inhomogeneity, which is comparable with the homogeneous absorption bandwidth as well as with the intermolecular resonance interaction, the coherent exciton states must be extensively perturbed.

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35

Most of the experimental results, obtained for a wide range of temperatures, can be well understood in terms of small excitons comprising of two to four Bchl (see

^{ref. 12}for the review).
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36

This conclusion is also supported by measurements of the superradiance in LH1 and LH2 complexes of

*Rhodobacter*(*Rb.*)*sphaeroides*from 4 K to room temperature^{13}leading to the same conclusion of a small radius of the exciton coherence.
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38

The inhomogeneous distribution among the LH2 complexes as well as within a separate complex is well demonstrated by spectroscopic experiments on single LH2 complexes supporting the conclusion that the absorption spectra of B850 at low temperatures is consistent with the coherent exciton model of a ring-like arrangement of pigments with a disordered distribution of the molecular transition energies.

^{15–18}
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39

The exciton dynamics in the strongly coupled B850 ring at room temperature as measured by transient absorption,

^{19}polarization decay,^{20,21}singlet–singlet annihilation^{21,22}and three-pulse photon echo,^{23,24}can be well understood by a combination of fast exciton relaxation followed by small exciton hopping between pigments with a mean hopping time of the order of a few hundred femtoseconds.
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Upon lowering the temperature, a new dynamic feature in the range of 1–100 ps is observed in membrane-bound LH2 complexes.

^{25–27}
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41

At low temperatures the stimulated emission/bleaching band broadens and splits into two bands in about 3 ps.

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42

The new red-shifted band continues to move further to the red and broadens in the tens of picoseconds.

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43

The results correlate with the low temperature Stokes shift, which was shown to be unusually large in LH2-only mutants of

*Rb. sphaeroides.*^{28}
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44

These data were interpreted in terms of fast exciton relaxation within a single disordered ring of B850, while the slower phases were assigned to the transfer among the inhomogeneously distributed rings.

^{26}
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45

Alternatively, a complex relaxation scheme has been proposed, where the initial ultrafast (of the order of 100 fs) exciton relaxation in a separate B850 ring is followed by mixing of the exciton states with charge-transfer (CT) states on a subpicosecond time scale and a subsequent evolution of the population in the charge-transfer state on the picosecond time scale corresponding to self-trapping (analogous to the polaron formation).

^{27}
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47

However, comparison of the exciton evolution in

*Rb. sphaeroides*membranes and in isolated LH2 complexes demonstrates that most of the effects on the time scale of picoseconds, such as unusually large Stokes-shift and the complex evolution of the emission band, are only observed in the membranes, containing the interacting LH2 complexes.^{31}
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48

For isolated LH2 changes in absorption spectrum are minor compared to the membrane-bounded form, while the Stokes shift is substantially smaller and the kinetics of the Stokes-shift evolution is much faster, taking place on a subpicosecond time scale.

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49

This Stokes shift of separate LH2 complexes recently was attributed to the self-trapped exciton formation,

^{32}however, the excitation dynamics within this model was not considered.
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Here we discuss the problem of the exciton localization and dynamics both in isolated rings of pigments and in an array of connected rings of LH2 complexes.

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51

Earlier a qualitatively similar but more simple model was used to describe the excitation dynamics in LH1 complexes and showed good correspondence to the experimental observations at room temperature and at 4 K.

^{33}
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Here we develop a sophisticated model based on the hopping of an exciton of small size, which is formed after some ultrafast relaxation of electronic coherences.

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The model accounts for spectral inhomogeneity in a single ring as well as among the ensemble of rings.

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It also includes dynamic formation of a Stokes shift in isolated chromophore.

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Hopping models were successfully applied for molecular polymers as being responsible for charge diffusion in the bulk.

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The concepts of correlated and uncorrelated distribution of on-site energies were successfully developed and applied.

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Here we investigate if such a hopping model contains the essential relaxation dynamics observed for isolated LH2 systems and connected aggregates of LH2.

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The analysis allows to judge the nature of excitations and their extension radius in the LH2 kinds of systems.

