Exciton transport in dynamically disordered molecular aggregates: influence on optical line shapes

ChemicalPhysics 177 (1993) 715-726 North-Holland

Exciton transport in dynamically disordered molecular aggregates: influence on optical line shapes P. Reineker, Ch. Warns, Th. Neidlinger Abteilung TheoretischePhysik, UniversitlitUlm, Albert-Einstein-Allee11, D-89069 Ulm, Germany

and I. Ban& Institute of Physics, Charles University,Ke Karlovu5, X21-16Prague, CzechRepublic Received 17 May 1993

For Frenkel excitons moving on a linear trimer we investigate the influence of dynamic disorder on their optical line shapes. The dynamic disorder models the intluence of vibrational degrees of freedom and is taken into account by fluctuations of the local excitation energy and of the transfer matrix element between neighbouring molecules. The fluctuations are represented by dichotomic Markov processes with coloured noise. We obtain a closed set of equations of motion for the correlation functions determining the optical line shape which is solved exactly as well as using various approximations. The line shapes are discussed for various sets of the model parameters. Furthermore we show that slower fluctuations need higher approximations in our factorization scheme. The limit of static fluctuations can only be covered by the full set of equations.

1. Introduction In recent years many investigations, both from the experimental and theoretical side, have been directed towards the understanding of the transfer of excitons and their optical line shapes in molecular systems under the influence of static and dynamic disorder. Experimentally many organic solids show [ l-31 a pronounced asymmetry in their optical line shapes at low temperatures. The observed exciton line shapes are broadened with the high energy portion being closer to Lorentzian, while the line falls off more rapidly on the low energy side of the peak. This asymmetry, having its origin in static disorder, disappears at higher temperatures, where phonon scattering dominates. In contrast, the subject of this paper is the investigation of the influence of phonons on optical lineshapes which is modeled by a stochastic process. Theoretical investigations of exciton dynamics in molecular aggregates have followed several directions. After a microscopic treatment of the excitonphonon interaction by Haken and Reineker [ 41, the 0301-0104/93/$06.00

microscopic theory was developed by Silbey and coworkers [ 5- 131. Before this development of a microscopic theory the stochastic Liouville equation (SLE) method has been introduced and expanded by Haken, Strobl, Reineker and their co-workers [ 14-19 1. Kenkre and co-workers [ 20-241 transferred the generalized master equation (GME) method to this problem. The relations between the various approaches have been discussed by Kenkre [ 21-231; many details are still under investigation and become transparent step by step [ 25-311. The latter investigations show that there is a wider agreement [ 29-3 1] between the various theories of the exciton dynamics than their diverse structure might lead one to believe. Here we consider a single exciton on a finite linear molecular chain (trimer) with the Hamiltonian H=Ho+HI, where in the tight-binding approximation Ho, describing the coherent exciton transfer, is given by HO=

,C,

Jmnahn.

0 1993 Elsevier Science Publishers B.V. All rights reserved.

(1)

P. Reineker et al. /Chemical Physics177 (1993) 715-726

716

Here a; and a, are creation and annihilation operators, respectively, for excitons at site m and Jmn is the transfer integral between sites m and n, containing Coulomb and exchange interaction integrals. In the case of nearest neighbour interaction we have Jmn=J(&+1,“+LL,)

(2)

*

In the Haken-Strobl stochastic model the influence of the phonons on Frenkel excitons was modeled by a &correlated Gaussian stochastic process which is taken into account by the Hamiltonian H,(t)=

c e,(t)a,‘a,+ ”

(3)

c JPPI,(0~~&l* m,n

Eq. (3) describes fluctuations of the local excitation energy e(t) and of the matrix element J,,(t) which are characterized by vanishing mean values and by two-time correlation functions in the following way:

(~n(O~m(~)> =4d2
(9) (10)

> =0 9

(Jmn(OJrs(.5) >

=(6,,,,6,+6,6,,)A~exp[

-&(t-r)]

.

