Excitonic and exciton-phonon interactions in molecular aggregates

(1)

Excitonic and exciton-phonon interactions in molecular aggregates

Roel Tempelaar Supervisors:

drs. Bas Vlaming prof. dr. Jasper Knoester

Zernike Institute of Advanced Materials University of Groningen

July 7, 2009

(2)

Abstract

This report describes the absorption spectra of molecular aggregates, in which both excitonic and exciton-phonon interactions play a role. The case of a single molecule is solved analytically using the Franck-Condon principle and

using the Lang-Firsov transformation. The two-particle approximation is introduced, and applied to the case of a pinwheel aggregate. The results are compared with the outcome by Spano [10,13]. Also, a comparison is made to the one-particle approximation for different excitonic and exciton-phonon

couplings.

(3)

Introduction

1.1 Molecular aggregates

Molecular aggregates are systems composed of monomers (smaller molecular units) that are held together by molecular interactions [1]. These aggregates show collective properties, that can be different from the properties of the single monomers.

As an example of an aggregate, we first mention the molecular dimer. A dimer consists of two molecules which interact with each other. Although this case of an aggregate is rather simple, it clearly reflects the aforementioned collective behaviour.

This report mainly focuses on cyclic aggregates. This is the case of a ring of N equally distanced molecules. Since this ring shows rotational symmetry, every molecule can be treated equally. Furthermore, periodic boundary conditions hold. This means that molecule n is identified with molecule n⁰= n + l · N for every integer l.

1.2 Optical response

A molecular aggregate can interact with an external electromagnetic field by changing its quantum state. The aggregate can either take energy out of the field (absorption) or lose energy by emitting electromagnetic radiation (emis- sion). The principle of conservation of energy states that the optical frequency of the field matches the difference in energy between the two corresponding quantum states. As a consequence, the optical spectrum is related to the possible quantum states of the aggregate. By doing spectroscopy, one obtains an image of the quantum mechanical transitions that are possible for the aggregate under consideration. The other way around, on a basis of the quantum states one can make predictions of the outcoming spectrum. This is done by applying

“Fermi’s Golden Rule”.

This report solely deals with absorption spectroscopy, in which case the aggregate gets excited (it makes a transition to a quantum state that is higher in energy). The aggregate is assumed to reside initially in state | ai, with corresponding energy Ea. In principle this can be any state. According to the

“Boltzmann distribution”, the probability for the aggregate to be in a particular

(6)

state | ai is expressed as [6]

P (a) = 1

Ze^−βE^a, (1.1)

in which case β ≡ 1/(kT ), T is the temperature, k is Boltzmann’s constant and Z ≡P

a⁰exp(−βEa⁰) is the partition function.

Upon interaction with the field, the aggregate makes a transition to state | bi, having an energy E_b > E_a. Fermi’s Golden Rule then gives the corresponding absorption spectrum [3]

IA(ω) =X

a,b

P (a) | Vba|²δ(Eb− Ea− ω), (1.2)

where the summation is taken over all possible states a and b. ω denotes the optical frequency of the field. Note that we take ¯h = 1.

The transition dipole moment Vba≡ hb | −M·e | ai represents the interaction between the field and the aggregate. e is the polarization vector of the field and M is the total dipole operator of the molecular system, which is given by

M ≡X

n

µ_n. (1.3)

Here, the summation is taken over all molecules that constitute the aggregate.

µ_n represents the dipole operator of molecule n. The oscillator strength of a transition to an excited state is given by the absolute value squared of the transition dipole moment.

In practice, the delta function in Eq. (1.2) will have a Gaussian or Lorentzian shape. If we assume the aggregate to consist of identical molecules, the peak width is determined by the inverse lifetime of the state | bi.

1.3 Frenkel exciton

The change in the quantum state is mainly ascribed to a change in the electronic configuration of the aggregate. In our treatment, the participating molecules are assumed to be identical. Furthermore, a single molecule is treated as a two-level electronic system, having an electronic ground state | gi and one electronic excited state | ei. This approach holds if the molecule has one particular electronic transition whose oscillator strength dominates all the others.

It is convenient to describe the electronic transition of the molecule in terms of the creation and annihilation operators, denoted by respectively c^† and c.

Electronic excitation is then expressed as

c^†| gi =| ei. (1.4)

In fundamental terms, an electron inside the molecule is excited to a higher orbit inside the molecule. This is equivalent with the creation of an electronic quantum called a “Frenkel exciton” [3].

Since electrons, being fermions, obey the “Pauli exclusion principle”, an electronic state cannot be created or annihilated twice (cc = 0, c^†c^†= 0). Also, the electronic creation and annihilation operators are related by the fermionic anti-commutation relation {c, c^†} = 1 [5].

(7)

In case of an aggregate composed of identical molecules, the electronic excited states of the system as a whole is defined as [10]

| ni ≡| eni Y

m6=n

| gmi ≡ c^†_n | gi, (1.5)

which means that molecule n is electronically excited, whereas all other molecules reside in the electronic ground state. Note that the creation operator c^†_n acts on molecule n only. The molecular aggregate has a collective electronic ground state | gi with corresponding energy E_g. If we assume the exciton energy to be equal for all molecules, the Hamiltonian corresponding to the electronic energy is expressed as

H =X

n

(Ee− Eg) | nihn |, (1.6) where Ee is the energy of the electronic excited states.

The basis of excited states, as put forward in the foregoing text, consists of one-exciton states only. In practice this means that during the interaction between the electromagnetic field and the aggregate, only one electronic excitation is created. The states are eigenstates in case of non-interacting molecules.

In case of interaction (an example of which will be given in the next section), the eigenstates are expressed as superpositions of these basis states.

1.4 Excitonic interaction

If the molecules which make up the aggregate are in close contact with each other, they will experience a coupling between their transition dipoles. As a result, the exciton can be transferred from one molecule to another. This so- called excitonic coupling is described by the following Hamiltonian [3]:

Hmn= Jmn| mihn | . (1.7)

Put into words, an exciton is created on molecule m, while annihilated on molecule n. Jmn is the excitonic interaction term. In the point-dipole approximation, this is given

J_mn=µ_m· µ_n

| rmn|³ − 3(µ_m· rmn)(µ_n· rmn)

| rmn|⁵ , (1.8)

where r_mn denotes the relative position vector between molecules m and n.

