Potential surfaces and delocalization of excitons in dimersW. J. D. Beenken, M. Dahlbom, P. Kjellberg, and T. Pullerits Citation: The Journal of Chemical Physics 117, 5810 (2002); doi: 10.1063/1.1502647 View online: http://dx.doi.org/10.1063/1.1502647 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/117/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Exciton delocalization, charge transfer, and electronic coupling for singlet excitation energy transfer betweenstacked nucleobases in DNA: An MS-CASPT2 study J. Chem. Phys. 140, 095102 (2014); 10.1063/1.4867118 Relativistic potential energy surfaces of initial oxidations of Si(100) by atomic oxygen: The importance of surfacedimer triplet state J. Chem. Phys. 136, 214704 (2012); 10.1063/1.4725542 Communication: Prediction of the rate constant of bimolecular hydrogen exchange in the water dimer using an abinitio potential energy surface J. Chem. Phys. 133, 111103 (2010); 10.1063/1.3481579 Photonic energy band structure of excitonic quantum dot dimer system J. Appl. Phys. 106, 054302 (2009); 10.1063/1.3211322 An interatomic potential for mercury dimer J. Chem. Phys. 114, 5545 (2001); 10.1063/1.1351877

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Potential surfaces and delocalization of excitons in dimersW. J. D. Beenken, M. Dahlbom, P. Kjellberg, and T. PulleritsDepartment of Chemical Physics, Lund University, Lund, Sweden

~Received 22 March 2002; accepted 2 July 2002!

In the present work we will demonstrate that the nuclear dynamics have a strong influence on thedelocalization of an exciton in a dimer, even if they do not effect the excitonic interaction. It will beshown that the internal nuclear conformation of the molecules forming the dimer depends criticallyon the delocalization of the exciton state in the dimer and vice versa. The resulting closed loopenforces a localization of the lower excitonic state, but, contrary to the commonly accepted view, adelocalization of the upper one. Qualitatively different time-evolution of the delocalization lengthfor the lower and upper excitonic state will be shown. Besides, it will turn out that the nuclearmotions inhibit a complete delocalization of the excitonic state in any case. To accomplish nuclearand exciton dynamics, the nonadiabatic coupling between the two excitonic states will be deduced.This causes a relaxation from the upper to the lower excitonic state, which limits the maximumreachable exciton delocalization. ©2002 American Institute of Physics.@DOI: 10.1063/1.1502647#

I. INTRODUCTION

Electronic excited states of interacting molecular sys-tems has been an active research area since early work byFrenkel.1 The concept of collective delocalized excitations,excitons, was developed. Phenomena like Davydov splittingin molecular crystals,2 motional narrowing in molecularaggregates,3–5 superradiant decay in molecular aggregates,6

and photosynthetic light-harvesting antenna systems,7 toname a few, have been explained by using the excitontheory. Particularly the excitons in antenna systems has beenrecently in the focus of active research.8 Such issues as ex-citon relaxation,9–12 excitation delocalization versuslocalization,13–16and the resulting spectroscopic signatures17

has been addressed.Already Frenkel recognized the important role of the

nuclear motions in the concept of excitons.1 First of all, evena weak coupling of the electronic and nuclear degrees offreedom causes dephasing and population relaxation amongexciton levels~cf. Ref. 18!. For strong electron–phonon cou-pling, further phenomena like self-trapping of the exciton,also called polaron formation, occur.19–21Various theoreticalapproaches addressing different aspects of exciton dynamicsin antenna systems have recently appeared. For example,Redfield relaxation theory, has been applied using modelfunctions22–24 as well as experimental data25 for the spectraldensity to describe exciton relaxation and corresponding ex-perimental observables in different antenna systems. Fo¨rstertransfer theory26,27 has been extended to account for thetransfer to delocalized states with very weak collective tran-sition dipole moments, which still can have a significantCoulombic interaction with the donor if one goes beyondpoint–dipole approach.28,29Theoretical studies have also ad-dressed the issue of polaron formation in photosynthetic lightharvesting.30,31 Particularly, for the peripheral light harvest-ing antenna~LH2! of purple bacteria all above processeshave been identified; for a review, see Ref. 32. Exciton re-

laxation in LH2 has been reported by a number ofauthors.10,11,33The low temperature long wavelength spectralfeatures of transient absorption measurements of LH2 havebeen assigned to the red stimulated emission indicating po-laron formation in these systems.34,35Very recently the sameinterpretation was given to explain the low temperature se-lectively excited fluorescence spectra of LH2.36

In order to describe simultaneously different stages ofexciton dynamics we have recently developed a method37

based on nuclear dynamics for explicit vibrational modescombined with the surface hopping approach.38 In thismethod the complete dynamic process from dephasing andexciton relaxation to polaron formation~self-trapping!, andeventually diffusion of the polaron can be described. Thisapproach seems to be appropriate for application on excitoni-cally coupled systems of arbitrary size, from the simpledimer to the extended photosynthetic antenna complexesmentioned above. Even the excitonic dynamics of the sim-plest possible system, the molecular dimer, are of interest.There exist a large variety of dimeric molecular systemsfrom interacting pair of guest molecules in a molecularmixed crystal39 to dimeric pigment complexes in biologicalsystems like a so called B820 antenna complex from purplebacteria40 and a special pair of the photosynthetic reactioncenter.41 Excited states and their dynamics in moleculardimers has been studied experimentally and theoretically bynumerous authors.42–48 The list of references can be signifi-cantly longer here.

