Kinetic theory of exciton–exciton annihilation Volkhard May Citation: The Journal of Chemical Physics 140, 054103 (2014); doi: 10.1063/1.4863259 View online: View Table of Contents: Published by the AIP Publishing Articles you may be interested in Exciton-exciton annihilation in organic lanthanide complexes J. Chem. Phys. 132, 024504 (2010); 10.1063/1.3280070 Ultrafast exciton-exciton coherent transfer in molecular aggregates and its application to light-harvesting systems J. Chem. Phys. 127, 075101 (2007); 10.1063/1.2754680 Dynamics of unidirectional exciton migration to the molecular periphery in a photoexcited compact dendrimer J. Chem. Phys. 122, 024708 (2005); 10.1063/1.1814054 Low-temperature kinetics of exciton–exciton annihilation of weakly localized one-dimensional Frenkel excitons J. Chem. Phys. 114, 5322 (2001); 10.1063/1.1352080 Pump–probe spectra of linear molecular aggregates: Effect of exciton–exciton interaction and higher molecular levels J. Chem. Phys. 109, 6916 (1998); 10.1063/1.477259

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Kinetic theory of exciton–exciton annihilation Volkhard Maya) Institut für Physik, Humboldt–Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany

(Received 6 December 2013; accepted 14 January 2014; published online 3 February 2014) Weakly excited states of dye aggregates and supramolecular complexes can be characterized by single or two exciton states. Stronger excitation results in the presence of multiple excited molecules, and complex processes of internal energy transfer dynamics take place. The direct consideration of all excited states is limited to systems with a few molecules only. Therefore, an approach is used based on transition operators among the molecular states of interest and resulting in a dynamic theory for excitation energy transfer in strongly excited molecular systems. As a first application of this theory a detailed description of exciton–exciton annihilation is given. The obtained novel nonlinear theory is related to the standard description. Possible further approximation schemes in the offered theoretical framework are discussed. © 2014 AIP Publishing LLC. [] I. INTRODUCTION

Weak electronic excitation of a supramolecular complex (SC) may result in the formation of a single Frenkel-exciton. For such a situation the probability remains very small to have two or more molecules excited or to have a single molecule in a higher lying electronic state. Respective weak excitation conditions are met under sun irradiation and are valid for biological photosynthesis and for photovoltaic systems. Present day laser technology easily overcomes this low optical excitation regime. Multiple excitons may be present and the process known as exciton–exciton annihilation (EEA) may become an essential part of the excitation energy transfer (EET) dynamics. EEA in SC or dye aggregates represents a two-step process (see Fig. 1). First, two excitations being in the S1 -state of the molecules have to move close together so that their excitation energy can be used to create a higher excited Sn -state (n > 1) at one molecule. This step usually named exciton fusion leaves behind the other molecule in the S0 groundstate. In a second step, an ultrafast internal conversion process brings back the molecule which is just in the higher excited Sn -state to the S1 -state. EEA has been often described by a rate equation for the exciton density n(r, t) at spatial position r where annihilation has been accounted for by −γ n(r, t)2 , with annihilation rate constant γ (see, for example, Ref. 1). Besides, various microscopic theories have been worked out.2–5 An inclusion of EEA into an exciton density matrix theory could be presented in Refs. 6–10 and later on in a consequent multiexciton density matrix approach in Refs. 11–13. The specificity of the multiexciton approach is to introduce EEA into a description based on delocalized exciton states. As a first step, one takes singly excited electronic states of the SC which lead to the ordinary exciton. Single-exciton states are followed by two-exciton states which are formed either by two simply excited molecules (excited into the a) Electronic mail: [email protected]


S1 -state) or by a single doubly excited molecule (put into the Sn -state). In the same way one can also construct higher excited multiexciton manifolds. Now, EEA appears as a simple nonradiative transition, a two-exciton state decays into a state of the single-exciton manifold. We also remind on the fact that the reverse process of exciton fission became of recent interest (see Fig. 2). Molecules are known in which first excited singlet state has an energy E(S1 ) twice as large as that of the first excited triplet state E(T1 ). Then, the singlet state may decay into two triplet excitations. This can be understood as the formation of two pairs of charge carriers (electrons and holes) what is of interest for the efficiency increase of organic solar cells14, 15 (the reverse process has been also discussed16 ). While the multiexciton density matrix theory accounts for details of the EEA and the formation of delocalized excited states in the SC, it is restricted in its practical use to doubly excited states (two exciton states). But strong optical excitation of SCs and molecular aggregates became of recent interest in proposing the so-called SPASER (see Refs. 17 and 18 for an overview). Whenever a SC is coupled to a metal nano-particle strong local field formation and subsequent intensive SC excitation may appear.19 So, it would be of interest to formulate a theory which is also valid for the presence of multiple excited states of a SC. In the following a methodology is presented which is ready to describe the EET dynamics in molecular systems if the number of excitations (excitons) is not small. All ingredients of the approach are not new, however, we did not become aware of their combination as described in the following: (i) introduction of transition operators (including transitions to higher excited molecular states), (ii) derivation of quantum master equation based equations of motion for operator expectation values, and (iii) decoupling of higher-order operator expectation values to arrive at closed kinetic equations. The respective SC Hamiltonian based on transition operators is introduced in Sec. II. There, we also derive equations of motion for transition operator expectation values. Section III and Appendices A and B offer a detailed derivation of nonlinear kinetic equations for EEA. Finally, in

