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Journal of Luminescence 112 (2005) 469–473

www.elsevier.com/locate/jlumin

The anisotropy of fluorescence in ring units II:transfer integral fluctuations

Pavel Hermana,�, Ivan Barvıkb, Michal Reiterb

aDepartment of Physics, University of Hradec Kralove, V. Nejedleho 573, CZ-50003 Hradec Kralove, Czech RepublicbFaculty of Mathematics and Physics, Institute of Physics of Charles University, CZ-12116 Prague, Czech Republic

Available online 13 October 2004

Abstract

The time dependence of the anisotropy of fluorescence after an impulsive excitation in the molecular ring (resembling

the B850 ring of the purple bacterium Rhodopseudomonas acidophila) is calculated. Fast fluctuations of the environment

are simulated by dynamic disorder and slow fluctuations by static disorder. Without dynamic disorder, modest degrees

of static disorder are sufficient to cause the experimentally found initial drop of the anisotropy on a sub-100 fs time

scale. In the present investigation we are comparing results for the time-dependent optical anisotropy of the molecular

ring for three models of the static disorder: Gaussian disorder in the local energies (Model A), Gaussian disorder in the

transfer integrals (Model B) and Gaussian disorder in radial positions of molecules (Model C). Both types of disorder—

static and dynamic—are taken into account simultaneously.

r 2004 Elsevier B.V. All rights reserved.

PACS: 82.39.�k; 82.53.Ps; 87.15.Aa

Keywords: Exciton transfer; Density matrix theory; Fluorescence

1. Introduction

We are dealing with the ring-shaped unitsresembling those from antenna complex LH2 ofthe purple bacterium Rhodopseudomonas acidophi-

la in which a highly efficient light collection andexcitation transfer towards the reaction center

e front matter r 2004 Elsevier B.V. All rights reserve

min.2004.09.078

ng author. Tel.: +420 493 331 186; fax: +420

ss: pavel.herman@uhk.cz (P. Herman).

takes place. Due to a strong coupling limit (largeinteraction J between bacteriochlorophylls) ourtheoretical approach considers an extended Fren-kel exciton states model.Despite intensive study, the precise role of the

protein moiety in governing the dynamics of theexcited states is still under debate [1]. At roomtemperature the solvent and protein environmentfluctuate with characteristic time scales rangingfrom femtoseconds to nanoseconds. The dynami-cal aspects of the system are reflected in the line

d.

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P. Herman et al. / Journal of Luminescence 112 (2005) 469–473470

shapes of electronic transitions. To fully charac-terize the line shape of a transition and thereby thedynamics of the system, one needs to know notonly the fluctuation amplitude (coupling strength)but also the time scale of each process involved.The observed linewidth reflect the combinedinfluence of static disorder and exciton couplingto intermolecular, intramolecular, and solventnuclear motions. The simplest approach is todecompose the line profile into homogeneous andinhomogeneous contributions of the dynamic andstatic disorder. Yet, a satisfactory understandingof the nature of the static disorder in light-harvesting systems has not been reached [1]. Inthe site excitation basis, there can be present staticdisorder in both diagonal and off-diagonal ele-ments. Silbey pointed out several questions: it isnot clear whether only the consideration of theformer is enough or the latter should be includedas well. If both are considered, then there remainsa question about whether they are independent orcorrelated.Time-dependent experiments of the femtose-

cond dynamics of the energy transfer and relaxa-tion [2,3] led for the B850 ring in LH2 complexesto conclusion that the elementary dynamics occurson a time scale of about 100 fs [4–6]. For example,depolarization of fluorescence was studied alreadyquite some time ago for a model of electronicallycoupled molecules [7,8]. Rahman et al. [7] were thefirst to recognize the importance of the off-diagonal density matrix elements (coherences) [9]which can lead to an initial anisotropy larger thanthe incoherent theoretical limit of 0.4. Alreadysome time ago substantial relaxation on the timescale of 10–100 fs and an anomalously large initialanisotropy of 0.7 was observed by Nagarjan et al.[4]. The high initial anisotropy was ascribed to acoherent excitation of a degenerate pair of stateswith allowed optical transitions and then relaxa-tion to states at lower energies which haveforbidden transitions. Nagarjan et al. [5] con-cluded, that the main features of the spectralrelaxation and the decay of anisotropy arereproduced well by a model considering decayprocesses of electronic coherences within themanifold of the excitonic states and thermalequilibration among the excitonic states. In that

contribution the exciton dynamics was not calcu-lated explicitly.In several steps [10–13], we have recently

extended the former investigations by Kumbleand Hochstrasser [14] and Nagarjan et al. [5]. Fora Gaussian distribution of local energies in the ringunits we added the effect of dynamical disorder byusing a quantum master equation in the Marko-vian and non-Markovian limits.In our present investigation we are comparing

the results for the time-dependent optical aniso-tropy of the molecular ring for three models of thestatic disorder: Gaussian disorder in the localenergies, Gaussian disorder in the transfer inte-grals and Gaussian disorder in radial positions ofmolecules.

