University of Groningen Exciton localization and optical … · absorption spectrum. In contrast, to calculate the nonlinear optical features [14–18] it is necessary to also invoke

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Exciton localization and optical spectroscopy of linear and tubular J-aggregatesBloemsma, Erik André

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Exciton localization and optical

spectroscopy of linear and tubular

J-aggregates

Zernike Institute PhD thesis series 2013-15

ISSN: 1570-1530ISBN: 978-90-367-6277-9 (printed version)ISBN: 978-90-367-6278-6 (electronic version)

The work described in this thesis was performed at the Zernike Institute for Ad-vanced Materials and the Centre for Theoretical Physics of the RijksuniversiteitGroningen.

Cover by courtesy of Klaas Bernd Over

Printed by GrafiMedia, University of Groningen

Copyright c© 2013 Erik A. Bloemsma

RIJKSUNIVERSITEIT GRONINGEN

Exciton localization and opticalspectroscopy of linear and tubular

J-aggregates

Proefschrift

ter verkrijging van het doctoraat in de

Wiskunde en Natuurwetenschappen

aan de Rijksuniversiteit Groningen

op gezag van de

Rector Magnificus, dr. E. Sterken,

in het openbaar te verdedigen op

vrijdag 14 juni 2013om 14:30 uur

door

Erik Andre Bloemsma

geboren op 27 februari 1983te Leeuwarden

Promotor: Prof. dr. J. Knoester

Beoordelingscommissie: Prof. dr. ir. C. H. van der Wal

Prof. dr. R. van Grondelle

Prof. dr. P. Reineker

Contents

1 Introduction 1

1.1 The discovery of J-aggregates . . . . . . . . . . . . . . . . . . . . . . 11.2 Natural light-harvesting systems . . . . . . . . . . . . . . . . . . . . 31.3 Amphiphilic dyes as building blocks for artificial light harvesting . . 61.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Disorder-induced exciton localization and violation of optical se-

lection rules in supramolecular nanotubes 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Frenkel exciton model for cylindrical aggregates . . . . . . . . . . . . 14

2.2.1 Geometry and Hamiltonian . . . . . . . . . . . . . . . . . . . 142.2.2 Absorption spectra . . . . . . . . . . . . . . . . . . . . . . . . 152.2.3 Selection rules in the absence of disorder . . . . . . . . . . . . 16

2.3 Characterization of exciton states in disordered cylinders . . . . . . . 182.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Localization of optically dominant states . . . . . . . . . . . . 202.4.2 Breaking of the selection rules . . . . . . . . . . . . . . . . . 252.4.3 Absorption spectra . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Signature of anomalous exciton localization in the optical response

of tubular J-aggregates 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Model and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 Physical origin: self-consistent approach . . . . . . . . . . . . 393.3.2 Formal approach: the coherent potential approximation . . . 41

3.4 Relation to one-dimensional toy models . . . . . . . . . . . . . . . . 433.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

vi Contents

4 Excitons in self-assembled double-walled tubular aggregates: molec-

ular structure, optical spectra and inter-wall coherences 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Aggregate geometry . . . . . . . . . . . . . . . . . . . . . . . 534.2.2 Hamiltonian and absorption spectrum . . . . . . . . . . . . . 574.2.3 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.4 Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Numerical results and comparison to experiment . . . . . . . . . . . 614.3.1 Inner wall absorption spectrum: homogeneous limit . . . . . 614.3.2 Inner wall absorption spectrum: inhomogeneous limit . . . . 644.3.3 Full spectrum and effects of coupling . . . . . . . . . . . . . . 65

4.4 Summary and concluding remarks . . . . . . . . . . . . . . . . . . . 70

5 Vibronic effects and destruction of exciton coherence in optical

spectra of J-aggregates: a variational polaron transformation ap-

proach 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 775.2.2 Polaron transformations . . . . . . . . . . . . . . . . . . . . . 795.2.3 Linear absorption spectrum . . . . . . . . . . . . . . . . . . . 82

5.3 Zero-temperature absorption spectrum . . . . . . . . . . . . . . . . . 845.3.1 General expressions . . . . . . . . . . . . . . . . . . . . . . . 845.3.2 Molecular dimer . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . 885.4.1 Comparison with the two-particle approximation . . . . . . . 885.4.2 Comparison with related polaron transformations . . . . . . . 925.4.3 Thermal effects . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.5 Summary and concluding remarks . . . . . . . . . . . . . . . . . . . 96

6 Photon emission statistics and photon tracking in single-molecule

spectroscopy of molecular aggregates: dimers and trimers 99

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 1036.2.2 Generating function formalism and photon statistics . . . . . 1056.2.3 Photon tracking . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.3 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3.1 Homogeneous dimer . . . . . . . . . . . . . . . . . . . . . . . 1086.3.2 Inhomogeneous dimer . . . . . . . . . . . . . . . . . . . . . . 114

Contents vii

6.3.3 Linear homogeneous trimer . . . . . . . . . . . . . . . . . . . 1176.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.5 Appendix: eigenstates and Generalized Bloch Equations for the dimer123

Samenvatting 127

Dankwoord 133

Bibliography 135

Chapter 1

Introduction

1.1 The discovery of J-aggregates

In 1936, Jelley [1] and Scheibe [2] independently observed that the absorption spec-trum of the dye pseudo-isocyanine (PIC for short) in aqueous solution changeddrastically upon increasing the dye concentration. The broad absorption bandof PIC dyes at 525 nm disappeared and instead a new absorption band emergedaround 570 nm. The most pronounced characteristic of this new band is its nar-rowness, ranging from a few tens of cm−1s at low temperatures to several hundredcm−1s at room temperature. Both Jelley and Scheibe attributed these strongspectral changes to the formation of supramolecular structures where the dyes areheld together by non-covalent interactions, such as hydrogen bonds, electrostaticforces (van der Waals, Coulomb), hydrophobic interactions and π − π stacking.Nowadays, molecular aggregates with a narrow absorption band that is shifted tolonger wavelengths (redshift, bathochromic) compared to the monomer absorptionare generally termed J-aggregates (J stands for Jelley) or sometimes Scheibe ag-gregates. On the other hand, aggregates with absorption bands that are shiftedto shorter wavelengths (blueshift, hypsochromic) are commonly referred to as H-aggregates.

Soon after these initial discoveries, it was realized by Frank and Teller [3] thatthe spectral changes upon aggregation of the PIC molecules could be explainedby the formation of charge-neutral collective electronic excitations. The conceptof such delocalized quanta of excitation was already introduced in the early 1930sby Frenkel [4] and Peierls [5] in the context of molecular crystals. To explain thisidea, consider an aggregate of N identical molecules where one of the molecules israised to an excited state. In the absence of intermolecular interactions, we thus

2 Introduction

have an N -fold degenerate excited state which is the direct product of the excitedstate of one molecule and the ground states of all other molecules. We refer to suchexcitations as molecular excitons. The coupling of different molecules, mediated bythe interaction between their optical transition dipoles, lifts the N -fold degeneracyresulting in new excited states where the excitations are shared coherently by (orin other words, are delocalized over) the molecules; that is, we have fixed phaserelations between the wave functions of the different molecules. Such coherentexcitations are known as collective excitons.

In principle, we may distinguish several types of excitons: Frenkel excitons,Wannier-Mott excitons and charge transfer excitons. In the former situation, theground state hole and excited state electron are always located on the same moleculeowing to strong Coulomb interactions between them and a weak orbital overlapbetween (neighboring) molecules. Such Frenkel excitons are typically found inmaterials with a small dielectric constant like molecular aggregates and crystals. Incontrast, for materials with a large dielectric constant, like semiconductors, electricfield screening can strongly reduce the Coulomb interactions. As a result, theelectron and the hole are positioned on different molecules. If the typical electron-hole separation corresponds to a large number of lattice spacings, this excitationis usually referred to as a Wannier-Mott exciton [6, 7]. When the separation isat most a few lattice spacings, the excitation is usually called a charge transferexciton [8, 9].

The foundations for the theoretical description of collective Frenkel excitons inmolecular crystals and aggregates and the implications on their optical propertiesand excitation energy transport dynamics have been developed by Agranovich [10–12] and Davydov [13] in the 1960s. A typical assumption in these exciton modelsis to describe each molecule as a two-level system, consisting of a molecular groundstate and a single excited state (see Fig. 1.1.1). Although in reality moleculespossess many different electronic transitions, this approximation is justified as longas one of these transitions is optically dominant and the frequency of the usedlight is close to this transition. The strong resonant excitation transfer interactionsbetween the molecules will mix the molecular excited states, resulting in delocalizedFrenkel exciton states. The energetically lowest excited states of the aggregate arethe one-exciton states (schematically depicted in Fig. 1.1.1), named after the factthat these states (coherently) share a single excitation quantum. This one-excitonband is sufficient to describe the linear optical observables of the aggregate, like theabsorption spectrum. In contrast, to calculate the nonlinear optical features [14–18]it is necessary to also invoke higher excited states such as the two-exciton states,which share two excitation quanta.

Based on these Frenkel exciton models, a number of collective optical propertiesof molecular aggregates can be understood [19]. For example, the redshift of the

1.2 Natural light-harvesting systems 3

Figure 1.1.1: Schematic illustration of the Frenkel exciton model for a linear J-aggregate. Each molecule is a two-level system with a ground state and a singleexcited state. Irregularities in the environment result in variations of the transitionenergies ǫn (and transition dipoles µn) of the molecules. The excitation transferinteractions J between molecules lead to coherent Frenkel exciton states, whichmay be classified into bands according to the number of shared excitation quanta.For J-aggregates, the states that dominate the linear optical response occur atthe bottom of the one-exciton band, which explains the redshift of the absorptionband compared to the single-molecule spectrum. The fast radiative decay andnarrowness of the absorption band also reflect the coherent nature of the excitonstates. Figure originally taken from Ref. [96].

absorption band observed for the PIC J-aggregates discussed above arises from thefact that the entire oscillator strength (i.e., the sum of the squares of all moleculartransition dipoles) from the ground state to the one-exciton band is collected inonly a few exciton states that lie at the bottom of the one-exciton band. Thisproperty also explains the origin of the ultrafast radiative emission observed forJ-aggregates, which is significantly faster than the single-molecule spontaneousemission [20,21]. This phenomenon is known as cooperative spontaneous emissionor exciton superradiance. In contrast, for H-aggregates the exciton states thatcontain the oscillator strength occur at the top of the exciton band, resulting in theblueshift of the absorption band compared to the monomer spectrum. Moreover,these type of aggregates usually will show little or no fluorescence. Finally, theobserved narrowness of the absorption band for J-aggregates also results from thedelocalized nature of the exciton states, which gives rise to exchange (or motional)narrowing of the disorder [22–24].

1.2 Natural light-harvesting systems

Historically, scientific interest in cyanine dyes and their aggregates was aroused bytheir great significance for application as spectral sensitizers in photographic pro-

4 Introduction

cesses [25], which even to date, is a topic of extensive research [26]. In the last twodecades, however, the field of molecular aggregates has been revolutionized by thediscovery that the light-harvesting complexes of photosynthetic systems of manyplants and bacteria consist of assemblies of strongly interacting, light-absorbingmolecules (chlorophylls) [27]. In photosynthesis, light-harvesting complexes arethe primary units responsible for capturing sunlight and, subsequently, transport-ing the excitation energy towards reaction centers where charge separation can takeplace. Their effectiveness relies on two important aspects: first, the self-assemblyof molecules into highly ordered, low-dimensional structures in order to maximizethe cross section for light absorption at particular frequencies and, second, theextremely fast excitation energy transfer within and between light-harvesting sys-tems.

The nature of the excited states in photosynthetic systems has generated muchscientific interest in the last decade [28]. In particular, an important issue is whetherFrenkel excitons could occur with a clearly delocalized nature. This quantumcoherence possibly plays a crucial role in the specific harvesting properties andenergy transport efficiency in these systems [29].

In general, a thorough understanding of exciton coherence requires detailedknowledge of the molecular structure, excitonic interactions, and the degree andeffects of inhomogeneities. The latter may include structural deformations, local in-homogeneities of the host medium that lead to variations in the excitation energiesand interactions of the molecules (static disorder), and coupling with vibrationaldegrees of freedom in the environment (dynamic disorder) [30–35]. The degree ofcoherence is thus determined by a complex interplay between (i) the intermolecularinteraction strength which tends to delocalize the exciton states (increase coher-ence), and (ii) the disorder which tries to localize these states (decrease coherence).

One of the best studied natural photosynthetic complexes occurs in purple bac-teria. Typically, these photosynthetic systems contain two types of light-harvestingantennae, known as the LH1 and LH2 complexes. The LH1 complex is believedto surround the reaction center, while the LH2 complexes are arranged aroundthe LH1 system [36]. Initially, sunlight is collected by the LH2 antennae wherethe energy is then rapidly transported to the LH1 system and, subsequently, tothe reaction center. This process occurs with an incredibly high efficiency: morethan 95% of the absorbed photons gives rise to a charge separation in the reactioncenter.

The structure of the LH2 complex of a specific purple bacterium known asRhodopseudomonas acidophila has been established in great detail using X-raycrystallography [37, 38]. This complex consists of 27 bacteriochlorophyll (BChl)molecules arranged in two closely-spaced concentric rings that contain, respectively,9 BChl molecules (the B800 ring, associated with the 800 nm absorption band) and

1.2 Natural light-harvesting systems 5

18 BChl molecules (the B850 ring, responsible for the 850 nm absorption band)positioned in a nine-fold symmetry. The close packing in the B850 ring leads tostrong electronic interactions between neighboring molecules, while the rather largedistances between adjacent BChl molecules in the B800 ring give rise to nearest-neighbor couplings that are typically an order of magnitude smaller than those inthe B850 ring [39].