Excitonic interaction leads to a mixing of the electronic excited states of the system.

1.5 Phonon

In addition to electronic excitation, a molecule can be vibrationally excited. The molecule absorbs energy and transfers it into vibrational motion of the nuclei involved [1]. This motion is described by the harmonic oscillator model. To keep things simple, we assume that one normal mode ω0 dominates all other

(8)

vibrational modes (this is called the “Einstein model”). The corresponding energy is [4,8]

E =1 2p²+1

2ω₀q². (1.9)

In this expression p, q and ω₀represent the oscillator momentum, displacement and frequency. The mass is taken to be unity.

The first energy term represents the kinetic energy, whereas the second term represents the potential energy. The potential well is characterized by a parabolic curve as a function of the displacement q, having a minimum at the position q = 0.

According to the theory of quantum mechanics, the harmonic oscillator eigenenergies are quantized. As a consequence, the harmonic oscillator is described by the phonon creation and annihilation operators, which are given by [4]

b^† ≡r ω0

2 (q − i p

ω₀), b ≡r ω0

2 (q + i p

ω₀). (1.10)

The Hamiltonian is then expressed as

H = ω₀(b^†b + 1

2). (1.11)

In the following we will omit the ω₀/2 term, as we are mainly interested in energy differences.

A vibrational quantum is called a phonon, and has an energy equal to ω0

(mind that ¯h = 1 is taken), which is typically smaller than the energy involved in electronic excitation. The vibrational state of the molecule is denoted | νi, where ν represents the number of phonons. The corresponding energy equals νωo.

As opposed to the Frenkel exciton, a vibrational quantum shows bosonic behaviour. This means that an arbitrary number of vibrational quanta can be excited. The phonon creation and annihilation operators obey the bosonic commutation relation [b, b^†] = 1 [4,5].

The case of both electronic and vibrational excitation is called a “vibronic transition”. Thus, a vibronic quantum consists of a phonon-exciton combina- tion.

1.6 Exciton-phonon coupling

In the previous section, vibrational quanta are described using the harmonic oscillator model. In case of exciton-phonon (EP) coupling there is a difference between the oscillator potential wells of the electronic ground and excited states.

For the excited state, the potential well is shifted over a distance d relative to the ground state one, therefore having a minimum at q = d. The physics behind this coupling is the principle that electronic excitation changes the (electronic) structure of the molecule which, as a result, influences the oscillator potential well of the nuclei.

Apart from the translation, both the electronic ground and excited state potential wells are equal. This means that the vibrational normal mode ω0

remains unchanged upon electronic excitation.

(9)

EP coupling is conveniently described by the so-called “Huang-Rhys factor”

λ which is related to the displacement by [8]

λ ≡ dr ω₀

2 . (1.12)

One can easily identify λ²with the potential energy corresponding to the electronic ground state at the position q = d, see Eq. (1.9).

(10)

Chapter 2

Franck-Condon principle

2.1 Introduction

Assume that the aggregate is described by the wavefunction φ_total. According to the “Born-Oppenheimer approximation”, the electronic and vibrational parts of the total wavefunction can be separated [1],

φ_total= φ_electronic⊗ φvibrational. (2.1) The total dipole operator M, defined in Eq. (1.3), is assumed to be inde- pendent of the initial and final vibrational quantum states of the aggregate.

As a result, the transition dipole moment contains an inner product of these vibrational states, which corresponds to the overlap in the corresponding wavefunctions. This sort of inner products is called “Franck-Condon factors”. Since the wavefunctions involved are just the quantum harmonic oscillator eigenfunctions, these factors can be easily evaluated.

Franck-Condon factors play an important role in vibrational transitions. To illustrate their use, the application to a single-molecule absorption spectrum is treated.

2.2 Franck-Condon factors

The “Franck-Condon principle” states that an electronic transition takes place so rapidly that the nuclei involved are unable to instantaneously respond to the change in the electronic environment. Consequently, the vibrational state is unchanged. In case of non-zero EP coupling, the vibrational potential well is shifted upon electronic excitation, as is shown schematically in Fig. 2.1. In accordance with quantum mechanics, the vibrational state can then be expressed as a superposition of the vibrational states related to the changed potential well.

For simplicity, let’s assume the case of electronic excitation in a two-level electronic system. It is convenient to make an explicit distinction between vibrational quanta in the electronic ground state and vibronic quanta (vibrational quanta in the electronic excited state). The former is denoted by ν, whereas ˜ν corresponds to latter. Just after the electronic excitation, the vibrational state

(11)

Figure 2.1: Energy levels in case of nonzero EP coupling. Picture adapted from [11].

| νi is written

| νi =X

˜ ν

fνν˜ | ˜νi, (2.2)

where the factors fνν˜ ≡ h˜ν | νi represent the Franck-Condon factors. Both | νi and | ˜νi are bases which describe the vibrational state of the system.

The normalized quantum harmonic oscillator eigenfunctions corresponding to the electronic ground state potential are given by [4]

φn(q) = AnHn(q)e^−ω⁰^q² A_n = (2ⁿn!√

π)^−1/2. (2.3)

H_n(q) represents the nth-order “Hermite Polynomial”, of which examples are tabulated elsewhere [4]. Consequently, the Franck-Condon factors are given by [14]

f˜νν = Z ∞

−∞

φν˜(q − d)^∗φν(q)dq

= rν!˜

ν!e^−λ²^/2λ^ν−˜^νL^ν−˜_ν_˜ ^ν(λ²), (2.4) where λ is the Huang-Rhys factor given by Eq. (1.12), and L^ν−˜_˜_ν ^ν(λ²) denotes the “Associated Laguerre Polynomial” [4].