(11)

The model generalized in this way is on the one hand much richer but on the other hand also considerably more complicated. Especially for dichotomic stochastic processes, used in our following treatment, A and AJ are the amplitudes of the local energy and transfer integral fluctuations, and ,l and IJ describe the average rates of the switchover between the two values. Furthermore multitime correlation functions of energy fluctuations at one site are calculated according to the following rule: (E(tl)~(tz)...E(t2,)~(~2~+1))

=o,

(12)

c~(tl)E(tz)...~(tzr-r)~(tzr))

(e,(%l(r))

=2&VlYo~(t-r)

(5)

,

=(~(tl)~(tz))...(~(tzr_1)~(tzrc)),

> =0 3

~Jmn(Wrs(~)>

(6) =2(4n,~m71 +&PI.A~,)~(~-~)

*

(7) Many results of this treatment, which covers the full regime between the limiting cases of purely coherent and purely incoherent motion are summarizedinref. [17]. It is argued [ 32 ] that the Haken-Strobl stochastic model retains its physical validity in narrow band systems and when the temperature is high compared to the Debye temperature and the temperature associated with the band width. In a sense the HakenStrobl stochastic treatment describes the limiting case of strong motional narrowing of optical lines by lattice vibrations. To overcome one of the simplifying assumptions of the Haken-Strobl model [ 33,341, namely the white noise treatment of the stochastic process resulting in the &shaped correlation function, the model was generalized by replacing the &correlated process by one with exponentially decaying correlation functions [ 35 1. Mean values and correlation functions are given by
1,

(4)

(e,(0)=0,


exp[ -J(t-7)

=0 ,

(8)

(13)

fl >t2>...>tzcr. For Gaussian stochastic processes the expressions for multitime correlation functions would be more complicated. Optical line shapes of excitons on linear molecular chains have been investigated [36-381 in the last years for dichotomic as well as for Gaussian coloured stochastic processes. In dimers [ 37,381, the use of dichotomic coloured noise for the calculation of the line shape resulted in a closed set of equations for correlation functions, which is easily solved at least numerically. In the slow modulation limit (A= 0) the peak structure could be explained by combining quantum mechanical and statistical considerations. With increasing value of the fluctuation rate (L+oo) the lines broaden, coalesce into an asymmetric line and finally, in the fast modulation limit, a single, motionally narrowed line with a Lorentzian line shape is formed. In the case of a linear chain optical absorption spectra have been approximately calculated [ 361 by a method based on the Mori formalism [ 39 1. A dynamical t-matrix scheme has been developed to arrive at a set of equations for the correlation functions which determine the excitonic optical line shape. The

111

P. Reineker et al. / ChemicalPhysics177 (I 993) 715-726

coupling among the correlation functions has its origin in the coherent interaction between molecular sites. The solution is based on the “RPA-like” decoupling which does not take into account the difference between Gaussian and dichotomic stochastic processes. Unfortunately, the approximation scheme holds only for a restricted range of parameters: A/kJz.

In addition to the investigation of the influence of the exciton-phonon interaction, another motivation to deal with the coloured noise is the consideration of the limit of the correlation time tending to infinity. In this limit (&NO) one ends up with the exciton motion on a static random lattice (i.e. the problem of Anderson localization). Hence one expects for finite correlation times to observe the interplay between static and dynamic (phonon) disorder. This static limit lies outside the applicability range of the “RPAlike” decoupling [ 36 1. The aim of the present paper, dealing with linear trimers, is to obtain a full closed system of equations for the correlation functions of optical absorption to cover exactly the range of small A.As in refs. [ 37-441 we shall model the influence of the exciton-phonon interaction by fluctuations of the local energies as well as by fluctuations of the transfer integrals. The number of correlation functions involved in the optical absorption calculations rises rapidly with the number of molecules involved. It is therefore worthwhile to investigate also various approximation schemes which produce results which are close to the exact ones. The first result presented here is a comparison of the influence of local energy and of transfer integral fluctuations on the optical line shapes of excitons. The second new result is the comparison of exact optical line shapes with those given by various approximations. The paper is set out as follows. The optical line shape theory is introduced in section 2. In section 3 we obtain for the dichotomic noise the full system of equations for the correlation functions of the linear trimer and show the comparison between exact results and different degrees of approximations for different types of fluctuations. The results are discussed and conclusions are given in section 4.

2. Line shape of optical absorption In the framework of linear response theory [ 45 ] and using the fluctuation dissipation theorem [ 391 the line shape of the optical absorption is given by the following expression: (p(t)&O)) .