By now, extensive experimental and theoretical reports have provided a clearpicture of the coherence effects in this LH2 complex. In particular, the electronicinteractions between BChl molecules in the B850 ring are sufficiently large to over-come the disorder resulting in the occurrence of Frenkel exciton states that aredelocalized over the entire ring [40, 41]. In contrast, excitations on the B800 ringare typically strongly localized on only 2 − 3 molecules owing to the weaker in-termolecular interactions in this ring [40, 42]. These exciton coherences stronglyinfluence the excitation energy transfer from the B800 ring to the B850 ring [43–46].

In addition to the light-harvesting systems of purple bacteria, also the Fenna-Matthew-Olson (FMO) complex of green bacteria has been the topic of a widevariety of studies in the last three decades (see Ref. [47] for a recent overview).These bacteria have been found to live in the lowest light environments of allknown photosynthetic organisms, which implies an extraordinarily high efficiencyof light harvesting and excitation energy transfer. Their photosynthetic systemscontain light-harvesting antennae, so called chlorosomes, that initially collect thesunlight. Unlike the ring geometry of the LH2 complexes of purple bacteria, thesechlorosomes typically consist of several single-layer and/or multi-layer cylindrical

structures with diameters on the order of 10 nm and lengths that can reach up tomicrometers [48–52], thereby being the largest antenna structures known to exist innature. The role of the FMO complex is to mediate the excitation energy transferfrom the light-harvesting chlorosomes to the reaction center.

The FMO complex of the bacterium Prosthecochloris aestuarii was the firstphotosynthetic complex for which the crystal structure was resolved, already backin 1975 [53]. Because of its relatively simple structure, which consists of threeweakly coupled identical molecular assemblies each containing 8 BChl molecules,FMO complexes are considered ideal model systems to study collective opticaland energy transport phenomena in nature and also inspired the rapid develop-ment of new experimental and theoretical techniques. In this respect, especiallytwo-dimensional electronic correlation spectroscopy has very recently proven to bea powerful new tool to extrapolate information on quantum coherences as it al-lows one to directly access these electronic excitation coherences and their timeevolution [54–57]. Based on these techniques, coherent energy transfers with sur-prisingly long-lived coherences were observed in FMO complexes [55–57]. Similareffects have now also been shown to exist in photosynthetic systems of marine algae

6 Introduction

and in conjugated polymers [58,59].

The above two examples clearly illustrate the complexity of nature’s light-harvesting systems and how quantum coherence might well be of crucial importancefor the various processes in photosynthetic systems.

1.3 Amphiphilic dyes as building blocks for artifi-

cial light harvesting

The discovery that the workings of natural light-harvesting complexes are basedupon the very same quantum mechanical concepts that also apply to syntheticcyanine dye aggregates have inspired the development of artificial molecular as-semblies that may, ultimately, find application as light-harvesting complexes innovel, man-made photosynthetic systems [60]. Recently, cyanine dye moleculeswith hydrophobic and hydrophilic side groups have attracted much attention ascandidates to realize highly ordered molecular structures in aqueous environments(see Ref. [61] for an excellent review). These aggregates are known as amphip-ipes [62,63], a term which relates to the amphiphilic nature of the cyanine dyes andtheir energy migration effects (amphiphiles with pigment interactions performingenergy migration). At the same time, it also points out their tubular morphology,which is the most commonly observed geometry for this type of J-aggregates.

One of the most intriguing and interesting types of amphipipes are the double-walled tubular J-aggregates formed by the amphiphilic cyanine dye known as C8S3,displayed schematically in Figs. 1.3.1(a) and 1.3.1(b) [64, 65]. The reason is thatthe cylindrical geometry is similar, both in shape and size, to the highly efficientlight-harvesting chlorosomes found in green sulphur bacteria [66–69]. Moreover,C8S3 dyes form mostly single tubules with a highly uniform structure, both withinthe tubule itself as well as between different tubules in an ensemble [70]. Thesefeatures make the C8S3 tubular aggregates ideal model systems, both to study on afundamental level the nature and dynamics of the excited states of low-dimensionalmolecular assemblies as well as for possible practical applications in artificial pho-tosynthetic units.

Similar to the natural light-harvesting systems, the close packing of the moleculesin the C8S3 nanotubes leads to strong excitation transfer interaction between themolecules, which in turn gives rise to coherent Frenkel exciton states. The red-shift and narrowing of the absorption bands of the nanotubes with respect to themonomer absorption, as shown in Fig. 1.3.1(c), indicate that these nanotubes canbe classified as J-aggregates [69, 71]. The coherent nature of the exciton statesis intimately connected with the specific details of the nanotubes’ structure on amolecular level, as well as with the degree of static and dynamic disorder in these

1.3 Amphiphilic dyes as building blocks for artificial light harvesting 7

Figure 1.3.1: Self-assembled double-walled tubular J-aggregates of amphiphilic cya-nine dye molecules. (a) The dye molecule C8S3 with hydrophilic sulfonate groups(red) and hydrophobic alkyl chains (light grey). (b) The double-layer tubular struc-ture (referred to as the inner and outer wall) owing to the amphiphilic nature ofthe dyes. (c) Absorption spectra of the dye molecules (black) and of the double-walled nanotubes (red). The narrowness and redshift of the nanotubes’ spectrumare characteristic for J-aggregates. Figure has been published in Ref. [139].

systems. Understanding how these factors determine the collective optical andenergy transport properties is crucial in the design of molecular assemblies withspecific functionalities.

One of the challenging tasks is to unravel the details of the spatial arrangementof the C8S3 nanotubes on the molecular scale and to determine the microscopicorigin of the various absorption bands. Cryo-transmission electron miscroscopy(Cryo-TEM) experiments have revealed that these aggregates possess a double-layer tubular wall structure [see Fig. 1.3.1(b)], resulting from the amphiphilic natureof the dyes [69–72]. Although Cryo-TEM techniques can be used to elucidate thediameters of the two cylinders that are stacked into each other (commonly knownas the inner and outer wall), the resolution is not sufficient to directly access the

8 Introduction

molecular arrangement within these cylinders. Instead, their microscopic structureand optical properties should be resolved by combining experimental techniques(these may include isotropic and polarization-dependent spectroscopies, nonlinearphoton echo and pump-probe techniques [20, 21, 73–78], two-dimensional correla-tion spectroscopy [54–59,79–83], and single-molecule spectroscopy [40,84–93]) withtheoretical models and simulations. Such combined experimental/theoretical stud-ies have recently already reported significant progress on these issues for severaltubular aggregates [71,94,95].

Another fascinating and currently one of the most heavily debated topics con-cerns the effect of the coupling between the two tubular assemblies (i.e., betweenthe inner and outer wall) in double-walled C8S3 nanotubes. In particular, it is ques-tioned whether this coupling results in two weakly coupled exciton systems with-out any significant coherence between the walls, or that these aggregates shouldbe viewed as one large excitonic system, where the exciton states can be coher-ently shared between the walls. In the former situation, the optical response canbe deduced from the separate contributions of the two tubular aggregates and thenature of energy transfer between the walls is considered to be incoherent. In theother case, however, the optical features arise from the collective properties of theentire system and energy transfer between the walls is of coherent nature. Dur-ing the last few years, several studies have addressed this issue with great care;for example, fluorescence and pump-probe experiments [71, 78], and very recentlyalso two-dimensional correlation spectroscopy [83], have been used in combinationwith extensive theoretical modeling. The results, however, are contradictory whichshows that a detailed understanding of the effects of coupling between molecularassemblies in close proximity is still missing.

1.4 Thesis outline

In this thesis we model the collective optical properties of molecular aggregatesassociated with extended electronic excitations. Important topics that will be ad-dressed in this thesis are: (i) the effect of disorder on the optical selection rules andquantum coherence of Frenkel excitons in aggregates of cylindrical geometry (chap-ters 2 and 3); (ii) the linear optical spectra, microscopic structure and inter-wallcoherences of double-walled C8S3 tubular aggregates (chapter 4); (iii) the spectralsignatures of exciton-vibration coupling and temperature-induced destruction ofexciton coherences in model systems, in particular small aggregates (chapter 5);(iv) the role of multi-exciton states in single-molecule spectroscopy (chapter 6).Below we will summarize the main results of each chapter in this thesis.

Chapter 2 starts with a study of the effects of disorder-induced exciton local-ization on the optical selection rules and spectra of cylindrical J-aggregates. In

1.4 Thesis outline 9

the absence of disorder, cylindrical aggregates exhibit three allowed superradianttransitions: one polarized along the cylinder axis and two (degenerate) polarizedperpendicular to this axis. Due to disorder, these optical selection rules breakdown, resulting in optically relevant exciton states that may have a wider rangeof orientations of the transition dipole. In particular, we found that the relevantparameter that governs the violation of the selection rules is the ratio of the excitonlocalization length and the cylinder circumference. For double-walled nanotubes,this means that the outer cylinder will likely show a larger variety of orientationsof the exciton transition dipoles than the inner cylinder. Also, as a consequence,the structure of the absorption spectra of the inner and outer tubes may differsignificantly.

In Chapter 3 we proceed with the role of disorder on the optical response oftubular J-aggregates. Here we show that the disorder scaling of the low-temperatureabsorption linewidth differs substantially from that for one-dimensional (linear)molecular aggregates. The origin of this lies in the long-range intermolecular inter-actions and additional dimension of the cylinders. These factors result in the strongsuppression of exciton localization in the tubular aggregates, similar to the weaklocalization limit (or, marginal regime) in the standard two-dimensional Andersonmodel. Our results provide a simple physical explanation for the high efficiencyof the light-harvesting complexes in green sulphur bacteria and the origin of thestrong linear dichroism typically observed for tubular J-aggregates.

Chapter 4 describes the main theoretical results from an experimental/theoreticalcollaboration which unravels the optical spectra and spatial structure of double-walled tubular aggregates of amphiphilic cyanine dye molecules. In particular, ourmodel for the microscopic structure consists of two-dimensional sheets of cyaninedyes with a two molecule per unit cell Herringbone arrangement rolled onto cylin-drical surfaces that represent inner and outer tubes. We demonstrate that thisstructural model can very well reproduce the exciton transitions observed in theexperimental spectra of C8S3 aggregates. Moreover, our results give new insightsinto the degree of coherence between supramolecular assemblies in close proximity.In particular, we will argue that while exciton coherence within each wall is im-portant, the degree of coherence between both walls is negligible. Understandinghow the microscopic details of the aggregate morphology relate to their opticalproperties and the degree of coherence within and between molecular assemblies isof vital importance for possible application in artificial light-harvesting units.

Chapter 5 is devoted to the effect of coupling between excitons and vibra-tional modes on the optical response and excited state coherence of molecular J-aggregates. We propose a method for calculating optical absorption spectra basedon a variational polaron transformation of the Holstein Hamiltonian. This modelallows us to derive semi-analytical expressions for the spectra that correctly cap-

10 Introduction

ture the optical properties associated with both limits of extremely weak and strongexciton-vibration coupling; that is, collective features in the former situation andmostly single-molecule characteristics in the latter case. Furthermore, within thismodel we can investigate the intermediate coupling regime, which is inaccessibleby means of standard perturbation techniques. Also, we show that the inclusionof temperature gives rise to the destruction of quantum coherence of the excitedstates, originating from the reduction of intermolecular interactions with increasingtemperature.

Finally, in Chapter 6 we present a theoretical model to describe the statisticsof the photon emission process of molecular aggregates, which is typically obtainedin single-molecule spectroscopy (SMS) experiments. SMS provides a useful experi-mental technique that allows one to address details of collective optical transitionsand exciton states that are usually hidden underneath the ensemble averaging inbulk experiments. Our results show that higher order moments of the photonemission process indeed reveal interesting information on the role of multi-excitonstates, even at low laser intensity. Knowledge of these states is also crucial tocorrectly interpret recent two-dimensional correlation spectroscopy experiments.

Chapter 2

Disorder-induced exciton

localization and violation of

optical selection rules in

supramolecular nanotubes

Using numerical simulations, we study the effect of disorder on the optical prop-erties of cylindrical aggregates of molecules with strong excitation transfer inter-actions. The exciton states and energy transport properties of such molecularnanotubes attract considerable interest for application in artificial light-harvestingsystems and energy transport wires. In the absence of disorder, such nanotubesexhibit two optical absorption peaks, resulting from three superradiant excitonstates, one polarized along the axis of the cylinder, the other two (degenerate) po-larized perpendicular to this axis. These selection rules, imposed by the cylindricalsymmetry, break down in the presence of disorder in the molecular transition ener-gies, due to the fact that the exciton states localize and no longer wrap completelyaround the tube. We show that the important parameter is the ratio of the excitonlocalization length and the tube’s circumference. When this ratio decreases, thedistribution of polarization angles of the exciton states changes from a two-peakstructure (at zero and ninety degrees) to a single peak determined by the orienta-tion of individual molecules within the tube. This is also reflected in a qualitativechange of the absorption spectrum. The latter agrees with recent experimental

12

Disorder-induced exciton localization and violation of optical selection

rules in supramolecular nanotubes

findings.1

2.1 Introduction

During the past decade, there has been a growing interest in molecular J-aggregatesof cylindrical geometry [65, 70, 71, 94, 97–106]. Examples of molecules that in so-lution self-assemble into aggregates of a tubular shape are carbocyanine moleculeswith hydrophobic and hydrophilic side groups [65, 70, 71, 98–100] and porphyrinderivatives [94, 101–104]. These molecules both yield aggregates with a diameterof the order of 10 nm and a length of up to microns, which explains why they areoften referred to as molecular nanotubes. Their shape and size resemble naturallight-harvesting systems in green sulphur bacteria [27, 48, 68, 97, 105–107], whilethese systems also have a comparable geometry to previously studied helical poly-mers [108–110]. Because of this and the strong intermolecular excitation transferinteractions, these synthetic J-aggregates are considered excellent candidates forartificial light-harvesting systems [111]. This perspective motivates many of therecent studies of the optical absorption and luminescence properties of the collec-tive exciton states in these systems, as well as the energy transport and relaxationcaused by these states [65, 69–71,98–100,112–115].