2.3 Single-molecule absorption spectrum

Franck-Condon factors completely determine the absorption spectrum of a single molecule, as will be shown in this section. The molecule is treated in a two-level electronic model, where the electronic ground and excited states are denoted respectively | gi and | ei, as outlined in Section 1.3. As a result of the Born- Oppenheimer approximation, the state vectors can be written as products of

(12)

the electronic and vibrational wavefunctions,

| g, νi ≡ | gi⊗ | νi

| e, ˜νi ≡ | ei⊗ | ˜νi. (2.5)

The molecule is assumed to initially reside in both the electronic and vibrational ground state | g, 0i. This condition is satisfied if kT ω₀ [10]. Since we are interested in energy differences only, we are free to identify the zero of energy with the ground state. ω₀₋₀denotes the energy corresponding to a transition from the ground state to the excited state | e, ˜0i, in which no phonons are present. The 0 − 0 subscript refers to the absence of vibrational quanta in both states.

Interaction takes place between the molecule and an external electromagnetic field, whose frequency is tuned in the range of ω0−0. As a consequence, the only possible transitions are those in which an electronic excitation is involved. To simplify things a bit, the field is polarized parallel to the molecule’s dipole. The resulting absorption spectrum follows from Eq. (1.2), identifying | ai ≡| g, 0i and | bi ≡| e, ˜νi:

IA(ω) =

∞

X

˜ ν=0

| he, ˜ν | µ | g, 0i |²δ(ω0−0+ ˜νω0− ω)

= | µeg |²

∞

X

˜ ν=0

| f˜ν,0|²δ(ω0−0+ ˜νω0− ω), (2.6)

where µ_eg ≡ he | µ | gi.

It is clear that the transition dipole moment is completely determined by the set of Franck-Condon factors for which ν = 0. From Eq. (2.4) it follows that

| fν,0˜ |²= λ^2˜^νe^−λ²

˜

ν! . (2.7)

so that the final expression for the absorption spectrum is IA(ω) =| µeg |²e^−λ²

∞

X

˜ ν=0

λ^2˜^ν

˜

ν! δ(ω0−0+ ˜νω0− ω). (2.8) Figs. 2.2(a)-2.2(d) demonstrate the single-molecule absorption spectrum, for the cases of λ²= 1.1, 2.2, 4.4 and 8.8. To obtain a nice spectrum, in Eq. (2.8) the delta function is substituted by a Gaussian line shape function W (ω^(j)− ω), where

W (ω) = 1

√πσe^−ω²^/σ², (2.9)

with a line width σ = 0.4ω0. Furthermore, the summation in Eq. (2.8) is carried out up to 21 phonons.

2.4 Discussion and conclusion

Figs. 2.2(a)-2.2(d) show a series of peaks, starting from energy ω₀₋₀, with interpeak distances of ω0. The peak heights appear to follow a Poisson distribution as a function over the energy, where the maximum of energy is reached roughly at peak number λ².

(13)

(a) λ²= 1.1 (b) λ²= 2.2

(c) λ²= 4.4 (d) λ²= 8.8

Figure 2.2: Single-molecule absorption spectra for four different EP couplings.

The energy is given in units of (ω − ω₀₋₀)/ω₀.

Obviously, the peak at energy ω0−0 corresponds to the bare exciton without any phonons. Similarly, the energy peak at energy ω0−0+ kω0 corresponds to the vibronic state | e, ˜ki. From Eq. (2.8) it can be checked that the series of peaks indeed follows a Poisson distribution.

(14)

Chapter 3

The Lang-Firsov transformation

3.1 Introduction

In the previous Chapter an expression for the absorption spectrum was derived for the case of a single molecule in which both vibrational excitations and an exciton, as well as their coupling, play a role. In order to succeed, the Franck- Condon principle was used. The same case can be solved using a completely different method. The starting point of this method is the “Holstein Hamilto- nian”, which is diagonalized using the “Lang-Firsov” transformation. Without even taking note of the Franck-Condon factors, this transformation allows us to derive an absorption spectrum expression similar to Eq. (2.8).

In other literature, the Lang-Firsov transformation is sometimes referred to as “polaron transformation”.

3.2 Holstein Hamiltonian for a single molecule

The Hamiltonian follows from the electronic energy Eq. (1.6) and vibrational energy Eq. (1.9),

H =| gihg | {1 2p²+1

2ω₀²q²}+ | eihe | {1 2p²+1

2ω₀²(q − d)²}

+ | eihe | {Ee− Eg}. (3.1) In this expression d represents the shift of the oscillator potential well caused by EP interaction. Using the Huang-Rhys factor and the phonon annihilation and creation operators, respectively Eq. (1.12) and Eq. (1.10), one obtains the following form of the single-molecule Holstein Hamiltonian:

H = (ω0−0+ λ²ω0)c^†c + λω0(b^†+ b)c^†c + ω0b^†b, (3.2) where ω0−0 = Ee and Eg is set to zero. (Recall that the zero phonon energy ω₀/2 is omitted.)

(15)

3.3 The transformation

For an arbitrary operator A, the transformed operator A is written as [5]

A ≡ e^SAe^−S, (3.3)

where S ≡ c^†cλ(b^†− b) ≡ c^†cF . Two properties turn out to be very useful in the further analysis. Firstly, for any two arbitrary operators A and B, we have A · B = A · B. Secondly, a transformed operator can be expressed as an expansion:

A = A + [S, A] +1

2[S, [S, A]] + ... (3.4) For the electronic annihilation operator we have

[S, c] = λc^†c(b^†− b)c − λcc^†c(b^†− b)

= 0 − λc(1 − cc^†)(b^†− b)

= −cF, (3.5)

where use was made of the fermionic nature of the operator concerned. Anal- ogously, [S, [S, c]] = cF², and as the succeeding higher order terms follow the exponential Taylor series, it follows that

c = e^−Fc ≡ Xc,

c^† = c^†e^F ≡ c^†X^†, (3.6)

where X ≡ exp[−λ(b^†− b)]. Note that the product of an electronic creation and annihilation operator is invariant under the transformation, i.e. c^†c = c^†c.

