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Journal of Luminescence 112 (2005) 420–423

www.elsevier.com/locate/jlumin

Abrupt exciton self-trapping in finite and disorderedone-dimensional aggregates

Gediminas Trinkunasa,�, Arvi Freibergb,c

aInstitute of Physics, Savanoriu pr. 231, LT-02300 Vilnius, LithuaniabInstitute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia

cInstitute of Molecular and Cell Biology, University of Tartu, Riia 23, 51010 Tartu, Estonia

Available online 14 October 2004

Abstract

We show that in finite, disordered one-dimensional aggregates the exciton self-trapping takes place at significantly

weaker exciton–lattice coupling than it is known for the regular infinite lattices. A series of abrupt changes of self-

trapped exciton energy and other parameters can take place at multiple critical couplings, depending on initial exciton

amplitude distributions. These states vary by the wave function localization at different segments of the aggregate. The

observed energy splitting suggests a coexistence of two or more types of self-trapped excitons with altered lattice

reorganization energy. This provides a general explanation for several recent spectroscopic observations on

photosynthetic antenna complexes.

r 2004 Elsevier B.V. All rights reserved.

Keywords: Fluorescence; Self-trapped excitons

In perfect one-dimensional aggregates a transi-tion from free exciton to small excitonic polaronor self-trapped exciton with increasing exciton–-lattice coupling is smooth [1]. For short-rangeexciton–lattice couplings, this result has beenconfirmed by a number of numerical simulations[2–4]. The self-trapped exciton features, however,differ considerably for finite molecular aggregatesincorporating diagonal disorder of excited stateenergies. This has been revealed by studying the

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

min.2004.09.040

ng author. Tel.:+3705 2661649; fax:+3705

ss: trinkun@ktl.mii.lt (G. Trinkunas).

ground-state properties of excitonic polarons in acircular one-dimensional aggregate simulating thelight harvesting complex LH2 [5] of purplephotosynthetic bacteria. The particular pigmentaggregate (see Fig. 1) has been chosen due tocomprehensive availability of structural and spec-troscopic data [6].The Hamiltonian for the exciton interacting

with the lattice distortion reads [7]

H ¼X

n

�njnihnj þXnam

tnmjnihmj

þ cX

n

qnjnihnj þ1

2

Xn

q2n: ð1Þ

d.

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B850

B800

Fig. 1. Arrangement of bacteriochlorophyll a molecule in the

LH2 light-harvesting complex from the purple bacterium Rps.

acidophila [5]. The closely spaced upper-ring molecules (centre

to centre distance of 0.9 nm) form the so-called B850 aggregate

under consideration in this work.

G. Trinkunas, A. Freiberg / Journal of Luminescence 112 (2005) 420–423 421

Here jni and hnj represent ket and bra vectors,respectively, for excitation localized on thelattice site n: The excited molecular energy �n isassumed to be a Gaussian random variable with amean �0 and standard deviation s: Matrix ele-ments tnm denote the intermolecule dipole–dipolecoupling energies between the pigments on thesites n and m ðnamÞ: qn represents the locallattice distortion at site n:C is the couplingconstant for the short-range excitation–lattice interaction. The first two terms in Eq. (1)represent the exciton energy, the third termstands for the excitation–lattice coupling energy,while the last term is the lattice potential energyat unit elastic lattice constants. Due to theadiabatic approximation the lattice kinetic energyis omitted.In rigid ðc ¼ 0Þ circular lattice without diagonal

disorder (�n ¼ �0) the exciton states are representedby Bloch-type wavefunctions that are character-ized by the wavenumber k: The optical propertiesof this structure are generally described by justthree states with k ¼ 0; �1; the latter two beingdegenerate. Introducing the spectral disorderremoves degeneracy of the exciton states andeffectively makes all the states optically allowed,while usually the states k ¼ �1 still dominate thespectrum (see, e.g., Ref. [8]).By expressing the eigenstates of the exciton–

lattice system on the basis of local state

vectors

jni ¼X

n

jnnjni (2)

with the amplitudes jnn; which satisfy thenormalization conditionX

n

jjnnj2 ¼ 1; (3)

the expectation value of Hamiltonian (1) can berepresented as a function

Jnðfjnng; fqngÞ ¼ hnjHjni ¼X

n

�njn

nnjnn

þX

n

XmðanÞ

tnmjn

nnjmn þ cX

n

qnjn

nnjnn

þ1

2

Xn

q2n: ð4Þ

One can further get an optimal distortion byfinding the extrema of Jn with respect to distor-tions fqng: At this specific distortion the energyfunction depends only on exciton amplitudes

