PhysicsLettersA 163 (1992)313—321 PHYSICS LETTERS ANorth-Holland

Trappingeffectsin excitonmotion in the frameworkof CTRW

PavelHehnanPedagogicalFaculty,HradecKralové, Czechoslovakia

and

IvanBarvikInstituteofPhysicsof theCharlesUniversity,KeKarlovu5, 12116Prague2, CzechoslovakiaandInternationalCentrefor TheoreticalPhysics,Trieste, Italy

Received30April 1991;revisedmanuscriptreceived27December1991;acceptedfor publication10 January1992Communicatedby A.A. Maradudin

Theinfluenceofatrap(sink)onanexcitontransferin molecularaggregatesis investigated.Pausingtimedistributionfunctions~v~,(t)andprobabilitydensitiesQ,,,,,(t)of thecontinuoustimerandomwalk theoryarecalculatedfrom memoryfunctionsenter-ing thegeneralizedmasterequations.Thepresenceof thesinkchangestheiranalyticalform. We usedtrimerasanexampletoshowthatalsolargetrappingratescouldnot giveaguaranteethatpausingtime distributionfunctionsandprobabilitydensitiesarealwaysnonnegative.

I. Introduction

During thelast twentyyearsmuchwork hasbeendonein the theoreticalinvestigationof coherenceand in-coherenceeffectsin the excitontransferin molecularaggregates.

Severalprincipaldirectionsin thetheoryhavebeenfollowed. After themicroscopictreatmentof theexciton—phononinteractionby Silbey [1], the generalizedmasterequationmethodhasbeendevelopedby Kenkre [2]andthe stochasticLiouville equationmethodhasbeenenrichedby Reineker[3].

Owing to greatefforts of manypeople,many detailsof the mutualrelationsbetweendifferent approachesbecomeclearstepby step [2—5].For example,the stochasticLiouville equationmodel [3] substitutingthebath (phonons)influencingFrenkelexcitonsby the externalpotentialstochasticfield may be rederivedasaspeciallimiting case(andgeneralizedto nonperiodicsystems)from thegeneralizedmasterequation(GME),taking fully thequantumcharacterof thebath into account[6,71. ThememoryfunctionsWmn ( t) enteringtheGME

~9Pm(t) C= ~ J [Wmn(tT)Pn(T)Wnm(tl~)Pm(T)]dt (l)

t fl~&m

for the siteoccupationprobabilitiesPm(t) maybe not onlycalculatedfrom first principles [4,5] but canalsobededucedfrom the stochasticLiouville equationmodel.

Kenkre, Montroll andShiesingerfirst pointedout [8,2] a formal equivalencebetweenthe GME andthecontinuoustime randomwalk (CTRW) problem.In ref. [8], someexplicit examplesof the connectionbe-tweenthememoryfunctionsWm,,(z) from (1) andthepausingtimedistributionfunctionsyfl( t) fromtheCTRWin a decoupledschemehavealsobeengiven.

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Theproblemis, however,that practically all paperswhich use theCTRW oftenstartwith someform of thepausing(waiting) timedistributionfunctionandtry todeducea correspondingformof thememoryfunctions.

KiafterandSilbey [9] tried toextendtheCTRWoverall transportregimesincludingthecoherent,diffusiveand“less than diffusive” one taking into accountKenkre’s [21 prescriptionfor memoryfunctions.The in-fluenceof a bath (phonons)wasexpressedby anexponentialprefactorconnectingthe memoryfunctionwitha purecoherentone,

Wmn(t)=exp(—at)w~(t). (2)

However,Barvik andSzöcs[10] theirresultsfrom thepointof viewof thememoryfunctionfor theexciton—phononsystem,which hasa two-channelform

Wmn(t)=Wh(1)+W~(t) , (3)

asderivedin refs. [4,5]. Twochannelsaremutuallyconnectedandtheir weightsin (3) dependon thepolaronbandwidth, temperature,phonon spectrumand strengthof the exciton—phononinteraction.Only the first(quasicoherent)channelhasKenkre’s form (2).

Barvik andSzöcscalculated[10] the Laplacetransformsof the pausingtime distributionfunction andtheprobability densityfor hoppingandused them for derivationof the averagehoppingtime andthe diffusionconstant.