Based on the previous work of Witkowski and Moffitt,49

Fulton and Gouterman demonstrated the mathematical treat-ment of the vibronic and excitonic coupling in a homodimerwith one vibrational mode per monomer.50 They applied thesymmetry of the problem to solve the quantum mechanicaleigenvalue problem for the vibrations and determine stickspectra43 for the corresponding vibrational progressions. Forthe same problem~homodimer! Hayashiet al.44 determinedthe adiabatic potential surfaces, explicitly. In the present

JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 12 22 SEPTEMBER 2002

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work we will extend these calculations of potential surfacesto the more general case of the heterodimer, where the tran-sition energies of the two molecular sites are different. Fur-thermore, different from the pervious works, we will use aquantum-classical approach to describe the nuclear motion ina dissipative surrounding by a Langevin equation.51,52 Ouraim is to correlate the nuclear motion on the potential sur-faces to the exciton delocalization, which is of relevance forabsorption, fluorescence, and energy transfer in molecularaggregates. Thereby it will arise that contrary to the com-monly accepted view the nuclear dynamics can result in anincrease of the delocalization for the upper exciton state. Thetime evolution of the delocalization will be demonstrated forthe two exciton states separately as well as for the case in-cluding the nonadiabatic coupling between them, which re-sults in exciton relaxation.

Finally we will give two examples from photosyntheticantennas containing dimeric chlorophyll subunits.

II. POTENTIAL SURFACES

According to the Born–Oppenheimer approximation theelectronic motion has been separated from the nuclear mo-tion. For the single molecule the electronic statesum& resultfrom solving the stationary Schro¨dinger equation, which alsoyields the electronic eigenenergiesEm(¯Rk¯). Both, thestates and the eigen-energies depend parametrically on thenuclear coordinatesRk . This dependence defines the adia-batic potential surfaceUm(¯Rk¯) of electronic stateum&.For sake of simplicity we will model the molecular elec-tronic system as a two-level system with a ground stateug&and an excited stateue&. For the ground state potential surfacethe minimum energy nuclear configuration isRk

(0) and thedisplacementsRk2Rk

(0) have been transformed to dimen-sionless normal coordinatesqj . Neglecting anharmonicityterms one obtains the adiabatic potential surface of the elec-tronic ground state as

Ug~¯qj¯ !5U011

2 (j

\vjqj2, ~1!

where vj is the oscillator mode frequency andU0

5Ug(¯Rk(0)¯) the minimum energy. For further simplifi-

cation we will assume that the excited state potential surfaceis given by the shifted ground state potential surface as

Ue~¯qj¯ !5U01\veg11

2 (j

\vj~qj222djqj!, ~2!

wheredj represents the origin-shift andveg the electronictransition frequency for the nuclear configurationRk

5Rk(0) , i.e., all qj50. We will limit the number of explicit

intramolecular modes to one per monomer site (j5 j ) inwhat follows. Thus the ground state potential surfaceUg(q1 ,q2) for a dimer is simply a paraboloid centered at(q1 ,q2)5(0,0).

For the excited states of the dimer we have to take intoaccount the excitonic interaction between the transition di-polesmi , usually described in the point–dipole approxima-tion as

\Ji j 5mimj

Ri j3 23

~miRi j !~mjRi j !

Ri j5 for iÞ j . ~3!

If the intermolecular distanceRi j is very small, one has to goto a description of the Coulomb interaction beyond thedipole–dipole ansatz53 and probably also to include the ex-change interaction term. This can be easily done in our ap-proach by a proper choice of the excitonic interaction matrixelementJi j . In what follows, we assume that the excitoniccoupling \Ji j does not depend on the vibrational modes.Combining the off-diagonal matrix elements\Ji j with thediagonal matrix elementsH j j represented by the potentialsurfacesUe, j (¯qi¯) of the excited sitej andUg,k(¯qk¯)for those sites in the ground state, the one-exciton Hamil-tonian in site representation is

Hi j 5\S veg,i11

2 (k

vkqk22v jdjqj D d i j 1\Ji j ~12d i j !.

~4!

Here, we have setU050. The diagonalization of the Hamil-tonianHi j results in the eigen-energies of collective excitedstates—the excitons. For the case of a dimer, with differentsite energiesveg1 and veg2 but the same vibrational modefrequenciesv and origin shiftsd, the two eigen-energies ofthe excitons yield as

U6~ q1 ,q2!5\veg1\v

2d2~ q1

21q222q12q2

6A~ q22q11D!21v2! ~5!

using the normalized dimensionless coordinatesq j5qj /d,the dimensionless excitonic interaction parameter

v52J12

vd2 ~6!

and the dimensionless parameterD for the site energy mis-match given as

D5veg22veg1

vd2 . ~7!