JnðfjnngÞ ¼X

n

�n �c2

2jjnnj

2

� �jjnnj

2

þXnam

tnmjn

njm: ð5Þ

The distribution of amplitudes fjnng can then befound by solving a set of equations

�njnn þX

mðanÞ

tnmjmn � c2jjnnj2jnn ¼ Enjnn; (6)

where the Lagrange multiplier En is the energy ofthe excitonic polaron. The system of Eqs. (6)represents a stationary version of the well-knowndiscrete nonlinear Schrodinger equation (discreteself-trapping equation) [9]. These equations con-stitute the nonlinear eigenvalue problem with thenonlinearity parameter c2. Due to the branching ofeigenstates with increasing nonlinearity parameter,the number of eigenstates increases rapidly and,therefore, the solution of Eqs. (6) is a cumbersomeprocedure even for regular chains [9]. Note thatwith c=0, Eqs. (6) become linear and theirsolution coincides with that of the commonmolecular chain exciton with diagonal disorder [8].In what follows, we concentrate on the ground-

state (n=0) properties of the excitonic polaron.

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G. Trinkunas, A. Freiberg / Journal of Luminescence 112 (2005) 420–423422

Determination of this lowest eigenstate is equiva-lent to finding the minimum of function (5) withthe constraint of amplitude normalization (3). Dueto nonlinearity of the function the minimumdepends on the initial (seeding) amplitude dis-tribution. Since the energy spacing between theadjacent k levels of disordered exciton is largerthan the characteristic frequency of the on-sitevibration, DEk4o (see Refs. [10,11] for theexperimental o value in LH2), the interstatemixing is suppressed. This allows the wavefunctionamplitudes of the disordered k-states to beconsidered as initial vectors for the self-trappedground state. Below, an upper index k relates thecharacteristics of the self-trapped exciton to theinitial conditions.The lowest eigenvalue Ek

0 is related to theexpectation value of Hamiltonian (1)

Ek ¼ min JðfjnngÞ

fjnng ð7Þ

via the relation

Ek0 ¼ Ek �

c2

2

XN

n¼1

jjkn0j

4 ¼ Ek �c2

2Lk; (8)

where Ek is the position of the zero phonon line(the phononless energy of the self-trapped exciton)and Lk is the delocalization length of the self-trapped exciton, defined as the inverse participa-tion ratio. Eq. (8) thus identifies c2=2Lk with thehalf spectral Stokes shift (state reorganizationenergy) and provides its dependence on the self-trapped exciton size.To investigate the ground-state properties of

excitonic polarons on exciton–lattice couplingconstant, g ¼ c2=4V (V ¼ maxðjtnmjÞ; the follow-ing three excitonic polaron characteristics havebeen simulated for a single diagonal disorderrealization with the random Gaussian site energydistribution (s ¼ 0:6V [8] ): the expectation energyEk; the delocalisation length Lk

0 ; and the super-radiance enhancement factor. The latter para-meter, defined as a ratio of the radiative decayrate of the aggregate to that of the single mole-cule, determines the superradiance coherence size

of the aggregate:

Fk ¼Xnm

ð~mn~mmÞjnkn0j

km0 (9)

(~m stands for the molecular transition dipolemoment vectors).The results of the calculations for the five lowest

k-state vectors are shown in Fig. 2 . It can be seenthat only in the case where k ¼ 0 all the threecharacteristics show a smooth self-trapping transi-tion typical for perfectly ordered one-dimensionalaggregates. Due to static disorder the transitionalready takes place at g � 0:2; the value muchsmaller than 0.7 known for the ordered aggregates[2–4] . Up to g � 0:3 the exciton characteristics areindependent of initial amplitude distributions. Atstronger couplings, the curves corresponding todifferent initial amplitude distributions abruptlysplit off from the k ¼ 0 curve towards largerparameter values. These abrupt changes occurringat individual critical g values are indicated in Fig. 2by dashed lines and labelled with particular k

numbers. It is important to notice that the splitbranches relate to the self-trapped exciton locali-zation on different segments of the aggregate. Thecentre of the exciton always coincides with thelowest site energy of the segment. The self-trappedexciton wavefunctions obtained for g=0.7 (whereall the five initial amplitude distributions reveal thesplit courses) are shown in Fig. 3 .At the relative diagonal disorder used