In homogeneous cylindrical aggregates, the rotational symmetry around theaxis imposes strong optical selection rules on the wave number k2 that charac-terizes the exciton’s Bloch wave function along a ring around the cylinder. Inparticular, it is easily shown that only states with k2 = 0 and k2 = ±1 maycarry optical oscillator strength [62, 97]. Further study shows that in case of longcylinders, for each of these wave numbers only one exciton state dominates theabsorption strength, as can readily be seen by imposing periodic boundary con-ditions in the axis direction [97, 116]. As the subbands with opposite sign of k2are degenerate, this yields a simple absorption spectrum with only two peaks, ofwhich the one with k2 = 0 is polarized along the cylinder axis, while the otherone is polarized perpendicular to this axis. These simple selection rules yield abasic explanation for the experimentally observed absorption and linear dichroismspectra of tubular aggregates of the carbocyanine molecule 3,3’-bis(2-sulfopropyl)-5,5’,6,6’-tetrachloro-1,1’-dioctylbenzimidacarbocyanine (C8S3 for short) [71].

In reality, the cylindrical symmetry is broken to some extent; this may be dueto the cylinder cross section not being perfectly round, due to defects within themolecular packing of the nanotube or due to the effect of local inhomogeneity ofthe solvent host. The latter gives rise to fluctuating local electric fields induced

1This chapter has been published as S. M. Vlaming, E. A. Bloemsma, M. Linggarsari Nietiadi,and J. Knoester, J. Chem. Phys. 134, 114507 (2011).

2.1 Introduction 13

by the host molecules, which in turn lead to fluctuations in Stark shifts in thetransition energies and the transition dipoles of the individual molecules withinthe aggregate [117]. As a consequence, in experiment disorder usually exists ineither the transition frequencies or the intermolecular resonance interactions, or inboth. Such disorder localizes the exciton states [118]; these localized states mayhave a much wider range of orientations of the transition dipole than just paralleland perpendicular to the cylinder axis. In the extreme case of disorder exceedingthe intermolecular transfer interactions, the exciton state gets localized on just onemolecule and the transition dipole of this state is simply given by the transitiondipole of this molecule, which in general will be neither parallel nor perpendicularto the axis. Recent experimental work has demonstrated that indeed the idealizedselection rules are an oversimplified view of reality, in particular in cylinders witha wider circumference [119]. Furthermore, it has also been demonstrated thatby scanning near-field optical microscopy the polarization dependent luminescenceof segments of individual nanotubes may be investigated [70], which opens theperspective to look into the optical selection rules and the polarization of excitonstates in much more detail.

The above situation motivates a detailed study of how broken cylinder sym-metry affects the selection rules. In this chapter we will do this for the case ofdisorder in the molecular transition energies (diagonal disorder), which is the typeof disorder that is most frequently used to model inhomogeneity in molecular ag-gregates [34]. Because of the large size of tubular aggregates, several recent papershave used the coherent potential approximation (CPA) to account for the effect ofdisorder on the spectra [71,94]. As the CPA does not allow for a proper descriptionof localization, the breaking of the selection rules and the distribution of dipole ori-entations cannot be studied within this approach. Rather, investigating the natureof the localized exciton states requires numerical simulations, a technique appliedin Refs. [120, 121]. A systematic study characterizing the breakdown of the se-lection rules was, however, not conducted. We have performed such a study byinvestigating the localization behavior of the exciton wave functions, the angulardistribution of the exciton transition dipoles, and the polarization dependent spec-tra for various disorder strengths and cylinder radii. From the results we concludethat the breakdown is governed by the ratio of the exciton localization length andthe cylinder circumference.

This chapter is organized as follows. In Sec. 2.2 we present the structural andexciton model for the aggregate and give expressions for the isotropic and polarizedabsorption spectra. In Sec. 2.3 we introduce various quantities that are used tocharacterize the exciton localization and the breakdown of the selection rules. Ournumerical results are presented and discussed in Sec. 2.4, while we summarize andconclude in Sec. 2.5.

14



2.2 Frenkel exciton model for cylindrical aggre-

gates

2.2.1 Geometry and Hamiltonian

Generally, the cylindrical structure of the aggregates can be obtained by rollinga two-dimensional lattice onto a cylindrical surface [97]. It is convenient to viewthis rolled lattice as a stack of rings, as shown in Fig. 2.2.1. Each cylinder consistsof N1 rings, each with N2 monomers, where each ring is rotated with respect toits neighbor over an angle γ. The distance between neighboring rings is h, whilethe rings have a radius R. The values of γ, h, R and N2 are determined by theunderlying two-dimensional lattice structure and the chiral vector that defines howthis lattice is rolled onto a cylinder [97]. Each molecule is labeled by the positionvector ~n = (n1, n2), where n1 labels the ring and n2 the position in the ring(i.e. the helix on which it is located, see Fig. 2.2.1). The angles α and β definethe orientation of the molecular transition dipole moments; β denotes the anglebetween the transition dipole moment and the cylinder axis, while α is the anglebetween the projection of the transition dipole moment on the ring plane and thetangent of the ring. For lattice structures with one molecule per unit cell, whichwe will restrict ourselves to, α and β are the same for all molecules. The aboveparameters fully define both the position and the transition dipole moment of eachmolecule,

~r~n = (R cos (n2φ2 + n1γ) , R sin (n2φ2 + n1γ) , n1h) , (2.2.1)

~µ~n = (−µ sinβ sin (n2φ2 + n1γ − α) , µ sinβ cos (n2φ2 + n1γ − α) , µ cosβ) ,(2.2.2)

where µ is the magnitude of the transition dipole of an individual molecule and wehave defined φ2 = 2π/N2.

We use a Frenkel exciton Hamiltonian to describe the electronically excitedstates [11, 13,97],

H =∑

~n

ω~nb†~nb~n +

∑′

~n,~m

J(~n− ~m)b†~mb~n, (2.2.3)

where ω~n is the excitation energy of molecule ~n, while b†~n and b~n are the Paulicreation and annihilation operators for the electronic excitation of molecule ~n.Furthermore, J(~n − ~m) is the excitation transfer interaction between molecules ~nand ~m, which depends on their (relative) positions and on their transition dipolemoments. The prime on the summation excludes the term ~n = ~m. Dependingon the system and the desired level of accuracy, the interactions are often takenas point dipole interactions or as extended dipole interactions. However, sincethe exact choice does not matter for what follows, we will keep the interactions

2.2 Frenkel exciton model for cylindrical aggregates 15

Figure 2.2.1: The cylindrical geometry in the stack of rings representation (left)and a cross section of the cylinder perpendicular to its axis (right). The dashedhelical line on the cylinder connects the molecules with n2 = 1. This figure hasbeen taken from Ref. [97].

unspecified for now. Finally, we account for disorder by taking the excitationenergies ω~n from a Gaussian distribution with mean ω0 and standard deviation σ.No correlation exists between the values of ω~n for different molecules. No disorderin the interactions is considered.

2.2.2 Absorption spectra

The observable quantities we are interested in are the linear absorption spectra forvarious polarization conditions. We assume interaction of the cylindrical aggregatewith linearly polarized light with a polarization direction ~e and frequency ω, anduse the Fermi golden rule to account for the interaction term between light andmatter. This leads to general expressions for a linear spectrum S(ω) of the form [97]

S(ω) =

⟨⟨∑

q

Xqδ(ω − Eq)

⟩⟩, (2.2.4)

where q labels the exciton eigenstates of the Hamiltonian Eq. (2.2.3), Eq denotethe corresponding eigenenergies (we set ~ = 1) and Xq are the oscillator strengthsof these states for the given polarization conditions. Finally, the double angularbrackets 〈〈...〉〉 denote an average over disorder realizations. The exciton eigenstates

16



may be written

|q〉 = b†q |g〉 =∑

~n

ϕq~nb†~n |g〉 , (2.2.5)

with |g〉 the ground state of the aggregate (all molecules in their ground state) andϕq~n the normalized coefficients of the eigenvector of the Hamiltonian correspondingto the eigenvalue Eq. Then, the (polarization-dependent) oscillator strengths Xq

can be written as

Xq =

⟨∣∣∣∣∣∑

~n

ϕq~n~µ~n · ~e∣∣∣∣∣

2⟩=∑

~n,~m

ϕq~nϕ∗q ~m 〈(~µ~n · ~e)(~µ~m · ~e)〉 ≡

∑

~n,~m

ϕq~nϕ∗q ~mX~n~m,

(2.2.6)where the single angular brackets 〈...〉 denote an average over the orientation of thecylinder relative to the polarization direction ~e.

We will consider three different polarization conditions. First of all, we willbe interested in the linear absorption of an isotropic solution, A0(ω). In this case,the cylinder orientations are distributed completely isotropically, and the spectrumdoes not depend on ~e. The two other linear absorption spectra we will calculate aretaken in an oriented sample, where the cylinders’ axes are all aligned. This maybe realized in streaming solutions, by spin coating or via microscopy experimentson single aggregates. Then, we can consider the absorption of light polarized inthe cylinder axis direction A||(ω) and absorption of light polarized perpendicularto the axis direction A⊥(ω). For all three cases, it is possible to explicitly evaluatethe orientational averages in Eq. (2.2.6) [97, 121],

X0~n,~m =

1

3

(X

||~n,~m + 2X⊥

~n,~m

),

X||~n,~m = µ2 cos2 β,

X⊥~n,~m =

1

2µ2 cos ((n2 −m2)φ2 + (n1 −m1)γ) sin

2 β. (2.2.7)

The factor of 2 before the perpendicular component in the isotropic absorptionstrength results from the fact that there are two mutually orthogonal perpendiculardirections contributing to it.

2.2.3 Selection rules in the absence of disorder

To derive the optical selection rules, we consider the homogeneous cylinder, i.e.ω~n = ω0 for all molecules ~n. In this case, we can explicitly use the cylindrical

2.2 Frenkel exciton model for cylindrical aggregates 17

symmetry in the ring direction and impose a Bloch form for the wave function inthat direction [97],

c†n1,k2=

1√N2

∑

n2

eik2φ2n2b†~n, (2.2.8)

where k2 is the transverse wave number that labels the Bloch state, i.e. we havek2 = 0,±1, ...,±(N2/2 − 1), N2/2 for N2 even and k2 = 0,±1, ...,±(N2 − 1)/2 forN2 odd. Using this transformation, the two-dimensional Hamiltonian [Eq. (2.2.3)]decouples into a set of one-dimensional Hamiltonians [97],

H =∑

k2

H(k2) =∑

k2

(ω0

∑

n1

c†n1,k2cn1,k2

+∑

n1,m1

J(n1 −m1; k2)c†n1,k2

cm1,k2

),

(2.2.9)where the effective interaction between rings separated by n1h is given by

J(n1; k2) =∑′

n2

J(n1, n2)e−ik2φ2n2 . (2.2.10)

It can be shown that only a few of the one-dimensional Hamiltonians in Eq. (2.2.9)have eigenstates that may contribute to the absorption spectrum [97]. More specif-ically, the k2 = 0 band gives contributions that are polarized in the cylinder axisdirection, while the degenerate k2 = ±1 bands give contributions that are polarizedperpendicular to the cylinder axis; all other bands are dark. Moreover, even withinthese three special bands only very few states have significant oscillator strength. Inparticular, for the k2 = 0 band there is one superradiant transition that dominatesthe optical response, while for the k2 = ±1 band there will in general be a few, en-ergetically similar, states that will contain a reasonable oscillator strength [97,116].The absorption spectrum is thus expected to be dominated by one peak polarizedin the cylinder axis direction, and one peak (or for very narrow linewidths, up tothree peaks close in energy [116]) polarized perpendicular to the cylinder axis. Thiscan be confirmed explicitly when one assumes periodic boundary conditions in thedirection of the cylinder axis (appropriate for long cylinders), where we find thatonly the above two peaks occur in the absorption spectrum [97].

When we account for disorder, the states will no longer be extended over theentire circumference of the cylinder, but they will localize over a smaller regioninstead. As a result of this breaking of the cylindrical symmetry, it is expectedthat the selection rules obtained for the homogeneous cylinder will no longer exactlyhold. The extent to which these selection rules are violated is detailed in Sec. 2.3and 2.4. Such deviations should be visible in a polarization-dependent absorptionspectrum taken in an oriented sample [69,71,122,123].

18



2.3 Characterization of exciton states in disordered

cylinders

As mentioned in Sec. 2.2.3, the presence of disorder induces localization of theexciton states. To quantify this behavior, a number of different measures ex-ist [33,34,115,121,124–127], each giving an estimate of the number of molecules thatcoherently participate in a given exciton state. We will consider here the autocor-relation function of the exciton wave function [121,125–127], which is particularlyuseful to probe not only the size but also the directionality of the localization onthe cylinder surface. For a given exciton state, it is defined as

Cq(~n) =∑

~m

∣∣∣ϕq ~mϕ∗q(~n+~m)

∣∣∣ , (2.3.1)

where the components of ~n and ~m should be consistent with the type of boundaryconditions we take. For open boundary conditions in the cylinder axis direction,we have: n1 = (−N1 + 1, ..., N1 − 1), n2 = (0, 1, ...N2 − 1), m1 = (1, ..., N1 − n1)for n1 ≥ 0, m1 = (1− n1, ..., N1) when n1 < 0, and m2 = (1, ..., N2).