140, 054103-1

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Volkhard May

J. Chem. Phys. 140, 054103 (2014)

Single molecules are represented by the Hamiltonian Hm where m (n) labels the different molecules in the SC. The complete Coulomb interaction among electrons and nuclei of two different molecules in the SC is considered by Vmn . Related basis states are given by adiabatic single molecule electronic states ϕ ma with a counting the levels. The basis states of the complete SC follow as  |ϕma . (2) |φA  =





FIG. 1. Energy level scheme demonstrating exciton-exciton (singlet-singlet) annihilation. Shown are the ground-state (S0 ), the first excited state (S1 ), and a higher excited state (Sn ) of a pair of molecules (the blue spheres indicate the degree of excitation). Left part: both molecules are in their first excited state; central part: de-excitation of the right molecule and higher excitation of the left molecule (exciton fusion); right part: internal conversion process moves the left molecule into its first excited state (exciton annihilation). Excitonexciton fission is highlighted if one moves from the central part to the left part.

Sec. IV we give a brief outlook on what is possible furthermore with the discussed technique. This replaces the presentation of some first simulation results which are postponed to a separate discussion. II. THE MODEL AND BASIC QUANTUM MASTER EQUATIONS

In the following, a SC model is introduced that is ready to describe EEA at arbitrary degrees of excitation. Moreover, the used density matrix equations are presented. Although the derivation of our SC model Hamiltonian is known from the literature (see Refs. 10 and 20–22), a brief presentation of the main steps and ideas would be useful for future generalizations. A. The Hamiltonian

We start with the following general form of the SC Hamiltonian:  1 Hsc = Hm + Vmn . (1) 2 m,n m

T2 S1 T1

S0 FIG. 2. Energy level scheme demonstrating exciton-exciton fission. Shown are the singlet ground-state (S0 ), the first excited singlet state (S1 ), the first excited triplet state (T1 ), and the second excited triplet state (T2 ) of a pair of molecules (the blue spheres indicate the degree of excitation). Left part: left molecule is in the first excited singlet state and the right one remains unexcited; right part: decay of the left molecule into its first excited triplet state and excitation of the right molecule into its first excited triplet state. Triplet-triplet fusion is highlighted if one moves from the right to the left part.

Such a product ansatz requires negligible electronic wave function overlap among adjacent molecules. An expansion with respect to these states turns the Hamiltonian, Eq. (1), into the form20–22  Hsc = Hm (a, b)|ϕma ϕmb | m,a,b


1 Jmn (ab, cd)|ϕma ϕmd | × |ϕnb ϕnc |. (3) 2 m,n

The Hm (a, b) are matrix elements ϕ ma |Hm |ϕ mb  of the single molecule Hamiltonian. They account for vibrational motion and non-adiabatic couplings. We reduce these expressions to electronic adiabatic energies Ema (vibrational contributions are neglected). The matrix elements of intermolecular Coulomb coupling (second term in Eq. (3)) read Jmn (ab, cd) = ϕma ϕnb |Vmn |ϕnc ϕmd   n(m) (x)n(n) bc (y) = d 3 x d 3 y ad . |x − y|


Although the matrix elements are defined by multi-electron wave functions, the whole expression can be reduced to a Coulomb-interaction with single particle densities23  (m) n(m) eZμ δ(x − Rμ ). (5) ab (x) = ρab (x) − δa,b μ∈m

The expression covers electronic contributions (first term on the right-hand side) and nuclear contributions (second term). (m) represents a single electron density (of molecule m) but ρab may connect different electronic states. As long as a weak optical excitation and/or a small excitation (exciton) density are present in theSC, its consideration by the overall ground-state  |φ g  = m |ϕ mg  and the singly excited states |φ m  = |ϕ me  n=m |ϕ ng  remains sufficient. The restriction results in Hsc ≈ H0 + H1 . Possibly, double excited states |φ mn  may be also considered. However, if multiple molecular excitations become of interest this expansion is not suitable. One should work with the original complete Hamiltonian, Eq. (3). Possibly, a certain restriction of the excited single molecule states ϕ ma (besides the groundstate ϕ mg ) may become useful. The standard multiexciton Hamiltonian is obtained if only the first excited state ϕ me (S1 -state) is considered (molecular two-level model). This implies that multiple excitations of the SC are exclusively formed by molecules being in their first excited state. Now, it is possible to introduce single molecule transition operators Bm+ = |ϕme ϕmg |,


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Volkhard May

J. Chem. Phys. 140, 054103 (2014)

Bm = |ϕmg ϕme |.


The transition operators fulfill the completeness relation 1 = Bm Bm+ + Bm+ Bm . Often, this expression is used as an anticommutator relation of a second quantized formalism (we do not follow this line here). The Coulomb-matrix elements Jmn (ab, cd) related to the molecular two-level model cover all combinations of a, . . . , d being identical either with g or with e. For example, electrostatic couplings and couplings between charges and electronic transitions result (cf. Refs. 20 and 23). To stay simple the set of Coulomb-matrix elements is restricted to those describing (m) excitation energy transfer (n(m) eg reduces to ρeg , cf. Fig. 3)  Jmn = Jmn (eg, eg) =

d 3x d 3y

(m) (n) ρeg (x)ρge (y)

|x − y|






With that, we arrive at the standard Frenkel-exciton Hamiltonian   Emg Bm Bm+ + Eme Bm+ Bm Hexc =




Jmn Bm+ Bn .