The exciton-energiesU6(q1 ,q2) represent two-dimensionalpotential surfaces for the nuclear motion in the dimer withrespect to the two explicit intramolecular modes. The physi-cal meaning of the two parametersv and D becomes clearonce one recognizes the denominatorsvd2 as the Stokesshift, which represents the coupling energy between the elec-tronic and vibrational degrees of freedom.

First we search for the stationary points~minima,maxima and saddle-points! of the potential surface. SinceU6(q1 ,q2) depends on the sumq11q2 like a harmonic po-tential centered at

q11q251, ~8!

the stationary points can only occur on the antidiagonalgiven by Eq.~8!. The dependence ofU6(q1 ,q2) on the dif-ference of the explicit coordinatesq22q1 is not trivial. Thenumber of stationary points depends critically on the inter-action parameterv as well as the site energy parameterD. InFig. 1 we have plotted thev-dependence of the stationary

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points of the lower exciton potential surfaceU2(q1 ,q2) forseveral positive site energy parametersD. In order to obtainthe positions of the stationary points, one has to choose thehorizontal line~horizontal arrows in Fig. 1! which representsthe given value of the interaction parameterv. Then one hasto search for the intersections with the line that represents thegiven site energy mismatch, e.g., the dashed line forD50.125. Together with the condition that all extrema have tolie on the line q11q251, see above, the obtained valuesq12q2 ~vertical arrows! give one the stationary points in the(q1 ,q2) plane. The first intersection point from the left rep-resents a minimum~thick line!, the second, provided threeintersections exist, a saddle-point~thin line!, and the third asecond minimum~thick line!.

If the parameters fulfill the equation

v2/31D2/351, ~9!

one obtains only two stationary points. In this case the sta-tionary point belonging to the second intersection in Fig. 1represents a flat-point since first and second derivative ofU2

in q22q1 become zero. The special casev51 and D50which represents the critical homodimer@see Fig. 2~a!# re-sults in a single minimum atq15q250.5, which is also a flatpoint. ForDÞ0 @e.g., in Fig. 2~h!# the minimum and the flatpoint do not coincide. In all supercritical cases, i.e., forv2/3

1D2/3.1 @e.g., in Figs. 2~b!, 2~c!, and 2~e!# one obtains onlyone single minimum, but no flat point. For the subcriticalcases, i.e.,v2/31D2/3,1, however, one obtains two minima@see Figs. 2~d!, 2~f! and 2~g!#. They are only forD50 @seeFig. 2~d!# symmetrical with respect to the diagonalq11q2

51. ForD.0 there exists always only one global minimum,while the other minimum is metastable.

For the potential surfaceU1(q1 ,q2), which belongs tothe upper exciton, only one single stationary point exists forall choices of the parametersv and D ~see Figs. 3 and 4!.This means that one obtains always a minimum. ForD.0 itis shifted to values ofq22q1.0, i.e., the direction awayfrom the global minimum of the lower exciton potential sur-face~cf. Fig. 2!. This is very important for the delocalizationand localization of the excitons, as one will see below. In thelimit v, D→0, the potential surface for the upper excitonstateu1& is formed by the truncated upper parts of two inter-secting paraboloids, resulting in biconical isoenergetic con-tours @see Fig. 4~f!#. The intersection-line given byq15q2

represents a sharp notch, and is coincident to the sharp ridgeon the lower exciton potential surface@cf. Fig. 2~f!#.

III. DELOCALIZATION

For an excitonically coupled dimer the upper and lowerone-exciton statesu1& and u2&, respectively, are representedby a superposition of the excited states of the sitesu1& andu2&as

u6&5c16u1&1c26u2&, ~10!

where the coefficientsci 6 are components of the eigenvec-tors of the excitonic Hamiltonian. They are given as

S c16

c26D5

1

AN6

S ~ q22q11D!6Av21~ q22q11D!2

v D~11!

normalized by

N65v21~~ q22q11D!6Av21~ q22q11D!2!2. ~12!

All coefficientsc16 andc26 depend only on the differenceof the dimensionless normal coordinatesq12q2 but not onthe sumq11q2 . This means that the character of the excitoncan be only changed by nuclear motions perpendicular to thediagonal of the (q1 ,q2) plane, given byq15q2 . We willdefine the delocalization length of the excitonL6 by theinverse participation ratio16 as

L6215(

jucj 6u4. ~13!

For the dimer the delocalization length of the one-excitonstates can be expressed as

L6~ q1 ,q2!

5112v2~ q22q11D6Av21~ q22q11D!2!2

v41~ q22q11D6Av21~ q22q11D!2!4. ~14!

