Study of Electron-Vibrational Interaction in Molecular Aggregates

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Study of Electron-Vibrational Interaction in Molecular Aggregates Using Mean-Field Theory: From Exciton Absorption and Luminescence to Exciton-Polariton Dispersion in Nanofibers Boris D. Fainberg J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00582 • Publication Date (Web): 22 Feb 2019 Downloaded from http://pubs.acs.org on February 28, 2019

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The Journal of Physical Chemistry

Study of Electron-Vibrational Interaction in Molecular Aggregates Using Mean-Field Theory: From Exciton Absorption and Luminescence to Exciton-Polariton Dispersion in Nano…bers. Boris D. Fainberg* Faculty of Sciences, Holon Institute of Technology, 52 Golomb St., Holon 58102, Israel Tel Aviv University, School of Chemistry, Tel Aviv 69978, Israel

Abstract

of cases photoemission from excitons is accompanied by the exciton annihilation and the photon creation where two (quasi-) particles (exciton and photon) can be considered separately. In contrast, in ordered materials with large oscillator strength possessing strong absorption, excitons that determine the medium polarization and photons (transverse …eld) are strongly coupled forming new elementary excitations: polaritons.10–13 Exciton polaritons (EPs) possess properties of both light and matter. Cavity EPs have a mass thanks to their excitonic part that enables us to consider them as an interacting Bose gas14 leading to Bose-Einstein condensation.15 The latter results in macroscopic coherence of the condensate and super‡uidity.16 In addition, polariton condensation enables us to realize low threshold polariton lasing without population inversion achieved with conventional nanosecond excitation.9 Recently topological insulators in EP systems organized as a lattice of coupled semiconductor microcavities in magnetic …eld were suggested17,18 and implemented.19

We have developed a model in order to account for electron-vibrational e¤ects on absorption, luminescence of molecular aggregates and exciton-polaritons in nano…bers. The model generalizes the mean-…eld electron-vibrational theory developed by us earlier to the systems with spatial symmetry, exciton luminescence and the exciton-polaritons with spatial dispersion. The correspondence between manifestation of electron-vibrational interaction in monomers, molecular aggregates and exciton-polariton dispersion in nano…bers is obtained by introducing the aggregate line-shape functions in terms of the monomer lineshape functions. With the same description of material parameters we have calculated both the absorption and luminescence of molecular aggregates and the exciton-polariton dispersion in nano…bers. We apply the theory to experiment on fraction of a millimeter propagation of Frenkel exciton polaritons in photoexcited organic nano…bers made of thiacyanine dye.

1

Furthermore, electron-vibrational interactions in molecular systems have a pronounced e¤ect on EPs resulting among other things in decay and their instability. One way of taking the decay into account is the introduction of complex frequencies with the imaginary part describing phenomenologically constant damping rates (Markovian relaxation).20,21 In general, taking the e¤ect of strong electron-vibrational interactions on the EPs into account is a non-trivial problem. The point is that in this case both the in-

Introduction

The emission of light by Frenkel excitons in organic excitonic materials, e.g. dye molecules, polymers and biological structures, is used in many photonic applications, including wave guiding, lasers etc.1–11 Frenkel excitons are formed by the Coulomb interaction between molecules, so that in the majority 1

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teraction with radiation …eld and electron-vibrational interaction should be considered as strong.22 La Rocca et al.21,23 studied polariton dispersion in organic-based microcavities taking a single highfrequency (HF) optically active (OA) intramolecular vibration into account introducing also complex exciton replicas frequencies (see above). However, in real situations the relaxation of molecular and excitonic systems is non-Markovian and cannot be described using constant decay rates resulting in the Lorentzian shape of spectra. Using such a description, one may simulate a separate spectrum of an exciton6 or even polaritonic luminescence3 using …tting parameters, but cannot describe the transformation of spectra when for example monomers form an aggregate etc.24 (see also25 ). The matter is that if the monomer spectrum has Lorentzian shape, the aggregate spectrum is simply shifted monomer spectrum.24 At the same time, other shapes that nonMarkovian theory leads to are able to describe the transformation of spectra including strong narrowing the J-aggregate absorption spectrum with respect to that of a monomer.24 It is worth noting that actual dissipative properties of the vibrational system are very important for EPs, in particular, for EP ‡uorescence propagating in organic nano…bers.3 The point is that a "blue" part of the ‡uorescence spectrum overlaps with the wing of the imaginary part of the wave number de…ning absorption and, therefore, is partly absorbed.3,24 In Ref.24 we developed a mean-…eld electronvibrational theory of Frenkel EPs in organic dye structures and applied it to the aggregate absorption and the experiment on long-range polariton propagation in organic dye nano…bers at room temperature.3,4 The theory is non-Markovian and is able to describe the transformation of absorption spectra on molecular aggregation. In the present work we generalize the theory developed in Ref.24 to the systems with spatial symmetry (like organic molecular crystals), to the exciton luminescence and the exciton-polaritons with spatial dispersion. The matter is that ordered structures include among other things also organic dye nano…bers1,3 that are synthesized by self-assembly of thiacyanine (TC) dye molecules in solution. We obtain the correspondence be-