Below we will somewhat change the notation by introducing the single-molecule excitation energies Em = Eme − Emg . To consider annihilation processes, a higher excited state ϕ mf with excitation energy Em = Emf − Emg is included. It should fulfill Em − Em ≈ Em . Then, the transition of two single excitations present in state ϕ me of molecule m and state ϕ ne of molecule n into a higher excitation ϕ mf or ϕ nf becomes possible. The related Hamiltonian is formulated by introducing new transition operators + = |ϕmf ϕme | Dm


Dm = |ϕme ϕmf |.



S0 Sn


The additional consideration of the states ϕ mf results in a number of new Coulomb-matrix elements. The expression  Kmn = Jmn (f g, ee) =

d 3x d 3y

(n) ρf(m) e (x)ρge (y)

|x − y|


is responsible for exciton fusion and is defined by the new transition densities ρf(m) e (see also Fig. 3). We also consider  Jmn = Jmn (f e, f e) =

d 3x d 3y

(n) ρf(m) e (x)ρef (y)

|x − y|

S0 FIG. 3. Energy level scheme demonstrating the different types of EET couplings used in the main text. Upper panel: first excited state to first excited state EET accounted for by Jmn ; middle panel: higher excited state to higher excited state EET accounted for by Jmn ; lower panel: higher excited state formation due to single excited state fusion accounted for by Kmn .

. (13)

It describes excitation energy transfer if the state ϕ nf is present (see Fig. 3). Since Kmn as well as Jmn are both determined by ρf(m) e they have to be considered at the same time. All further matrix elements which arise if the higher excited state is considered are neglected. For example, matrix elements of type Jmn (fg, fg) occur which include transition densities (transition dipole moments) between the ground-state and the higher excited state. It is assumed that they are small enough to be neglected.

According to the foregoing discussion, we end up with the SC Hamiltonian used in the following:10

Hsc = Hexc +



+ Em Dm Dm +

+ Jmn Dm Dn


  + Kmn Dm Bn + H.c. .



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Volkhard May

J. Chem. Phys. 140, 054103 (2014)

The energy levels already notice that the completeness relation is restricted to two excited states ϕ ma :  + |ϕma ϕma | = Bm Bm+ + Bm+ Bm + Dm Dm . (15) 1= a=g,e,f

 Using thisrelation we may replace m Emg Bm Bm+ in Eq. (9) + by Eg − m Emg (Bm+ Bm + Dm Dm ) where Eg = m Emg is the ground-state energy.Accordingly, the first sum in Hexc can be replaced by Eg + m Em Bm+ B m (remember Em = Eme + − Emg ). In the same way the sum m Em Dm Dm in Eq. (14) results (usually one sets Eg = 0). B. Dissipative dynamics

For a direct description of EET dynamics one may introˆ duce the density matrix ρAB (t) = φA |ρ(t)|φ B . It is given via matrix elements of the (reduced) density operator ρ(t) ˆ with respect to the SC product states, Eq. (2). Concerning the used three levels per molecule, the number of density matrix elements explodes if the number of molecules increases slightly. In such a situation, it is advisable to avoid a state expansion of ρ. ˆ A tractable alternative is given by equations of motion for the following type of expectation values: ˆ =  O. ˆ tr{ρ(t) ˆ O}


Oˆ has to be understood as an arbitrary operator formed by the Bm+ (Bm ) and Dn+ (Dn ). The introduced bracket notation (without explicit presentation of the time-argument) will be used in the following. In this way we obtain, for example, the excited state population as Pm (t) = Bm+ Bm  = tr{ρ(t)|ϕ ˆ me ϕme |}.


The ground-state population m (t) = Bm Bm+ 


+ Bm+ Bm , ρ(t) ˆ + − 2Bm ρ(t)B ˆ m . 2 m


Dissipation due to the decay of higher excited states follows from  rm  

+ + Df ρ(t) Dm ˆ = Dm , ρ(t) ˆ + − 2Dm ρ(t)D ˆ m . (22) 2 m To get an equation of motion for the expectation value of Oˆ we note the quantum master equation (20). Then, some rearrangements result in ∂ ˆ = i [H, O] ˆ −  − D˜ O. ˆ tr{ρ(t) ˆ O} ∂t ¯


The new type of dissipative superoperator separates again into D˜ e and D˜ f . We have D˜ e Oˆ =


ˆ m Bm+ Bm , Oˆ + − 2Bm+ OB 2 m



+ + ˆ Dm ODm . Dm , Oˆ + − 2Dm 2 m


and D˜ f Oˆ =

Calculating the commutator (anti-commutator) relations entering Eq. (23) results in equations of motion for the expectaˆ (and related functions). tion values O


does not need to be determined since we can deduce it from the completeness relation, Eq. (15), as m = 1 − Pm − Nm . Since vibrational contributions to Hsc have been neglected, we assume that they form a dissipative environment. A system-bath description is chosen where ρˆ has to be considered as a reduced density operator which undergoes dissipative dynamics according to i ∂ ρ(t) ˆ = − [HS , ρ(t)] ˆ − − Dρ(t) ˆ . ∂t ¯

ˆ = De ρ(t)


If we express  the trace by the basis states, Eq. (2), we get Pm (t) = A, B ρ AB (t)φ B | |ϕ me ϕ me | |φ A  and realize that all those states φ A and φ B contribute where molecule m is in its first excited state. In the same manner, we define the higher excited state population + Nm (t) = Dm Dm .