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## Formulation of the model

59

The exciton energy spectrum of a ring of pigment molecules is modulated by the random fluctuations of surrounding proteins, which have a broad range of spectral density.

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We assume the presence of fast and slow fluctuations of surrounding.

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The fast ones are responsible for ultrafast dephasing of exciton states.

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62

Slow (static) fluctuations introduce static energy disorder

^{34,35}..
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63

In the absence of fast fluctuations (this limit is accessible for instance at low temperatures) for a weak disorder (

*σ*_{inhom}< the coupling between the pigments) the spectral shift of all the energy levels is slightly perturbed by the disorder and depends linearly on its value, while for higher values of*σ*_{inhom}this dependence does not persist.^{17,35,36}
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64

Disorder mixes different exciton states and changes the phase relationships between molecular excited states contributing to a particular exciton wavefunction.

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65

The resulting redistribution of the dipole strengths between the exciton transitions is used as a possible explanation of the various properties of the LH1 and LH2 absorption spectra

^{23,37,38}and of the spectra observed in single molecule spectroscopy.^{15–18}
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66

This effect of disorder is especially pronounced in extended one-dimensional systems where its infinitesimal value results in Anderson localization of the excitons, while for the two- and three-dimensional systems a critical value of the disorder has to be reached before this type of the localization is realized.

^{39}
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For finite one-dimensional systems an additional competition between the exciton localization length and the size of the molecular aggregate is expected.

Type: Hypothesis |
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68

In Fig. 1 we present the wavefunction,

*c*_{nk}, for an arbitrary exciton state,*k*, in a ring of 18 molecules (*n*enumerates these molecules) with each pigment characterized by two electronic states corresponding to its lowest electronic transition.
Type: Observation |
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69

The average difference between the transition energies for two adjacent molecules is assumed to be equal to the resonance interaction, which is typically assumed for LH1 and LH.2

^{1}
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Fig. 1 shows an example for a particular realization of the disorder how the exciton may be localized on some molecules in exciton states (as measured by the |

*c*_{nk}|^{2}values) constituting the effect of Anderson localization.^{39}
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71

In the case of small disorder (in comparison with the resonance interaction) the Anderson localization is predominantly observed for the states at the edges of the exciton band, while the states in the middle of the band remain delocalized.

^{39}
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In addition to the static disorder, the interaction of the exciton with fast intramolecular and protein vibrations must be included.

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This causes a dephasing of the excited molecules participating in a certain exciton state and results in the homogeneous bandwidth of the corresponding optical transition.

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74

This type of interaction, generally called dynamic disorder, operates on the time scale of exciton evolution.

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The exciton states, localized with respect to the Anderson localization, can undergo a subsequent intramolecular relaxation step resulting in molecular Stokes-shift formation (opposed to the line shift due to exciton relaxation within the exciton band, which we do not identify as the Stokes shift) and/or even exciton self-trapping in the case of large exciton–photon coupling.

^{1}
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The observed broad absorption band can be attributed to this dephasing time being of the order of tens of femtoseconds.

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78

This conclusion is further supported by modelling the ultrafast spectral evolution in LH1 using a model of coherent excitons with disorder and by analysing the femtosecond time resolved measurements.

^{40,41}
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Thus, the exciton dynamics in a ring like LH1 and LH2 with intrinsic diagonal disorder, is complex with various stages of exciton relaxation.

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The optical absorption most probably can be attributed to the transition into coherent exciton states.

Type: Conclusion |
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Because of the inherent disorder, these exciton states are localized (in the sense of the Anderson localization) on a particular part of the aggregate.

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The size of the exciton localization depends on the value of the disorder and varies depending on position of the exciton state in the band.

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83

A subsequent dephasing and/or self-trapping of the exciton (on the time scale of tens of femtoseconds) takes place because of the exciton–phonon interaction.

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84

The rates of the exciton relaxation can be estimated by means of the Redfield theory in terms of the second order perturbation of the exciton–photon and/or vibronic coupling.