The autocorrelation function measures the correlation between the wave func-tion at positions separated by the vector ~n; because of normalization of the wavefunctions it obeys Cq(~0) = 1, while, generally, it will fall off for growing |~n|. Forthe cylindrical structures, Cq(~n) has a two-dimensional argument, and as a resultwe can obtain information on how an exciton state is extended on the cylindri-cal surface. From the autocorrelation function one can define a measure for thenumber of molecules that coherently participate in a given exciton state, whichwe will denote by Ncoh. For the cylindrical aggregates, Ncoh has previously beendefined in Ref. [121] as the total number of molecules ~n with Cq(~n) > 1/e. Here wewill not use that definition of Ncoh, but rather define it as proposed in Ref. [127]:Ncoh ≡

∑~n Cq(~n). Clearly, states that are localized on a single molecule (strong

disorder) give Ncoh = 1. For states that are completely delocalized (no disorder) wefind, using periodic boundary conditions along the cylinder axis, that Ncoh = N .Comparing the above two different definitions of Ncoh, it is clear that the latterdefinition is less sensitive to fluctuations in the correlation function (due to thesummation). It should be noted that in the same way one can also define a mea-sure for the localization of exciton states in certain preferential directions, as willbe detailed in Sec. 2.4.

The extent to which the optical selection rules are violated, is expected to berelated to the amount of localization in the ring direction. If a state remainsextended over (close to) the entire circumference of the cylinder, one expects theselection rules to be (approximately) obeyed. On the other hand, if a state is

2.4 Numerical results 19

localized on a small part of the circumference of the cylinder, large deviations fromthe selection rules are expected.

To quantify the violation of selection rules for a given exciton state q, we con-sider the transition dipole moment vector corresponding to it, which is given by

~µq =∑

~n

ϕq~n~µ~n. (2.3.2)

We use the angle βq between the transition dipole moment vector of state q and thecylinder axis z as a measure of the extent to which the selection rules are violated,

βq = arccos

( |~µq · z||~µq|

). (2.3.3)

For a homogeneous cylinder, all states with nonzero oscillator strength will haveeither βq = 0 for the states polarized in the cylinder axis direction (k2 = 0),or βq = 90 for the states polarized perpendicular to the cylinder axis (k2 =±1). Deviations from the selection rules imply deviations of βq from these twovalues. In the limit of extremely large disorder (compared to the intermolecularinteractions), we expect to have strongly localized exciton states that resemble themolecular excited states, which have a transition dipole moment with a fixed angleβq ≈ β~n = β. As a result, for large disorder values, the distribution of excitonorientation angles βq should tend to a delta function centered around β.

It should be noted that we will not be concerned with the distribution of βq

itself, but rather focus on the closely related angle resolved oscillator strength,which we define as

F (η) =∑

q

|~µq|2 δ(η − βq). (2.3.4)

For given angle η, F (η) simply sums the oscillator strengths of all exciton statesthat satisfy βq = η. In this way, the states which are optically dominant will alsoplay the most important role in characterizing the extent to which selection rulesare violated in disordered cylinders.

2.4 Numerical results

The cylindrical aggregates of choice that we apply our theory to are C8S3 aggre-gates. As has been shown in Refs. [69–71], C8S3 molecules can form double-walledcylindrical structures where the two walls are weakly coupled. Depending on thedetails of the preparation, different radii for the walls can be obtained. In par-ticular, for the Direct Route (DR) synthesis, the walls have radii of roughly 5.4nm and 7.8 nm respectively [71], while for the Alcoholic Route (AR), the radii are

20



reduced to approximately 3.2 nm and 6.5 nm [72]. As a result, this type of cylin-drical structure naturally shows a large variation in radius, which as we will showleads to large differences in localization behavior and optical properties. Whilethe precise structural parameters for the different walls and preparation routes arenot identical, they presumably are close, and in our simulations we do not varyany of these structural parameters in order to isolate the radius dependence. Inparticular, we choose the DR inner wall parameters from Ref. [71]; that is, we havea bricklayer lattice with lattice constants a = 2.0 nm and d = 0.4 nm, and a latticeshift of s = 0.488 nm. We assume a single-molecule transition dipole moment ofµ = 11.4 Debye and use extended dipole-dipole interactions with a charge sepa-ration distance of l = 0.7 nm [71]. We use a rolling angle of θ = 50.7 (β = θ,α = 0), which is slightly larger than the value used in Ref. [71]. The advantageis that our value of θ yields a chiral vector that is rather commensurate with thelattice structure (i.e., it intersects with many lattice points), which facilitates astudy of radius dependence and also simplifies the visualization of the wave func-tions. In our simulations, the cylinders have a length of about 63 nm, which issufficiently long to eliminate finite size effects for all but the lowest disorder values.One should note that our results are generic and do not depend on the exact choiceof the various parameters.

2.4.1 Localization of optically dominant states

Before addressing the disorder averaged localization behavior, we first consider atypical disorder realization. In Fig. 2.4.1 typical wave functions of the opticallymost active state are shown for different disorder strengths and for cylinders withradius R = 4 nm (corresponding approximately to the inner wall of an AlcoholicRoute C8S3 aggregate) and R = 8 nm (corresponding approximately to the outerwall of a Direct Route C8S3 aggregate). The wave functions are plotted as afunction of the coordinates z and φ. Here, z is simply the distance in the cylinderaxis direction, while φmeasures the rotation with respect to a reference line parallelto the cylinder axis (i.e., not relative to the helix in Fig. 2.2.1). In other words,a constant φ represents a vertical line on the cylinder surface; φ ranges over aninterval of length 2π, which corresponds to one circumference.

Comparison of Fig. 2.4.1(a) with 2.4.1(b) and Fig. 2.4.1(c) with 2.4.1(d) clearlyshows that increasing the disorder while keeping the radius fixed leads to strongerlocalization, as one expects. It is worthwhile to note that for the weak disorder case,Fig. 2.4.1(a), the wave function is rather delocalized. It wraps around the entirecylinder, with coefficients ϕq~n that have the same sign practically everywhere,leading to in-phase absorption for all molecules and showing that the state inFig. 2.4.1(a) is indeed a superradiant state.


0

20

40

60

0

−0.1

0

0.1

0.2

0.3

0.4

zφ

φq (

z,φ

)

020

4060

0

−0.2

−0.1

0

0.1

0.2

0.3

zφ

φq (

z,φ

)

0

20

40

60

0

−0.1

0

0.1

0.2

0.3

zφ

φq (

z,φ

)

0

20

40

60

0

−0.02

0

0.02

0.04

0.06

0.08

zφ

φq (

z,φ

)

π

2π

π

2π

π

2π

π

2π

a) b)

d)c)

Figure 2.4.1: Exciton wave functions for the state with highest oscillator strengthfor different disorder magnitudes and radii in a typical realization. [(a), (b)] R = 4nm, σ = 200 cm−1 and 650 cm−1, respectively. [(c), (d)] R = 8 nm, σ = 650cm−1 and 1000 cm−1, respectively. The cylinders have the same length of 63.4nm; this corresponds to cylinders consisting of N1 = 50 rings with total numberof monomers equal to N = 2000 (R = 4 nm) or N = 4000 (R = 8 nm). Thecoordinate z measures the position in the axis direction, while φ is the azimuthalangle, measured relative to a line parallel to the cylinder axis.

Furthermore, several interesting features of the optically dominant state followfrom Fig. 2.4.1. First, all wave functions have a helical character. This behav-ior results from the tendency of exciton states to extend in the direction of thestrongest interactions. Second, as we anticipated in Sec. 2.2.3, fixing the disordervalue while increasing the radius will also lead to stronger localization relative tothe cylinder circumference in the ring direction. Specifically, while the absoluteexciton localization length should be quite similar, the localization length with re-spect to the cylinder circumference decreases. This can be observed by comparingFigs. 2.4.1(b) and 2.4.1(c). Finally, an interesting observation on the wave functionsshown in Figs. 2.4.1(b-d) is the fact that there tend to be multiple regions wherethe wave function has appreciable values. This is in contrast to the one-dimensional

22



−50

0

500

0

0.2

0.4

0.6

0.8

1

zφ

C(z

,φ)

−50

0

500

0

0.2

0.4

0.6

0.8

1

zφ

C(z

,φ)

−50

0

500

0

0.2

0.4

0.6

0.8

1

zφ

C(z

,φ)

−50

0

500

0

0.2

0.4

0.6

0.8

1

zφ

C(z

,φ)

π

−π−π

π

π

−π−π

π

d)c)

a) b)

Figure 2.4.2: Autocorrelation functions for the state with highest oscillator strengthfor different disorder magnitudes and radii in a typical realization. The disorderrealization and parameters are the same as in Fig. 2.4.1. Contrary to the irregularstructure of the wave functions, the autocorrelation functions are rather smooth,which makes it the preferable quantity to extract localization lengths.

case [128], but is fully consistent with the multifractal nature of the excited statesthat has been predicted previously in two-dimensional (and higher-dimensional)lattices [129,130].

In Fig. 2.4.2 we present the autocorrelation functions corresponding to the wavefunctions of Fig. 2.4.1. As in the case of the wave functions, the autocorrelationfunction is plotted as a function of z and φ. The autocorrelation functions clearlycorroborate the features already seen for the wave functions; that is, the helicalorientation and the increased amount of (relative) localization upon both increas-ing disorder strength and increasing radius. However, the smoother structure ofthe autocorrelation function compared to the wave function (resulting from thesummation over ~m in Eq. (2.3.1)) makes it the preferred quantity to extract the(relative) localization lengths.

So far we have discussed the qualitative features of the exciton state with thehighest oscillator strength. To make the above observations more quantitative,


we will calculate the exciton localization length in the direction of the strongestinteractions, i.e. along the helical orientation seen in Fig. 2.4.2, relative to thetube’s circumference. This quantity, hereafter abbreviated as relative localization,will be denoted by Λ and for a given exciton state q is expressed as

Λ ≡ Lφ

2πR. (2.4.1)

Here Lφ =∑′

zCmax

q (z, φ), with Cmaxq (z, φ) denoting the maximum value of

the autocorrelation function at position z and the restricted summation includesonly values of Cmax

q (z, φ) above a threshold value C0. Thus, Lφ is a measure forthe extent of the excitonic wave functions in the helical direction. It should benoted that we have introduced a threshold value C0 in the definition of Lφ, thusneglecting contributions from small values of Cmax

q (z, φ). The reason for this isthat for these small values (which occur at large values of |z|), the autocorrelationfunction is no longer oriented in the helical direction due to the fractal nature ofthe wave functions. Rather, these small values Cmax

q (z, φ) appear to be randomlydistributed along the different rings. Therefore, values below C0 have not beenincluded in the calculation of Lφ. For the threshold value we always take C0 = 0.3,which is slightly lower than the value 1/e used in Ref. [121].

As we are only interested in the optically relevant states, for each disorderrealization we consider only those states that contain at least 10% of the maximumoscillator strength of that realization. For each of those states, and for a numberof disorder realizations, the localization measure Λ was obtained, resulting in adistribution for Λ. Figure 2.4.3 presents the means 〈Λ〉 and standard deviations(denoted as error bars) of this distribution for various values of the disorder strengthand cylinders with radius R = 4 nm and R = 8 nm. It clearly shows that, for bothradii, an increase of the disorder leads to a decrease of 〈Λ〉. Furthermore, for a givenvalue of σ, 〈Λ〉 for the cylinder with radius R = 4 nm is approximately twice aslarge as for the cylinder with radius R = 8 nm (this ratio increases somewhat withincreasing disorder strength). This is not surprising, because for a particular valueof σ, the absolute localization Lφ is roughly the same for both radii, as is expectedfrom the fact that the underlying two-dimensional lattice structure is identical. Itshould be noted that the fluctuations in Λ are of the same order of magnitude as itsmean value, which has also been found previously in one-dimensional lattices [131].

An interesting aspect of Fig. 2.4.3 is the scaling of the localization with disorder.Previous studies addressing localization behavior in aggregate systems [33,34,128,132] have established both by analytical arguments and numerical simulations thatthe scaling relation is well-described by a power law dependence. We found thatthe data in Fig. 2.4.3 may also be accurately fitted to a power law, 〈Λ〉 = aσb, withcorresponding exponents for both cylinders that are large (b ≈ −2.2 for R = 4 nm

24



550 600 650 700 750 800 850

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

σ (cm−1

)

Λ

fit b = −2.22 ± 0.05

fit b = −2.67 ± 0.1

R = 4 nm

R = 8 nm

Figure 2.4.3: Mean and standard deviations (denoted by error bars) of the relativelocalization lengths Λ of optically dominant states. For each disorder value, 75realizations were taken into account. For each realization only states that containat least 10% of the maximum oscillator strength of that realization are accountedfor. The data points are fitted by a power law with exponent b ≈ −2.2 for R = 4nm and b ≈ −2.7 for R = 8 nm. For a particular disorder value, the average relativelocalization is approximately a factor of 2 bigger for the cylinder with small radius(R = 4 nm), which implies that the absolute localization length is the same forboth cylinders.

and b ≈ −2.7 for R = 8 nm) in comparison to the well-known scaling behavior inone dimension, where the exponent is approximately given by b ≈ −2/3 [33,34]. InRefs. [128, 132], it was shown that the exponent can be obtained by equating thetypical energy spacing between the lowest energy states on a localization segment tothe disorder induced scattering rate between these states. Unfortunately, as a resultof the two-dimensional nature of the cylindrical surface and the long-range dipolarinteractions that are present, both the energy spacing and the disorder inducedscattering rate scale with the localization size in a very similar way, making itimpossible to use the simple arguments of Refs. [128, 132] to analytically predictthe disorder scaling of the localization seen in Fig. 2.4.3. Also, the range of disordervalues that can be studied numerically is limited either due to the system size (incase of small disorder) or due to the number of necessary realizations (for largedisorder values) so that it is not possible to examine how the power law behaviorholds in these limiting cases. A more detailed investigation of the scaling behavior


0 20 40 60 800

2

4

6

8

10x 10

6

η (degrees)

F(η

)

0 20 40 60 800

5

10

15x 10

5

η (degrees)F

(η)

0 20 40 60 800

0.5

1

1.5

2x 10

6

η (degrees)

F(η

)

0 20 40 60 800

0.5

1

1.5

2

2.5x 10

6

η (degrees)

F(η

)

(a)

(c)

(b)

(d)

Figure 2.4.4: Angle-resolved oscillator strength distributions for different disordervalues and radii. Parameters are the same as in Fig. 2.4.1. For each panel, 200realizations of disorder were taken into account. Deviations from the ideal selectionrules that exist for a homogeneous cylinder become more pronounced for increasingdisorder and larger radius. The fluctuations still apparent in the distributions aremost probably of statistical nature and will thus disappear with increasing numberof realizations.

in situations where the scattering rate scales in a similar way with the localizationsize is subject to ongoing research.

