ARTICLE IN PRESS

Journal of Luminescence 108 (2004) 85–89

*Correspond

49-551-3890.

E-mail addr

0022-2313/$ - se

doi:10.1016/j.jlu

Energy relaxation and transfer in excitonic trimer

Pavel He$rmana,*, Ivan Barv!ıkb, Martin Urbaneca

aDepartment of Physics, University of Hradec Kr !alov!e, V. Nejedl!eho 573, CZ-50003 Hradec Kr !alov!e, Czech RepublicbFaculty of Mathematics and Physics, Institute of Physics of Charles University, CZ-12116 Prague, Czech Republic

Abstract

Two models describing exciton relaxation and transfer (the Redfield model in the secular approximation and $C!apek’s

model) are compared for a simple example—a symmetric trimer coupled to a phonon bath. Energy transfer within the

trimer occurs via resonance interactions and coupling between the trimer and the bath occurs via modulation of the

monomer energies by phonons. Two initial conditions are adopted: (1) one of higher eigenstates of the trimer is initially

occupied and (2) one local site of the trimer is initially occupied. The diagonal exciton density matrix elements in the

representation of eigenstates are found to be the same for both models, but this is not so for the off-diagonal density

matrix elements. Only if the off-diagonal density matrix elements vanish initially (initial condition (1)), they then vanish

at arbitrary times in both models. If the initial excitation is local, the off-diagonal matrix elements essentially differ.

r 2003 Elsevier B.V. All rights reserved.

PACS: 71.35.Aa; 31.10.+Z; 31.15.Md

Keywords: Exciton transfer; Relaxation; Density matrix theory

1. Introduction

The very symmetric arrangement [1] of BChlmolecules with short distances between thembrought up the usage of the exciton concept inthe description of the energy transfer in the LH2ring-shaped subunit of the antenna complex ofRps. acidophila. The energy transfer in this purplebacteria is likely to involve initially a coherentstage, in which the excitation is delocalized overseveral molecules. Localization takes place due tothe static and dynamic disorder [2]. The excited-

ing author. Tel.: +420-495-061-186; fax: +420-

ess: pavel.herman@uhk.cz (P. He$rman).

e front matter r 2003 Elsevier B.V. All rights reserve

min.2004.01.016

state dynamics are measured using, e.g., one- andtwo-color pump-spectroscopy, transient gratingand three-pulse photon-echo spectroscopy. Studieshave been conducted at both low temperatures androom temperature.

While the influence of the static disorder can betheoretically taken into account by repeating thecalculation for another set of the stochastic localenergies, one is generally forced to deal with a verycomplicated Liouville equation for the wholesystem-bath density matrix s to describe theexciton interacting with phonon bath. That iswhy various types of convolutional or convolu-tionless dynamical equations for the excitondensity matrix r (bath degrees of freedom aretraced off) have been often used to describecoherence effects in the exciton transfer.

d.

ARTICLE IN PRESS

P. He$rman et al. / Journal of Luminescence 108 (2004) 85–8986

In the past, two different approaches have beendeveloped:

Working on the basis of exciton eigenstates ofthe molecular aggregate, the Redfield equations inthe secular approximation (henceforth referred toas the Redfield equations) have been often used [2].

On the other hand, the static and dynamicdisorder destroys correlations between the phasesof distant molecular sites and typical electronicproperties are therefore local, and an intuitivephysical picture should then be based on theproperties of wave packets. A local real-spacedescription (the dynamical equations for theexciton density matrix in the local site basis) couldbe more appropriate [3].

The main message of our work is that it could berisky only to transform the Redfield equationsfrom the eigenstate basis to the local-site basis.

In a previous paper [5], we investigated excitontransfer and relaxation in a symmetric dimer. Wehave shown how the proper dynamical equations(first developed by $C!apek) for the exciton densitymatrix rðtÞ written in the local state basis resemble,but substantially enrich, the equations used in theHaken–Strobl–Reineker Stochastic Liouvilleequations method (HSR-SLE) [4]. In this paperwe continue our analysis (from Ref. [6]) byconcentrating on exciton transfer and relaxationin a symmetric trimer with an energy spectrum (atwo-fold degenerate upper level and a non-degenerate lower level) similar to the three-levelmodel used for an explanation of the opticalproperties of the LH2 subunit [7].

2. Hamiltonian

We shall be dealing with just one exciton in thesymmetric cyclic trimer interacting with thephonon bath. The Hamiltonian then consists ofthree parts

H ¼ H0 þ Hb þ Hint; ð1Þ

where H0 is the exciton part, Hb the bath part andHint describes the local and linear exciton phononcoupling.