Type: Method |
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85

According to this theoretical approach the corresponding matrix elements of the tensor qualifying the exciton relaxation are proportional to the overlap factor

*S*_{k1k2,k3k4}, where:^{42}Thus, it follows that exciton relaxation is expected to be faster in the center of the exciton band where exciton states are delocalized and overlap of wavefunctions is significant and it slows down at the edges in the case of weak disorder.
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86

In the case of strong disorder the overlap of the wavefunctions of different exciton states is small because of the exciton localization (see Fig. 1), and fast exciton relaxation between these states is very unlikely.

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87

Thus, exciton localization is expected to be typical in one-dimensional aggregates due to static and dynamic disorder resulting in hopping-type of the exciton dynamics instead of coherent delocalized exciton relaxation.

Type: Hypothesis |
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## Modelling

88

Based on the previous section, we consider the dynamics of a small exciton localized on a few pigment molecules.

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89

We define our system as a circular aggregate of

*N*identical molecular complexes.
Type: Model |
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ConceptID: Mod4

90

Each complex is considered as a two level system with the site dependent transition energy

*E*_{x}^{A}for site*x*.
Type: Model |
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91

We consider the dynamic formation of the intramolecular excitation relaxation resulting in the development of molecular Stokes shift,

^{43}which we take into account by means of the following procedure.
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ConceptID: Mod5

92

Let us define the total Stokes shift of a site

*x*as*E*_{S}.
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93

Since the vibrational relaxation is responsible for the Stokes shift formation, the energy interval

*E*_{S}is divided into*M*equal portions, determining a ladder of energies,*ε*_{S}=*mE*_{S}/*M*, where*m*= 0,…,*M*(in accord with the harmonic approximation for the bath).
Type: Model |
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94

Thus, after the annihilation of the excitation on the site

*x*the emitted energy is*E*D*x*=*E*_{x}^{A}−*ε*_{S}, which is a function of the number*m*(the superscripts D or A refer to the complex donating the exciton (D) and accepting the exciton (A), respectively).
Type: Model |
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95

The time evolution of the Stokes shift is determined by rates

*k*_{shift}, which are the transition rates from state*m*to the subsequent state*m*+ 1.
Type: Model |
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ConceptID: Mod6

96

We assume

*k*_{shift}being independent of*m*.
Type: Model |
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97

For qualitative consideration the upward transition can be ignored at low temperatures.

Type: Method |
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98

The number of steps

*M*is the parameter of a harmonic bath and corresponds to the number of vibrational states involved in the vibrational relaxation.
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99

It can be explicitly determined by defining one vibrational mode responsible for the exciton intramolecular relaxation, which can be crucial in the case of high frequency vibrational motion when

*M*is small.
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ConceptID: Mod7

100

However for a small vibrational frequency a large number of vibrational states is participating the relaxation and the importance of

*M*becomes negligible.
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101

We limit ourselves to this large

*M*limit.
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102

The transition energies of the sites in the ring are assumed to be disordered.

^{44,45}
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103

The disorder effect is introduced by taking the site transition energy

*E*_{x}^{A}as a random value from a Gaussian distribution, characterized by the mean value associated with the particular ring, 〈*ε*〉_{ring}, and by a dispersion of the distribution*σ*_{intra}^{2}.
Type: Model |
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ConceptID: Mod8

104

This kind of disorder is denoted as

*intra-ring*disorder reflecting local inhomogeneities in the system.
Type: Model |
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105

This kind of definition also allows to introduce the

*inter-ring*disorder reflecting large scale inhomogeneities by defining the ring-dependent energies 〈*ε*〉_{ring}as a Gaussian random numbers characterized by zero mean and the width*σ*_{inter}^{2}.
Type: Model |
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ConceptID: Mod9

106

According to these definitions, a ring is constructed in the following way: (1) the mean energy value of the ring, 〈

*ε*〉_{ring}, is selected randomly in accord with the Gaussian distribution with the dispersion*σ*_{inter}^{2}, (2) the site energies of the ring,*E*_{x}^{A}, are selected in accord with the Gaussian distribution with the dispersion*σ*_{intra}^{2}and the selected mean value 〈*ε*〉_{ring}.
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107

And this is repeated for each new ring.

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108

We also include the correlations in intra-ring diagonal disorder of each ring using the following procedure.