(s ¼ 0:6V ), the features just described showed upin 25% of the aggregates out of an ensemble of1400 aggregates.We have shown that in finite, disordered one-

dimensional aggregates the self-trapping transitiontakes place at significantly smaller exciton–latticecoupling value than it is known for the regularinfinite lattices. A series of abrupt changes of self-trapped exciton energy and other parameters cantake place at multiple critical couplings, dependingon initial exciton amplitude distributions. Pro-vided the latter follow the ones of the disorderedexciton wave functions, which seems to be aphysically reasonable assumption, the presence ofa few distinct self-trapped exciton states per singleaggregate is likely. These states differ by the wavefunction centre localization at the molecular site

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Fig. 2. Ground state properties (A—the expectation energy,

B—the delocalization length, and C—the superradiance ratio)

of excitonic polarons in diagonally disordered aggregates as a

function of exciton–lattice coupling constant g. Dashed vertical

lines indicate abrupt changes of the course discussed in the text.

The numbers at dashed lines are for k and label the initial

amplitude distributions. The vertical dotted line points the

coupling strength used to calculate the wavefunctions in Fig. 3.

Fig. 3. A. Self-trapped exciton wavefunctions for the exciton–-

lattice coupling constant g=0.7. The numbers (k-values) under

the wavefunction cusps label different initial amplitude

distributions. B. Distribution of site excitation energies

corresponding to data presented in panel A.

G. Trinkunas, A. Freiberg / Journal of Luminescence 112 (2005) 420–423 423

characterized by the lowest excitation energy (anattractor).The observed splitting of energies suggests a

coexistence of two or more sorts of self-trappedexcitons with different lattice reorganization en-ergies. Similar diversity concerns other character-istics of self-trapped excitons. This provides aviable general explanation for several recentspectroscopic observations on LH2 complexes atcryogenic and ambient temperatures: a structuredfluorescence excitation spectrum [12], temporaljumping of the fluorescence band maximum [13],and a dual fluorescence band [10,11,14,15].The Lithuanian and Estonian Science Founda-

tions (Grants nos. T-04185 and 5543, respectively)supported this work.

References

[1] I. E. Rashba, Self-trapped excitons, in: M. D. Sturge (Ed.),

Excitons, North-Holland, Amsterdam, 1982, p. 543.

[2 ] V.V. Kabanov, O. Yu. Mashtakov, Phys. Rev. B 47 (1993)

6060.

[3 ] S. Higai, A. Sumi, J. Phys. Soc. Japan 63 (1994) 4489.

[4] K. Noba, Y. Kayanuma, J. Phys. Soc. Japan 67 (1998)

3972.

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

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

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

[6] H. van Amerongen, L. Valkunas, R. van Grondelle,

Photosynthetic Excitons, World Scientific, Singapore,

2000.

[7] T. Holstein, Ann. Phys. 8 (1959) 325.

[8] A. Freiberg, K. Timpmann, R. Ruus, N.W. Woodbury,

J. Phys. Chem. B 103 (1999) 10032.

[9] J.C. Eilbeck, P.S. Lomdahl, A.C. Scott, Physica 16D

(1985) 318.

[10] M. Ratsep, A. Freiberg, Chem. Phys. Lett. 377 (2003) 371.

[11] K. Timpmann, M. Ratsep, C.N. Hunter, A. Freiberg,

J. Phys. Chem. B 108 (2004) 10581.

[12] A.M. van Oijen, M. Ketelaars, J. Kohler, T.J. Aartsma,

J. Schmidt, Science 285 (1999) 400.

[13] D. Rutkauskas, V. Novoderezhkin, R.J. Cogdell, R. van

Grondelle, Biochemistry 43 (2004) 4431.

[14] A. Freiberg, M. Ratsep, K. Timpmann, G. Trinkunas,

N.W. Woodbury, J. Phys. Chem. B 107 (2003) 11510.

[15] A. Freiberg, M. Ratsep, K. Timpmann, G. Trinkunas,

J. Lumin 108 (2004) 107.