2. Model

In the following, we assume that only oneexcitation is present on the ring after an impulsiveexcitation [14]. The Hamiltonian of an exciton inthe ideal ring coupled to a bath of harmonicoscillators reads

H0 ¼X

m;nðmanÞ

Jmnayman þ

Xq

_oqbyqbq

þ1ffiffiffiffiffiN

pX

m

Xq

Gmq _oqay

mamðbyq þ b�qÞ

¼ H0ex þ Hph þ Hex�ph: ð1Þ

H0ex represents the single exciton, i.e. the system.

The operator aym (am) creates (annihilates) an

exciton at site m: Jmn (for man) is the so-calledtransfer integral between sites m and n: Hph

describes the bath of phonons in the harmonicapproximation. The phonon creation and annihi-lation operators are denoted by by

q and bq;respectively. The last term in Eq. (1), Hex�ph;represents the exciton–bath interaction which isassumed to be site–diagonal and linear in the bathcoordinates. The term Gm

q denotes the exciton–phonon coupling constant.Inside one ring the pure exciton Hamiltonian

H0ex (Eq. (1)) can be diagonalized using the wave

vector representation with corresponding deloca-lized ‘‘Bloch’’ states and energies. Considering a

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P. Herman et al. / Journal of Luminescence 112 (2005) 469–473 471

homogeneous case with only nearest-neighbortransfer matrix elements Jmn ¼ J12ðdm;nþ1 þ

dm;n�1Þ and using Fourier-transformed excitonicoperators (Bloch representation)

ak ¼X

n

aneikn; k ¼2pN

l;

l ¼ 0;�1; . . . ; �N=2; ð2Þ

the simplest exciton Hamiltonian in ~k representa-tion reads

H0ex ¼

Xk

Ek ay

kak; with Ek ¼ �2J12 cos k:

(3)

Influence of static disorder is modelled by aGaussian distribution

(A)

For the uncorrelated local energy fluctuations�n (with a standard deviation D)

HAs ¼

Xm

�nayman:

(B)

For the uncorrelated transfer integral fluctua-tions dJnm with a standard deviation DJ

HBs ¼

Xm;nðmanÞ

dJmnayman:

We are using nearest-neighbor approxima-tion.

(C)

For the uncorrelated fluctuations of radialpositions of molecules (with standard devia-tion Dr and hrni ¼ r0)

rn ¼ r0 þ drn

leading to HCs ; which adds to the Hamiltonian

of the ideal ring

H ¼ H0 þ HXs : (4)

All of the Qy transition dipole moments of thechromophores (bacteriochlorophylls (BChls)B850) in a ring without static and dynamicdisorder lie approximately in the plane of the ringand the entire dipole strength of the B850 bandcomes from a degenerate pair of orthogonallypolarized transitions at an energy slightly higherthan the transition energy of the lowest excitonstate.

The dipole strengths ~ma of the eigenstate jai

of the ring with static disorder and ~ma ofthe eigenstate jai of the ring without static dis-order read

~ma ¼XN

n¼1

can~mn; ~ma ¼

XN

n¼1

can~mn; (5)

where can and can are the expansion coefficients of

the eigenstates of the unperturbed ring and thedisordered one in site representation, respectively.In the case of impulsive excitation, the dipolestrength is simply redistributed among the excitonlevels due to disorder [14]. Thus the impulsiveexcitation with a pulse of sufficiently wide spectralbandwidth will always prepare the same initialstate, irrespective of the actual eigenstates of thereal ring. After impulsive excitation with polariza-tion ~ex the excitonic density matrix r [11] is givenby [5]

rabðt ¼ 0;~exÞ ¼1

Að~ex � ~maÞð~mb � ~exÞ;

A ¼Xa

ð~ex � ~maÞð~ma � ~exÞ: ð6Þ

The usual time-dependent anisotropy of fluores-cence

rðtÞ ¼hSxxðtÞi � hSxyðtÞi

hSxxðtÞi þ 2hSxyðtÞi;

SxyðtÞ ¼

ZPxyðo; tÞdo ð7Þ

is determined from

Pxyðo; tÞ ¼ AX

a

Xa0

raa0 ðtÞð~ma0 � ~eyÞð~ey � ~maÞ

� ½dðo� oa00Þ þ dðo� oa0Þ�: ð8Þ

The brackets hi denote the ensemble average andthe orientational average over the sample.The crucial quantity entering the time depen-

dence of the anisotropy in Eq. (7) is the excitondensity matrix r: The dynamical equations for theexciton density matrix obtained by Capek [15] read

d

dtrmnðtÞ ¼

Xpq

iðOmn;pq þ dOmn;pqðtÞÞrpqðtÞ: (9)

In long time approximation coefficient dOðt ! 1Þ

becomes time independent.