This non-radiative excited state decay should proceed from the higher excited state to the first excited one. The respective rate is kmf→me = rm . We also include decay of the first excited state described by the rate kme→mg = km which, however, is ˆ of smaller than rm . Accordingly, the dissipative part Dρ(t) ˆ and Df ρ(t). ˆ Eq. (20) separates into De ρ(t) The easiest way to account for these decay processes is to reduce the general structure of the dissipative contributions to expressions which do not couple diagonal and off-diagonal density matrix elements (in the used local representation). This specification also known as the secular approximation leads to the Lindblad-form of dissipation (it guarantees positivity of the diagonal density matrix elements). Accordingly, dissipation due to the first excited state decay is described by


It should be understood as a standard quantum master equation neglecting any non-Markovian effect and being of second order with respect to the system reservoir (bath) coupling. The latter should only induce internal conversion processes.


Identifying Oˆ with different types of transition operators EET dynamics can be simulated. The choice of the operators is dictated by the questions we would like to address. Note also that the representation of the SC Hamiltonian in local excited states is general and allows to describe any type of spatially delocalized or partially delocalized excitation. However, we have in mind a situation which is not dominated by delocalized states (exciton formation). We focus on the case of a sufficiently weak excitonic coupling and characterize the system dynamics by molecular excited state populations Pm , Eq. (17), and Nm , Eq. (18), describing first excited state population and higher excited state population, respectively. Concerning the first quantity, we use here the representation by

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J. Chem. Phys. 140, 054103 (2014)

Bm+ Bm  and get the following equation of motion: ∂ B + Bm  ∂t m + Dm  = − km Bm+ Bm  + rm Dm   i Jmn Bm+ Bn  − Jnm Bn+ Bm  − ¯ n + − Jmn Dm Dn  + Jnm Dn+ Dm  ∗ + + Kmn Bn+ Dm  − Kmn Dm Bn   ∗ + Knm Bm+ Dn  − Knm Dn+ Bm  .


The first two terms on the right-hand side describe depopulation due to nonradiative decay into the ground-state and repopulation due to the nonradiative decay of the higher + Dm ), respectively. The excited state (with population Dm terms proportional to Jmn account for excitation energy motion among the first excited states. Excitation energy transfer among the higher excited states is considered by terms proportional to Jmn . Transitions among the first and the higher excited states are characterized by the terms proportional to Kmn . Since all these couplings are assumed to be small their consideration via rate expressions of second order in these couplings will be sufficient. A. Rate equations for Pm and Nm

In order to derive rate equations for Pm and Nm all the other functions (time-dependent expectation values) appearing in Eq. (26) have to be determined in lowest order with respect to the different types of coupling. For further use we introduce at m = n Wmn (t) = Bm+ Bn ,


+ Dn , Zmn (t) = Dm


+ Rmn (t) = Dm Bn .



A simple notation of the equation of motion for Pm , Eq. (26), results ∂ 2  Jmn Wmn Pm = −km Pm + rm Nm + Im ∂t ¯ n 2  2  − Im Jmn Zmn − Im Kmn Rmn ¯ ¯ n n 2  − Im Knm Rnm . ¯ n

∂ ˜ mn Zmn Zmn ≈ i ∂t i + Jnm ((Pm − Nm )Nn − Nm (Pn − Nn )), (33) ¯ and ∂ ˜ mn Rmn Rmn ≈ i ∂t i ∗ + Kmn ((Pm − Nm )Pn − Nm (n − Pn )). (34) ¯ The solutions of these inhomogeneous differential equations are readily obtained. In the t case of Wmn it takes the ˜ − t0 ))+ t0 d t¯ exp(i ω(t ˜ − t¯))I (t¯). The form Wmn (t0 ) exp(i ω(t initial value of Wmn is Wmn (t0 ) and I denotes the inhomogeneity of the equation of motion. For the following it suffices to set the initial value zero and to ignore the appearance of the inhomogeneity at times earlier than t. This would be valid if exp(i ω(t ˜ − t¯)) decays fast compared to any change of the populations determining the inhomogeneity (approximation of instantaneous dephasing). Accordingly, the equations for Wmn , Zmn , and Rmn are solved by neglecting the respective time derivative. We introduce the solutions into Eqs. (30) and (31) and arrive at a closed set of rate equations ∂ Pm = −km Pm + rm Nm ∂t   − Pm γmn (n − Pn ) + (m − Pm ) γmn Pn n

+ Nm


mn (Pn − Nn ) − (Pm − Nm )


− Pm

mn Nn


ϒnm (Pn − Nn ) + (m − Pm )


− (Pm − Nm )


ϒmn Pn + Nm

ϒnm Nn


ϒmn (n − Pn )


(35) (30)