2.4.2 Breaking of the selection rules

In the previous section we established the localization behavior of exciton states forcylinders with different radii. Now we will address the consequences of localizationfor the optical selection rules. As mentioned in Sec. 2.3, it is straightforward to cal-culate to what extent selection rules are violated for various choices of the disorder.For each realization, we simply calculate all eigenstates, their transition dipoles,

26



and then evaluate the angular distribution function F (η) for the oscillator strength,defined in Eq. (2.3.4). The results are shown in Fig. 2.4.4. Obviously, these resultsare mainly determined by the optically dominant states, as the contribution ofstates with no or very little oscillator strength is negligible.

Figure 2.4.4(a) has been obtained for the cylinder with radius R = 4 nm, witha (small) disorder value σ = 200 cm−1 (smaller than occurs in experiment [71]). Asexpected for small disorder values, the states with appreciable oscillator strengthstill have dipoles oriented close to the cylinder axis or perpendicular to it (corre-sponding to transitions to the k2 = 0 states and the k2 = ±1 states, respectively).Indeed, this is clearly seen in Fig. 2.4.4(a). Increasing the disorder to a more rea-sonable value [71] of σ = 650 cm−1 and keeping the radius fixed, we see clearlyfrom Fig. 2.4.4(b) an increase of oscillator strength in orientations which are notallowed in the homogeneous case, and thus an increase in the extent to which theselection rules are violated. This is in agreement with Fig. 2.4.3, which showed thatthe relative localization decreased, and thus deviations from the selection rules areexpected to become larger upon increasing the disorder strength.

Increasing the radius of the cylinder to R = 8 nm while keeping the disor-der value fixed at σ = 650 cm−1 leads to the oscillator strength distribution ofFig. 2.4.4(c). It is clearly seen here that the selection rules are broken to a muchlarger extent than for the cylinder with smaller radius R = 4 nm, as shown inFig. 2.4.4(b). This finding is also in agreement with the results of Fig. 2.4.3, whereit was shown that upon increasing the radius at constant disorder strength, therelative localization becomes smaller, and thus deviations from the selection rulesbecome more pronounced. Eventually, upon further decrease of the relative local-ization [Fig. 2.4.4(d)], convergence sets in towards a distribution centered aroundthe monomer dipole orientation β (which is the rolling angle, β = θ = 50.7),where in the extreme case of very large disorder all oscillator strength should becollected.

Besides the above observations, we note that the oscillator strength distributionsin Fig. 2.4.4 show an asymmetry between small and large angles. There is almostno oscillator strength at an orientation angle η = 0, while the weight at largerangles η is enhanced considerably. The reason for this is purely geometrical; thereexist only very few orientations of ~µq such that η ≈ 0. More specifically, the phasespace volume of possible orientations grows proportional to sin(η), simply becausethe area of the part of the sphere with an inclination angle between η and η + dηis given by A(η) = 2π sin ηdη. This explains the skewing of the oscillator strengthdistributions in Fig. 2.4.4 towards larger angles.

We have so far qualitatively discussed the role of localization for the violationof the optical selection rules and established that a decrease of the localizationrelative to the tubes’ circumference leads to a larger degree of violation of the


0.35 0.4 0.45 0.5 0.55 0.60.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

M

Λ

fit b = −1.84 ± 0.16

R = 4 nm

R = 8 nm

Figure 2.4.5: Mean and standard deviations (denoted by error bars) of the relativelocalization lengths Λ versus the measure M for cylinders with different radii andvarious disorder values. For R = 4 nm: σ = 650− 1000 cm−1 in steps of 50 cm−1;for R = 8 nm: σ = 500 − 750 cm−1 in steps of 50 cm−1. For each data point, 75realizations of disorder were taken into account. The data points are fitted to apower law with exponent b ≈ −1.8. This figure shows that different cylinders withthe same relative localization 〈Λ〉 also have the same amount of violation of opticalselection rules.

optical selection rules. To further support our findings that the relative localizationΛ is the parameter that governs the breakdown of the selection rules, we considerthe following measure, denoted by M , to quantify the extent to which the selectionrules are broken

M =70∑

η=20

F (η)/90∑

η=0

F (η). (2.4.2)

In this way, we have a measure for the violation of selection rules that equals zerofor the homogeneous cylinder (σ = 0 cm−1), increases with increasing disorder,and tends to unity in the limit of extremely large disorder (if 20 ≤ β ≤ 70).Obviously, the chosen range of 20 ≤ η ≤ 70 is rather arbitrary, and differentintervals for η may be considered to define the measure M . The consequences ofthis are discussed at the end of this section.

In Fig. 2.4.5 we show for various values of the disorder and two cylinders withdistinct radii the measure M and the corresponding mean relative localization 〈Λ〉(error bars denoting the standard deviations). Clearly, for both cylinders a decrease

28



of 〈Λ〉 leads to an increase of M as we have anticipated before. Most interestingly,cylinders with different radii that have roughly the same relative localizations 〈Λ〉(and thus different disorder strengths) also have the same measure M . This isexpressed by the fact that all data points (of both cylinders) follow the same curvein Fig. 2.4.5. This curve may be fitted to a power law 〈Λ〉 = aM b with exponentb ≈ −1.8. The range of measures M for which the data points for different radiifollow the same curve is expected to break down at large values of the disorder,i.e. at large values of M . A signature of this in Fig. 2.4.5 seems to be that for thelarger values of M , the data points for the cylinder with radius R = 8 nm have atendency to lie below the curve. The reason for this trend becomes obvious if oneconsiders the limit of very large disorder. In this case, all states are localized on asingle molecule and the transition dipoles are simply given by β. This means thatboth cylinders have the same measure M = 1 although their relative localizationsΛ differ by a factor of 2.

As mentioned above, the definition for the measure M [Eq. (2.4.2)] is somewhatarbitrary in the way the range of η is chosen. We have compared our findings withother intervals for η, ranging from 15 ≤ η ≤ 75 up to 30 ≤ η ≤ 60. It was foundthat although the exact value of the exponent of the power law fit does depend onthe chosen interval, our main conclusion that the fitted curve is independent of theradius of the cylinder holds for all definitions of M .

2.4.3 Absorption spectra

To connect more closely to experiment, we have also calculated the absorptionspectra for disordered cylindrical aggregates. In Fig. 2.4.6, we show the variousabsorption spectra A0(ω), A||(ω) and A⊥(ω) for cylindrical aggregates of two dif-ferent radii: R = 4 nm and R = 8 nm, both with the molecular arrangement usedthroughout this chapter and a disorder strength of σ = 650 cm−1, which is quitetypical of the experimentally found value for C8S3 aggregates [71]. The differencein structure in the absorption spectra is clear. For the cylinder with a small radius,a clear two-peak structure can still be observed in the isotropic absorption spectrum[Fig. 2.4.6(a)]: a low-energy peak corresponding to transitions that are approxi-mately polarized in the axis direction, and a higher energy peak correspondingto transitions that are approximately polarized perpendicular to the cylinder axis.This is corroborated by the polarization dependent absorption spectra shown inFig. 2.4.6(c) and also in agreement with the results obtained for the angle resolvedoscillator strength distribution [Fig. 2.4.4(b)]. In contrast, for a system with thesame disorder value but a larger radius (R = 8 nm), the selection rules are brokento a much larger extent. The peaks polarized parallel and perpendicular to thecylinder axis overlap to a large degree [Fig. 2.4.6(d)] and the isotropic spectrum


500 550 600 6500

1

2

3

4

5

6x 10

5

λ (nm)

A0 (

λ)

500 550 600 6500

5

10

15x 10

5

λ (nm)A

0 (

λ)

500 550 600 6500

5

10

15x 10

5

λ (nm)

A⊥(λ

), A

||(λ

)

500 550 600 6500

0.5

1

1.5

2

2.5

3x 10

6

λ (nm)

A⊥(λ

), A

||(λ

)

A||(λ)

A⊥(λ)

A||(λ)

A⊥(λ)

R = 8 nmR = 4 nm

R = 8 nmR = 4 nm

(a) (b)

(d)(c)

Figure 2.4.6: Absorption spectra for two cylinders with distinct radius and fixeddisorder σ = 650 cm−1. [(a), (b)] Isotropic absorption spectra A0(ω) for radii R = 4nm and R = 8 nm, respectively. [(c), d)] Corresponding polarization dependentspectra A||(ω) and A⊥(ω) for the two radii.

shows hardly any structure [Fig. 2.4.6(b)]. The reason is that the absorption comesfrom a large bulk of states with widely varying polarizations [Fig. 2.4.4(c)], insteadof two distinct groups of states with polarizations near either 0 or 90.

Thus, the general conclusion is that, keeping the disorder strength fixed, theabsorption spectrum looses structure upon increasing the cylinder radius, which isa consequence of an enhanced breaking of the optical selection rules in the widercylinder.

The above conclusion is in qualitative agreement with the experimental findingsreported in Ref. [119]. There, the absorption spectra of the inner and outer wallcylinders of C8S3 aggregates were separated through preferentially perturbing themolecules within the outer wall cylinder by means of chemical oxidation, effectivelyswitching off the optical response of the outer wall at a higher rate. The resultinginner wall spectrum exhibits a rich spectral structure with multiple (at least four)

30



well separated transitions, which can likely be interpreted as two Davydov-splitdoublets. The low energy doublet in the absorption spectrum of the inner wallshows two well-separated peaks, in contrast to the outer wall absorption whichhardly shows any spectral structure (likely only two transitions close in energywith one appearing as a shoulder of the other).

2.5 Summary

The unavoidable presence of disorder in cylindrical molecular aggregates will lead tolocalization of the collective electronic excited states and a breakdown of the opticalselection rules that exist for perfect cylindrical symmetry and which are oftenused to explain the experimental absorption spectra. Using numerical simulationswe have shown that localization indeed occurs, leading to deviations from thehomogeneous situation, but depending on the disorder magnitude and the radiusof the cylinder, the selection rules may still apply to a large extent. The relevantquantity to determine this is the relative localization length Λ, i.e. the excitonlocalization length relative to the total cylinder circumference. As a result, bothincreasing the disorder (and thus decreasing the localization length) and increasingthe radius (and thus the circumference) will lead to stronger deviations from thehomogeneous selection rules. We have shown and quantified this by analyzing thedistribution of polarizations of the exciton states with nonzero oscillator strength,and using this distribution to define a measure M to quantify the extent to whichthe selection rules are broken. We have found that both an increase of the disorderstrength and an increase of the cylinder radius result in larger values of M . Mostimportantly, we have shown that plotting 〈Λ〉 against M , a curve is found that doesnot depend on the radius of the cylinder. This proves that, indeed, the relativelocalization governs the breaking of the selection rules.

The structure of the wave functions and their autocorrelation functions obvi-ously show localization as well. In particular, both an increasing disorder strengthand an increasing radius will show stronger localization relative to the circum-ference. We find that both the wave functions and the autocorrelation functionsare helically oriented, in the direction of the strongest interactions. An interest-ing observation is that the exciton wave functions tend to have multiple regionswhere the amplitude is nonzero; this is consistent with the multifractal nature oftwo-dimensional exciton states as suggested by Schreiber and co-workers [129,130].

Finally, the localization and the resulting breaking of the selection rules have di-rect consequences for the (polarization-dependent) absorption spectra. Quite gen-erally, keeping the disorder strength fixed, the absorption spectrum looses structureupon increasing the cylinder radius, which is a consequence of an enhanced break-ing of the optical selection rules in the wider cylinder. The separate peaks in the

2.5 Summary 31

smaller-radii cylinders have polarization directions either mostly parallel or mostlyperpendicular to the cylinder axis. For the structureless spectra of the larger-radiicylinders this polarization information is lost, because the underlying exciton stateshave a wide range of polarization orientations. These findings are in qualitativeagreement with recent experiments [119].

32



Chapter 3

Signature of anomalous

exciton localization in the

optical response of tubular

J-aggregates

We demonstrate that the disorder scaling of the low-temperature optical absorp-tion linewidth of tubular J-aggregates sharply contrasts with that known for one-dimensional J-aggregates. The difference can be explained by an anomalous local-ization of excitons originating from both the long-range intermolecular interactionsand the quasi two-dimensional geometry of the tubes. This regime approaches theweak localization limit of the standard Anderson model. Moreover, we demonstratethat the optical properties of excitons in tubular aggregates with dipole-dipole in-teractions show striking similarities to one-dimensional toy models with inter-siteinteractions falling off with distance r as r−ξ, with 3/2 ≤ ξ ≤ 2. Our results pro-vide a simple explanation for the origin of the strong linear dichroism and weakexciton-exciton scattering in tubular J-aggregates observed in experiments.1

1This chapter is based on E. A. Bloemsma, S. M. Vlaming, V. A. Malyshev, and J. Knoester,in preparation.