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tween manifestation of electron-vibrational interaction in monomers, molecular aggregates and excitonpolaritons in nano…bers. With the same description of material parameters we calculate both the absorption and luminescence of molecular aggregates and the exciton-polariton dispersion in nano…bers. We apply the theory to experiment on fraction of a millimeter propagation of Frenkel exciton polaritons in photoexcited organic nano…bers made of TC dye.3,4 The paper is organized as follows. We start with the derivation of the mean-…eld equations in ordered structures taking electron-vibrational and dipoledipole intermolecular interactions in condensed matter into account. Then we solve these equation in the momentum representation, Section 2.1. In Section 3 we calculate polarization, susceptibility and dielectric function in the k space. In Section 4 we calculate the relaxed luminescence of aggregates for weak excitation. The exciton luminescence and absorption spectra in our mean-…eld theory obey Stepanov’s law.11,26 Our theory describes both narrowing the Jaggregate absorption and luminescence spectra, and diminishing the Stokes shift between them with respect to that of a monomer. In Section 5 we apply the theory to experiment on fraction of a millimeter propagation of EPs in photoexcited …ber-shaped Haggregates of TC dye at room temperature3 bearing in mind the correspondence between manifestation of electron-vibrational interaction in monomers, molecular aggregates and EP dispersion in nano…bers, and in Section 6, we brie‡y conclude.

2

Derivation of Equations for Expectation Value of Excitonic Operator in Ordered Structures

In this section we shall consider an ensemble of molecules with two electronic states n = 1 (ground) and 2 (excited) in a condensed matter described by the exciton Hamiltonian Hexc = H0 + Hint 2


(1)

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Here the molecular Hamiltonian, H0 , is given by H0 =

2 X

[Ej + Wj (Q)]

X m

j=1

jmjihmjj

noting that if one wants to introduce photons in a cavity, he uses the cavity eigenmodes for the expansion instead of plane waves. ^ Among other things the interaction, the term " D ^ is responsible for the creation of EPs. In an exE" periment related to a linear absorption by excitons one can consider the electromagnetic …eld classically. In that case one can use the same formula (4) considering E(r; t) and Eq as classical function. In this work we also consider a luminescence experiment where the electromagnetic …eld may be decomposed into two modes: classical (incoming …eld), and quantum (the scattered …eld mode generated by spontaneous emission), Eq.(4). In any case, the …eld frequency, ! q , is close to that of the transition 1 ! 2. Consider structures that are symmetric in space like organic molecular crystals. Such structures include also organic dye nano…bers1,3 that are synthesized by self-assembly of TC dye molecules in solution. We de…ne the exciton annihilation bk and creation byk operators in the momentum representation11,27

(2)

where E2 > E1 ; Ej is the energy of state j; Wj (Q) is the adiabatic Hamiltonian of the vibrational subsystem of a molecule interacting with the twolevel electron system under consideration in state j. The dipole-dipole intermolecular interactions in the condensed matter are described by the interaction Hamiltonian27,28 XX Jmn bym bn (3) Hint = ~ m n6=m

where Jmn is the resonant exciton coupling, bm = jm1i hm2j is the operator that describes the annihilation of an excitation in molecule m at level 2, and bym = jm2i hm1j is the operator that describes the creation of an excitation of molecule m to level 2. We adopt here the Coulomb cauge for the electromagnetic …eld, according to which the Coulomb interaction between molecules is conditioned by the virtual scalar and longitudinal photons.11,27 In addition, the interaction conditioned by the transverse ^ photons exists, quantum electromagnetic …eld E(t), X ^ t) = 1 f eq Eq exp[i(q r E(r; 2 q