We complete this equation by the one for Nm ∂ 2  Nm = −rm Nm + Im Jmn Zmn ∂t ¯ n 2  + Im Kmn Rmn . ¯ n

A closed set of equations for Pm and Nm is obtained if Wmn is determined linearly in Jmn , Zmn linearly in Jmn , and Rmn linearly in Kmn . The approximations necessary to do this are explained in Appendix B. The exact equations of motion are given in Appendix A. We list here the approximate equations of motion whose solutions can be directly inserted in Eqs. (30) and (31) (for the definition of complex transition frequencies see Appendix A): ∂ Wmn ≈ i ω˜ mn Wmn ∂t i + Jnm ((m − Pm )Pn − (n − Pn )Pm ), (32) ¯

and ∂ Nm = −rm Nm ∂t   − Nm mn (Pn − Nn ) + (Pm − Nm ) mn Nn n

− Nm (31)



ϒmn (n − Pn ) + (Pm − Nm )

ϒmn Pn .



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Volkhard May

J. Chem. Phys. 140, 054103 (2014)

Note the appearance of three types of transition rates. The rate of S1 -state EET reads (kmn = (km + kn )/2) γmn =

2|Jnm |2 kmn . 2 (Em − En )2 + ¯2 kmn


It is determined by the S1 -state energy difference of both involved molecules. The rate responsible for higher excited state EET takes the form (rmn = (rm + rn )/2) mn =

2|Jnm |2 (kmn + rmn ) . [(Em − Em ) − (En − En )]2 + ¯2 (kmn + rmn )2 (38)

Here, the differences appear for the excitation energies between the first and the higher excited state. And, finally, we have the rate of exciton fusion (note ϒ mn = ϒ nm ) ϒmn =

2|Kmn |2 (kmn + rm /2) . (Em − Em − En )2 + ¯2 (kmn + rm /2)2


This rate is governed by the difference of the excitation energies if one molecule moves from its first excited state into its higher excited state and the other molecule from its groundstate into its first excited state. Due to the rough description of intra- and inter-molecular vibrations by internal conversion rates entering the description of dissipation, Eqs. (21) and (22), the above given rates do not consider detailed balance. This is a general defect which, however, will be accepted for the following considerations (possible improvements are well known24 ). B. Interpretation of the rate equations

We discuss the different terms entering the rate equations (35) and (36) (the contributions proportional to km and rm have been already identified as terms related to internal conversion processes). So,  we focus on the second line of Eq. (35). The term −Pm n γ mn (n − Pn ) is responsible for first excited state EET from molecule m to molecule n. Since arbitrary excitation is possible, the final state of the transfer is weighted by the population difference n − Pn . This term guarantees that EET does not take place if the probability to have molecule n in its ground-state coincides with the probability to have it in the first excited state. In the low excitation regime, i.e., if P n , Nn n ≈ 1 we arrive at the standard expression −Pm n γ mn for EET. Energy transfer from all molecules  n to molecule m is accounted for by the term (m − Pm ) n γ mn Pn . Here, the population difference of molecule m appears. The standard (low excitation regime) expression is n γ mn Pn . Next we consider the third line of Eq. (35). The respective terms are determined by the rate mn and their interpretation has to be related to EET among the higher excited states. If energy is transferred from state ϕ mf to state ϕ nf molecule m moves to its first excited state. This process is accounted for by Nm n mn (Pn − Nn ). In all energy accepting molecules n the population difference Pn − Nn appears because these molecules undergo the transition from the first to the higher excited state.  The reverse process is accounted for by the term −(Pm − Nm ) n mn Nn . Molecule m becomes excited due to

EET from other molecules n. Those are represented by their higher excited state population while for molecule m the population difference Pm − Nm is present. These processes covered by the third line of Eq. (35) also appear in the second line of the rate equation (36) for the higher excited state population Nm . Of course, the sign has been changed. Exciton fusion is described by the terms  of the fourth and fifth lines in Eq. (35). The term −Pm n ϒ nm (Pn − Nn ) characterizes the decay of the molecule’s m first excited state and the excitation of the molecules n into the higher excited state (again the population difference Pn − Nn appears). The reverse of this process is covered by the term (m  − Pm ) n ϒ nm Nn (the higher excited state of the molecules n decays and molecule m is moved into its first excited state). The alternative transition which can takeplace at molecule m is covered by the term −(Pm − Nm ) n ϒ mn Pn . It describes exciton fusion with the molecules n which all become de-excited (from the first excited state to the groundstate). Molecule m becomes higher excited (the respective population difference Pm − Nm appears). Population increase of the first excited state of molecule m due to the de-excitation of the higher excited state and excitation of themolecules n in their first excited state is described by Nm n ϒ mn (n − Pn ) (inverse of the fusion process). Both last mentioned terms (fifth line of Eq. (35)) also appear in Eq. (36) (third line) but with an alternated sign describing decrease and increase of the higher excited state population Nm . In summary, we can state that the chosen decoupling of higher (more complex) expectation values results in reasonable kinetic equations for the first and higher excited state population. All processes we have to expect are incorporated. Some simplifications are considered in the following. C. Limit of instantaneous higher excited state decay

We discuss the case where the life-times 1/rm of the higher excited states of the molecules fall below all other characteristic times. Before considering this limit we change to the weak excitation case with 1 ≈ m Pm Nm . The equation for Pm reduces to ∂ Pm = −km Pm + rm Nm ∂t   − γmn (Pm − Pn ) + mn (Nm Pn − Pm Nn ) n


(ϒmn + ϒnm )Pm Pn +


 (ϒmn Nm + ϒnm Nn ), n

(40) and for Nm we get the equation  ∂ Nm = −rm Nm − mn (Nm Pn − Pm Nn ) ∂t n  − ϒmn (Nm − Pm Pn ).