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All details of a calculation leading to the time-convolutionless dynamical equations for the ex-citon density matrix are given elsewhere [13] andwe shall not repeat them here. The full timedependence of dOðtÞ is given through timedependent parameters [15]

ApmnðtÞ ¼

Z t

0

i_

N

Xk

o2kðG

m�k � Gn

�kÞX

r

Grk

�Xa;b

hb j rihr j aiha jmihp jbie�i=_ðEa�EbÞt

� f½1þ nBð_okÞ� eiokt þ nBð_okÞ e

�ioktgdt: ð10Þ

Obtaining the full time dependence of dOðtÞ isnot a simple task. We have succeeded to calculatemicroscopically full time dependence of dOðtÞ onlyfor the simplest molecular model namely dimer[16]. In case of molecular ring we should resort tosome simplification [13].In what follows we use Markovian version of

Eq. (10) with a simple model for correlationfunctions Cmn of the bath assuming that each site(i.e. each chromophore) has its own bath com-pletely uncoupled from the baths of the other sites.Furthermore it is assumed that these baths haveidentical properties [2,17]. Then only one correla-tion function CðoÞ of the bath is needed

CmnðoÞ ¼ dmnCðoÞ

¼ dmn2p½1þ nBðoÞ�½JðoÞ � Jð�oÞ�: ð11Þ

Here JðoÞ is the spectral density of the bath [17]and nBðoÞ the Bose–Einstein distribution ofphonons. The model of the spectral density JðoÞoften used in the literature is

JðoÞ ¼ YðoÞj0o2

2o3c

e�o=oc : (12)

Spectral density has its maximum at 2oc: We shalluse (in agreement with Ref. [2]) j0 ¼ 0:4 and oc ¼

0:2:

Fig. 1. The time and D dependence of the anisotropy

depolarization for three models of the static disorder is given.

In the left column the results without the exciton–bath

interaction are shown, in the right column the interaction with

the bath is taken into account in the Markovian treatment of

the dynamic disorder with the j0 ¼ 0:4 and for temperature T ¼

0:5 (in dimensionless units).

3. Results and conclusions

The anisotropy of fluorescence (Eq. (7)) hasbeen calculated using dynamical equations for theexciton density matrix r to express the timedependence of the optical properties of the ring

units in the femtosecond time range. Details arethe same as in Ref. [13]. In Ref. [14], which doesnot take the bath into account, the anisotropy offluorescence of the LH2 ring decreases from 0:7 to0:3� 0:35 and subsequently reaches a final valueof 0:4: One needs a strength of static disorder ofD � 0:4� 0:8 to reach a decay time below 100 fs.Our numerical results are presented graphically

in Fig. 1. We use dimensionless energies normal-ized to the transfer integral J12 and the renorma-lized time t: To convert t into seconds one has todivide t by 2pcJ12 with c being the speed of light incms�1 and J12 in cm�1: Estimation of the transferintegral J12 varies between 250 and 400 cm�1: Forthese extreme values of J12 our time unit (t ¼ 1)corresponds to 21.2 or 13.3 fs.In Fig. 1 the time and static disorder D

dependence of the anisotropy depolarization forthree models of the static disorder is given. In

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model B D ¼ DJ and in model C D ¼ Dr: In the leftcolumn the results without the exciton–bathinteraction are shown, in the right column theinteraction with the bath is taken into account inthe Markovian treatment of the dynamic disorderwith the j0 ¼ 0:4 and for the temperature T ¼ 0:5(in dimensionless units). To convert T into kelvinsone has to divide T by k=J12 with k being theBoltzmann constant in cm�1 K�1 and J12 in cm�1:Rahman et al. [7] were the first who recognized

the importance of the off-diagonal density matrixelements (coherences) [9] which can lead to aninitial anisotropy rð0Þ larger than the incoherenttheoretical limit of 0.4.Without dynamic disorder modest degrees of

static disorder are sufficient to cause the experi-mentally found initial drop of the anisotropy on asub-100 fs time scale. Difference between theGaussian static disorder in the local energies(Model A) and the Gaussian static disorder inthe transfer integrals (Model B) calculationsexpressed by the time interval in which theanisotropy depolarization reaches r ¼ 0:4 (theincoherent theoretical limit) is almost as much as100% for the same value of the static disorder D ¼

DJ : It means that the same drop of the anisotropymay be caused even by the diagonal static disorder(model A) with D or by the static disorder in thetransfer integrals with DJ ¼ 0:5D: This differencebetween Model A and B calculations is still presentalso in the case, when the exciton interaction withthe bath is taken into account. In model C thestrength of the static disorder Dr ¼ 0:16 haspractically the same effect as D ¼ 0:8 in model A.

Acknowledgements

This work has been funded by the projectGACR 202-03-0817.

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