! q t)] + h:c:g;

Then the system Hamiltonian takes the form X ^ E ^ H = Hexc + ~ ! q ayq aq D

bk

=

byk

=

1 X p bm exp( ik rm ) N m 1 X y p bm exp(ik rm ) N m

(6) (7)

and the lattice Fourier transform of intermolecular interaction10,11,13 X (4) J(k) = Jmn exp[ik (rn rm )]) (8) n6=m

where N is the number of interacting molecules. Then 1 X bn = p bk exp(ik rn ) (9) N k P It should be noted that J(0) = p n6=m Jmn = where p is the parameter of intermolecular interaction used in Ref.24 In the absense of vibrations, the unitary transformation, Eqs. (6), (7), (8) and (9), enables us to diagonalize the electronic part of the excitonic Hamiltonian, Hexc , considering bk as Bose operators

(5)

q

p Here eq Eq exp(iq r) = i2 2 ~! q aq uq (r) is the …eld amplitude, aq is the annihilation operator for mode q, eq is the unit photon polarization vector, V is the photon quantization volume, uq (r) describes a space dependence of the …eld amplitude where ^ uq (r) = eq exp(iq r)V 1=2 for plane waves, and D is the dipole moment operator P of a molecule. The ^ = D (bm + bym ) with D latter can be written as D m

bk0 byk

the electronic transition dipole moment. It is worth 3


byk bk0 =

kk0

(10)


single vibronic transition related to a HFOA vibration. Generalization to the case of a number of vibronic transitions with respect to a HFOA vibration will be made later. Consider the expectation value of excitonic operai ^ d bk [Hint ; bk ] = iJ(k)bk (11) tor bm dt ~ hbm ( )i T r[bm m ( ; t)] (12) Now let us take the vibrational subsystem of molecules into account. Since an absorption spectrum of a where m;ij ( ; t) is the partial density matrix of mole24,32,33 Diagonal elements of the density malarge molecule in condensed matter consists of over- cule m. trix ( ; t) describe the molecule distribution in lapping vibronic transitions, we shall single out the m;jj states 1 and 2 with a given value of at time t. contribution from the low frequency (LF) OA vibraThe complete density matrix averaged over the stotions f! s g to Wj (Q): Wj (Q) = WjM + Wjs where chastic process which modulates the molecule enWjs is the Hamiltonian governing the nuclear degrees ergy levels, is obtained by integration of ( ; t) of freedom of the LFOA molecular vibrations, and m;ij R over , h i (t) = ( ; t) d , where quantiWjM is the Hamiltonian representing the nuclear dem ij m;ij grees of freedom of the HFOA vibrations of a mole- ties h m ijj (t) are the normalized populations of the corresponding electronic states: h m ijj (t) nm;j , cule. n + n = 1. Combining Eqs.(6) and (12), one m;1 m;2 The in‡uence of the vibrational subsystems of can introduce the expectation value of b k molecule m on the electronic transition within the range of de…nite vibronic transition related to HFOA 1 X vibration ( 1000 1500cm 1 ) can be described as hbm ( )i exp( ik rm ) (13) hbk ( )i = p N m a modulation of this transition by LFOA vibrations f! s g.29 We suppose that ~! s kB T . Thus f! s g 24,34 and averaging in is an almost classical system. In accordance with where hbm ( )i = m;21 ( ; t), the density matrix is carry out with respect to the the Franck-Condon principle, an optical electronic vibrational subsystem of the m-th molecule. transition takes place at a …xed nuclear con…guraLet us write the equation for the expectation value tion. Therefore, the quantity u1s (Q) = W2s (Q) of bk corresponding to operator equation (11). If one W1s (Q) hW2s (Q) W1s (Q)i1 representing electron^ int gives considers only intramolecular vibrations, H vibration coupling is the disturbance of nuclear mothe following contribution to the change of the expection under electronic transition where hij stands for tation value of excitonic operator b in time in the m the trace operation over the reservoir variables in 24 site-representation the electronic state j. Electronic transition relaxation stimulated by LFOA vibrations is described by @ i ^ i ^ int ; bm ] ) = the correlation function Km (t) = h m (0) m (t)i of hbm ( )i h[Hint ; bm ]i T r([H @t ~ ~ the corresponding vibrational disturbance with charX i Jmn h^ nm1 ( ) n ^ m2 ( )ihbn i acteristic attenuation time s 28,30,31 where m n6=m u1s =~. In other words, LFOA vibrations lead to a stochastic modulation of the frequency of elec(14) tronic transition 1 ! 2 of molecule m according to ^ m1 = bm bym , n ^ m2 = bym bm is the exciton ! ~ 21 (t) = ! 21 E1 ) + where n m (t) where ! 21 = [(E2 hW2s (Q) W1s (Q)i1 ]=~ is the frequency of Franck- population operator, hnm1 ( )i = m;11 ( ; t), and Condon transition 1 ! 2, and m (t) is assumed to be hnm2 ( )i = m;22 ( ; t)Rmay be neglected R for weak exGaussian-Markovian process with h m (t)i = 0 and citation. Here hbn i = hbn ( )id = n;21 ( ; t) d exponential correlation function Km (t) = K(t) = is the complete density matrix averaged over the stoK(0) exp( jtj= s ). For brevity, we consider …rst a chastic process. that is correct for weak excitation. In that case using the Heisenberg equations of motion, one obtains that ^ int gives the following contribution to the change of H the excitonic operator bk in time