I(w)=Regdteia’

(14)

0 ( ) denotes the thermal equilibrium expectation value (quantum mechanical and statistical average). Assuming that in thermal equilibrium the number of excitons is negligible, the equilibrium density operator is given by

(15)

Po,cx= IO>(01 ,

with ]0) denoting the excitonic ground state. p(t) is the optical dipole moment operator in the Heisenberg picture and can be expressed via the exciton creation and annibilation operators in the following way: p(t)=

CPn[dw+~nwl n

(16)

>

where p” is the optical transition matrix element at site n. The time dependence of the Heisenberg operator is given by a,‘(t)=U-l(t)a,+U(t),

(17)

a,(t)=V’(t)a,U(t)

)

(18)

where the time development operator U( t) (h = 1) tY(t)=lfexp(

-i i H(r) dr)

(19)

fulfills the equation of motion i$ U(t)=H(t)U(t)

.

(20)

The dipole moment correlation function becomes W(MO)>=

;~/W”(
= ~+&.W~,.(t)>

*

IO>> (21)

In the following section we calculate the optical line shapes of a linear trimer. The optical transition di-

P. Reineker et al. /Chemical Physics177 (1993) 715-726

718

pole moments of the molecules are assumed to be equivalent, P?lcn=P,

Table 1 Eigenenergies and corresponding oscillator strengths. Trimer, J_*=J,=J

(22)

and to be oriented e.g. along the z direction. The optical line shape of a finite linear molecular chain is then given by the following expression, Zz(~)=$Re~dteiW’

C (U”,Jt)).

m.n

0

(23)

citon transfer on the linear trimer with the tight binding Hamiltonian

The equation of motion for the matrix elements tY,,,( t) of the time evolution operator for 5,+l(t)=J,,~+,(t)=J,+,,~(t) isgivenby

Ho=

$ hm(O =

The energy eigenvalues of Ho are Ej=Wcos(lcj/4),

-iJ[ Un+dO + Un-l,dt) 1

-ia(t)U,,,(O

i.e. 0 and f $

-i[J,+~(t)U,+,,,(t)+J”-1(t)U,-~,,(t)19

J, with corresponding eigenfunctions

(24)

I&=

(25)

Optical absorption from the excitonic ground state to the state 1yj) has an oscillator strength

with the initial condition &I,,(O) =&,, * Introducing Xl(t)=

c Z-&m(t)3

(26)

m

Fi= f (“iI sin(rrjrz/4 ))’ and the optical line shape is given by

the equation of motion transforms into

ZJO)-/J’

,

i

$Gu)=

with Ej and Fj from table 1.

-i~,(f)x,(t)-il[X,+,(t)+X,-,(t)l -i[J.+,(t)x,+,(t)+J"-l(t)X,-l(t)l,

(27)

3.1. Trimers withlocalenergyfluctuations Taking into account only local (energy) fluctuations in the Hamiltonian

with the initial condition X.(0)=1.

(28) c-l(t)

The optical line shape then reads

H,(t)=

00 Zz(o)=p2Re

C FG(o-E')

s 0

dteiU’C (X,(t)). n

(29)

(

0 0

0

0

co(t) 0

0 e,(t)

we get the following set of equations: $X_,(t)=-ic_I(t)X_I(t)--iTXo(f),

(30)

3. Results $X,(t) In the following, we shall take into account the ex-

=

-i~o(t)Xo(t>--il[X,(t)+X-1(t)l

,

(31)

P. Reineker et al. / Chemical Physics177 (1993) 71% 726

OX,=-ie,(t)X,(t)-irX,(t).

(32)

Carrying out the stochastic average in the equations of motion, new correlation functions occur on the right side. To obtain equations of motion for them we apply a theorem by Shapiro and Loginov [ 35 ]:

z

(E~X, > =-i(~o~lxl

>

-i.J(QXo >-GEoX, > ,

&x-l>= -i(e--LelX--L) -il(t,Xo)

g

719

-Qe,X-,

>3

$(elX,>=-i<~Oc%>

= ( c(t) $ @,[~I -n(dwJ~l> >

9

-il(~,X_,)-iJ(E,X,)-I(~,Xo)

(33)

,

~(r,x,>=-id*
together with rules ( 12) and ( 13) for dichotomic noise. Using computer algebra (symbolic programming in MAPLE [ 461) we arrive at a closed system of coupled equations for 24 correlation functions:

~(c_leOx-,)=-id’
~(X-I)=-i(c_IX_I)-iJ<%),

~(c_,eax~)=-id’-il(c_19X_,)

-2rZ(e_,eoX-1))

-il(~_,gx,)-2~(~_~~oxo),

$(X0)=-i(eOXO)-iJ(X-,)-iJ(X,),

z (E_~z~X~)=-~<~_~~~~~X~)-~J(~_~~XO) $<~l)=-i(cx,>-ir(&),

-2A > $ (E_*X_1)=-iLP(X-,)

$ (E-IEIX-_I)=

-i.J

-2A
= -i(c-IeOXO>

-iJ
>

$ (~_,~,XO>=-i(E_l~O~lXO)

-iJ(E_1XI)-;l(e_1XO))

-ir(E_~E,x-~)--iJ(E_,~~X~)-21(~_,EIXo))

$=-i(r-It,XI)

&,e,Xd=

-iJ(E_,X0)-A(E_1Xr),

-2Gc,eX1>

$(cOX-,)=-i
;

-A(eoX--1) -iLP(X,)

(~oeX-*

-id2(~_,X,)-il(E_1~1Xo) 9 * > =-1(E_~~O~~X_~)-J(~O~~XO)

3 -W~oGX-I

-iJ(

> ,

toX_, ) 2

-1JCEoX1 >-GQXo>

-id*(~,X_,)-il(t_,EIXO)

~oQXo)=

-iLP(t,Xo)-iJ(~Ot,X_,)

9 -iJ(66&

> -2qto~,xo>

,

P. Reineker et al. /Chemical Physics 177 (I 993) 715- 726

720

-ir(E_,~,t,X,)-3n(C_,~o~~~~).

(34)

These equations are then Laplace-transformed

and the resulting linear system of equations is solved numerically. In fig. 1a the exact line shape I,( o/J) is shown for J= 1, A= 0.2 and various values of A*. It is seen that with increasing amplitudes of the energy fluctuations and given values of J and I ( 1) the line becomes more strongly structured, (2 ) its width increases and (3) because of the disorder, transitions are allowed which in the completely ordered case are forbidden (e.g. the line at w=O).

8 20""""";

I,(w) : A2=’ . ^r

,5 _I h=U L

IO

-L

-2

0

2

i!

-I

c

L w

exact

--

RPA

2

.

RPA

3

20

I,(w) 15

10

5

0 -L

-2

0

2

L w

Fig. 1. Optical line shapes I,( w/J) of the trimer with local energy fluctuations obtained by solving the complete system of equations (full) compared with those obtained from a reduced set of equations after decoupling according to eq. (35 ) (dotted) and (36) (dashed) for(a)J=land1=0.2and(b)J=landA=l.

721

P. Reineker et al, /Chemical Physics 177 (I 993) 715-726

’

10'

A2=0 25

I,(W)

!

I,(w)

I3-

h=l

20

’

’

’

0

, 2

’

’

A2=l A=1

6-

2-

0,

0

-1

,

-L

2

0

-2

exact ’

’

-

RPA

2

I,

I, -2

RPA

, L w

3

’

h=l L-

Fig. 1. Continued.

The full line in fig. lb gives the line shapes for the same parameters of Jand A2 as in the previous figure, however, now for a larger value of A, i.e. a faster decay of the correlation functions on account of faster fluctuations. The faster fluctuations, but still keeping 1
>
*

(35)

We then arrive at an approximate set of 21 coupled equations. The cruder approximation

= (c,(h2(0

>(&JO >

(36)

results in a set of 12 equations which lead to the line shape given by the &shed line. It is obvious that (35) represents a considerable improvement of the line shape as compared to the decoupling using ( 36 ) . In the case of static disorder (LO), one can analytically compute the eigenvalues of H=& + Hi and the corresponding oscillator strengths as shown previously for Ho. Taking into account all 8 combinations of + A for the three site energies ei, we obtain 14 different eigenvalues corresponding to 14 absorption frequencies. Because these expressions are rather lengthy, we show numerical values for Ej and Fi obtained for A= 2 only (table 2 ) .