Similarly, the transformed vibrational annihilation operator is derived, where use is made of the bosonic commutation relation:

[S, b] = λc^†c(b^†− b)b − λbc^†c(b^†− b)

= λc^†c([b^†, b] − bb + bb)

= −λc^†c, (3.7)

[S, [S, b]] = 0, and as a result all higher order terms in the expansion vanish.

Consequently,

b = b − λc^†c,

b^† = b^†− λc^†c. (3.8)

Now that the transformed creation and annihilation operators are derived, one can easily transform the Hamiltonian as a whole:

H = e^S{(ω0−0+ λ²ω0)c^†c + λω0(b^†+ b)c^†c + ω0b^†b}e^−S

= (ω₀₋₀+ λ²ω₀+ λω₀(b^†+ b))c^†c + ω₀b^†b

= (ω0−0+ λ²ω0+ λω0b^†+ λω0b − 2λ²ω0c^†c)c^†c +ω0b^†b − λb^†ω0c^†c − λbω0c^†c + λ²ω0c^†cc^†c

= ω₀₋₀c^†c + ω₀b^†b. (3.9)

The result is a diagonalized Hamiltonian in which the electronic and vibrational parts are completely decoupled. Although this Hamiltonian takes up a rather simple form, the derivation of the absorption spectrum is not so straightforward, since the transition dipole moment must be transformed as well.

(16)

3.4 Green’s function

In obtaining the single-molecule absorption spectrum, again our point of departure is Fermi’s Golden Rule, Eq. (1.2),

IA(ω) =X

a,b

P (a) | hb | µ | ai |²δ(ωba− ω), (3.10)

where ωba≡ Eb− Ea and µ is assumed to be parallel to the field polarization.

The delta function can be written explicitly as [8]

δ(ωba− ω) = 1 2π

Z ∞

−∞

dte^−i(ω−ω^ba^)t, (3.11)

so that

IA(ω) = 1 2π

Z ∞

−∞

dte^−iωtX

a,b

1

Ze^−βEâha | µ | bihb | eîE^b^tµe^−iEâ^t| ai

= 1

2π Z ∞

−∞

dte^−iωthµ(0)µ(t)i. (3.12)

According to the “Heisenberg notation” µ(t) ≡ e^iHtµe^−iHt. Note that Eq. (1.1) is used. Beside that, the “dynamics correlation function” is introduced, being the trace over the matrix product e^−βHµ(0)µ(t),

hµ(0)µ(t)i = 1

ZTr(e^−βHµ(0)µ(t)). (3.13) The dipole operator is related to the electronic creation and annihilation operators by the expression µ(t) = µ(c(t) + c^†(t)). Combined with Eq. (3.13), this leads to four terms, of which two terms cancel due to the fact that in a two-level system an exciton cannot be created or annihilated twice. As a result hµ(0)µ(t)i = µ²hc^†(0)c(t)i + µ²hc(0)c^†(t)i, (3.14) A closer look at the origin of the dynamics correlation function shows that the first term contains the factor ha | c^†, which we assume to be zero at this point (this is indeed the case, as will be shown later on). We conclude that also this term cancels.

It turns out to be convenient to rewrite the second term in the time-ordered form. Use is made of the step function θ, which obeys θ(x) = 1 if x > 0, θ(x) = 1/2 if x = 0 and θ(x) = 0 elsewhere. One can always multiply the expression by (θ(t) + θ(−t)) as it equals 1 for all t:

hµ(0)µ(t)i = µ²hc(0)c^†(t)iθ(−t) + µ²hc(0)c^†(t)iθ(t). (3.15) The cyclic properties of the trace are used to reformulate both terms:

Tr(e^−βHce^iHtc^†e^−iHt) = Tr(e^−βHe^−iHtce^iHtc^†). (3.16) The second term is modified even further by using the following:

Tr(e^−βHe^−iHtce^iHtc^†) = Tr(ce^−iHtc^†e^iHte^−βH)^†

= Tr(e^−βHe^iHtce^−iHtc^†)^†. (3.17)

(17)

Accordingly, the final form of the dynamics correlation function is expressed as hµ(0)µ(t)i = µ²hc(−t)c^†(0)iθ(−t) + µ²hc(t)c^†(0)i^†θ(t). (3.18) Substituted back into the absorption spectrum expression, we now have

I_A(ω) = µ² 2π

Z ∞ 0

dte^−iωthc(t)c^†(0)i^†+µ² 2π

Z 0

−∞

dte^−iωthc(−t)c^†(0)i

= µ² 2π

Z ∞ 0

dt{e^−iωthc(t)c^†(0)i^†+ e^iωthc(t)c^†(0)i}

= µ² πIm{

Z ∞ 0

dte^iωtG(t)}. (3.19)

G(t) is known as the “Green’s function”, and it is constituted as

G(t) = −ihc(t)c^†(0)i. (3.20)

The next step is to apply the Lang-Firsov transformation to this Green’s function.

3.5 The transformed Green’s function

By multiplying the Green’s function by e^−Se^S = 1 and using the cyclic properties of the trace, the transformed Green’s function is obtained,

G(t) = −i1

ZTr(e^−βHe^iHtce^−iHtc^†e^−Se^S)

= −i1

ZTr(e^Se^−βHe^iHtce^−iHtc^†e^−S)

−i1

ZTr(e^−βHe^iHtXce^−iHtc^†X^†), (3.21) where c = Xc and c^† = c^†X is written explicitly for sake of convenience. Next we point out the following set of equations,

e^iHtXce^−iHt= e^−itω⁰⁻⁰cX(t) (3.22) X(t) = exp[−λ(b^†e^iω⁰^t− be^−iω⁰^t)], (3.23) which allows us to rearrange the Green’s function and separate the vibrational and electronic parts,

G(t) = −ie^−itω⁰⁻⁰1

ZTr(e^−βHce^iHtXe^−iHtc^†X^†)

= −ie^−it(ω⁰⁻⁰⁾ 1 Zel

Tr[e^−H^elcc^†] 1 Zph

Tr[e^−H^phX(t)X^†(0)], (3.24) where

Hel≡ c^†c(ω0−0), Hph≡ ω0b^†b. (3.25) The electronic part_Z¹

elTr[e^−H^elcc^†] equals 1 [5]. The next step is to consider the vibrational part, which is denoted F (t),