4


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We emphasize that factorization adopted in sum with respect to m. As a result we get Eq.(14) corresponded to neglect of all correlations @ among di¤erent molecules.24 At the same time the ) L11 ]hbk ( )i [ + i (! 21 @t factorization corresponds to a random phase approxp i (0) imation35 that enables us to split the term h(^ nm1 N D21 ek Ek exp( i! k t) 11 ( ) (19) = 2~ n ^ m2 )bn i into the product of populations and polarization (hbn i). It is in this sence that the factorization where we used formula P exp[i(q k) r ] = m m can be understood in the momentum representation. N qk .11,27 y Furthermore, neglecting bk bk0 in Eq.(10) for weak Combining Eqs.(16) and (19), we …nally get excitation, one can write bk0 byk ' kk0 or @ [ + i (! 21 ) L11 ]hbk ( )i y y @t bk0 bk bk bk0 ' n ^ 1k kk0 (15) p N (0) D21 ek Ek exp( i! k t) J(k)hbk i] 11 ( ) = i[ where n ^ 1k ' 1. Using Eq.(14) and Eqs.(6), (7), (8), 2~ (9), (13) and (15), we get (20) @ (0) hbk ( )i iJ(k)hbk i 11 ( ) (16) @t R (0) where hbk i = hbk ( )id and 11 ( ) is the equilibrium value of 11 ( ). ^ int to the Eq.(16) describes the contribution of H change of the expectation value hbk ( )i in time in the momentum representation. In addition, the change of hbk ( )i is determined by the vibrational relaxation. If one considers an absorption experiment and the corresponding polariton problem, the relevant vibrational relaxation occurs in the ground electronic state. In that case the density matrix of a monomer molecule m;21 ( ; t) = hbm ( )i obeys the equation24,32,33,36–38

2.1

Consider …rst the slow modulation limit when K(0) 2s >> 1. In that case the term L11 on the lefthand side of Eq.(20) can be discarded,24,33,37 and we get @ ~ hbk ( )i = @t

i (! 21

!k

D21 eEk

p N ~ ) hbk ( )i + i[ 2~ (0) J(k)h~bk i] 11 ( ) (21)

where h~bk ( )i = hbk ( )i exp (i! k t). In the steadystate regime, Eq.(21) leads to p

i Wa (! k ) 2~N D21 eEk h~bk i = 1 + i Wa (! k )J(k)

@ + i (! 21 ) L11 ] m;21 ( ; t) @t i (0) where D21 eq Eq exp[i(q rm ! q t)] m;11 ( ) Z 2~ (17) Wa (! k ) = i

[ =

Solution of Eq.(20)

where operator

1 1

d

(0) 11

(! k

! 21 + )= ;

(22)

(23)

is the monomer spectrum, (! k ! 21 + ) = P i (! ! + ), P is the k 21 @ @ ! k ! 21 + + K(0) 2 ] L11 = s 1 [1 + (18) symbol of the principal value. The imagi@ @ nary part of " iWa (! k )" with sign minus, describes the di¤usion with respect to coordinate Im[ iWa (! k )] =ReWa (! k ) Fa (! k ), describes in the e¤ective parabolic potential U1 ( ). the absorption lineshape of a monomer molecule, and Bearing in mind that hbm ( )i = m;21 ( ; t), we the real part, Re[ iWa (! k )] =ImWa (! k ), describes multiply both sides of Eq.(17) by exp( ik rm ) and the corresponding refraction spectrum. For the 2