Our further investigationindicated [11] that goingin a directionoppositeto the usualone, i.e. whenwestartwith somereliableform of thememoryfunction Wmn (t) yieldingphysicallyreasonablesolutionsin (1),oneobtainsthe waiting (pausing)time distribution functionswhich might (at variancewith their physicalmeaning)becomenegative.We showedthis on an exampleof symmetricdimer andtrimer interactingwitha bath.A crucial propertyof the exciton transferwhich influencesthe nonnegativityof its pausingtime dis-tribution functionis the incoherence(a~ 0).

We recentlyturnedour attention [121 to theproblemwhich is commonandwastreatedmany timesin thepast.The systemwe addressedis a molecularaggregatein which guestmolecules(i.e. traps)are introduced,let us say,interstitially. Theexperimentswe are interestedin herecreateexcitonsin thehostsystemthroughillumination, anappropriatefrequencyrangebeingchosento ensurethatonly thehostis excited.Theexcitonscandecayradiativelyandalsomove within the host. If theyarrivein the sphereof influenceof the traps,theymaybecaptured.If theyarecaptured,theylaterdecayradiativelyin a differentfrequencyrange.Forsimplicityweneglectthe radiationdecayin the host.

To takeinto accountthe effectofthetrapon excitondynamics,Kenkresuggestedtheso-called“sink” model.In the restrictedversionof thismodel theexciton probabilitydecaysto the trap at constantratey wheneverthe exciton occupiesa hostsiteinfluencedby the trap. Kenkresuggestedto appendto (1) a sink term

2~dmsyPm(t), (4)

wherethe summationextendsonly overthe sitess influencedby the trap.~ápek andSzöcs[131 drawattentionto the fact, that (1) togetherwith (4) couldnotgive a guaranteethat

occupationprobabilitiesPm(t) ? 0. Using the simpleexampleof dimer theyhaveshownthat a consistentde-scription hasto be basedon the sink termin the Hamiltonianof the system,i.e. in the equationfor the am-plitudesof the probabilities.Usageof sucha full-sink modelleadsto thetransformationof thecoherentmem-ory functionsWmn(t) in (1) to takeinto accountthe presenceof the sink. In dimerwith a sinkat theendonehasto use the coherentmemoryfunction

w12(t)=exp(—yt)w[~(t) . (5)

Thesimplestandnaturalgeneralizationof their result (5) would bea prescriptionto multiply the memoryfunctionsWms(t) or Wsm(t) in (1) with a prefactorexp(—yt).

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We generalized[12] our direct method [141for the calculationof coherentmemory functionsfor finitesystemsto showon a possiblemostsimpleexample(trimer) the influenceof the sink on coherentmemoryfunctionsconnectingdifferent sitesof the finite system.The coherentmemory functionsfor a finite systemwithout anytrapare periodicwith time [14]. The memoryfunctionsfor a finite systemwith thetrapchangetheir formappreciably[12]. Wequestioned[121 Kenkre’sresults [2] in which notransformationof memoryfunctionstakesplacemainly for large y andfinite systems.

Theaim ofour paperis tocontinuetheinvestigation[10,111of themutualconnectionbetweenthememoryfunctionsin the GME andthepausingtime distribution functionsandtheprobabilitydensitiesin the CTRW.We follow the influenceof the trap in a simplesystem(trimer) usingthecorrectcoherentmemoryfunctions,which havebeenderivedrecently[12]. Westartfromthe coherentmemoryfunctionswithoutanytrap leadingto thepausingtimedistribution functionswhich are notnonnegative[11]. Fora largetrappingratey only thepausingtimedistributionfunctionsbecomenonnegative.Theinfluenceofthesink is in thisrespectcomparableto thatof the incoherence.

Thepaperis setas follows. In section2 we usethe coherentmemoryfunctions [121 by obtainingof whichthe influenceof the sink wastakeninto accountto calculatethe pausingtime distribution functionsandtheprobability densitiesfor the trimer in which the sinkwasplacedin the center.In section3 resultsof ourcal-culationsare givenandconclusionsfor further investigationare drawn.