First excited state EET is described now by the standard expression −  n γ mn (Pm − Pn ). Higher excited state EET acts via the term n mn (Nm Pn − Pm Nn ). It increases the actual

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Volkhard May

J. Chem. Phys. 140, 054103 (2014)

state population Pm via decay of the higher excited state in the course of EET to another molecule n (provided the latter is in the first excited state). The reverse process decreases Pm . The same expression but with a changed sign enters the equation for Nm . EEA (the last two terms in Eq. (40)) affects Pm in two ways, via the transition into the higher excited state and a transition into the ground state of molecule m and the reverse process. Therefore, we meet the expression − n (ϒ mn + ϒ nm )Pm Pn . In a similar way, Pm is increased via the transition of molecule m or molecule n from their higher excited state. Of course, this doubling of terms is absent in Eq. (41) for Nm . In the case of very large rm we can approximate the equation for Nm according to  rm Nm ≈ ϒmn Pm Pn . (42) n

All terms ∼Nm except the one proportional to rm have been neglected. The same approximation is applied to the equation for Pm and we get   ∂ Pm = −km Pm − γmn (Pm − Pn ) − ϒnm Pm Pn . ∂t n n (43) The equation combines in a simple form and in a standard way nonradiative decay of excited states, EET and EEA (first, second, and third terms on the right-hand side, respectively). In a continuum limit with P(r, t) replacing Pm (t) these nonlinear rate equations turn into a diffusion equation with the annihilation induced nonlinearity ∼P2 (r, t). IV. CONCLUSIONS AND OUTLOOK

The standard Frenkel exciton Hamiltonian completed by a higher excited electronic state per molecule has been used to study EET kinetics if many molecules of the complex or aggregate are excited. Emphasis has been laid on EEA in the presence of weak excitonic coupling where Frenkel exciton formation is less important. To circumvent an expansion with respect to the multitude of possible excited states the theory has been formulated in using transition operators Bm+ from the electronic ground-state to the first excited state and transition + from the first excited state to a higher excited operators Dm one. The quantum master equation for the molecular system established the basis to derive equations of motion for expectation values of transition operator combinations. Considering + Dm , equations of motion for the populations of Bm+ Bm and Dm the first and higher excited states, respectively, could be deduced. A rather simple and direct decoupling of higher-order expectation values offers a nonlinear microscopic theory for EEA. It is just the advantage of this approach that an appropriate decoupling is obvious. The offered material focuses on the derivation of general nonlinear rate equations. Therefore, we finally comment on some other possible extensions of the introduced computation scheme. While the formulation of the exciton Hamiltonian by transition operators is well-established there seems no formulation of EET kinetics in the literature as it was pre-

sented here.25 This is somewhat astonishing since the general approach is well-established in studying photonic systems (see, for example, Refs. 26–28). There, it is common to derive equations of motions for photon operator expectation values and to truncate the subsequent hierarchy of higher order expectation values with different techniques. In the light of these approaches, it would be of interest to avoid the simple decoupling used here and to establish new equations of motions for the fourth operator expectation values. To arrive at the rate equations of EEA we assumed weak excitonic coupling. It would be of particular importance to go beyond this limit and to account for exciton state formation. This does not require an explicit introduction of exciton energies and states but an indirect consideration via a nonperturbative treatment of the excitonic coupling. Moreover, it would be of interest to include charge transfer states. Those may be accounted for by the following more involved transition operators. Therefore, positively and negatively charging of molecule m is described by + = |ϕm± ϕmg |, Cm±


where ϕ m − corresponds to an excess electron (in a HOMOLUMO scheme it would be placed in the LUMO) and ϕ m + to a missing electron (hole; placed in the HOMO). Then, a charge transfer excitation from molecule n to molecule m + + Cm+ and the general methodology preis described by Cm− sented here can be used. A. Coupling to the radiation field

While the above mentioned future investigations are of interest for special systems the consideration of the coupling to the radiation field would be of general importance. We finally comment on it. The coupling to the radiation field includes transitions from the ground-state into the first excited one (transition dipole moment dm ) and from there to the higher excited state (transition dipole moment mm ). Direct transitions from the ground state into the higher excited state will be ignored. Accordingly, the Hamiltonian which couples of the molecular system to the radiation field takes the following standard form:  + (dm Bm+ + mm Dm ) + H.c. (45) Hfield (t) = −E(t) m

The external field induced contributions to the rate equations are

  ∂ i = − E(t) d∗m Bm  − dm Bm+  Pm ∂t ¯ field


∂ Nm ∂t


  i + + E(t) m∗m Dm  − mm Dm  ¯


  i + = − E(t) m∗m Dm  − mm Dm  . ¯


In both expressions transition amplitudes appear. We quote respective equations of motion. The one for Bm+  takes the