The first part (where awn; an are creation and

annihilation operators of exciton on site n and J is

the transfer integral) is

H0 ¼ Jðaw1a2 þ aw

1a3 þ aw2a3 þ aw

2a1

þ aw3a1 þ aw

3a2Þ; ð2Þ

which gives for Jo0 eigenenergies EI ¼ �2jJ j;EII ¼ EIII ¼ jJ j and eigenstates:

jIST ¼1ffiffiffi3

p ð1; 1; 1Þ;

jIIST ¼1ffiffiffi3

p ð1; e2pi=3; e�2pi=3Þ;

jIIIST ¼1ffiffiffi3

p ð1; e�2pi=3; e2pi=3Þ: ð3Þ

3. Dynamical equations for the exciton density

matrix

To follow exciton transfer and relaxation onehas to obtain the time dependence of the excitondensity matrix r:

3.1. Local-site representation

$C!apek’s dynamical equations for the exciton-density matrix r with time-dependent coefficientsionm;qpðtÞ derived in local basis are [8]

d

dtrnmðtÞ ¼

Xpqionm;qpðtÞrqpðtÞ;

ionm;qpðtÞ ¼ iOnm;qp þ idOnm;qpðtÞ: ð4Þ

The time-independent part iO is given by thecoherent part of the Liouville superoperator. Weshall consider also the part idOnm;qpðtÞ; whichdetermines dissipative and relaxation processes,to be, after some initial time interval, timeindependent [8].

We suppose that each site has its own indepen-dent bath, but with otherwise the same properties.In this case we parameterize [6] idO using threecoefficients A; C and D: The temperature depen-dences of these parameters are entirely different.Parameter A is independent of temperature,whereas parameter CE½1þ 2nBð3jJ jÞ� and para-meter DEnBð3jJ jÞ (here nBð_oÞ is a Bose–Einsteindistribution of phonons).

ARTICLE IN PRESS

P. He$rman et al. / Journal of Luminescence 108 (2004) 85–89 87

The HSR-SLE [4], which has been many timessuccessfully used in the exciton transfer problem,but very often also out of the range of itsapplicability, has in the case of a weak linear localexciton–phonon interaction a dissipation part withonly one nonzero parameter C ¼ 2g0 (in $C!apek’snotation).

3.2. Eigenstate representation

The earliest way used to treat the relaxation dueto the exciton–bath interaction is perhaps theRedfield model [9,10], which uses an expansion inpowers of the exciton–bath coupling, i.e. it is of theweak-coupling type. To make the equations moresimple, Redfield himself suggested the so-calledsecular approximation [9,10] leading, for ourHamiltonian, to independent time developmentof each off-diagonal exciton density matrixelement in the eigenstate basis. This Redfieldmodel has been successfully used in descriptionof the energy relaxation between extended states.However, interpretation of recent optical experi-mental data [11,12] requires knowledge of the timedevelopment of the whole exciton density matrix rin both extended and local state representations.

Fig. 1. Time dependences of the real parts of the density matrix

elements rI ;II ; rI ;III and rII ;III for g0 ¼ C=2 ¼ 1 for the case of

initial excitation to the local state j1S in (a) the low-temperature

limit, (b) the high-temperature limit. The results for the HSR-

SLE model are shown only in the high-temperature limit.

4. Results

To compare these different models of excitontransfer in the local-site representation and in theeigenstate representation, we have to transformthe equations of $C!apek and HSR-SLE into theeigenstate basis and the Redfield equations back tothe local-site representation.

From such a comparison, we conclude:(i) There are nonzero elements Rij;ji for iaj in the

Redfield model contrary to the $C!apek and HSR-SLEmodels. This means (in the HSR-SLE notation)

%gija0: But the parameters %gij should be equal to zerofor linear and local exciton–phonon coupling [13].

(ii) In the Redfield model, there are nonzeroelements Rii;jj which open a second incoherentchannel (in addition to the quasicoherent one) ofexciton transfer.

To display more thoroughly the differencesbetween the three above-mentioned models we

compare solutions of the corresponding dynamicalequations of motion for the exciton density matrixr with two different initial conditions:

(1)

the eigenstate jIIS is initially occupied:rII ;II ð0Þ ¼ 1 and ra;bð0Þ ¼ 0 for aaII or baII ;

(2)

the local state j1S is initially occupied:r1;1ð0Þ ¼ 1 and rm;nð0Þ ¼ 0 for ma1 or na1:

The most interesting results of the time develop-ment of the exciton density matrix elementsobtained in the eigenstate and in the local-siterepresentations are displayed in Figs. 1 and 2.