The delocalization length for both one-exciton states areequal and depends only onuq12q2u/v. In Fig. 5 the lines ofconstant delocalization lengthL6 are shown on the (q1 ,q2)plane for the homodimer, i.e.,D50, in the critical casev51. One can see that full delocalization is reached along thediagonal q15q2 , where L6(q1 ,q2)52. At increasingq2

2q1 , the delocalization length decreases towards anasymptotic value ofL6(q1 ,q2)51 for uq12q2u→`. Thecritical parameterv is only scaling the dependency of thedelocalization length on the distanceuq12q2u. For v.1 the

FIG. 1. Positions of the stationary points on the lower potential surfaceU2(q1 ,q2) on the antidiagonalq11q251 given in dependency on the co-ordinate differencesq22q1 and the critical interaction parameterv for sev-eral site energy parameter values:D50 ~solid line!, D50.1 ~dashed line!,D50.5 ~dotted line!, andD51 ~dotted–dashed line!. The thick lines repre-sent minima, the thin lines represent saddle-points. The arrows illustrate themethod described in the text to find the stationary points forv50.5 andD50.1.

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delocalization ridge is broader, while forv,1 it is narrowerthan in the critical case. Forv→0, this means the vibrationalcoupling is much stronger than the excitonic, one obtainsL6(q1 ,q2)51 for all points in the (q1 ,q2) plane except ofthe lineq15q2 , whereL6(q1 ,q2) jumps abruptly to a valueof 2. In the limit v→` one obtainsL6(q1 ,q2)→2 for allpoints in the (q1 ,q2) plane, meaning an ubiquitous delocal-ization of the exciton. For the case of the heterodimer, i.e.DÞ0, the maximum valueL652 is reached for the diagonalq11q252D. For D.0 this means a shift of all contours inFig. 5 to the left upper corner, forD,0 to the right bottomcorner. Despite this shift, the slope ofL6(q1 ,q2) is the sameas in Fig. 5.

If one compares the value of the delocalization lengthL6(q1 ,q2) at the global minimum of the one-exciton poten-tial surfaces with that at the ground state minimumL6(0,0)one can state:

~i! For the lower exciton state the delocalization lengthwill always decrease when approaching the global minimum~see Fig. 6!. For D!v,1, which means the subcritical caseof nearly a homodimer, one obtains a significant loss of de-localization. In this case, the initial nuclear conformationcorresponds to an almost fully delocalized exciton state.However, the global minimum of the potential surface,which the nuclear configuration moves to, corresponds to anearly localized exciton, as mentioned above. Especially, inthe extreme caseD50 ~homodimer! and v→0 ~vanishingexcitonic coupling! the initial total delocalization of the ex-citon will be destroyed completely, i.e.,DL2521. For othervalues of the parametersv and D the loss of delocalizationlength is smaller. On the one hand, in the supercritical casev.1.D ~homodimerlike!, for both, the initial and finalnuclear configuration, the distance to the diagonal given byq15q2 is too similar to cause significantly different delocal-

FIG. 2. The lower exciton potential surfacesU2(q1 ,q2) ~panels a–h! for several combinations of the parametersv andD displayed in the central diagram,where the shaded area assigns the subcritical cases withv2/31D2/3,1. Isocontours for each 0.05\vd2 up to\vd2 above the minimum are not shown. Darkershading means lower value.

5813J. Chem. Phys., Vol. 117, No. 12, 22 September 2002 Excitons in dimers

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ization lengths. On the other hand, in the supercritical caseD.1.v ~extreme heterodimer!, there exists no initial delo-calization to be lost.

~ii ! For the upper exciton state the delocalization lengthwill always increase, as shown in Fig. 7. The reason for thisbehavior is that the single minimum of the upper potentialsurface lies always between the diagonal given byq25q1

and the line of maximum delocalization length given byq1

5q21D. The reason for the smaller gain of delocalization inthe casev@D is that the initial state has nearly the maxi-mum delocalization lengthL152. For D.v and D.1 thesmaller gains in delocalization results from the fact that theexcitonic interaction is not able to compensate the site en-ergy mismatch.

IV. NUCLEAR DYNAMICS

In the previous section, we have studied the delocaliza-tion of an one-exciton state in the relaxed nuclear configura-tion. Here we will demonstrate, how the system moves fromthe initial to the final nuclear conformation. Furthermore, wewill show the fluctuating nuclear dynamics around the mini-mum of the potential surfaces, particularly the zero-pointmotion. These fluctuations are of great importance influenc-ing the effective delocalization length. In the subcritical casefor the lower exciton they can cause transitions of the nuclearconfiguration from one minimum of the potential surface tothe other. To take thermal as well as zero-point motion intoaccount, we will use a quantum-classical description of thenuclear motion on the potential surface. This approach iseasier to handle than the nuclear wave-packet propagationand sufficient for dissipative quantum systems based on har-monic oscillator potentials.51,52,54

In dimensionless normal coordinatesq j the Hamiltonianfor the nuclear motion is given as

H665\vd2

2 (j

p j21U6~¯q j¯ !, ~15!

where thep j represent the normalized dimensionless nuclearmomenta, quantum-mechanically given as

p j51

d2

]

]q j. ~16!