34

Signature of anomalous exciton localization in the optical response of

tubular J-aggregates

3.1 Introduction

Self-assembled low-dimensional nanostructures of organic molecules are known tohave special optical and energy transport properties, among which are enhancedspontaneous emission [20, 21], strong nonlinear susceptibilities [14–16, 18], andhighly efficient energy transfer [11,66,133], rendering them ideal for possible use infuture optoelectronic devices. Double-walled nanotubular J-aggregates formed bythe synthetic amphiphilic cyanine dye C8S3 [65] are currently of particular interest:their highly uniform supramolecular structure closely resembles the natural light-harvesting antennae (chlorosomes) in green sulphur bacteria [27,48,68,70,105–107].The strong interactions between the molecular transition dipoles result in collective(extended) optical excitations in the nanotubes, known as Frenkel excitons. Theirlocalization (coherence) length depends on the scattering on static (and dynamic)disorder imposed by the host medium. The complex interplay between intermolecu-lar resonance transfer interactions, static disorder, and dynamic degrees of freedomdetermines the fascinating optical and transport properties of the nanotubes.

In the absence of disorder, the absorption and linear dichroism spectra of tubu-lar J-aggregates exhibit two red shifted absorption bands resulting from three al-lowed superradiant transitions: one polarized along the cylinder axis and one de-riving from two degenerate states polarized perpendicular to this axis [62,97,116].In general, disorder-induced exciton scattering results in states that have a widerrange of polarization directions [134]. Remarkably, measurements of polarizationdependent absorption spectra of C8S3 tubular J-aggregates reveal that, even atambient temperatures, the optical response is still governed by transitions withpreferred parallel and perpendicular orientation, despite a significant degree ofdisorder [69, 71]. These findings indicate that exciton localization in cylindricalaggregates is strongly suppressed. This is also consistent with recent experimen-tal data on weak exciton-exciton scattering in molecular nanotubes, revealed bydouble-quantum two-dimensional electronic spectroscopy [135].

In this chapter, we study the localization properties of excitons in molecularnanotubes and how these are reflected in the linear optical response. To achievethis, we calculate the disorder scaling of the low-temperature absorption linewidth.This relation is sensitive to the particular form of the density of states (DOS) inthe optically relevant region, which plays a major role in the localization propertiesof the excitons. In particular, we demonstrate here that the disorder scaling ofthe low-temperature absorption linewidth (FWHM) of C8S3 tubular J-aggregatesdiffers substantially from what is known to date for one-dimensional (1D) molecularJ-aggregates. The physical origin of this can be traced back to the long-range inter-molecular interactions and quasi two-dimensional structure of the tubes, giving riseto an unusual exciton energy dispersion relation and DOS. More specifically, in the

3.2 Model and results 35

absence of disorder, the DOS of the nanotubes vanishes near the lower exciton bandedge (where the optically dominating states reside), in contrast to the divergentbehavior of the DOS for 1D J-aggregates in this region [23,33,34]. The latter resultsin strongly localized excitons, while the nanotubes’ vanishing DOS gives rise toseverely weaker scattering of the excitons. Interestingly, this regime turns out to beclose to the weak localization limit in the standard Anderson model [118,136,137].

To further quantify the apparent relation between the disorder scaling of opticalquantities, like the absorption linewidth etc., and the energy dependence of theDOS, we exploit in the last part of this chapter an artificial 1D exciton model, wherethe long-range inter-site interactions decay with inter-site distance r according tothe power law r−ξ, where 3/2 ≤ ξ ≤ 2. We demonstrate that the disorder-inducedscaling laws can be derived directly from the explicit form of the disorder-free DOSin the optically relevant region. Moreover, when the 1D toy model DOS matchesthat of the tubular J-aggregates, we find that the absorption line shapes in bothexciton models show a high degree of similarity.

This chapter is organized as follows. In Sec. 3.2 we present the theoretical modelfor the nanotubes and numerically obtain the disorder scaling of the absorptionlinewidth. In Sec. 3.3 we discuss the physical origin of this disorder scaling andapply the coherent potential approximation to explain the numerically found scalingexponent. In Sec. 3.4, using the toy model, we establish the general relation betweenthe DOS in the optically relevant region and the disorder scaling of the linewidth,while we provide conclusions in Sec. 3.5.

3.2 Model and results

Our model of the tubular aggregate consists of a 2D sheet of molecules wrappedaround a cylindrical surface. It can be shown that such an aggregate may beconsidered as a stack of N1 equidistant rings of radius R, each ring containingN2 uniformly distributed molecules with adjacent rings rotated relatively to eachother by a helical angle γ (i.e., see Fig. 2.2.1) [97]. Furthermore, the moleculartransition dipoles follow the cylindrical symmetry and are specified by angles αand β (Fig. 2.2.1); here α denotes the angle between the projection of the dipolevector on the ring plane and the local tangent of the ring, whereas β gives the anglebetween the dipole vector and the cylinder axis [97]. Each molecule is identifiedby its position vector n = (n1, n2), with n1 indicating the ring on which it residesand n2 labeling the position in the ring.

The optical excitations of the tubular aggregate are described by a Frenkelexciton model that accounts for molecular excitation energies En with Gaussiandisorder (mean ω0 and standard deviation σ) and non-fluctuating intermoleculartransfer interactions Jnm determined by extended transition dipoles [138]. The

36



−3000 −2500 −2000 −1500 −10000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

E (cm−1

)

A(E

)

σ=600 cm−1

Figure 3.2.1: Isotropic absorption spectrum of tubular J-aggregates for disorderstrength σ = 600 cm−1 and model parameters taken from Ref. [71]: a radius ofR = 5.455 nm, a single-molecule transition dipole of µ = 11.4 Debye, a chargeseparation distance of L = 0.7 nm, dipole angles α = 0, β = 47.4 and a helicityangle of γ = 6.74.

corresponding Hamiltonian reads

H =∑

n

En |n〉〈n|+∑′

n,m

Jnm |n〉〈m| , (3.2.1)

where |n〉 denotes the state in which molecule n is excited while all others are inthe ground state. The prime on the summation indicates the exclusion of termswith n = m.

To establish the disorder scaling of the optical linewidth, we performed numer-ical simulations of the absorption spectrum for tubular aggregates with diagonal(energy) disorder. The zero-temperature linear absorption spectrum is given by

A (E) =⟨∑

q Oqδ (E − Eq)⟩, where Oq is the oscillator strength of the exciton

state q, averaged over all possible orientations of the cylinder, Eq gives the energyof state q and the brackets 〈. . . 〉 denote the average over all random energies ofthe molecules (see Ref. [97] for details). The spectra are obtained by numericaldiagonalization of Eq. (3.2.1) for 103 disorder realizations and using the procedureoutlined in Ref. [140] to minimize fluctuations in the spectra. In all calculations,we simulated cylinders of N = 6000 molecules to avoid finite size effects and, forexplicitness, considered the set of parameters that was used in Ref. [71] to fit the

3.2 Model and results 37

450 500 550 600 650 700

40

60

80

100

120

140

σ (cm−1

)

W|| (

cm

−1)

Data points

W||=(1.3×10

−6)σ2.83

Figure 3.2.2: Numerically obtained linewidths (FWHM) W|| (symbols) of the en-ergetically lowest absorption band of the tubular aggregate for various disorderstrengths σ together with the best power law fit W|| (σ) ∝ σ2.83 (straight line).The tubular aggregate parameters are identical to those used in Fig. 3.2.1, and aresummarized in Table 3.2.1 as Structure I.

measured absorption spectrum of the inner wall of C8S3 aggregates. That is, wetake a radius of R = 5.455 nm, a single-molecule transition dipole of µ = 11.4 De-bye, a charge separation distance of L = 0.7 nm, dipole angles α = 0, β = 47.4

and a helicity angle of γ = 6.74. This parameter set is summarized in Table 3.2.1as Structure I.

In Fig. 3.2.1, we present the simulated absorption spectrum for disorder magni-tude σ = 600 cm−1. The spectrum reveals the signatures characteristic for tubularJ-aggregates; that is, there are two narrowed superradiant exciton transitions redshifted compared to the monomer spectrum (we take ω0 = 0). Here, the low-est energy J-band (E|| = −2450 cm−1) stems from transitions polarized mainlyalong the tube axis and lies at the lower exciton band edge, while the other band(E⊥ = −2000 cm−1) is mainly polarized perpendicular to this axis and is locatedabove the band edge.

To calculate the width (FWHM) of the energetically lowest absorption line,we first decompose the spectrum into the sum of two Lorentzian lineshapes tofilter the contribution of the high energy absorption band to the width of thelow energy absorption peak. The width of the energetically lowest absorptionband is then taken as the FWHM of the corresponding Lorentzian. The results

38



450 500 550 600 650 700

50

100

150

200

σ (cm−1

)

W|| (

cm

−1)

Data points

Bβ=3.29±0.09

450 500 550 600 650 700

40

60

80

100

120

140

160

σ (cm−1

)W

|| (

cm

−1)

Data points

BR=3.15±0.18

450 500 550 600 650 700

40

60

80

100

120

140

σ (cm−1

)

W|| (

cm

−1)

data points

Bγ=2.83±0.11

450 500 550 600 650 700

40

60

80

100

120

140

σ (cm−1

)

W|| (

cm

−1)

Data points

BN

2

=2.84±0.17

(c)

(a) (b)

(d)

Figure 3.2.3: Numerically obtained linewidth scalings (best power law fit [straightlines] of the data points [symbols]) of the energetically lowest absorption band forfour distinct sets of parameters for the tubular aggregate geometry (see main textfor details), summarized in Table 3.2.1: (a) the dipole orientation β (Structure II),(b) the cylinder radius R (Structure III), (c) the helicity angle γ (Structure IV ),and (d) the commensurability N2 (Structure V ).

for different disorder values in the interval of σ = 425 − 700 cm−1, shown assymbols, together with the best power law fit W|| (σ) ∝ σB (straight line) of thedata points, are displayed in Fig. 3.2.2. We point out that the disorder intervalis bound from below due to finite size effects and from above because of overlapbetween the two absorption bands. The observed trend that W|| (σ) increases forlarger σ stems from the fact that the number of dye molecules that coherentlyshare an excitation decreases if σ increases. The resulting power law exponentB = 2.83 implies that the rate at which the line broadens differs drastically fromthat of 1D molecular J-aggregates, where (assuming nearest-neighbor interactions)W1D ∝ σ4/3 [33, 34,132].

To corroborate further on the obtained power law exponent, we calculated thedisorder scaling of the linewidth for several other molecular arrangements, in ad-

3.3 Discussion 39

Model R (nm) N2 β (degrees) γ (degrees) scaling exponent BI 5.455 2 47.4 6.74 2.83II 5.476 2 −1.1 16.7 3.29III 3.619 2 45.3 10.1 3.15IV 5.410 2 43.1 93.4 2.83V 5.428 53 51.6 3.0 2.84

Table 3.2.1: Summary of the sets of model parameters used for the various tubularaggregate structures discussed in this chapter. The parameters that were changeddrastically in each model are printed in bold.

dition to the specific structure used above (i.e., Structure I). For each geometry,one of the model parameters R (radius), N2 (commensurability), γ (helicity), orβ (dipole orientation) was varied drastically, while the other parameters were keptsimilar to those of Structure I. The specific values for the parameters R, N2, γ,and β used in the different structural arrangements are given in Table 3.2.1, whilethe results for the disorder scalings of the absorption linewidths (FWHM) are dis-played in Fig. 3.2.3. These results clearly reveal in all cases a power law exponentthat is significantly larger than that of 1D J-aggregates. Interestingly, the closesimilarity between the numerically obtained exponents suggests that the observed,approximately cubic, disorder scaling of the linewidth is almost universal; that is,we found only a minor dependence on the cylinder radius and orientation of themolecular dipoles, while varying the helicity and commensurability of the ringsresulted in identical scaling exponents.

3.3 Discussion

3.3.1 Physical origin: self-consistent approach

In general, the disorder scaling of optical quantities, like the linewidth, is closelyrelated to the localization of excitons. The latter may be estimated by comparingthe size scaling of energy spacings δE (i.e., the DOS) of exciton states in the bare(σ = 0) exciton spectrum with the size scaling of the disorder-induced scatteringrate σeff = σ/N1/2 between them. Here the scattering rate reflects the effect ofexchange narrowing: due to their delocalized nature, exciton states feel an effectivedisorder σeff which is smaller than the bare disorder value σ [23]. If σeff ≪ δE,the exciton states will be mixed weakly and, therefore, remain delocalized over theentire system. In this case, the absorption linewidth scales linearly with σ. On the

40



other hand, if σeff ≫ δE, disorder-induced scattering between the states will resultin their localization with typical size given by Ncoh. The latter may be interpretedas the effective number of coherently bound molecules.