P. Reineker et al. /Chemical Physics177 (1993) 715-726

722

Table 2 Eigenenergies and corresponding oscillator strengths for A= 2 t-1

f0

t+1

E-1

EO

E+’

F-1

FO

F+’

+2 +2 +2 +2 -2 -2 -2 -2

+2 +2 -2 -2 +2 +2 -2 -2

+2 -2 +2 -2 +2 -2 +2 -2

3.41 3.10 2.45 2.25 3.10 2.45 2.25 -0.59

2.00 1.15 2.00 -1.15 1.15 -2.00 -1.15 -2.00

0.59 -2.25 -2.45 -3.10 -2.25 -2.45 -3.10 -3.41

2.91 2.38 2.72 1.61 2.38 1.91 1.61 2.91

0.00 0.01 0.00 1.39 0.01 0.00 1.39 0.00

0.09 0.61 0.28 0.01 0.61 1.09 0.01 0.09

Optical absorption from the excitonic ground state to these states is again given by Z,(O)-/J~

c F'G(o-E') i

-iJ(X-1) $(X0)=

,

-i(.Z_,X_,)

with

-il(X,

-i(J,X,

>

>,

-$(X,)=-iJ(&)-i
F’=

I I Cc’,

In

,

$

I

where c’, are the coeffkients in thejth eigenfunction ofH. We can now compare this absorption spectrum for static disorder with the one obtained from the dynamic disorder model with very small 1, e.g. 1= 0.025 in fig. 2 (left column, A= 2 ) . Absorption maxima are at the same energies and show corresponding intensities, the effect of the noise modulation of the site energies being in the broadening of the line shapes.

(J_,X_,)

=

-iJ(J_,Xo)

-i&(X,)

-b
>,

-g (J-,X0>= -iJ(J_,X_,) -iJ(J_,X,)-iAj(X_,) -i
(J-,X,)

> -& =

-i
,

-ir(J_,Xo>

-&
>

,

>

,

3.2. Trimers with transfer integraljluctuations ;

Taking into account only the transfer integral fluo tuations in the Hamiltonian H ,=

(

0

J-l(t)

J-,(t)

0

0

Z,(t)

-i
0

J,(t)

3

g

0 )

-i(J_,X,)

=

-iJ(J,X_,

) -iJ(J,X,

>

-i(J_,J,X_,)-i&(X,)-&(J,X0),

we obtain, by the same procedure as in the previous subsection, a closed system of equations for the 12 correlation functions necessary to calculate the line shape: z (X_, >= -iJ(X,)

(J,X_,)=-iJ(J,Xo>

,

~<~,x,)=-irc~,xo~-id:
g (J-,J,X_,) -iAf(J,X,)

= -iJ(J_,J,Xo) -2rl.r(J_,J,X_,

>,

723

P. Reinekw et al. /Chemical Physics 177 (1993) 715-726

A2=4 A=0.025

5

4

A,‘=4

Fj 3

static

2

1

I

I

I

-2

0

2

0

I

I I

-1

’ E

*

I

-2

0

2

’

E

Fig. 2. Optical line shapes Z,(w/J) of trimers with local energy fluctuations (left column) and transfer integral fluctuations (right column) for .I= 1 anddkl: ~4, in the slow modulation case (1=1J=0.025, top row) and in the static case (bottom row).

$ (J-*J1Xo>= -iJ(J__IJIX-*

) -iJ(J_,J,&

>

(Jn,(t)Jn*(t)Xnl(t)>=(Jn,(t)Jn2(t)>(Xn~(t)> . (38)

-id~(J,x_,)-id:(J_,x,)-21,(J_,J,xo), ; (J-,Jl~l> -zn,(J_,J,x,

= -iJ(J_,J1X,)-id3(J_,X,) >.

(37)

The line shape is given by the full line of fig. 3a for J= 1, AJ= 0.2 and various values of A;. For this small value of II we obtain fairly structured lines which in general broaden for increasing values of A,. Fig. 3b gives the line shapes for the same values of J and A) and AJ= 1. The faster fluctuations result in a broadening of the lines and thus the structure is smeared out. The dashed line gives the line shape if the following decoupling approximation is used:

Instead from the exact set of 12 coupled equations, the line shape is calculated from 9 approximate equations. In the case of LJ= 1, shown in fig. 3b, the approximation works reasonably well, however for slow fluctuations (A,=0.2) it cannot be applied. In the static limit (A= 0) there are two possibilities: (a) either J_ L and J1 are the same and the eigenenergies are the same as for Ho alone with renormalized J, (b) orJ_,=J+A,, J,=J-A,andviceversa. Inthis case the eigenvalues are 0 and +_,/mj. All three optical transitions are ahowed in case (b), while only transitions to states with nonxero energy are allowed in the case (a) and contribute to the absorption.