It has to be noted that the Hamiltonian for the nuclear motionin general will not be diagonal in the exciton representation,since the operator( j p j

2 may not be diagonal in this repre-sentation~see below!. For the time being, we will neglect theoff-diagonal matrix elementsH12 . This means use the adia-batic approximation. The effects resulting from the off-diagonal matrix elementsH12 , consequently called nona-diabatic couplings, will be discussed subsequently.

For the quantum-classical description of the nuclear mo-tion on the potential surfaceU6(¯q j¯), one can use thequantum-statistical expectation values of the coordinatesqi

and the momentapi . Their time evolutions are ruled by thecanonical equations given as

]qi

]t5

1

\d2

]H66

] pi5v pi , ~17!

] pi

]t52

1

\d2

]H66

]qi5F6,i~ q1 ,q2!. ~18!

The second canonical equation defines the forcesF6,i(q1 ,q2) as the negative gradient of the potential surfaceU6(¯q j¯). For the one-exciton states these are given as

F6,1~ q1 ,q2!5vS 1

22q1D6

v

2

q22q11D

A~ q22q11D!21v2~19!

and F6,2(q1 ,q2), analogously. It is the second term in Eq.~19!, which is specific for the one-exciton states,u1& andu2&. It can give rise to both localization and delocalization ofthe exciton. Depending on its sign it can amplify or compen-sate the generally delocalizing first term in Eq.~19!. Espe-cially, if it overcompensates the first term this will result inthe subcritical cases where two minima occur on the poten-tial surface. Due to the sign of the second term in Eq.~19!,this can happen only for the lower exciton stateu2&, asshown in Fig. 2.

To describe the nuclear motion in terms of the explicitcoordinatesqi including a dissipative bath of modes as sur-rounding, one can use the Langevin equation,51 which isgiven as

]2qi

]2t2vF6,i~ q1 ,q2!52g

]qi

]t1v f i~ t !. ~20!

The left-hand side of Eq.~20! represents the canonical equa-tion for the explicit mode as given in Eqs.~17! and ~18!,including the forces resulting from excitonic polaron forma-tion, as given in Eq.~19!. The right-hand side representsadditional forces resulting from the interaction between theexplicit modes with the surrounding thermalized bath. Thefirst term on the right-hand side, containing the parameterg,gives a damping of the explicit mode motion by friction, the

FIG. 3. Positions of the stationary point~minimum! on the upper potentialsurfaceU1(q1 ,q2) on the antidiagonalq11q251 given in dependency onthe coordinate differencesq22q1 and the critical interaction parameterv.For further explanation see text and Fig. 1.

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second term represents fluctuating forces. Without the fluc-tuating forcesf i(t), the damping term in Eq.~20! results innuclear motion towards one of the minima of the potentialsurface. However, the fluctuating forcesf i(t), cannot beswitched off, since they are an expression of the microscopicthermodynamical kinetics. According to the dissipation fluc-tuation theorem51,55 the variances of the fluctuating forces,given as^ f i

2&, are connected with the damping by

^ f i2&5

2g

t* ^ pi2&5

2g

t* d2 S n~v,T!11

2D , ~21!

where t* represents the bath correlation time. The Bose–Einstein distributionn(v,T) and the summand12 take intoaccount the thermal and zero-point energy, respectively. Thelatter means that even forT→0 K the fluctuating forces willresult in a nuclear zero-point motion. It has to be noted that

we use normalized dimensionless momenta@see Eq.~16!#.Therefore the fluctuating forcesf i scale withd21.

Because of the fluctuating forcesf i(t) the nuclear con-formation (q1 ,q2) moves like a Brownian particle on thepotential surface. Consequently the~de!localization of theexciton is now a statistical quantity. Using the distributionN6(q1 ,q2 ;t), which gives the probability to find a nuclearconformation (q1 ,q2) at time t, the effective delocalizationlength is given by

L6,eff~ t !5E E L6~ q1 ,q2!N6~ q1 ,q2 ;t !dq1dq2 . ~22!

In order to determine the time evolution of the effectivedelocalization lengthL6,eff(t) we have performed MonteCarlo simulations of 400 000 trajectories. In Fig. 8, choosinga displacement ofd52 and considering only the zero-point

FIG. 4. The upper exciton potential surfacesU1(q1 ,q2) for several combinations of the parametersv andD displayed in the central diagram. For the sakeof comparison the critical linev2/31D2/351 is indicated. For further explanations see text and Fig. 2.

5815J. Chem. Phys., Vol. 117, No. 12, 22 September 2002 Excitons in dimers

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motions (T→0 K) the final distributionsN2(q1 ,q2 ;t→`)are displayed for the lower exciton state. As one can see, thecontours follow those of the corresponding potential surface~cf. Fig. 2!. For sections parallel to the diagonalq15q2 , theslope of the distribution is a Gaussian as one can expect forthe ground state of a harmonic oscillator potential. The noisein the distributions reflects the fluctuations of the motion.