F (t) ≡ (1 − e^−βω⁰)

∞

X

ν=0

e^−βνω⁰hν | X(t)X(0) | νi (3.26)

X(t) = e^−λ(b^†^e^iω0t^−be^−iω0t⁾, (3.27)

(18)

where the vibrational partition function is written explicitly 1

Z_ph = (

∞

X

ν=0

e^−βνω⁰)⁻¹= 1 − e^−βω⁰. (3.28) The expression for F (t) can be simplified by acting with the creation and annihilation operators on the bra and the ket states respectively. To this end, the part X(t)X(0) is reformulated using “Feynman’s theorem on the disentangling of operators”. This theorem states that for two operators A and B, which both commute with their commutator C = [A, B], the following holds [5]:

e^A+B = e^Ae^Be^−1/2[A,B]. (3.29) Leaving the proof aside, we use this theorem by making the following choice of variables:

A ≡ −λb^†e^iω⁰^t (3.30)

B ≡ λbe^−iω⁰^t. (3.31)

From X(t) = e^A+B, it follows that

X(t)X(0) = e^−λ²e^−λb^†^e^iω0te^λbe^−iω0te^λb^†e^−λb. (3.32) The middle annihilation and creation operators in this expression can be inter- changed, using the basic commutation rules. By doing so, one gets for F (t):

F (t) = (1 − e^−βω⁰)e^−λ²^(1−e^−iω0t⁾

∞

X

ν=0

e^−βνω⁰hν | e^λb^†^(1−e^iω0t⁾e^−λb(1−e^−iω0t⁾| νi.

(3.33) The vibrational annihilation operator acts on the vibrational ket state in the following way [5]:

b | νi =√

ν | ν − 1i. (3.34)

A similar expression holds for the creation operator acting on the bra state.

This helps us to reformulate the expression for F (t), setting u ≡ λ(1 − e^−iω⁰^t):

hν | e^u^∗^b^†e^−ub| νi =

ν

X

l=0

(−|u|²)^l (l!)²

ν!

(ν − l)!

= Lν(|u|²), (3.35)

where L_ν is the Laguerre polynomial of order ν [4].

Next, the following identity is used [5]:

(1 − z)

∞

X

ν=1

Lν(|u|²)z^ν = e^|u|²^z/(z−1). (3.36) Identifying z ≡ e^−βω⁰, this equation for the Laguarre polynomials leads to the following formula for the vibrational part:

F (t) = e^−Φ(t) (3.37)

Φ(t) ≡ λ²[Y (1 − e^iω⁰^t) + (Y + 1)(1 − e^−iω⁰^t)] (3.38)

Y ≡ 1

e^βω⁰− 1. (3.39)

As a result, the transformed Green’s function takes up a rather simple form:

G(t) = −ie^−itω⁰⁻⁰e^−Φ(t). (3.40)

(19)

3.6 Absorption spectrum

As in Section 2.3, the temperature is assumed to satisfy kT ω0 (so that the molecule starts in both the electronic and vibrational ground state). In this limit, β → ∞ and as a consequence Y → 0, which leads to a simplification of Φ, Φ(t) = λ²(1 − e^−iω⁰^t). (3.41) Using this, and substituting the transformed Green’s function into Eq. (3.19), we now have for the absorption spectrum expression

I_A(ω) = µ² π Im[−i

Z ∞ 0

dte^iωtexp(−itω₀₋₀− λ²+ λ²e^−iω⁰^t))]

= 1

πRe Z ∞

0

dte^−λ²exp(iωt − itω0−0)exp(λ²e^−iω⁰^t). (3.42) Using the power series expansion of exp(λ²e^−iω⁰^t), one obtains

I_A(ω) = µ²e^−λ²

∞

X

ν=0

λ^2ν

ν! δ(ω₀₋₀+ νω₀− ω). (3.43)

3.7 Results

The Lang-Firsov transformation leads to a single-molecule absorption spectrum equivalent to Eq. (2.8). Of course, that is not so surprising, since the transformation should not change the underlying physics.

This method may appear to be somewhat excessive, especially in the light of the much simpler method outlined in Chapter 2. Nevertheless, the advantage of this approach lies in the point of departure, the Holstein Hamiltonian, which can be easily extended for the case of molecular aggregates.

3.8 Aggregate transformation

The Lang-Firsov transformation nicely diagonalizes the one-molecule Holstein Hamiltonian. In the case of a molecular aggregate, the situation is troubled by the dipole-dipole coupling, which also contributes to off-diagonal Hamilto- nian elements. Scherer and Fischer [2] have dealt with this case analytically.

Their approach is to apply a polaron transformation similar to the Lang-Firsov transformation, after which the Hamiltonian is formulated into block-diagonal form.

It turns out to be rather tedious to treat the aggregate case in this way.

Much simpler is the so-called “two-particle approximation”, that is treated in the next Chapter.

(20)

Chapter 4

Two-particle approximation

4.1 Introduction

Instead of simplifying the Holstein Hamiltonian using transformations, the aggregate absorption spectrum can be calculated in a brute-force way. In this case the untransformed Hamiltonian is diagonalized numerically, which leads to the eigenvectors and eigenenergies.

This numerical methods suffers from the fact that for a growing number of molecules the corresponding Hilbert space increases very rapidly, especially when a large number of phonons is taken into account, which is necessary for a large EP coupling. In case of an N-molecule aggregate, the number of states equals [10]

N (N + ν_max)!

N !νmax! , (4.1)

where νmaxdenotes the maximum of the number of phonons (that is, the maximum of ν + ˜ν). However, the Hilbert space dimension can be limited using an approximation.

The “two-particle approximation” (TPA) is a convenient way of dealing with the Holstein Hamiltonian numerically. This Chapter gives an outline of this approach. In the next Chapter, several simple examples are given concerning the application of the TPA.