P. Reineker et al. / Chemical Physics 177 (1993) 715-726

124

8 20;

’

’

’

’

’

’

20""""'

i

I,(w)

exact

--

"

AJ2=1

RPA

2

10

5

0

Fig. 3. Optical line shapes I.(w/J) of the trimer with transfer integral fluctuations obtained by solving the complete system of equations (full) compared with those obtained from a reduced set of equations (dashed) for (a) J= 1 and d,,= 0.2 and (b) J= 1 and AI= 1.

Expressions for Ej and Fj for the latter case are given in table 3. Again comparing this absorption spectrum for static disorder with the one obtained from the dynamic disorder model with very small Iz,, e.g. &=0.025, in fig. 2 (right column, A,=2), we observe exact correspondence.

4. Discussion and conclusions We have calculated the optical absorption line shape of Frenkel excitons on linear trimers. The interaction between the molecules is taken into account by nearest neighbour transfer matrix elements and the influence of the phonons and of the static disorder is modeled by a stochastic process with dichotomic coloured noise which gives rise to fluctuations of the lo-

cal excitation energies and of the transfer integrals. It is assumed that fluctuations at different sites and between different molecular pairs are independent and that their correlation functions decay exponentially. The optical line shape of linear trimers has been determined from an exact closed system of coupled linear equations. The parameters in these equations are the transfer matrix element J between nearest neighbours, the strengths d and d, of the fluctuations of the local excitation energies and of the transfer matrix elements, and the decay constants 1 and & of the correlation functions of these fluctuations. The number of equations would increase rapidly with the number N of molecules in the chain. Therefore, we have developed a procedure in MAPLE (symbolic programming) which, for arbitrary N, allows to obtain the complete set of equations for the correlation

P. Reineker et al. /Chemical Physics 177 (1993) 715-726

03

-2

-L

RPA

--

1.j

0

2

2

"'I"',"

I,(w) - aJ2=L -j_ A,=1

i\

2-

-L

-2

0

2

Fig. 3. Continued. Table 3 Eigenenergies and corresponding oscillator strengths. Trimer, J_,=J+A,, J,=J-A, EJ

FJ

1

JrJ7q

QJ+Jz @q,’ 4(J*+A:)

2

0

3

-JzmJ

24: J2+A: (U-Jzpg,~ 4(J2+A:)

i

functions. MAPLE generates the LATEX output of eqs. ( 34 ) and ( 37 ) . Furthermore, MAPLE produces the coefficient matrix of these equations in FORTRAN. In this paper we have presented, for lack of space, the equations obtained in this way and their solution for the simplest example, namely a trimer.

Results for a linear hexamer do not change qualitatively our conclusions. Results of the calculation are shown in figs. 1 and 3. It is seen that for small values of 1 and &, i.e. slow fluctuations, the line shape is strongly structured. These structures can be interpreted by the superposition of the spectra in the static limit. With increasing values of I and J.,, the lines become broader and the structure of the spectra is smeared out. Furthermore, we have derived approximate sets of equations by applying a factorization procedure for the correlation functions. The figures show that the approximation is more appropriate for larger values of 2. A more detailed discussion [ 36 ] shows that the crucial point for the validity of the optical line shape approximation is the relation d versus A and AJ versus La respectively. For the periodic linear molecular chain

726

P. Reineker et al. /Chemical Physics 177 (1993) 715-726

and for local energy fluctuations the “RPA-like” decoupling

=(E”,(t)~n2(f))
(39)

is valid only for A/l< $. Results presented in fig. 1 show that decoupling of the higher correlation functions

= (Q,(Wl2(Wz3(0)

(L(t)>

(40)

holds to smaller values of il as the “RPA-like” decoupling. Finally it should be remarked that the calculation of the optical absorption line shape in the static limit (LO and lz,+O) requires the use of the full system of equations. Resorting to any approximations could lead to erroneous results.

Acknowledgement IB gratefully acknowledges the support by Deutscher Akademischer Austauschdienst (DAAD ) .

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