The time evolution of the effective delocalization lengthsLeff2 of the lower exciton are shown in Fig. 9 for the param-eter sets used in Figs. 2 and 8. The timet is presented in unitsof the inverse vibrational frequencyv21. The exciton local-

izes with time, as expected~cf. Fig. 6!. In the case of thehomodimer~a, b, d! the initial delocalization length is lowerthan two due to the finite width of the initial distributionN2(q1 ,q2 ;0). For supercritical homodimers~a, b!, wherethe excitonic coupling is stronger than the electron–vibrationcoupling (v>1), the delocalization length does not change alot. A localization occurs only because the final distributionN2(q1 ,q2 ;`) is broader than the initialN2(q1 ,q2 ;0) ford.1.

For the subcritical homodimer~d! as well as for het-erodimers with moderate site energy mismatch~c, h! the lo-calization of the exciton with time is much more significant,since here the whole distributionN2(q1 ,q2 ;t) drifts awayfrom the line of maximum delocalization, towards the poten-tial surface minima. If both, excitonic coupling and site en-ergy mismatch are greater than the electron–vibration cou-pling ~e! the delocalization length starts already at a lowvalue, i.e., the exciton is localized all the time. In the oppo-site casev5D→0 ~f; not shown in Fig. 9!, the effectivedelocalization lengths is triviallyLeff251, since the area forL2(q1 ,q2).1 is infinitesimally small.

The time evolution of the effective delocalization lengthsfor the upper exciton state (Leff1) are shown in Fig. 10.Contrary to the commonly accepted view that dynamic dis-order leads to a localization of the excitation, here the delo-calization length increases, particularly for the heterodimers~c, e, h!. This means that the nuclear dynamics~dynamicdisorder! can compensate the localization effect of the siteenergy mismatch~static disorder!.

Last but not least, in Fig. 11 the temperature dependenceof Leff2(t) for two different homodimers (D50) is shown.The differences between the supercritical~b! and the sub-critical ~d! case can be explained by the broadening of thepeak~s! of the respective distributionsN2(q1 ,q2 ;t) if thetemperature increases. In the supercritical case~b! a broader

FIG. 5. Delocalizations lengthL6(q1 ,q2) plotted over the (q1 ,q2)-planefor the critical case of the homodimerv51. Contours are displayed in stepsof DL6(q1 ,q2)50.1. The shading becomes darker for lower values ofL6(q1 ,q2). The maximum valueL6(q1 ,q2)52 is reached along the diag-onal q15q2 , the minimumL6(q1 ,q2)51 is asymptotic foruq22q1u→`.

FIG. 6. Decrease of the delocalization lengthL2(q1 ,q2) for the lower ex-citon state by polaron formation, fromDL250 ~white! until DL2521~black!, in dependence of the interaction parameterv and the site energyparameterD.

FIG. 7. Increase of the delocalization lengthL1(q1 ,q2) for the upper exci-ton state by polaron formation, fromDL150 ~white! until DL150.5 ~dark-est shading!, in dependence of the interaction parameterv and the siteenergy parameterD.

5816 J. Chem. Phys., Vol. 117, No. 12, 22 September 2002 Beenken et al.

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peak centered at~0.5, 0, 5! results in a smaller effectivedelocalization length, as one can imagine by mapping thedistribution N2(q1 ,q2 ;t) on the delocalization functionL2(q1 ,q2) ~cf. panel b in Fig. 8 with Fig. 5!. In the subcriti-cal case, the final distributionN2(q1 ,q2 ;`) contains twopeaks, approximately centered at~1, 0! and ~0, 1!. If thesebecome broader, they will fill the area between them wherethe delocalization length is close to the maximum value two~cf. panel d in Fig. 8 with Fig. 5!. Therefore the temperaturedependence of the final effective delocalization length isweaker in the subcritical case than in the supercritical case.Especially, in a certain range of temperatures the final delo-calization may even increase with temperature. One can seethat for T50.5\v/k ~dashed curve d in Fig. 11! the finaleffective delocalization length is slightly larger than forT50 K ~solid line!. For higher temperatures~T.1\v/k;dashed–dotted curves d!, however, the final effective delo-

calization length will always decrease with temperature,though in a lower degree in the subcritical than in the super-critical case.

V. NONADIABATIC COUPLING

In the previous section we have described the nuclearmotion on the two-dimensional adiabatic potential surfaces,and the effect of these dynamics on the delocalization. Onehas to consider, however, that the one-exciton state may notremain on the initial potential surface. A switching betweenthe upper and lower potential surface may occur and is ofgreat interest for the polaron formation process. This switch-ing is caused by nonadiabatic terms of the excitonic Hamil-tonian.

Due to Eqs.~10! and~11! it is obvious that not only theelectronic energies represented by the potential surfaces

FIG. 8. Final distributionsN2(q1 ,q2) for the lower energetic exciton for several combinations of parametersv andD at temperatureT→0. The distributionsare normalized to their maximum value 1~black!. For further explanation, see text and Fig. 2.