2. GME andCTRW

In general,the densityoperatorp satisfiesthe Liouville—von Neumannequation

it3p/c9t=Lp. (6)

Here,L is the Liouville superoperatorand /1 = 1.Onecanusethe Nakajima—Zwanzigequationfor a projectionon a relevantpart of the densityoperator

ÔPD(t) =—iDLPD(t)— JdrDLexP(_irQL)QLDPD(t—r), Q=1—D. (7)

Thekernelof the integralis theso-called(superoperator)memoryfunction (MF). Theinitial termis omit-ted, the properchoseninitial conditionsare assumed.

Choosinga properform of D, thematrix elementsof theoperatorPD= Dp becomesiteoccupationproba-bilities Pm(I) andthe elementsof the superoperatorMF are the memoryfunctionsin thecommonsense.Theformalismof the projectionsuperoperatorD employedin thederivationof (7) from (6) enablesus to followthe site occupationprobability andthustheexcitonmigration. It generallyinvolvesthe influenceof phononsor anotherbath.

As in (7) only the positive time is used,we completethe definitionof the MF in the following manner,

w(t) = —O(t)DLexp(—itQL)QLD, (8)

where0(i) is the (Heaviside)stepfunction.Thus we are ableto definetheFourier transformationof (8),

w(z)= $ w(t)exp(izt)dt, Imz?0, (9)

which may be analyticallycontinuedto the lower halfplane,

w(z)=—iDL[Q(z—L)Q]1LD. (10)

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In the absenceof phonons(or anotherkind of bath)we takethe following form of D,

D=~Im><mI(Im><ml,...), (11)

where I m> is the completeorthonormalsetof Wannier-likefunctions,M= 1, 2, ..., N is the site index.Thefull-sink model gives the HamiltonianH,

H11=�1—iyd15, (12)

LT L~T* T 4’

“jk~~~kj—~jk,

To find a connectionbetweenthe OME andthe CTRWlet us proceedasin refs. [2,11]. TheFouriertrans-formationof (7) may be rearrangedas

Pm(Z) ~(l_ _iZ+2yö~$+~�rn)W~(Z))~)m(t0) exp(izt0)+n(~m) 1Z+2Yôms±L(#m) Wrm(Z)

(13)

whereWmn(Z) are given in ref. [12].Denoting

~ti~ (z)= 2Yôms+>~r(�m) Wrm(Z) (14)m 1Z+2Ydms+~r(,~m) Wrm(Z)

and

—+ Wmn(Z) 15Qmn(Z) 1Z+2Ydms+>~r(�m)Wrm(Z) (

andperformingthe reciprocaltransformation,oneobtainsthe exactCTRW counterpartof (7) as

/ I—tO (—to

Pm(t)(l $ yi~,(r) dt)Pm(10)+ ~ $ Qmn(t)Pn(1T) di, (16)n(�m)

where w~,(r)is the pausingtime distribution function,i.e. the integralJ’~°~tí~,(r) di equalsthe probabilityof performingthe hop off the sitem within the time t — t0. Q,,,( r) is the probabilitydensitythat the hopper(exciton) originallybeing at siten performsa hop to m afterthe time interval (waiting time) i.

For trimerwith the sink in the center,Im H22 = — iy, oneobtains

P( )_ 2y+w12(z)+w32(z)Z — —iz+2y+w12(z)+w32(z)

and

— — w21(z)+w31(z) (17)w1(z)—w3(z)——iZ+w21(Z)+W31(z)~

Let usdefine

d=—4(y6—l8y4+81y2—216)

Wefound that

d>0, .J~=i~Ji~fory<yo

and

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Volume 163,number4 PHYSICSLETFERSA 23 March1992

d<0, ~/~~is realandpositivefor y>Yo,

where

yO=\/6+3[(3+2%J~)”+(3_2~j~)”3J=3.65.