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109

Let

*ε*_{x}^{(0)}denotes the zeroth generation energy of the site*x*obtained by drawing the independent random numbers as described above.
Type: Model |
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ConceptID: Mod11

110

The zeroth generation of the site energies is a set of uncorrelated Gaussian random numbers characterized by width

*σ*_{intra}.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod11

111

The first generation of the site energies

*ε*_{x}^{(1)}is created by averaging the zeroth order energy values of neighbouring sites.
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met8

112

This procedure introduces nearest-neighbour correlations into the energy picture.

Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met8

113

By repeating the averaging, higher order correlations of larger extension are introduced, thus, giving the (

*n*+ 1)-th generation of the site energies accordingly:The normalization factor*Z*^{(n)}conserves the width of the energy distribution and is calculated from the following relation:where*N*is the number of sites in the lattice.
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met9

114

Before making correlations, the mean value of the energies in the 0’th generation is shifted to 0, while after the creation of correlated energies this average value is restored.

Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met10

115

The correlated intra-ring and inter-ring disorder reflect the natural biological systems where the surrounding proteins span over large distances exceeding the extension of the aggregate.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac14

116

Thus, intra-ring disorder comes from local differences in nearest surrounding of different pigments (like interactions with the side groups of surrounding protein).

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac14

117

The long-range interactions with charges in the protein and distortion of the shape of aggregate due to mechanical tensions build correlations between different pigments inside the ring and lead to inter-ring disorder.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac14

118

The dynamic parameter of our system is the exciton hopping rate,

*k*_{0}, between complexes with the same transition energy.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

119

The exciton hopping rates are modulated by energy disorder,

*i.e.*the excitation hopping rate from the*x*th complex onto the*y*th one,*k*_{x→y}, is determined as follows:where*R*(|*x*−*y*|) is a distance-dependent factor,*E*D*x*and*E*A*y*are the energies corresponding to complexes*x*or*y*, respectively, while,*k*_{B}is the Boltzmann constant and*T*is the temperature.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

120

To calculate exciton transfer over large distances, we assume the Förster mechanism of exciton transfer.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod13

121

In this case the distance-dependent function

*R*(|*x*−*y*|) is given by:where |*x*−*y*| is the distance between complexes*x*and*y*and the distance between nearest neighbours is 1.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod13

122

In order to account for the exciton transfer between different rings (for instance, the complex of rings located in the membrane), a triangular two-dimensional lattice of rings is considered as the simplest case.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod13

123

We define the distance between the centers of the nearest-neighbour rings in the lattice as

*a*_{r}, while the radius of the ring is defined as*a*_{R}with*a*_{r}− 2*a*_{R}≥ 1.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod13

124

In this case the exciton hopping rate between the sites of different rings can be calculated by using eqns. (4) and (5) when

*x*and*y*are vectors pointing to the sites in different rings.
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met11

125

By taking into account the overall excitation decay rate,

*k*_{d}, on each complex, a Monte-Carlo simulations are performed to study the exciton dynamics in this system during the excitation lifetime.
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met12

126

Each event in the system is simulated by assuming probabilistic nature of the process with the event probability defined by the ratio of the rate of the corresponding process

*k*_{i}with to the total rate of all possible processes,*k*_{∑}, originating from that particular system state.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod14

127

The total rate

*k*_{∑}defines the timescale of the event, while the actual time of the event is calculated by drawing random number of exponential distribution with the mean*k*_{∑}^{−1}.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod14

128

This procedure is repeated after each event and, thus, the random hopping is simulated.

Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met12

129

We assume the following set of parameters for the simulations.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod15

130

According to the estimates based on various experimental observations the hopping time is of the order of 100 fs at room temperature, thus, we assume, (

*k*_{0}exp(−*E*_{S}/(*k*_{B}*T*_{0})))^{−1}= 100 fs, where*T*_{0}= 293 K.^{1,40,46}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod15

131

From the single molecule spectroscopy data it follows that the exciton dephasing time is of the order of 50–100 fs;

^{16–18}therefore, the total Stokes shift formation time has to be of the same order or slower, thus, we assume*k*_{shift}^{−1}= (100/*M*) fs.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod15

132

A typical value for the resonance interaction between pigments is of the order of 300 cm

^{−1}.^{1,27,47}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac15

Type: Method |
Advantage: None |
Novelty: Old |
ConceptID: Met13

134

The absorption bandwidth at 10 K for the disconnected LH2 complexes is of the order of 270 cm

^{−1}(FWHM).^{31}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac16

135

As it was already mentioned, the inhomogeneous broadening has two origins,

*i.e.*the intra-ring broadening and the inter-ring broadening.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac17

136

These values can be assumed to be equal to

*σ*_{intra}= 106 cm^{−1}in accord with FWHM_{intra}= 300 cm^{−1}obtained by adjusting the bandwidths and the shift of the fluorescence spectra for disconnected LH2 complexes, while*σ*_{inter}= 54 cm^{−1}in accord with the total inhomogeneous bandwidth equal to 120 cm^{−1}.^{31}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod16

137

The natural decay of the excitation in an isolated complex is of the order of nanoseconds, and

*k*_{d}^{−1}was set to 1 ns in the calculations.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod17

138

We start our simulation by mimicking optical excitation conditions.

Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met14

139

Two kinds of initial conditions are used for the simulations.

Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met14

140

1.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod18

141

The

*nonselective*excitiation is defined by placing initial excitation randomly at any site without the Stokes-shifted states involved.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod18

142

This case is analogous to broad band optical pulse excitation.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod18

143

Due to the Gaussian diagonal disorder, there will be the Gaussian distribution of energies of initial excitations.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod18

144

2.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod19

145

The

*selective*excitation is defined by assigning the optical excitation energy dependent probability*P*(*ε*_{x},*ω*_{ex}) to each site*x*, which is a probability to place the initial excitation on site*x*, where*ω*_{ex}is the optical field frequency.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod19

146

We assume Gaussian spectrum of the optical field.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod19

147

Then the excitation probability is given by

*P*(*ε*_{x},*ω*_{ex}) ∝ exp(−(*ε*_{x}−*ω*_{ex})^{2}/(2*σ*_{ex}^{2})).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod19

148

The width of the optical excitation is defined by full width at half maximum (FWHM = 2.355

*σ*_{ex}).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod19

149

The overall probability of finding the excitation at particular energy is also weighted by a probability of having a site with this energy coming from the diagonal disorder.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod19

150

This case corresponds to the narrow band optical excitation, which can select particular area in the energy distribution (hole burning regime).

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod19

151

After the initial creation of the excitation we perform Monte Carlo simulations of system dynamics: the hopping motion of the exciton is simulated and its energy dependence on time is recorded until the excitation decays due to its finite lifetime.

Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met15

152

By repeating the simulations for many independent excitations the statistical distribution of excitation energy as a function of time,

*ρ*(*E*,*t*), is revealed.
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met15

153

It is worth noting that this distribution is directly related to the time evolution of the fluorescence spectrum.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac18

154

Assuming the transition dipole magnitude being independent of the transition energy, the relation between

*ρ*(*E*,*t*) and the time-dependent fluorescence spectrum is given by:where*Γ*_{h}is the homogeneous linewidth of a particular exciton transition, which is usually much smaller than the inhomogeneous distribution of transition frequencies, represented by*ρ*(*E*,*t*).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod20

## Simulation of the hopping motion: the case of isolated rings

155

Let us assume that the coherence size of the exciton equals 2, thus, we will consider the incoherent exciton hopping in a ring of nine complexes characterized by uncorrelated diagonal disorder.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod21

156

The Monte-Carlo simulations give fast (of the order of hundred femtoseconds) equilibration of excitons in the ring at room temperature upon selective excitation into the lower edge of the absorption band (see Fig. 2) as might be expected.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res11

157

Similar dynamics is observed when the system is excited into the maximum of the absorption band or to higher energies.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res12

158

The relaxation process is nonexponential due to a distribution of hopping distances, and the leading term originates from the Stokes shift and nearest-neighbour hopping.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res13

159

In the case of selective excitation into the maximum and to the very blue edge of the absorption band the final shape of the exciton distribution behaves as a one-peak function as a result of the exciton hopping in the aggregate resulting in predominant population of the lower energy sites and the molecular Stokes shift (see Fig. 3a and b).