4.2 Holstein Hamiltonian for a molecular aggre- gate

Consider an aggregate consisting of N identical molecules. The corresponding Holstein Hamiltonian follows from the single-molecule one, Eq. (3.2), combined with the one-exciton states given by Eq. (1.5) and the excitonic interaction term given by Eq. (1.7),

H = (ω0−0+ λ²ω0) | nihn | +λω0

X

n

(b^†_n+ bn) | nihn | +ω0

X

n

b^†_nbn

+X

m,n

J_mn| mihn | . (4.2)

(21)

Note that the summations over n and m are taken over (1, 2, ..., N ). In the following this will always be the case, unless explicitly noted otherwise. b^†_n and bn denote the vibrational creation and annihilation operators, Eq. (1.10), acting on molecule n. These operators correspond to the harmonic oscillator potential well of the electronic ground state. Since EP coupling causes a shift in the electronic excited state potential well, it is convenient to introduce shifted phonon operators [14],

˜b^†_n= b^†_n+ λ (4.3)

˜bn = bn+ λ, (4.4)

which create and annihilate a phonon in the electronic excited state potential.

Using these operators, the Hamiltonian is written

H =X

n

ω0−0| nihn | +ω0

X

n

˜b^†_n˜bn| nihn | +ω0

X

n

b^†_nbn(1− | nihn |)

+X

m,n

J_mn| mihn | . (4.5)

4.3 State vectors in TPA representation

At this point, we assume that the aggregate starts off in the collective electronic and vibrational ground state

| gi ≡Y

n

| gn, 0i. (4.6)

TPA limits the space of states up to two-particle states. These states are a special case of k-particle states, which involves 1 vibronic state and k − 1 vibrational states. This means that in the aggregate 1 molecule is both electronically and vibrationally excited, whereas k − 1 molecules are vibrationally excited while residing in the electronic ground state.

One-particle states (k = 1) actually form a subset of two-particle states, and are denoted as

| n, ˜νi ≡| e_n, ˜νi ⊗ Y

m6=n

| gm, 0i. (4.7)

Two-particle states (k = 2) are defined as

| n, ˜ν; m, νi ≡| e_n, ˜νi⊗ | g_m, νi ⊗ Y

l6=m,n

| gl, 0i, (4.8)

where necessarily ν ≥ 1 to keep these states different from one-particle states.

TPA strongly reduces the number of states compared to the complete basis set (CBS). If the maximum number of phonons is taken to be νmax, the Hilbert space dimension is [10]

N (ν_max+ 1)[(N − 1)νmax

2 + 1], (4.9)

which is small compared to CBS, see Eq. (4.1). As an example, in case of νmax = 4, the number of TPA states drops well below the number of CBS

(22)

N 1 2 3 4 5 6 7 8 9 10 CBS 5 30 105 280 630 1260 2310 3960 6435 10010

TPA 5 30 75 140 225 330 455 600 765 950

Table 4.1: Number of CBS and TPA states for N molecules, in the case of ν_max= 4.

states for N > 2, as can be seen in Table 4.1. Note that CBS corresponds to (ν_max+ 1)-particle states.

The TPA eigenstates are defined as

| Ψ^(j)i ≡X

n ν_max

X

˜ ν=0

c^(j)_n,˜_ν| n, ˜νi +X

n

X

m6=n ν_max−1

X

˜ ν=0

ν_max−˜ν

X

ν=1

c^(j)_n,˜_ν,m,ν | n, ˜ν; m, νi, (4.10) where proper normalization requires that

X

n νmax

X

˜ ν=0

| c^(j)_n,˜_ν |²+X

n

X

m6=n νmax−1

X

˜ ν=0

νmax−˜ν

X

ν=1

| c^(j)_n,˜_ν,m,ν |²= 1. (4.11)

To express the Holstein Hamiltonian in the TPA representation, the operation of the Hamiltonian on one- and two-particle states is evaluated, to start with the former:

X

n

ω₀₋₀| nihn | n⁰, ˜νi = ω₀₋₀| n⁰, ˜νi (4.12)

ω0

X

n

˜b^†_n˜bn| nihn | n⁰, ˜νi = ˜νω0| n⁰, ˜νi (4.13)

ω0

X

n

b^†_nbn(1− | nihn |) | n⁰, ˜νi = 0 (4.14)

X

m,n

J_mn| mihn | n⁰, ˜νi =X

m

J_mn⁰ | m, ν = 0; n⁰, ˜νi.

The last equation needs a closer look. As explained in Section 1.4, excitonic interaction causes the creation of an electronic excitation on molecule m. Ac- cording to the Born-Oppenheimer approximation, this molecule stays in the vibrational ground state ν = 0 which can be expressed as a superposition of the vibrational states ˜ν using the Franck-Condon factors f_˜_ν0, recall Eq. (2.2). At the same time, an exciton is annihilated on molecule n, which then finds itself in a superposition of the vibrational states ν⁰. As a result

X

m,n

J_mn| mihn | n⁰, ˜νi =X

m νmax

X

˜ ν⁰=0

νmax−˜ν⁰

X

ν⁰=1

J_mn⁰ | m, ˜ν⁰; n⁰, ν⁰if˜ν⁰0f_νν_˜ ⁰

+X

m ν_max

X

˜ ν⁰=0

Jmn⁰ | m, ˜ν⁰ifν˜⁰0fν0˜ . (4.15)

Note that the summation is limited to one- and two-particle states.

(23)

The operation of the Hamiltonian on two-particle states is as follows:

X

n

ω0−0| nihn | n⁰, ˜ν; m⁰, νi = ω0−0| n⁰, ˜ν; m⁰, νi (4.16)

ω₀X

n

˜b^†_n˜b_n| nihn | n⁰, ˜ν; m⁰, νi = ˜νω₀| n⁰, ˜ν; m⁰, νi (4.17)

ω0

X

n

b^†_nbn(1− | nihn |) | n⁰, ˜ν; m⁰, νi = νω0 (4.18)

X

m,n

Jmn| mihn | n⁰, ˜ν; m⁰, νi = X

m6=m⁰

Jmn⁰ | m, 0; n⁰, ˜ν; m⁰, νi +J_m⁰_n⁰ | m⁰, ν; n⁰, ˜νi.