5



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”slow modulation” limit, quantities Wa (! k ) and Moreover, we can take also HFOA intramolecular Fa (! k ) are given by vibrations into account, in addition to the LFOA vibrations f! s g discussed thus far. In that case Wa (! k ) s is given by24 (see Section 2 of the Supporting Infor! k ! 21 1 w( p ) (24) mation) Wa (! k ) = 2 K(0) 2K(0) where w(z) = exp( z 2 )[1 + ier…(z)] is the probability integral of a complex argument,39 and s 2 1 (! 21 ! k ) Fa (! k ) = exp[ ] (25) 2 K(0) 2K(0)

Wa (! k ) =

s

(1; 1 + xa ; K(0) 2s ) xa

exp( S0 coth

0)

1 X

Il (

l= 1

exp(l 0 )

S0 sinh

)

0

(1; 1 + xal ; K(0) 2s ) (27) xal

where xal = K(0) 2s + i s (! 21 ! k + l! 0 ). We consider one normal HF intramolecular oscillator of frequency ! 0 whose equilibrium position is shifted under electronic transition, and S0 is the dimensionless parameter of the shift, 0 = ~! 0 =(2kB T ), Il (x) is the modi…ed Bessel function of …rst kind.39 Eq.(27) is the extension of Eq.(26) to the presence of the HFOA intramolecular vibrations. For 0 >> 1 we obtain

It might be well to point out that the magnitude Wa (! k ) is proportional to the molecular polarizability, and the expression in the square brackets on the right-hand side of Eq.(21) may be considered as the interaction with the local …eld in the k-space divided by ~. Therefore, Eq.(22) can be used also beyond the slow modulation limit when Wa (! k ) is given by36,40 (see Section 1 of the Supporting Information) Wa (! k ) =

s

(26) Wa (! k ) =

where xa = K(0) 2s + i s (! 21 ! k ), (1; 1 + xa ; K(0) 2s ) is a con‡uent hypergeometric function.39 In that case one cannot neglect the term L11 describing relaxation in the ground electronic state (see Section 1 of the Supporting Information). In this relation one should note the following. The ”slow modulation”limit, Eqs. (24) and (25), is correct only near the absorption maximum. The wings decline much 4 slower as (! 21 ! k ) .36 At the same time, the ex~ pression for hbk i, Eq.(22), has a pole, giving strong increasing h~bk i, when 1=[J(k) ] =ImWa (! k ): If parameter of the dipole-dipole intermolecular interaction jJ(k)j is rather large, the pole may be at a large distance from the absorption band maximum where the ”slow modulation” limit breaks down. This means one should use exact expression for the monomer spectrum Wa that is not limited by the ”slow modulation”approximation, and properly describes both the central spectrum region and its wings.24 Eq.(26) is the exact expression for the Gaussian-Markovian modulation with the exponential correlation function K(t) = K(0) exp( jtj= s ).

s

exp( S0 )

1 X Sl

0

l=0

3

l!

(1; 1 + xal ; K(0) 2s ) xal (28)

Polarization, Susceptibility and Dielectric Function in kSpace

The positive frequency component of the polarization per unit volume at point r can be written as P+ (r;t) = N D12 hbm i = P(k;! k ) exp[i(k r

! k t)] (29)

where N D12 i Wa (! k )D21 ek Ek 1 P(k;! k ) = N D12 p h~bk i = 2~ 1 + i Wa (! k )J(k) N (30) N is the density of molecules, and we used Eqs.(9) and (22). Knowing P(k;! k ), one can calculate the susceptibility 6


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(k;! k ) =

N D12 D21 i Wa (! k ) ~ 1 + i Wa (! k )J(k)

and the dielectric function "(k;! k ) 4 (k;! k )]35

=

(31) "0 [1 +

N D12 D21 i Wa (! k ) ] ~ 1 + i Wa (! k )J(k) (32) where "0 = n20 , n0 is the background refractive index of the medium, and the vector product in the numerator of Eqs.(31) and (32) is a dyadic product. If we assume for simplicity that the excitons have an isotropic e¤ective mass, then "(k;! k ) = "0 [1 + 4