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Volkhard May

J. Chem. Phys. 140, 054103 (2014)

start with Wmn = Bm+ Bn :

form (ω˜ m = Em /¯ + ikm /2) ∂ B +  = i ω˜ m Bm+  ∂t m i  Jnm (Bm Bm+ − Bm+ Bm )Bn+  + ¯ n

∂ B + Bn  = i ω˜ mn Bm+ Bn  ∂t m i  Jkm (Bm Bm+ − Bm+ Bm )Bk+ Bn  + ¯ k

i  + + Jmn Dm Bm D n  ¯ n +

i  + + Kmn Dm Bm Bn  ¯ n

+ Knm Dn+ (Bm Bm+

− Jnk Bm+ Bk (Bn Bn+ − Bn+ Bn ) + + + Jmk Dm Bm Dk Bn  − Jkn Bm+ Dk+ Bn Dn  + + ∗ + Kmk Dm Bm Bk Bn  − Knk Bm+ Bk+ Bn Dn 

+ Kkm (Bm Bm+ − Bm+ Bm )Dk+ Bn   ∗ − Kkn Bm+ Dn (Bn Bn+ − Bn+ Bn ) .

Bm+ Bm )

i − E(t)d∗m Bm Bm+ − Bm+ Bm  ¯ i + + − E(t)mm Dm Bm . ¯


+  reads (ω˜ m = (E− m The equation of motion for Dm − Em )/¯ + i(km + rm )/2)

∂ + ˜ m Dm D +  = i  ∂t m i  + + − Jmn Dm Bm Bn  ¯ n

i  ∗ + + K (Dm Dm − Dm Dm )Bn+  ¯ n mn

i  ∗ + + K Dn Dm Bm  ¯ n nm

i + + + E(t)dm Dm Bm  ¯ i + + − Dm Dm . − E(t)m∗m Dm Dm ¯

Note the introduction of the complex transition frequencies ω˜ mn = (Em − En )/¯ + i(km + kn )/2. Next, we change to + Bn  and obtain Zmn = Dm ∂ + ˜ mn Dm D + Dn  = i Dn  ∂t m i  + + − Jmk Dm Bm Bk D n  ¯ k + + Bk Bn D n  − Jkn Dm + + − Jkm (Dm Dm − Dm Dm )Dk+ Dn 

i  + + Jnm Dn+ (Dm Dm − Dm Dm ) ¯ n +


+ + Jnk Dm Dk (Dn Dn+ − Dn+ Dn ) ∗ + + − Kmk (Dm Dm − Dm Dm )Bk+ Dn  ∗ + + + Kkm Dk Dm Bm D n  + + Knk Dm Bk (Dn Dn+ − Dn+ Dn )  + − Kkn Dm Bn Dn Dk+  .


The newly defined complex transition frequencies read ˜ mn = [(Em −Em )−(En −En )]/¯ + i(km +rm + kn + rn )/2.

+ Bn  Finally, we have for Rmn = Dm (49)

A decoupling as presented earlier in the discussion would make the equations ready for a detailed application. To conclude, photoinduced processes in molecular systems resulting in a rather high density of excitations can be properly treated within the present scheme. Of course, the ultimate proof has to be given by related numerical simulations. Respective studies are under progress.

∂ + ˜ mn Dm D + Bn  = i Bn  ∂t m i  + + − Jmk Dm Bm Bk Bn  ¯ k + + Jnk Dm (Bn Bn+ − Bn+ Bn )Bk  + + − Jkm Dk+ (Dm Dm − Dm Dm )Bn  + + + Jkn Dm D k Bn D n  ∗ + + − Kmk (Dm Dm − Dm Dm )Bk+ Bn 


∗ + + Kkn Dm Dk (Bn Bn+ − Bn+ Bn )

Financial support by the Deutsche Forschungsgemeinschaft through Sfb 951 is gratefully acknowledged.

∗ + + + Kkm Dk Dm Bm Bn   ∗ + + + Knk Dm Bk Bn Dn  .


In Sec. III we introduced the equations of motion for + Dm  which include the new Pm = Bm+ Bm  and Nm = Dm functions (expectation values), Eqs. (27), (28), and (29). We present exact equations of motion for the latter quantities and


˜ mn The complex transition frequencies take the form = [Em − Em − En ]/¯ + i(km + rm + kn )/2. The derived equations of motion include a number of four operator expectation values which cannot be expressed by those functions defined so far, i.e., the set of equations is not closed. This becomes only possible in the framework of

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Volkhard May

special approximation schemes. Here, we only consider the simplest variant of such schemes. APPENDIX B: LOWEST-ORDER APPROXIMATION OF SOME EQUATIONS OF MOTION + Noting the equations of motion for Bm+ Bn , Dm Dn , + and Dm Bn  introduced in Appendix A, new expectation values with four transition operators appear. We introduce a simple factorization of the four-operator functions and a computation becomes possible. Let us start by considering the first four-operator function on the right-hand side of Eq. (A1). A possible approximation takes the form