ARTICLE IN PRESS

Fig. 2. Time dependences of the occupation probabilities r11;r22 and r33 for g0 ¼ C=2 ¼ 1 for the case of initial excitation to

the local state j1S in (a) the low-temperature limit, (b) the high-

temperature limit. Results for the HSR-SLE model shown only

in the high-temperature limit.

P. He$rman et al. / Journal of Luminescence 108 (2004) 85–8988

4.1. The eigenstate initially occupied

The diagonal and off-diagonal parts of excitondensity matrix r in the eigenstate representationdevelop in time independently in the $C!apek modeland the Redfield model. Then, in the case of theinitial occupation of the eigenstate jIIS (or morecommonly in the case of an initially diagonaldensity matrix in the eigenstate basis), the timedevelopment of the whole density matrix r is thesame as in the $C!apek model as in the Redfield oneand both models describe properly the energy

relaxation to the lower eigenstates. Long-timeasymptotics of the eigenstate occupation probabil-ities reveal a proper Boltzmann distribution.

In high-temperature limit, the occupation prob-ability of the eigenstate jIIS decreases morequickly in the HSR-SLE model in comparisonwith the Redfield one. On the other hand, theoccupation probability of the eigenstate jISincreases more quickly in the Redfield model.The real parts of the nondiagonal exciton densitymatrix elements in the local-site representationapproach zero more quickly in the $C!apek model incomparison with the HSR-SLE results. Results forthe imaginary parts are diverse.

4.2. The local state initially occupied

The time development of the eigenstate occupa-tion probabilities is again the same in the $C!apekmodel and in the Redfield model. But this is notthe case for the off-diagonal exciton density matrixelements in the eigenstate basis. In the Redfieldmodel, contrary to the $C!apek one, each off-diagonal matrix element develops with timeindependently.

In the local-site representation the excitontransfer from site 1 to sites 2 and 3 is quicker inthe Redfield model due to a spurious incoherentchannel. Differences between the $C!apek and Red-field results for all elements of the exciton densitymatrix persist up to the moment of equilibrium.

In the high-temperature limit, $C!apek’s resultsfollow closely to the HSR-SLE results in the initialstage of the exciton transfer.

In this paper we have used a simple system—trimer as a typical example which we can solve (atleast partially) analytically. The obtained resultsjustify our application of the $C!apek model to a morecomplicated system, namely to the exciton transferand relaxation in the LH2 subunit of the antennasystem of the bacterial photosynthetic units [14].

Acknowledgements

This work has been partially funded by theproject GA $CR 202-03-0817 of the Czech GrantAgency.

ARTICLE IN PRESS

P. He$rman et al. / Journal of Luminescence 108 (2004) 85–89 89

References

[1] G. McDermott, S.M. Prince, A.A. Freer, A.M.

Hawthornthwaite-Lawiess, M.Z. Papiz, R.J. Cogdell,

N.W. Issacs, Nature 374 (1995) 517.

[2] V. Sundstr .om, T. Pullerits, R. van Grondelle, J. Phys.

Chem. B 103 (1999) 2327.

[3] S. Mukamel, S. Tretiak, T. Wagenstreiter, V. Chernyak,

Science 277 (1997) 781.

[4] P. Reineker, in: G. H .ohler (Ed.), Exciton Dynamics in

Molecular Crystals and Aggregates—Springer Tracts in

Modern Physics, Vol. 94, Springer, Berlin, Heidelberg,

1982, p. 111.

[5] V. $C!apek, I. Barv!ık, P. He$rman, Chem. Phys. 270 (2001)

141.

[6] P. He$rman, I. Barv!ık, Chem. Phys. 274 (2001) 199.

[7] M.V. Mostovoy, J. Knoester, J. Phys. Chem. B 104 (2000)

12355.

[8] V. $C!apek, Z. Phys. B 99 (1996) 261.

[9] A.G. Redfield, IBM J. Res. Dev. 1 (1957) 19.

[10] A.G. Redfield, in: J.J. Waugh (Ed.), Advances in Magnetic

Resonance, Vol. 1, Academic Press, New York, London,

1965, p. 1.

[11] V. Nagarjan, R.G. Alden, J.C. Williams, W.W. Parson,

Proc. Natl. Acad. Sci. USA 93 (1996) 13774.

[12] V. Nagarjan, E.T. Johnson, J.C. Williams, W.W. Parson,

J. Phys. Chem. B 103 (1999) 2297.

[13] V. $C!apek, I. Barv!ık, Phys. Rev. A 46 (1992) 7431.

[14] P. He$rman, I. Barv!ık, U. Kleinekath .ofer, M. Schreiber,

Chem. Phys. 275 (2002) 1.