In order to estimate Ncoh, we use the concept of a hidden structure in theLifshits tail [128, 132], the existence of which has been confirmed explicitly for 1Dlocalized Frenkel excitons by numerical simulations [128, 131, 141]. It consists ofseveral, almost non-overlapping, segments of typical size Ncoh ≪ N , each segmentcontaining two (or more) localized states that resemble low energy homogeneousBloch states undergoing level repulsion with an effective energy separation δE∗.The latter can be estimated from the bare level spacings by replacing N with thetypical localization size Ncoh. Similar, the effective mixing between states on thesame segment is given by σ∗

eff = σ/√Ncoh. For a disorder strength σ, Ncoh is

determined from a competition between δE∗ and σ∗eff . If σ

∗eff > δE∗, the scattering

between the states tends to further reduce Ncoh, thereby effectively increasing δE∗

at a higher rate than σ∗eff . When σ∗

eff < δE∗, disorder is of perturbative nature andresults in the increase ofNcoh, where now σ∗

eff increases faster than δE∗. Estimationof Ncoh thus follows in a natural way from the equality δE∗ = σ∗

eff [128, 132]. Theabsorption linewidth, being determined by the typical energy spacing in the Lifshitstail set by Ncoh, is then given by W = σ/

√Ncoh.

In the standard 1D Anderson model (assuming nearest-neighbor interactionsonly) the bare level spacings near the band edge diminish with increasing sys-

tem size as δE ≈ 4π2N−2, i.e., the DOS diverges as D (E) ∝ |E − E0|−1/2.

Thus, in the thermodynamic limit (N → ∞) the equality σeff ≫ δE alwaysholds, resulting in strongly localized exciton states. From the above explainedself-consistency arguments, we find the well-known relations for the coherence

number N1Dcoh ≈

(4π2)2/3

(σ/ |J |)−2/3and the disorder scaling of the linewidth

W1D ≈(4π2)−1/3

(σ/ |J |)4/3 [132]. In two dimensions, the Anderson model withnearest-neighbor coupling leads to a level spacing in the optically relevant regionthat decreases linearly with system size, i.e., δE ≈ 4π2N−1, resulting in a con-stant DOS near the exciton band edge. Because the exciton separation in the2D case diminishes with increasing N at a lower rate than for the 1D model,excitons in two dimensions are more extended than those in one dimension, forthe same disorder strength. This is directly reflected in the coherence number

N2Dcoh ≈

(4π2)2

(σ/ |J |)−2. Consequently, we find the following disorder scaling of

the linewidth W2D ≈(4π2)−1

(σ/ |J |)2.Comparison of the linewidths in one and two dimensions shows two important

features. First, for the same disorder strength the linewidth is typically smaller intwo dimensions, which results from the weaker localization of the excitons. Second,the disorder scaling exponent of the linewidth in 2D is larger, which indicates that

3.3 Discussion 41

here the line broadening is more sensitive to variations of the disorder strengthσ. We point out that this dimensionality dependence of the optical response isnot confined to the cases discussed above. In fact, a 3D model with nearest-neighbor coupling reveals a cubic disorder scaling of the linewidth and even weakerlocalization of the excitons compared to two dimensions [33,142].

We now address the effects of the long-range dipole-dipole interaction (LRI)on the absorption linewidth. In 1D, accounting for these interactions beyond thenearest neighbor, this leads to a slight decrease of the linewidth (for fixed disorderstrength) and, moreover, results in a minor increase of the disorder scaling expo-nent [34,128,131,143]. These effects, although present, do not significantly changethe optical response in 1D. For the standard 2D Anderson model with dipolar LRI,the energy spacing in the optically relevant region scales as δE ∝ N−1/2 (conse-quently, the DOS vanishes linearly with energies approaching the band edge) [144].In this case, σeff and δE scale similarly with system size N , indicating a specialregime of localization. This situation is known as the marginal regime or the weakAnderson localization limit [118, 136, 137]. The self-consistency arguments predictthat in this case the scaling exponent of the linewidth approaches infinity [145].These arguments, although no longer sufficient to determine the explicit value ofthe scaling exponent in this regime, do reveal that, unlike the 1D case, the pres-ence of dipolar LRI in two dimensions has more pronounced effects on the opticallinewidth and disorder scaling. In particular, the existence of weakly localized ex-citons yields typically very narrow absorption lines which are highly sensitive tochanges in the disorder.

The self-consistent approach presented here qualitatively explains the numeri-cally found large linewidth scaling exponent of tubular aggregates (i.e., see Sec. 3.2).That is, its origin can be traced back to the quasi two-dimensional structure of thetubes and the presence of long-range intermolecular interactions. In the followingSection, we will adapt a more formal approach, known as the coherent potentialapproximation. This semi-analytical approach yields a direct relation between thevalue of the disorder scaling exponents of the absorption linewidths and the explicitform (i.e., energy dependence) of the DOS in the optically relevant energy region.

3.3.2 Formal approach: the coherent potential approxima-

tion

To gain insight into the value of the numerically obtained scaling exponent of thetubular aggregate absorption linewidth, we apply the coherent potential approxi-mation (CPA), thereby closely following the procedure outlined in Ref. [142]. The

42



200 400 600 800100

150

200

250

300

|E−E0| (cm

−1)

D(E

−E

0)

Data points

D(E−E0)=(16.3±5.5)|E−E

0|0.44±0.06

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

|E−E0| (cm

−1)

D(E

−E

0)

(a) (b)

Figure 3.3.1: (a) Numerically calculated DOSD (E − E0) in the absence of disorderfor the tubular aggregate with parameters identical to those used in Fig. 3.2.1(Structure I in Table 3.2.1). Exciton energies are binned into intervals of lengthδ = 140 cm−1. (b) Disorder-free DOS (symbols) near the lower band edge together

with the best power law fit D (E − E0) ∝ |E − E0|0.44±0.06(straight line).

spectral line shape in this case is given by

L (E) =1

π

Im [V (E)]

(E − Re [V (E)])2+ (Im [V (E)])

2 , (3.3.1)

where Re [V (E)] and Im [V (E)] are the real and imaginary parts, respectively,of the energy dependent coherent potential V (E) representing the line shift (halfwidth) of the spectrum. They can be found by solving a self-consistent equation,which, in the weak disorder limit, takes the form

V (E) = σ2G (E − V (E)) . (3.3.2)

HereG (E) = N−1∑

k [E − E0 − Ek]−1

is the disorder-free exciton Green function.The energies Ek are obtained from diagonalizing Eq. (3.2.1) (for σ = 0) and E0

introduces a shift of the energy spectrum such that its zero point lies at the bareexciton band bottom.

To solve Eq. (3.3.2), the Green function is separated into its real and imagi-nary parts, G (E) = Re [G (E)] + iIm [G (E)], where we explicitly assume that thereal part Re [G (E)] is a constant. The imaginary part Im [G (E)] can be iden-tified with the disorder-free DOS, defined as D (E) = N−1

∑k δ (E − Ek). In

3.4 Relation to one-dimensional toy models 43

Fig. 3.3.1, we display the numerically calculated DOS (shown as squares) of thetubular aggregates near the lower exciton band edge together with the best powerlaw fit D (E) ∝ |E − E0|P (straight line) for the same parameters used to generateFig. 3.2.2, i.e., those of Structure I given in Table 3.2.1. The results reveal thatthe DOS falls off near the band bottom, approximately in a square root dependentfashion (P = 0.44±0.06). This behavior of the DOS leads to weak scattering of theexciton states in this region, which can hold even for a moderate disorder strength.

From the above, it follows that the Green function can to good approximation

be expressed as G (E) = −C + iD |E − E0|1/2, where C,D > 0. Inserting this intoEq. (3.3.2) and solving the resulting self-consistent equation yields the coherentpotential near the band bottom V (E0). To lowest order in σ, it is given by

V (E0) = −Cσ2 + iC1/2Dσ3. (3.3.3)

From the CPA approximation, we thus find a σ3 dependence on the linewidth.This is in good agreement with the numerically found disorder scaling of the lowenergy absorption linewidth (W|| ∝ σ2.83) for the tubular aggregates based on theparameter set of Structure I (shown in Fig. 3.2.2).

3.4 Relation to one-dimensional toy models

The above results confirm that there exists a relation between the energy depen-dence of the disorder-free DOS and the disorder scaling of the optical absorptionband at low temperature. We will now further quantify and generalize this apparentrelation by means of a simple one-dimensional toy model. It consists of a 1D latticeof N sites with uncorrelated site energies drawn from a Gaussian distribution withmean zero and standard deviation σ, and non-fluctuating couplings between sitesn and m of the form Jnm = J/ |n−m|ξ, where J denotes the nearest-site couplingstrength. In the absence of disorder, the eigenstates of this model (assuming pe-riodic boundary conditions) are Bloch waves |K〉 = N−1/2

∑n exp (iKn) |n〉, with

K = 2πkN−1 denoting the normalized wave number. The energy dispersion rela-tion reads EK = −2 |J |∑n>0 cosKn/nξ, where we explicitly take J < 0 to ensurethat the optically active state (i.e., the state labeled by wave number K = 0) hasthe lowest energy. Near the lower band edge (K → 0), the energies may be approx-imated by (1 < ξ < 3) EK ≃ E0 + |J |AξK

ξ−1 [145, 146]. Here E0 = −2 |J | ζ (ξ)is the lower band edge in the thermodynamic limit (N → ∞), ζ (ξ) the Riemannzeta function and Aξ a positive constant of the order of unity [147,148].

The disorder-free DOS in the vicinity of the lower band edge for this toy modelyields the following power law behavior

D (E) ∝ |E − E0|1

ξ−1−1

. (3.4.1)

44



1 1.1 1.2 1.3 1.4 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

σ/|J|

Wξ

ξ = 19/10

Bξ = 2.13±0.04

ξ = 9/5

Bξ = 2.36±0.1

ξ = 5/3

Bξ = 2.86±0.1

ξ = 8/5

Bξ = 3.35±0.13

Figure 3.4.1: Numerically obtained absorption linewidths (FWHM) Wξ (σ) (sym-bols) in the 1D toy model for different values of the interaction exponent ξ and forvarious disorder strengths σ/|J |. The best power law fits Wξ (σ) ∝ σBξ are shownas straight lines.

This relation shows that the DOS is very sensitive to the value of ξ: we finddivergent (2 < ξ < 3), constant (ξ = 2) or vanishing behavior (1 < ξ < 2)near the band bottom. We point out that ξ = 3/2 corresponds to the marginal(weak localization) regime. Below this value, both the DOS and its derivativevanish resulting in the existence of extended states (even at moderate values of thedisorder) that exhibit a localization-delocalization transition with respect to thedisorder strength σ [148]. Henceforth, we will focus on values of ξ in the interval3/2 ≤ ξ ≤ 2; here, the DOS itself vanishes, although all states remain localized.

An approximate solution for the coherent potential near the band bottomV (E0) for the DOS in Eq. (3.4.1) can be obtained from Eq. (3.3.2) in a similarway as before. Keeping only lowest order terms in σ, it now reads

V (E0) = aξσ2 + ibξσ

2ξ−1 , (3.4.2)

where aξ, bξ are constants that depend on ξ. Thus, the CPA gives a disorder-induced linewidth scaling Wξ ∝ σ2/(ξ−1) which shows strong dependence on ξ,ranging from a σ2 behavior (for ξ = 2) to a σ4 dependence (for ξ = 3/2). InFig. 3.4.1 we present the results of the numerically calculated absorption linewidths(symbols) for interaction exponents ξ = 8/5, 5/3, 9/5, 19/10 and various disorder

values σ = 1 |J | − 3/2 |J |, together with the power law fits Wξ (σ) ∝ (σ/ |J |)Bξ

3.4 Relation to one-dimensional toy models 45

−2600 −2400 −2200 −2000 −1800 −16000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E (cm−1

)

A(E

)

σ = 0.8|J|

σ = 450 cm−1

σ = 1.1 |J|

σ = 625 cm−1

Figure 3.4.2: Absorption spectra of the tubular aggregate (parameters identical toFig. 3.2.1) for disorder values σ = 450 cm−1 and σ = 625 cm−1 together with thecorresponding spectra of the 1D toy model for interaction exponent ξ = 5/3 anddisorder strengths σ = 0.8 |J | and σ = 1.1 |J |, respectively. Due to rounding errorsin σ/|J |, the latter spectra have been shifted by 5 cm−1 (for σ = 0.8 |J |) and 18cm−1 (for σ = 1.1 |J |) to match the exact position of the tubular absorption peak.

(straight lines). The spectra were calculated for lattices of N = 6 · 103 − 104 sitesand 102 − 103 disorder realizations, depending on ξ. The linewidths have beenobtained by fitting each disorder-averaged spectrum to a Lorentzian function.

Figure 3.4.1 shows that, for each σ, the linewidths broaden with increasingvalue of ξ. This trend is consistent with the fact that the typical localizationlength decreases for higher values of ξ, resulting from diminishing strength of thelong-range interactions. Furthermore, the calculated linewidth scaling exponentstogether with their CPA predicted values (given in brackets) are B8/5 = 3.35(3.33), B5/3 = 2.86 (3.0), B9/5 = 2.36 (2.5) and B19/10 = 2.13 (2.22), which showoverall good agreement. We note that the slight discrepancy observed betweenour numerical findings and the CPA predicted values are due to neglecting higherorders of σ in Eq. (3.4.2). This has been verified explicitly by limiting σ to bothsmaller and larger values. For smaller values, this yields scaling exponents thatget closer to the expected CPA values, while in the opposite case the exponentsslowly tend towards linearity, as expected for completely localized exciton (i.e.,single molecule) states.