(Bm Bm+ − Bm+ Bm )Bk+ Bn  ≈ Bm Bm+ − Bm+ Bm Bk+ Bn  = (m − Pm )Wkn . (B1) If the system is externally excited by a laser pulse electronic transitions are driven and we can assume that expectation values of the type Bm+  exist (see also Sec. IV A). Then, and for k = m the expression Bk+ Bn  could be factorized to give Bk+ Bn . However, in the present considerations we ignore this possibility. We also indicate that in the case of Bm+ Bm  such a factorization would be misleading since the expression |ϕ me ϕ me | is augmented to |ϕ me ϕ mg ||ϕ mg ϕ me |. Accordingly, the replacement of Bm Bm+ − Bm+ Bm in the above given equation by its expectation value would be a reliable approximation. Before considering the factorization of other types of four operator expectation values we briefly remind how to get rate equations for Pm and Nm . This avoids to discuss the factorization of expectation values which do not contribute. Therefore, + Dm . Rate expressions for we take a look at Eq. (31) for Dm 2 . So, for excitation energy motion shall be of the order Jmn + example, Dn Dm  has to be determined exclusively of first order with respect to Jmn . Accordingly, we need to factorize in Eq. (A2) only terms proportional to Jmn and neglect the other ones. The following approximation is taken: ∂ + ˜ mn Dm D + Dn  ≈ i Dn  ∂t m i  + + + Jkm Dm Dm − Dm Dm Dk+ Dn  ¯ k  + − Jnk Dm Dk Dn Dn+ − Dn+ Dn  . (B2) Since the right-hand side further depends on off-diagonal expectation values like Dk+ Dn  the solution becomes of higherorder in Jmn . We ignore off-diagonal contributions and further approximate ∂ + ˜ mn Dm D + Dn  ≈ i Dn  ∂t m i + + + Jnm Dm Dm − Dm Dm Dn+ Dn  ¯  + Dm Dn Dn+ − Dn+ Dn  . (B3) − Jnm Dm Now, the solution is of first order in Jmn . + Dm  we next consider Moving back to Eq. (31) for Dm the terms proportional to Kmn . Here, we need to determine

J. Chem. Phys. 140, 054103 (2014) + Bn , Eq. (A3), of first order in Kmn . In a first approxiDm mation step, we focus on respective terms on the right-hand side: ∂ + ˜ mn Dm D + Bn  ≈ i Bn  ∂t m i  ∗ + + + Kmk (Dm Dm − Dm Dm )Bk+ Bn  ¯ k ∗ + − Kkn Dm Dk (Bn Bn+ − Bn+ Bn )

 ∗ + + ∗ + + − Kkm Dm Bm Dk Bn  − Knk Dm Bk Bn D n  . (B4) The first and second terms of the k-summation are treated as already described. Concerning the third and fourth terms we take a full factorization. The third term results in + + + + Bm Dk Bn + Dm Dk Bm+ Bn  + Dm Bn Bm+ Dk . Here, Dm + + Bm  represents a new type of function (transition ampliDm tude from the ground-state to the higher excited state) which will be ignored. The second term of the factorized expression includes Bm+ Bn  with m = n. Therefore, it produces a nonzero contribution in Jmn and can be neglected. Finally, the third term in the factorization leads to higher orders in Kmn and is also neglected. Accordingly, the used factorized form of Eq. (B4) is ∂ + ˜ mn Dm D + Bn  ≈ i Bn  ∂t m i ∗ + + + Kmn (Dm Dm  − Dm Dm )Bn+ Bn  ¯  ∗ + − Kmn Dm Dm (Bn Bn+  − Bn+ Bn ) . (B5) Note the removal of the k-summation by the restriction to k = n (first term) and to k = m (second term). To change from the exact Eq. (A2) to a rate equation + + for Dm Dm , we derived approximate equations for Dm Dn  + and Dm Bn . It remains to approximate Eq. (A1) to also turn Eq. (26) for Bm+ Bm  into a rate equation. According to the procedure explained above, we arrive at ∂ B + Bn  = i ω˜ mn Bm+ Bn  ∂t m i + Jnm (Bm Bm+  − Bm+ Bm )Bn+ Bn  ¯  − Jnm Bm+ Bm (Bn Bn+  − Bn+ Bn ) . (B6)

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054103-10 11 B.

Volkhard May

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J. Chem. Phys. 140, 054103 (2014) 24 The

present study reduces electronic energy dissipation to simple excited state decay rates km and rm . Consequently, the related electronic level broadening determines the various EET rates, Eqs. (37), (38), and (39). Since any consideration of intra- and inter-molecular vibrations is absent, the rates do not account for thermal distributions of vibrational coordinates in the energy donor and energy acceptor. More correct rates may be introduced if the expressions (Eqs. (37), (38), and (39)) are replaced by Golden Rule type formulas (see also Ref. 20). If one includes vibrational states from the very beginning these rates are automatically derived in the limit of fast vibrational relaxation (respective computations will be published elsewhere). 25 According to the author’s overview only the nearly 45 years old work of A. Suna, Ref. 2, includes in its appendix some remarks which point in this direction. 26 D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, Phys. Rev. Lett. 102, 163601 (2009). 27 M. Richter, A. Carmele, A. Sitek, and A. Knorr, Phys. Rev. Lett. 103, 087407 (2009). 28 A. Carmele, M. Richter, W. W. Chow, and A. Knorr, Phys. Rev. Lett. 104, 156801 (2010).

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Kinetic theory of exciton-exciton annihilation.

Weakly excited states of dye aggregates and supramolecular complexes can be characterized by single or two exciton states. Stronger excitation results...
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