Let us denote

C= (iig,,Jj~+y)~~3+(..L,~J~_y)I/3 D= (j~J+y)”3— (jig~/~_y)1/3,

for d>0 and

o=jfii~, C=2(y2+ö2)”6sin[~,t—.~arctan(y/ô)] , D=2(y2+ô2)”6cos[~~—~arctan(y/ö)]

for d<0.In the time domain

~(t) = 2 exp( —‘t— ~Dt) {4CD[y(D—y) +2] exp(~Dt)+C(3C2y+5D2y+4y2D—8D)cos(~J~Ci)

3C(C2+3D2)

+.J~[2(2—y2)(C2+D2)+yD(C2—D2)] sin(~J~Ct)), (18)

4exp(—yt—~Dt){2C(D—y)[exp(~Dt)—cos(~,,J~Ct)]

Q2

1=Q23= 3C(3D2—C2)

+,J~(C2+2D+D2) sin(~J~Ct)), (19)

for d>0and

2 exp(—yt— lDt) ‘4CD[y(D—y) +2] exp(~Dt)+C( — 3C2y+5D2y+4y2D—8D)cosh(~J~Ci)

3C(3D2—C2) ~~ [2(2—y2)(D2—C2)+yD(C2+D2)J sinh(3.~~,/~Ci)), (20)

4 exp(—vi— ~Dt) {2C(D—y)[exp(~Dt) —cosh(~j~Ct)]Q21=Q

23= 3C(3D2—C2)

+.J~(—C2+2D+D2) sinh(~/~ Ct)), (21)

for d<0.Further

________ l—~/~ sin(V2_~J~_1y2t))~w~(t)=~exP(_~vt)( 1 ~ sin(~2+V~_~y2‘~,J~+~J~—~y2

(22)

1 ~Yt)( 2+~~fl ____________________ 2—~J~ sin(~J~_~y2 t)),Q12(t)= ~exp(— ~/2+~,/~_~y2 sin(~2+,J~_~v2t)—

(23)

1 sin(~.J2+~./~_~y2t) — 1 sin(~,J2_~/~_~y2t)).

Q~3(t)=—~exP(_~vt)(~~2

(24)

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for y~2—%,/~,

sin(I2+~_~y2t)— ~~y2 t)).sinh(~(t)= ~exP(_~Yt)(~f

(25)

2+\/~ _____________ 2—~hQ12(I)= ~exp(_~Yt)(~~ sin(~2+~_~y2 ~

(26)

1 _____________________________ _____________________________1 2 sin(f2+~_~y2 t)— 1 5iflh(~/~_2+~y2t))~Q13(t)= — ~exP(_~Yt)(~/~

(27)

for 2—J~<y<2+~%/~,

-~ 0 ~C VV\JVVVV\0—1 —1

05

05 I I CC -05

05

2

1=10 iL t I I

C0

058

1 =50u50

I I I4o I I I 0

401=20 301- o=20,0

201:101 __________________________

0 ~..II 0 —I~-ET1 w380 119 m399 119 lSBT

5 10 15 20 5 10 15 20Fig. I. Pausingtimedistribution functionç~(C) for trimerwith Fig. 2. Pausingtime distributionfunction~ (t) for trimerfromthesink — 2)’P2(t). coherent(a= 0) to incoherent(a= 20)regimes.

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( l+,.J~ ________sinh(~J~y2_2_~,/~t)_ l—.~/~ sinh(\/\h_2+~y2i))yi~(t)= ~exP(_~Yi)çf2~

(28)

i I 2+~h 2V~ sinh(~/~J~_2+~y2t))~Q12(t)= 7~exp(_ivt) ~,,f ~,r5h1(\/22~/~~ ~J~/~_2+~v2

(29)

Q~3(t)=~ 1 sinh(~,J~J~_2+~y2t))~—2 +

(30)

for y>2+.,/~.

3. Conclusions

Thispaperaddressesaspecificproblemin theexcitontransferin molecularaggregates,namelythe influence

3=0 041=0

C ___________________________________U VVVVV\M-0.4—1 -08

051 0.5

06Q8~ 1 ~ ~ I I I I I

CC

0.20/.10 ~ I I =1.0 I I I I I

C-C’

0.3

0.2 2L;;__i___i—_- I I I I Ii = 5.0 ~Tr = 500.1

I 0

C(~1b’

2)8

320320.0

0 [~i. ‘ I I I I I4C -t ii I I I I5 13 15 20 5 10 15 20

Fig. 3. Pausingtime distributionfunctionwi’(t) for trimer with Fig.4. ProbabilitydensityQ21(:)=Q2(t) fortrimer with thesink

thesink —2yP2(t). —2yP2(t).