J(k) = J(0) +

~k2 2m

(33)

where the exciton e¤ective mass, m , may be both positive and negative. One can see that the susceptibility, Eq.(31), has a pole when the imaginary part of the monomer spectrum, ImWa (! k ), is equal to 1=[J(k) ], i.e. at the frequency of the exciton with the same wave vector as the exciting …eld. In other words, we deal with spatial dispersion. It is worth noting that the structure of the dispersion curves J(k) occurs on the scale of jkj (0:1 1 nm) 1 , which is much larger than the scale of an optical wave vector. For this reason in practice one may often neglect the spatial dispersion and simply calculate the exciton resonances for J(0), as we did in Ref.24 and where we obtained a good agreement between theoretical and experimental absorption spectra of H-aggregates. In contrast, spatial dispersion may be of importance for J-aggregates due to small bandwidth of their spectra, and also in microcavities where the exciton e¤ective mass may be much smaller than electron mass,11 and then the second term on the right-hand side of Eq.(33) strongly increases.Fig.1 shows the absorption spectra of a J-aggregate that are proportional to the imaginary part of the susa (! k ) ceptibility Re 1+i W Wa (! k )J(k) , Eq.(33), calculated for

Figure 1: Absorption spectra (in terms of s = ) of a J-aggregate calculated for J(k) = J(0) (solid ~k2 line), and J(k) = J(0) + 2m (dashed line) in the p case of slow modulation ( K(0) s = 10:9 >> 1) and J(0) s = 42. Dimensionless parameter is = s (! k ! 21 ). Other parameters are m = 0:1mel , mel is the electron mass, jkj = n2 = ; n = 3:16 is the refraction index, = 0:5 .

2

~k J(k) = J(0)24 (solid line), and for J(k) = J(0)+ 2m (dashed line). The monomer spectrum Wa (! k ) is

7



calculated using Eq.(26). One can see that due to small bandwidth of the J-aggregate absorption with respect to that of the monomer (see Fig.1 of Ref.24 ), the spatial dispersion can lead to marked broadening the J-aggregate spectrum. In contrast, the role of the spatial dispersion may be overestimated in works not considering the vibrational contribution.3,41

4

Luminescence

In this section we shall use our mean-…eld theory for the calculation of the relaxed luminescence of aggregates for weak excitation. To describe this process, we shall consider a quantum electromagnetic …eld of the spontaneous emission 1 1 es Es exp(ks r i! s t)+ es Esy exp(ks r i! s t) 2 2 (34) in addition to the incident classical …eld of frequency !. Eq.(34) is Eq.(4) for q = ks . This process is depicted by the double-sided Feynman diagrams28,37,42 where due to condensed matter the lightmatter interaction described by the Rabi frequency, R = (D ek )Ek =~, should be replaced by the e¤ective Rabi frequency, ef f (t) = R =[1 + i Wa (! k )J(0)]24 (see also Section 3). In the site representation the photon emission rate of mode k obeys the equation

Es (r; t) =

@ y ha ak i = @t k

p i 2 ~! k y ^ ^ h[ (D E)ef f ; ak ak ]i = 2 D Figure 2: Double sided Feynman diagrams for relaxed ~ ~ y~ luminescence. X hak bm iuk (r) Re (35) 1 i Wa (! k )J(0) m

where in general the trace in this expression is over the vibrational as well as the …eld degrees of freedom, and we used relation eq Eq exp(iq r) = p i2 2 ~! q aq uq (r) between the amplitude of electric …eld, Eq , and the annihilation operator for mode q, aq . Here ~bm = bm exp (i! k t) and hbm i = m;21 that should be calculated in the third order with respect to the light-matter interaction (see Fig. 2). 8


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Diagrams (1) and (2) of Fig.2 give contributions (3) into h 21 i. Adopting for a while the picture of fast vibrational relaxation when the equilibrium distribution into the excited electronic state has had time to be set during the lifetime of this state, one gets for the contribution described by the lower parts of these diagrams

Eq.(37) can be reduced to Eq.(1) of the Supporting Information using notation 2 = ! st with the only di¤erence that ! 21 should be replaced by ! 21 ! st . Then we obtain for the line shape of a monomer ‡uorescence

Wf (! k ) = 22

( )=

(2) n2 1=2

(2 K(0))

exp[

(

1

Z

1

exp[i(! k

! 21 + ! st )t + gs (t)]dt

0

2

(38) (36) instead of Eq.(5) of the Supporting Information, and

! st ) ] 2K(0)

(2)

where n2

Study of Electron-Vibrational Interaction in Molecular Aggregates

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