5817J. Chem. Phys., Vol. 117, No. 12, 22 September 2002 Excitons in dimers

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U6(q1 ,q2) but also the exciton statesu6& themselves de-pend on the normal coordinatesq1 and q2 . Therefore onehas to consider that the representation of the HamiltonianHin the adiabatic basis$u1&, u2&% does not only consist of thediagonal matrix elementsH665U6(q1 ,q2), but containsalso off-diagonal matrix elements given as

H125 i\v(i , j

ci 1

]ci 2

]q jp j , ~23!

and represent the nonadiabatic coupling between the excitonstatesu1& and u2&. Since they depend on the nuclear mo-mentap j , they cannot be included in a conservative poten-tial, but are connected with such phenomena as energy relax-ation and internal conversion. The mixing of the adiabaticexciton states by nonadiabatic couplingsH12 can be con-sidered as stochastic instantaneous jumps of the system fromone potential surface to the other. The rate of this surfacehopping k6→7 is related to the non-adiabatic couplinguH12u2 via Fermi’s Golden Rule,

k6→752p

\ E uH12u2dS U62U711

2\v

3 (j 51,2

~pj822d2pj

2! D dp18dp28 , ~24!

where pj8 and pj represents the~nonscaled, dimensionless!final and the initial nuclear momenta, respectively. The deltafunction in Eq.~24! means that after the hopping the excesspotential energyU62U7 has to be taken by kinetic energyof the nuclear motion. The change of the nuclear momenta,p82p is parallel to the nonadiabatic coupling vector,38,56 i.e.,

pj82pj}(i

ci 1

]ci 2

]q j. ~25!

For a dimer this means that the momentum change happensin the direction~1, 21!. Thus after hopping from the upperto the lower exciton state the~scaled, dimensionless! mo-menta will rise fromp1,2 to p1,28 as

p1,28 5p11 p2

26

p12 p2

2A11

4Av21~ q22q11D!2

~ p12 p2!2 .

~26!

At low temperatures one needs only the surface hopping ratefor the jump from the upper to the lower exciton state givenby

k1→25pv

2 S vu p22 p1uv21~ q22q11D!2D 2

. ~27!

This hopping rate depends on the critical interaction param-eterv, momentump22 p1 , and the relevant nuclear coordi-nate q12q21D, the latter both parallel to the antidiagonalgiven by 8. One can see that the ratek1→2 increases forapproaching to a nuclear configuration whereq12q25D.For v→0 one obtains a singularity there. However, in thecasev50 one should describe the motion by two diabaticharmonic potential surfaces, one for each of the non-interacting sites.

FIG. 9. Time evolution of the effective delocalization lengthLeff2 in theoverdamped caseg52v for displacementd5A8 and temperatureT→0 K. The parameter sets (v,D) are the same~a–h! as used in Figs. 2 and8, respectively. The timet is given in units ofv21.

FIG. 10. Time evolution of the effective delocalization lengthLeff1 for theupper excitonic stateu1&. Parameters and time scale as in Fig. 9.

FIG. 11. Time evolution of the effective delocalization lengthLeff2 of thelower excitonic stateu2& for temperaturesT50 ~solid!, 0.5\v/k ~dashed!,\v/k ~dotted!, 2\v/k ~dashed–dotted!, and 4\v/k ~dashed–dotted–dotted!. For parameter sets~b,d! and time scale, see Fig. 9.

5818 J. Chem. Phys., Vol. 117, No. 12, 22 September 2002 Beenken et al.

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In Fig. 12 the time evolution of the effective delocaliza-tion lengthL6(t) of a dimer initially prepared in the upperexciton state performing a surface hopping is shown for thesets~a–h! of the parametersv andD as used in Fig. 4. Onecan see that initially the delocalization length rises due to thegenerally delocalizing forces in the upper exciton state asmentioned above. After a while the system has jumped to thelower exciton state, and localization takes effect. For the het-erodimers~c, e, h! the final effective delocalization length~cf. Fig. 9! has been reached significantly later than for thecomparable homodimers~a, b, d!. The reason is that thenuclear system first searches for the minimum on the upperpotential surface, which is situated in a position apart fromthe diagonal q22q11D50. Therefore the hopping ratek1→2 is smaller than forD50, where minimum of the up-per exciton potential surface and maximum of the hoppingrate coincide. Furthermore, after the hopping to the lowerexciton state in the heterodimer case (DÞ0) the nuclear sys-tem needs to follow a longer track to reach the final mini-mum potential surface which is diametrically situated to theminimum of the upper exciton potential surface. For thecritical case~h! with v50.5 andD'0.225 the latter effectmay be more important than the delayed hopping.