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res14

160

The final width of the exciton distribution is much broader compared to the original excitation spectrum due to the thermally assisted hopping motion.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res15

161

In the case of selective excitation into the very red part of the absorption band the population of the excitations is equilibrated also on the same time scale but the final distribution is characterized by a broad band with the red-shifted maximum with an additional broad wing to higher energies (see Fig. 3c).

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs3

162

The first originates from the population of initially excited molecules that are Stokes shifted, while the broad wing corresponds to the distribution of transition energies in the ensemble due to exciton equilibration at room temperature.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res16

163

This complex shape of the distribution is obtained because of the competition of two processes: thermal activation and Stokes shift.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res17

164

These two competing processes cause the dependence of the fluorescence spectrum on the excitation wavelength as shown in Fig. 3.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res18

165

For nonselective excitation the final distribution of the population mainly resembles the Stokes shift,

*E*_{S}(see Fig. 4).
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res19

166

The dependence of the final population distribution on the excitation conditions reflect that the different rings are disconnected and the complete equilibration of population in the ensemble of rings cannot be achieved, while the equilibration inside each ring is very fast.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con8

167

The effects due to disorder are substantially enhanced at low temperatures, because then the excitation will not be able to escape when it is trapped in some local minimum in the one-dimensional arrangement of the system.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res20

168

Thus, in the case of uncorrelated disorder and with selective excitation even in the vicinity of the maximum of the absorption band the excitation population finally is distributed as a double peaked structure.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con9

169

This is because in some systems in the ensemble the excitations are able to move to other minima thereby generating a broad red-shifted band in the population distribution within several hundreds of femtoseconds, while others, corresponding to excitation in some local minimum, are responsible for the population of the Stokes shifted band, which appears within hundred femtoseconds.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con10

170

This situation can be modified by assuming correlated disorder and/or by increasing the number of coherently coupled pigments in the ring (

*i.e.*decreasing the number of sites per ring).
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met16

171

However, in all cases the value of the Stokes shift and the rate of its formation have an important influence on the equilibrated excitation distribution in the ensemble of single rings.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con11

172

Modelling the low temperature dynamics with

*N*= 5 and assuming correlated disorder with a correlation radius corresponding to the ring size results in the spectral evolution shown in Fig. 5.
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met17

173

The dynamics of the Stokes shift formation is taken into account as it was described in the previous section by assuming

*M*= 50 thereby allowing for energy transfer through “hot” states.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod22

174

The possibility to escape from the initially excited complex is much larger in the case of excitation into the maximum of the absorption band leading to a much more red-shifted fluorescence band as compared to the initial excitation wavelength (see Fig. 5b).

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res21

175

Almost complete escape of the excitation from the initially excited complex is observed when excitation occurs into the blue part of the absorption spectrum (see Fig. 5a), and the fluorescence is strongly red shifted (almost 400 cm

^{−1}).
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs4

176

In the case of the selective excitation into the low energy wings of the absorption, the hopping motion is not possible, and a single Stokes shifted fluorescence band is obtained (see Fig. 5c).

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res22

177

Nonselective excitation results in a broad fluorescence band, which is shifted 200 cm

^{−1}from the absorption maximum as shown in Fig. 6.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res23

## Simulation of the hopping motion: the case of connected rings

178

In photosynthetic membranes the LH2 rings are organized into larger light-harvesting systems and the exciton can be transferred between the rings.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac19

179

However, in trying to understand the resulting exciton dynamics one has to take into account the fact that the exciton interring transfer rate is much slower than the rate of nearest-neighbour exciton transfer inside the ring (as has been experimentally confirmed, for instance, in the case of the LH2-LH1 transfer

^{1}).
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac19

180

Therefore, the distance between the nearest pigments located on different rings has to be postulated larger than the nearest-neighbour distance

*a*= 1 between pigments in the ring.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac19

181

Here we assume the following relation between different distances:

*a*_{r}= 2*a*_{R}+ 2*a*, implying that the distance between neighbouring sites on different rings is at least two times larger than the distance*a*.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod23

182

The internal structure of each of the rings is assumed to be the same as in the case of the disconnected rings.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod24

183

Thus, for the calculation of the exciton dynamics at room temperature each ring consists of nine sites and the disorder is not correlated.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod24

184

In this case two distinct components can be distinguished in the exciton equilibration dynamics.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res24

185

A fast one, taking less than 1 ps, which corresponds to the exciton equilibration within the ring resulting in an exciton distribution that is similar to that observed for the disconnected rings (see Fig. 7).