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Volume163, number4 PHYSICSLETTERSA 23 March1992

of the trap (here taken in a full-sink model [15]) on the pausingtime distribution function iu~(t)and theprobability densityQ~m(I) of the continuous time random walk theory (CTRW).

Weused memory functions Wmn( t) [12] connecting different sites of the system in the derivation of theCTRW-counterpart of the GMEformalism.

Wearrived at severalconclusions:(1) Application of the full-sink model [15] for the descriptionof thecoherentexciton transferin the pres-

enceof the trap requires[12] for a sufficiently largetrappingratev a morepronouncedmodificationof thegeneralizedmasterequation(1) thansuggestedby Kenkre [21. Oneshouldgenerallynotonly appendthesinkterm (4). One has to use the proper form of the memory functions Wmn(I), in the calculationof which theinfluence of the sink was taken into account[12]. This bringsalsoa newformfor thepausingtimedistributionfunction W~,(t) and the probability densityQm~(t)owing to a tight connection given by (14) and (15).

(2) The transformation of the pausing time distribution functions ~ (t) and the probability densities Qm,~(1)

which take into accountthe influence of the sink is so principal that their analytical form is changedpronouncedly.

(3) In the time dependence of ~ (I) it is reflected that the coherent memory functions are generally losing[121 their periodiccharacterwith increasingof thetrappingratey and memory effects arepreservedforshortertimes.

(4) The resultswhich are presentedon figs. 1—5 showthat the presenceof the sink, the influenceof whichis in this respectsimilartotheinfluenceofthebathwhich isdescribed(2) in thequasicoherentexcitontransferregimeby the commonexponentialprefactor[2,3], couldnot give a guarantee(for sufficiently largetrappingrates) that the pausing time distribution functions ~ (I) andtheprobabilitydensitiesQm,, (1) are always pos-

C-~II

06~ I

(,,1j

2)

20 I - I I I I I I(=10 2

1.5 -

0.5 -

C- I I I I I I I I Fig. 5. Probability density Q~3(t)for trimer with the sink

5 10 15 20 —2yP2(t).

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Volume 163,number4 PHYSICSLETTERSA 23 March1992

itive. We seethat for instanceQ2 (I) = Q23 (I) can be negativefor sometimesfor arbitraryy.Weconcludethat in someregimesthe pausingtime distribution functionsandthe probabilitydensitiesare

negativefor sometimesso that their usual interpretationasprobabilitiesbecomesdubious.Hence,weare led to the conclusion[11] that thoughthemathematicalform of (35) as the CTRW-coun-

terpartof the GMEis doubtless, its physicalinterpretationandconsequentlyits practicalusein e.g. theMonteCarlo modelling of the excitation transfer may get into trouble.

Our conclusions are based on resultsfor finite systems.Forinfinite systemsthe application of the two-chan-nel memory functions (3) [10] would be desirable.

Acknowledgement

One of us (I.B.) would like to thank Professor Abdus Salam, the InternationalAtomic EnergyAgency andUnesco for hospitality at the International Centre for Theoretical Physics, Trieste, where a part of this workwas done.

References

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Berlin, 1982).[31P.Reineker,in: Springertractsin modernphysics,Vol. 94.Excitondynamicsin molecularcrystalsandaggregates(Springer,Berlin,

1982).[4] V. ~ápekandI. Barvik,J. Phys. C18 (1985) 6149.[5] V. SzöcsandI. Barvik, J.Phys. C21 (1988) 1533.[6]V.~ápek,Z.Phys.B60(1985)101.[7] V. ~ápekand V. Szocs, Phys. Stat. Sol. (b) 131 (1985) 667.[8] V.M. Kenkre, E.W. Montroll andM.F. Shlesinger,J. Stat. Phys. 9 (1973) 45.

[9) J. Klafter and R. Silbey, Phys. Lett. A 76 (1980)143.[1011. Barvik and V. Szocs, Phys. Lett. A 125 (1987) 339.[11] V. ~ápekand I. Barvik, Czech. J. Phys. B37 (1987) 997.[121 P. Hetman and I. Barvik, Phys. Rev., to be published.[13] V. ~ápek andV. Szöcs, Phys. Stat. So!. (b) 125 (1984) Kl37.

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