To put the last equation into words, an exciton is created on molecule m, which possibly is identical to molecule m⁰. Simultaneously, an exciton is annihilated on molecule n⁰. Again the vibrational states are adapted to match the electronic states, and the summations are restricted to one- and two-particle states:

X

m,n

J_mn| mihn | n⁰, ˜ν; m⁰, νi = X

m6=m⁰ νmax−ν

X

˜ ν⁰=0

J_mn⁰ | m, ˜ν⁰; m⁰, νif_˜_ν⁰₀f_ν0_˜

+

ν_max

X

˜ ν⁰=0

ν_max−˜ν⁰

X

ν⁰=1

Jm⁰n⁰ | m⁰, ˜ν⁰; n⁰, ν⁰ifν˜⁰νf˜νν⁰

+

ν_max

X

˜ ν⁰=0

Jm⁰n⁰ | m⁰, ˜ν⁰if˜ν⁰νfν0˜ . (4.19)

4.4 Hamiltonian in TPA representation

The Hamiltonian can now be expressed in the TPA representation. It is convenient to write the total Hamiltonian [14]

H = H^1p+ H^2p+ H^1p−2p, (4.20) where H^1p couples one-particle states to one-particle states, H^2p couples two- particle states to two-particle states and accordingly H^1p−2pcouples one-particle states to two-particle states. The expression of these parts follow from Eq.’s (4.12) - (4.19).

H^1p=X

n ν_max

X

˜ ν=0

X

n⁰ ν_max

X

˜ ν⁰=0

((ω₀₋₀+ ˜νω₀)δ_nn⁰δ_{ν ˜}_˜_ν⁰+ J_n⁰_nf_˜_ν0f_ν_˜⁰₀) | n⁰, ˜ν⁰ihn, ˜ν | (4.21)

H^2p = X

n

X

m ν_max

X

˜ ν=0

ν_max−˜ν

X

ν=1

X

n⁰

X

m⁰ ν_max

X

˜ ν⁰=0

ν_max−˜ν⁰

X

ν⁰=1

((ω0−0

+(˜ν + ν)ω0)δnn⁰δmm⁰δν ˜˜ν⁰δνν⁰ + Jn⁰nδn⁰mδmn⁰f˜ν⁰νfνν˜ ⁰

+Jn⁰nδmm⁰δνν⁰fν˜⁰0f˜ν0) | n⁰, ˜ν⁰; m⁰ν⁰ihn, ˜ν; m, ν | (4.22)

(24)

H^1p−2p = X

n νmax

X

˜ ν=0

X

n⁰

X

m⁰ νmax

X

˜ ν⁰=0

νmax−˜ν⁰

X

ν⁰=1

J_n⁰_nδ_nm⁰f_˜_ν⁰₀f_˜_νν⁰ | n⁰, ˜ν⁰; m⁰, ν⁰ihn, ˜ν |

+X

n

X

m ν_max

X

˜ ν=0

ν_max−˜ν

X

ν=1

X

n⁰ ν_max

X

˜ ν⁰=0

Jn⁰nδn⁰mfν0˜ f˜ν⁰ν | n⁰, ˜ν⁰ihn, ˜ν; m, νi.

(4.23) This Hamiltonian H is diagonalized numerically, to obtain the eigenenergies ω^(j), as well as the coefficients c^(j)_n,˜_ν and c^(j)_n,˜_ν,m,ν that determine the eigenstates defined in Eq. (4.10).

4.5 Absorption spectrum

Recalling Eq. (1.2), the absorption spectrum is expressed as I_A(ω) =X

j

| hΨ^(j)| M | gi |²δ(ω^(j)− ω), (4.24)

where again the external field is assumed to be polarized parallel to the aggregate’s total dipole moment.

According to the Franck-Condon principle, the transition dipole moment contains vibrational overlap factors between the ground state g and the excited state Ψ^(j). Since two-particle states include a vibrational excitation in the electronic ground state, these give rise to a factor fν0 = 0. As a consequence, only the one-particle coefficients contribute to the absorption spectrum. Using Eq. (1.3),

hΨ^(j)| M | gi = X

n⁰

X

n ν_max

X

˜ ν=0

c^(j)_n,˜_νhn, ˜ν | µn⁰ | gi

= X

n ν_max

X

˜ ν=0

c^(j)_n,˜_νµ, (4.25)

where hn, ˜ν | µn | gi = µ is taken for all molecules. The spectral function then becomes

I_A(ω) =| µ |²X

j

|X

n νmax

X

˜ ν=0

c^(j)_n,˜_νf_ν0_˜² |²δ(ω^(j)− ω). (4.26)

(25)

Chapter 5

Applications of TPA

5.1 Introduction

The two-particle approximation strongly reduces the Hamiltonian’s dimensions, making this method suitable for numerical calculation of the absorption spectrum. TPA is applied to the case of a pinwheel aggregate by Spano [10,13].

These calculations are redone, and the outcome is compared to his results. Sub- sequently, a comparison is made with the “one-particle approximation”, also for the case of a pinwheel, varying firstly the excitonic coupling strength with constant EP coupling, and then varying the EP coupling strength with constant excitonic coupling.

5.2 Pinwheel aggregate absorption spectrum

A pinwheel aggregate consists of 4 molecules that are oriented in a specific way, as depicted in Fig. 5.1. Contrary to what might appear at first sight, the molecules are not located at the corners of a square. The setup rather has the shape of a rhomb, where the distances between opposing molecules varies over the two different pairs. In the figure, the arrows denote the transition dipole components parallel to the plane of the drawing. Moreover, there is a dipole component perpendicular to the plane of the drawing, which is equal for all molecules. In obtaining the absorption spectrum, the electromagnetic field is polarized in this perpendicular direction.

The phonon energy is taken ω0 = 1400cm⁻¹. In the following, all energies will be expressed in terms of ω₀, whereas the zero of energy is set equal to ω₀₋₀. The nearest-neighbour excitonic interaction is denoted J₀. Due to the pinwheel structure, there are two different next-nearest-neighbour excitonic interaction terms. They are denoted J₁and J₁⁰, dependent on the orientation.