VI. CONCLUDING DISCUSSION

In the present work the dynamics of the delocalization ofexcitons for the homo- as well as heterodimers with onevibrational degree of freedom per monomer have been stud-ied. The minima of the adiabatic potential surface, whichhave been determined in the usual way,44 has been mapped tothe delocalization length described as the inverse participa-tion ratio. Instead of solving the nuclear eigenvalueproblem,43,44 we have simulated the nuclear motion on thepotential surfaces by using a quantum-classical approach.The resulting distribution of classical trajectories is compa-rable to the nuclear wave packet dynamics on a single po-tential surface in a dissipative surrounding.51,52This distribu-tion enables one to calculate an effective delocalizationlengthLeff(t). This quantity is a good theoretical measure ofthe delocalization, since it shows independently of the sym-

metry of the exciton state how many sites participate in theexcitonic states in average. Generally it cannot be directlymeasured. Various other quantities have been suggested tocharacterize the extent of exciton delocalization. The com-parison of them is not always straightforward.16 For ex-ample, the superradiant enhancementLsr depends beside onthe extent of delocalization also on the orientations and val-ues of the transition dipoles of the dimer and the populationof the excitonic states. Only in special cases, for example adimer with equal transition dipoles in head-to-tail conforma-tion and only the lower exciton state populated, the superra-diant enhancementLsr(t) directly measures the delocaliza-tion dynamics, e.g., in time resolved fluorescenceexperiments.

Our simulations predict that after excitation from theground-state~here assumed to be ad-pulse! the lower exci-ton state is localizing in any case, while the upper one can bedelocalizing in the subsequent time evolution by the nucleardynamics. Including the nonadiabatic coupling between thepotential surfaces, by using the surface hopping method,38

we have shown that after exciting the upper exciton the de-localization length is first increasing according to theexciton-delocalizing tendency of the nuclear dynamics on theupper potential surface, and later decreasing due to the relax-ation to the lower exciton potential surface, where thenuclear dynamics gives rise to exciton localization. Thereby,a 30 cm21 mode would result in exciton~de!localization dy-namics on a picosecond time scale.

Finally we would like to relate our theoretical results toa few examples from real life. The B820 subunit of thepurple bacterial core antenna LH1 is a dimer of bacteriochlo-rophyll ~BChl!.57 The excitonic coupling in B820 has beenestimated to be around 230 cm21. In order to evaluate theelectron-vibration coupling we take a closer look at low tem-perature fluorescence site selection studies of B820.58,59 Inthese studies the Huang–Rhys factorS5 1

2d2 was estimated

to be 0.5, i.e.,d'1. For the phonon frequencyv we take theaverage of the phonon function, which is about 150–200cm21 and results in an interaction parameterv'2.5. If onerelates the site energy parameterD to the variance of thetransition-energy differences between the two BChls of thedimeric subunit, one obtainsD'0.5. This combination ofparametersv and D is located above the cases b and c~cf.Fig. 2! and corresponds clearly to a supercritical case withonly one minimum on the lower potential surface. Conse-quently the excitons in the B820 subunit can be described asalmost delocalized.

As another example, we take the light harvesting com-plex two ~LHCII ! of green plants. There exists no cleardimeric system in LHCII. Nevertheless, the structure sug-gests that several pairs of chlorophyll molecules~Chl-a andChl-b! are close together.60 For the sites, assigned as Chl-a2and Chl-b2, one could estimate a distance of about 10 Å. Fora nearly in-line geometry of the transition dipoles, a exci-tonic coupling between Chl-a2 and Chl-b2 wascalculated61,62 to values between 120 and 180 cm21. At leastthe site Chl-a2 has been identified to be involved influorescence.63 Therefore from fluorescence site selectionmeasurements64 of LHCII the relevant Huang–Rhys factor

FIG. 12. Time evolution of the effective delocalization lengthL6(t) for asurface hopping system initially prepared in the upper excitonic stateu1&.Parameters and time scale as in Fig. 9.

5819J. Chem. Phys., Vol. 117, No. 12, 22 September 2002 Excitons in dimers

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times phonon-frequency for the Chl-a2 can be evaluated asSv'200 cm21. With this value one obtains an interactionparameterv between 0.6 and 0.9. This means that the Chl-a2/b2 dimer, unlike the B820 subunit, could represent a sub-critical case. However, for a Chl-a/b heterodimer the differ-ence between the transition energy of Chl-a and Chl-b isaround 400 cm21. Thus the site energy parameterD is nearlyfour times larger than in the case e~cf. Fig. 2!, which repre-sents an almost localized exciton. Hence, the Chl-a2/b2 het-erodimer can be described in the same manner as a pair ofuncoupled Chl-a and Chl-b molecules. Due to their experi-mental observation of a significant redistribution of oscillatorstrength to the fluorescent subband, Schubertet al.62 havesuggested that the Chl-b2 site might be occupied by a Chl-amolecule instead of a Chl-b, i.e., the Chl-a2/b2 dimer is ac-tually a homodimer (Chl-a2/a28). In this case the excitoniccoupling between the two sites is increased to around 220cm21, resulting in v'1.1. This is just above the criticalvalue 1. Using a site energy parameterD'0, as they sup-pose, one obtains an effective delocalization of initiallyLeff(0)'1.8 andLeff(t→`)'1.5. The first value has to becompared with the size enhancement of the absorption di-pole, the latter with that of the emission.

It has to be noted that our estimates for both parameters,v andD, are rather rough. This is particularly problematic inthe vicinity of the critical case, where slightly different pa-rameters will result in a completely different character of theexciton states.

ACKNOWLEDGMENTS

This work was supported by the Swedish ResearchCouncil and the Wenner–Gren Foundations.

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