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res24

186

However on a longer time scale, much slower inter-ring equilibration manifests itself leading to complete exciton equilibration in the ensemble of rings on this time scale (see Fig. 8).

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res25

187

Thus, for connected rings at room temperature a total equilibration is reached within tens of picoseconds and the final distribution is independent of the excitation wavelength.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con12

188

The final width of the exciton distribution is much broader compared to the excitation spectrum due to thermally assisted hopping motion.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res26

189

In the case of nonselective excitation, the evolution of exciton distribution is similar to the case of the disconnected rings as shown in Fig. 9.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res27

190

To study the exciton dynamics in the system of connected rings at low temperatures the same parameters of the rings as used in the previous section for isolated rings were applied,

*i.e.*,*N*= 5 and with correlated disorder.
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met18

191

For selective excitation in the low energy wing of the absorption band a single Stokes shifted exciton distribution is rapidly reached (see Fig. 10).

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs5

192

In the case of excitation in the maximum of the absorption band, as well as in the blue side of the absorption band, a much larger shift of the exciton distribution is observed as compared to red-side excitation.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res28

193

This feature is very similar to the case obtained for disconnected rings, however because of the hopping motion of excitons over larger distances a larger shift of the exciton distribution compared to the case of the disconnected rings is reached.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res29

194

Nonselective excitation yields a broad fluorescence band, which is very similar to the case of the disconnected rings as shown in Fig. 11.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res30

## Discussion

195

It is well known that in extended one-dimensional systems in the presence of any small value of the disorder the exciton states are localized.

^{39}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac20

196

However, the radius of the localized exciton is dependent on the ratio between the dispersion of the disorder and the exciton bandwidth.

^{1}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac20

197

For a finite size of the molecular aggregate, as is the case in a ring-type molecular arrangements discussed here, the value of the localization radius relative to the size of the aggregate provides an additional parameter that determines the transition between delocalized-to-localized exciton states for different values of the disorder.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod25

198

As is shown in Fig. 1 the exciton wavefunctions display a clear localized behaviour for all the exciton states in the system of 18 molecules characterized by typical parameters of the B850 ring of Bchl of

*Rps. acidophila*.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs6

199

It is noteworthy that the value of the disorder is assumed to be of the same order of magnitude as the value of the resonance interaction.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod26

200

Smaller values of the disorder would lead to a variation in the localization radius for the different exciton states, with the states at the edge of the exciton band characterized by a smaller radius of localization and with an increase of the radius of localization towards the center of the exciton band becoming commensurable with the size of the aggregate for smaller values of the disorder.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod26

201

The interaction of the exciton with local vibrational modes gives rise to an additional localization.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod26

202

For substantial values of the disorder when the Anderson-type of localization is significant for all the exciton states, the interaction with local vibrations would result in a small additional shrinking of the exciton localization radius.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod26

203

In the opposite case for small values of the disorder the exciton interaction with local vibrations may cause substantial exciton localization.

^{32}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod26

204

In conclusion, after exciton generation by light it evolves according to the delocalized (coherent) exciton representation

^{40}losing the coherent phase matching because of the exciton interaction with local vibrations and with phonons and resulting in the localization of the exciton on this time scale of the exciton relaxation.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod26

205

The subsequent exciton migration can then be considered as using such a localized picture.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod27

206

Fast exciton equilibration in ring-type structures in the case of nonselective excitation can be well described by means of the hopping model as demonstrated by modelling of the exciton dynamics in LH.1

^{33}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod27

207

Qualitatively the fluorescence band formation based on such a model can be understood as follows.

Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp3

208

At infinitely high temperatures it would be Stokes shifted only.

Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp3

209

By lowering temperature additional red shift of the fluorescence band is expected because of the Boltzmann factor weighting distributions of site energies.

Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp4

210

However, some differences are expected in the evolution of the exciton population upon selective excitation.

Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp5

211

Our application of the hopping model to LH2 aggregates show that the exciton equilibration in disconnected rings depends on the excitation conditions even at room temperature as shown in Fig. 3.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res31

212

This dependence is most pronounced for excitation to the red of the absorption band resulting in a bimodal distribution of the excitation population.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res32

213

One mode originates from rings in which the red most complex is excited.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res33

214

The second mode of the distribution of the population is related to the inhomogeneous distribution of transition energies of the ensemble of the systems.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res34

215

This effect is even more pronounced at low temperatures (see Fig. 5).

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res35

216

In the case of the disconnected rings a substantial amount of the exciton population remains in the initially excited complex, while only the remaining part gives rise to a broad distribution due to down hill energy transfer to even lower energy sites in the ensemble.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res36

217

When exciton migration between rings in a system of connected rings is induced in the model the final exciton distribution becomes independent of the excitation wavelength at room temperature as shown in Fig. 8.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res37

218

However, at low temperature equilibration of the excitation distribution is restricted because of the large Stokes shift and the fast rate of its formation, and equilibrated state is not reached in the case of selective red-side excitation on the time scale under consideration (Fig. 10).

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res38

219

Equilibration can be reached on a much slower time scale as a result of long distance excitation hopping between the suitable molecular complexes.

^{33}
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res39

220

In the case of correlated disorder the amount of excitons remaining in the initially excited state decreases due to an increased probability of down hill energy transfer and the formation of the broad red-shifted distribution.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res40

221

The amount of excitons remaining in the initial excited state is also sensitive to the size of the localized exciton with more down hill energy transfer for the larger number of the coherently coupled pigments (smaller values of sites per ring

*N*).
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res41

222

The spectral distribution of the final population displays itself in the fluorescence spectra with the homogeneous bandwidths as modulating factors.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res42

223

Aggregation of the separate rings into clusters increases the number of accessible sites for the exciton hopping, leading to two time scales of the exciton dynamics.

Type: Result |
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ConceptID: Res43

224

A fast one (subpicosecond time scale) represents the exciton migration in the ring resembling the case of the isolated rings, while a slow phase is related to the exciton equilibration in the whole system reached even at low temperatures.

Type: Result |
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ConceptID: Res44

225

This kind of organization of the rings explains the observation of a biphasic spectral evolution in membranes containing aggregated LH.2

^{27}
Type: Conclusion |
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ConceptID: Con13

Type: Conclusion |
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ConceptID: Con14

227

Indeed, at low temperatures position of the fluorescence is independent (or very weakly dependent) of the excitation wavelength when exciting to the blue side of the absorption band, while the fluorescence follows the excitation wavelength at the red-side of the absorption band.

Type: Result |
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ConceptID: Res45

228

In addition, the fluorescence bandwidth is independent of the excitation wavelength except upon excitation in the very red edge.

Type: Result |
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ConceptID: Res46

229

The same conclusions also follow from our model calculations as presented in Fig. 5.

Type: Result |
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ConceptID: Res47

230

Correlation of the disorder suggests some global distortion of the ring-type aggregate, possibly similar to that suggested on the basis of single molecular spectra of LH.2

^{15–17}
Type: Conclusion |
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ConceptID: Con15

231

Most probably the exact exciton dynamics is determined by a model, which is in between the stochastic exciton hopping and the coherent exciton relaxation.

Type: Conclusion |
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Novelty: None |
ConceptID: Con16

232

It might be expected that the correlation of the disorder results in the relaxation between some coherent exciton states at the very initial moment after the excitation of the system.

Type: Conclusion |
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ConceptID: Con17

233

Additional effect could be obtained from the exciton interaction with long-wavelength phonons, which have also to be taken into consideration.

Type: Conclusion |
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Novelty: None |
ConceptID: Con18