As in Section 2.3, in the spectral expression Eq. (4.26), the delta function δ(ω^(j)− ω) is replaced by a Gaussian line shape function W (ω^(j)− ω) having a line width of σ = 0.4ω0, see Eq. (2.9).

A Huang-Rhys factor is used such that λ²= 1.1, for which it suffices to set the maximum number of phonons taken into account to 4 [9]. Two different sets of excitonic couplings are evaluated. In the first case J0= 0.59ω0, J1= 0.34ω0

and J₁⁰ = 0.32ω0. Fig. 5.2 demonstrates the outcome obtained by Spano [10] (a

(26)

Figure 5.1: Schematic picture of a pinwheel aggregate. The arrows denote projections of the transition dipoles on the plane of the drawing. Nearest- neighbour excitonic couplings J0 are equal for all molecules, whereas next- nearest-neighbour couplings J₁ and J₁⁰ are different along axis a and axis b.

Picture taken from [10].

Figure 5.2: Calculated absorption spectra for a pinwheel using TPA. The solid curve represents the result by Spano [10], which is taken by converting the article picture to numerical data. The dashed curve shows our result. The energy is given in units of (ω − ω₀₋₀)/ω₀. The excitonic couplings are J₀ = 0.59ω₀, J₁ = 0.34ω₀ and J₁⁰ = 0.32ω₀. The EP coupling is taken λ² = 1.1. In our comparison, we assume the gas-to-crystal transition energy shift D to be zero.

program is used to convert the curve picture into numerical data). In the same figure, our result for the same set of parameters is shown.

The second set of excitonic couplings consists of J0 = 0.62ω0, J1= 0.28ω0

and J₁⁰ = 0.48ω0. Both the outcome by Spano [13] and our outcome are depicted in Fig. 5.3.

5.3 Comparison between TPA and OPA

The Hamiltonian Hilbert space can be reduced even more by applying the “one- particle approximation” (OPA), in which case the space is spanned by one- particle states only, recall Eq. (4.7). Firstly, OPA is applied to the case of the pinwheel, using the same parameters as was the case in Section 5.2, but for different excitonic interactions. The latter is set J₀= 0.62ω₀· β, J1= 0.28ω₀· β and J₁⁰ = 0.48ω₀· β, varying β. The outcome is compared to the TPA result.

(27)

Figure 5.3: Calculated absorption spectra for a pinwheel using TPA, by Spano [13] combined with our result. The energy is given in units of (ω − ω0−0)/ω0. The excitonic couplings are J0 = 0.62ω0, J1 = 0.28ω0 and J₁⁰ = 0.48ω0. The EP coupling is taken λ²= 1.1.

Figs. 5.4(a)-5.4(d) demonstrate the calculated absorption spectra for β = 1, 0.75, 0.50 and 0.25.

The same comparison is made, but this time varying the Hyang-Rhys factor, setting β = 1 constant. Figs. 5.5(a)-5.5(d) demonstrate the calculated absorption spectra for λ² = 1.1, 2.2, 3.3 and 4.4. In order to obtain convergent data for TPA, the maximum number of vibrational quanta is set νmax= 25.

5.4 Discussion and conclusion

In our comparison with the results obtained by Spano, we find a good agreement for both sets of excitonic couplings. In Fig. 5.3 both curves appear to coincide completely. In Fig. 5.2 one curve seems to slightly deviate from the other. This might be caused by inaccuracy in the conversion process from Spano’s curve picture to numerical data. We can strongly presume that both our and Spano’s method of applying TPA do correspond.

In our first comparison between TPA and OPA, a significant difference can be observed in case of β = 1. Upon decrease of β, both curves will converge to one another, as can be seen in Figs. 5.4(a)-5.4(d). A convergence can also be observed when the Huang-Rhys factor is increased, see Figs. 5.5(a)-5.5(d).

However, a remark must be made at this point. When β is decreased by a factor of 4, the convergence is much faster than is the case when λ² is increased by a factor of 4.

β = 1 corresponds to the so-called “intermediate coupling regime”, in which the total excitonic coupling has approximately the magnitude of EP coupling, J ≈ λ²ω0. In this case, an exciton can easily be transferred between neighbour- ing molecules, resulting in both a vibronic and a vibrational excitation. This is not described in OPA, so that this approximation doesn’t suffice. This explains the difference between the OPA and TPA curves in Fig. 5.4(a) and Fig. 5.5(a).

Small values of β corresponds to the “weak coupling” regime, in which the total excitonic coupling is small compared to EP coupling. When β drops to

(28)

(a) β = 1 (b) β = 0.75

(c) β = 0.5 (d) β = 0.25

Figure 5.4: Calculated absorption spectra for a pinwheel, using OPA (solid curve) and TPA (dashed curve), for various values of β. The energy is given in units of (ω − ω0−0)/ω0. The EP coupling is taken λ²= 1.1.

zero, excitonic interaction does no longer contribute. In the absence of excitonic coupling, the aggregate behaves the same as a single molecule. The corresponding absorption spectrum (which is discussed in Section 2.4, recall Fig. 2.2(a)) is completely described in OPA. Even for β = 0.25 we already observe a good agreement between the OPA and TPA curves, as can be seen in Fig. 5.4(d).

Upon increase of the Huang-Rhys factor, one would also expect to reach the “weak coupling regime”. As it turns out, the convergence is slower this time. A possible explanation for this, is the fact that the coupling regimes don’t only depend on the ratio of electronic coupling to EP coupling, but also on the absolute values of these couplings (in terms of ω0) [7].

In conclusion, OPA breaks down upon increase of the ratio of electronic coupling to EP coupling. If the electronic coupling is increased even further, so that β > 1, TPA starts to break down too. In this coupling regime, there is need for an expansion of TPA to the “three-particle approximation”, details of which can be found elsewhere [12].

Excitonic and exciton-phonon interactions in molecular aggregates