We now turn our attention to the case ξ = 5/3, for which the DOS [Eq. (3.4.1)]

46



vanishes as E1/2. The observed scaling exponent B5/3 = 2.86 is nearly identicalto that obtained for the tubular aggregates (W|| ∝ σ2.83), due to the fact thatDOS energy dependence of both systems is similar. This demonstrates that thetoy model can in principle mimic the optical features of this complex system verywell. To corroborate on this further, we show in Fig. 3.4.2 the absorption spectraof the tubular aggregates for two disorder values σ = 450 cm−1 and σ = 625 cm−1

together with the corresponding two spectra of the toy model (σ = 0.8 |J | andσ = 1.1 |J |). The disorder values for the toy model were obtained in the followingway. We first calculated the redshifts of the exciton absorption bands relative tothe single-site transition in the homogeneous limit (σ = 0). The value of |J | thenfollowed from the equality of the redshifts (i.e., we find |J | = 520 cm−1). Next,we calculated the disorder-induced shift of the cylinder absorption band for bothvalues of the disorder. The corresponding toy model disorder values are thosefor which the disorder-induced shifts of both systems are equal. The results inFig. 3.4.2 reveal a high degree of similarity concerning both the absolute value ofthe width as well as the overall lineshape, which provides final support that theoptical responses of both systems are similar.

3.5 Conclusions

Summarizing, we have investigated theoretically the disorder scaling of the linewidth(FWHM) of tubular J-aggregates at low temperature. We found numerically thatthe disorder scaling exponent of the FWHM is drastically increased compared tothat of 1D molecular J-aggregates. In addition, this exponent is almost indepen-dent on specific details of the structural parameters of the tubular aggregate, likethe radius, dipole orientation, helicity, and commensurability. We have shown thatthe large value of the scaling exponent originates from the combination of thequasi two-dimensional geometry of the cylinders and the anisotropic long-rangeintermolecular dipolar interactions. This gives rise to an unusual exciton energydispersion and corresponding DOS in the optically relevant region (close to thebare exciton band edge); the DOS vanishes roughly in a square-root dependentfashion, thus reducing substantially the exciton scattering. As a result, the exci-ton localization is strongly suppressed in tubular aggregates and approaches theweak localization limit of the standard Anderson model. Furthermore, we haveestablished a general relation between the specific form of the vanishing DOS andthe disorder-induced scaling relations. As a result, the exciton model for tubularJ-aggregates can be mapped onto one-dimensional toy models with highly similaroptical features, provided that the behavior of the DOS close to the band edgeis identical. This correspondence may provide a helpful tool for future researchconcerning the optical and exciton coherence properties of disordered nanotubes.

3.5 Conclusions 47

To end, we like to emphasize that our results provide an explanation to theorigin of the strong linear dichroism of C8S3 tubular J-aggregates observed in ex-periments. Specifically, measurements of polarization dependent absorption spectrareveal that, despite a significant degree of disorder, the optical response is still gov-erned by transitions with preferred parallel and perpendicular orientation [69, 71],as dictated by the optical selection rules in the absence of disorder. Also, we havealready shown in the previous chapter that, even if the disorder value exceeds theintermolecular interaction strengths, the selection rules still remain valid to a goodapproximation. These features indicate that the exciton states should remain quiteextended at large disorder. Our findings strongly support that exciton localizationis indeed severely suppressed in cylindrical J-aggregates and provide a deeper un-derstanding of the localization mechanism in these systems. In addition, they alsoshed light onto the recently observed weak exciton-exciton scattering in cylindricalaggregates using double-quantum two-dimensional electronic spectroscopy [135].Finally, we point out that the suppression of disorder-induced localization is ex-pected to lead to enhanced excitation energy transport, which is advantageous forapplication in artificial light harvesting.

48



Chapter 4

Excitons in self-assembled

double-walled tubular

aggregates: molecular

structure, optical spectra

and inter-wall coherences

The optical properties of self-assembled double-walled tubular aggregates formedby the amphiphilic cyanine dye molecule C8S3 prepared in a water/methanol solu-tion are studied theoretically. These materials have recently attracted much inter-est owing to their great similarity to natural light-harvesting antenna complexes,rendering them ideal for applications in artificial excitation energy collection andtransport systems. We propose a novel model for the molecular arrangement in-side these aggregates, which is based on rolling two-dimensional lattices with twomolecules per unit cell Herringbone-like structures, onto cylindrical surfaces thatrepresent the inner and outer cylinders. We show that this model very well re-produces the experimental absorption spectra. In particular, our results providestrong support that the two high energy transitions should be associated with ex-citon transitions of the aggregate. The detailed molecular structure allows us toaddress the role of excitation transfer interactions between the inner and outercylinders, which is of vital importance for understanding the nature of the optical

50

Excitons in self-assembled double-walled tubular aggregates: molecular

structure, optical spectra and inter-wall coherences

excitations as well as for future applications.1

4.1 Introduction

Self-assembled aggregates of organic molecules are abundantly found in naturalphotosynthetic complexes of many bacteria and higher plants [27, 37, 38, 48, 68,105–107]. Their fascinating linear and nonlinear optical features as well as theirunique excitation energy transfer properties make them ideal building blocks forutilization in artificial light harvesting, optoelectronics and energy transport sys-tems [11,26,28,42,66,111,133,149–160]. Moreover, they are of scientific interest asmodel systems to obtain deeper insights into the nature and dynamics of collectiveexcitations (excitons) in low-dimensional molecular assemblies. Understanding howthe molecular structure of these materials determines their intriguing properties iscrucial for ultimately designing molecular assemblies with specific functionalities.

Recently, much attention has been devoted to a new family of 5,5’,6,6’- tetra-chlorobenzimidacarbocyanine dyes with attached hydrophobic alkyl chains and hy-drophilic acid side groups [61, 63]. The reason is that cryo-transmission electronmicroscopy (cryo-TEM) revealed that the aggregates formed from these dyes showa large variety in their morphology, depending strongly on the hydrophilic and hy-drophobic side groups as well as the the addition of alcohols. Among the variousmorphologies are planar, spherical and tubular structures [98, 100]. These struc-tures give rise to strong differences in their optical properties, owing to the influenceof the side chains and the environment on the details of the spatial arrangementof the molecules within the aggregate. A common feature for all these aggregatesis that their optical absorption spectra are red shifted as a whole compared to themonomer spectrum, a spectral signature characteristic for J-aggregates. To date,however, it remains a challenge to directly relate the specific optical features of theaggregate to its structural details on the molecular scale, as the spatial resolution ofcryo-TEM images is not sufficient to resolve the underlying details of the moleculararrangement in the aggregate [98,100,161].

In this chapter, we address this issue for a particular type of amphiphilic cya-nine dye, namely the dye 3,3’-bis(2-sulfopropyl)-5,5’,6,6’-tetrachloro-1,1’- dioctyl-benzimidacarbocyanine (C8S3 for short) which in solution self-assembles into ag-gregates of cylindrical geometry [64,65]. The reason for our interest in these specificaggregates is that their tubular structures show strong similarities with the photo-synthetic antenna complexes (chlorosomes) found in green bacteria [48–52,66–68].

1Parts of this chapter have been published as D. M. Eisele, C. M. Cone, E. A. Bloemsma, S.M. Vlaming, C. G. F. van der Kwaak, R. J. Silbey, M. G. Bawendi, J. Knoester, J. P. Rabe, andD. A. Vanden Bout, Nature Chemistry 4, 655 (2012).

4.1 Introduction 51

540 560 580 600 6200

0.5

1

1.5

2

2.5

λ (nm)

A(λ

) [a

rb.

units]

550 560 570 580 590 600 6100

0.2

0.4

0.6

0.8

1

1.2

λ (nm)

A(λ

) [a

rb.

units]

C8S3 aggregates Inner Wall aggregate(a) (b)

I: 599.6

II: 589.4

III: 581

V: 558.8

V: 568.2

B: 580.0

A: 598.8

D: 558.4

C: 568.2

Figure 4.1.1: Experimental spectra of C8S3 aggregates. (a) Absorption spectrumfor the double-walled C8S3 aggregates. Absorption bands I, II, and IV are po-larized mainly parallel to the cylinder axis, while bands III and V are polarizedperpendicular to this axis. (b) Absorption spectrum of the inner cylinder obtainedafter oxidation of the double-walled aggregates. Bands A and C have parallel po-larization, while bands B and D are polarized perpendicular to the cylinder axis.Spectra are published in Ref. [139].

Moreover, C8S3 dyes form mostly single tubules with a highly uniform struc-ture [70], unlike for instance the closely related C8O3 dye which often are found toassemble into (helically twisted) bundles of several tubules [100,161].

Owing to the amphiphilic nature of the molecules, C8S3 aggregates possess adouble-layer tubular wall structure (commonly known as the inner and outer wall)with a typical wall-to-wall separation of about 3−4 nm [69–72], which explains whythey are often referred to as double-walled aggregates. An intriguing observationis that the optical properties of C8S3 aggregates are drastically influenced by thesolvent in which they are prepared [69]. Two different ways can be distinguished:in the direct route the dye is directly dissolved in pure water as the solvent, whereasin the alcoholic route the dye is first dissolved in pure methanol before adding it topure water. Aggregates prepared by the two routes yield absorption spectra (shownfor the alcoholic route in Fig. 4.1.1(a), taken from Ref. [139]) consisting of severalstrongly absorbing exciton transitions that are partially narrowed and red shiftedwith respect to the monomer absorption (not shown here) [69,71,139]. In particular,common to both routes are two dominant absorption bands, labeled I and II inFig. 4.1.1(a), positioned at wavelengths λI ≈ 600 nm (λI ≈ 602 nm) and λII ≈ 589nm (λII ≈ 592 nm), respectively, for the alcoholic (direct) route. These bands arepolarized mostly parallel to the long axis of the tubes [69,71,162]. Additionally, for

52



the aggregates prepared via the direct route a third absorption band can be seen,centered around λIII ≈ 580 nm, which is mainly polarized perpendicular to thetubes’ long axis [69, 71]. In contrast, for the aggregates prepared via the alcoholicroute three absorption bands, labeled III − V in Fig. 4.1.1(a), can be discerned inthe spectrum in addition to the two main bands I and II: transition III appearsas a shoulder of band II, located roughly at λIII ≈ 581 nm, while bands IV andV are centered at wavelengths λIV ≈ 568 nm and λV ≈ 559 nm, respectively. Herethe polarizations of bands III and V are mostly perpendicular to the tubes’ longaxis, while band IV is preferentially oriented parallel to this axis [69, 162].

The above explained differences in the absorption spectra strongly suggest dis-tinct molecular arrangements for the C8S3 aggregates prepared via the direct andthe alcoholic route. In previous work by Knoester and co-workers [71], the spectralproperties of C8S3 aggregates prepared via the direct route have been successfullyexplained using an exciton model for cylindrical aggregates. In this model, themolecular arrangement is obtained by wrapping a planar brick-layer lattice, whereeach unit cell of the lattice is occupied by one cyanine dye molecule, onto a cylin-drical surface [62, 71, 97]. To account for the double-layer structure of the C8S3aggregates, two such cylinders (i.e., the inner and outer wall) with the appropri-ate radii are arranged in a concentric way. The collective optical excitations ineach cylinder, resulting from the strong excitation transfer interactions betweenthe various molecules, were described by a Frenkel exciton Hamiltonian [3, 4]. Animportant assumption here is that the coupling between the inner and outer wallis weak enough to neglect the occurrence of coherent exciton states shared byboth. Evidence for the validity of this assumption has been provided by previouspump-probe experiments [78], while further support has been obtained from theo-retical estimations of the inter-wall interaction strengths [71]. Within this model, itcan then readily be shown that each cylindrical aggregate exhibits two absorptionbands resulting from three allowed superradiant transitions; one polarized alongthe cylinder axis and two (degenerate) polarized perpendicular to this axis [97].Consequently, the two dominant absorption bands I and II in the spectra of thedirect route C8S3 aggregates, have been assigned to exciton transitions of the innerand outer cylinders, respectively, while transition band III is believed to containexciton contributions from both the inner and outer wall cylinders.

Recently, the point of view that the C8S3 aggregates consist of two weaklycoupled exciton manifolds, and thus that their optical response can be describedas a simple superposition of two single cylinder contributions, has been disputedby a two-dimensional electronic spectroscopy study combined with theoretical sim-ulations based on a Frenkel exciton model that takes into account all resonanceinteractions between molecules, both within and between the cylinders [83]. Theresults suggest the formation of coherent exciton states shared by both walls, re-

4.2 Theoretical model 53

sulting in a single collective optical response of the C8S3 aggregates.In this chapter, the main results of which are reported in Ref. [139], we present

the first detailed microscopic model of double-walled C8S3 aggregates prepared ina water/methanol solution (alcoholic route) and their optical properties, developedthrough a combination of experimental and theoretical techniques, and provide athorough analysis of the importance of excitation transfer interactions between theinner and outer cylinders. In particular, simple redox chemistry can be used as ahelpful tool to address these issues [139, 163]. Through chemical oxidation of thedyes preferentially located in the outer cylinder of the C8S3 aggregates, therebyessentially destroying the optical response of this wall, it is possible to isolatethe absorption spectrum associated with the inner cylinder [139]. The resultingspectrum, shown in Fig. 4.1.1(b), thus provides us with a well-defined referencepoint for the investigation of the molecular arrangement of the C8S3 aggregates.In particular, the spectrum reveals a total of four absorption bands for the innerwall, which strongly hints at a two molecule per unit cell molecular arrangementof the cylinders. Moreover, comparison of the spectra before and after oxidation(Fig. 4.1.1(a) and (b), respectively) clearly showed that the absorption associatedwith transition II vanishes entirely upon oxidation, consistent with the assumptionthat this transition arises from excitations on the outer wall. Although the othertransitions I, III − V revealed a decrease in absorption strength upon oxidation,their energy positions and shape showed no appreciable changes [Fig. 4.1.1(b)],which strongly supports the idea that the optical response of the double-walledC8S3 aggregates is determined by the separate absorption contributions of theinner and outer cylinders. Also, it is evident that transitions IV and V , previouslyof unknown origin, arise from excitations of the C8S3 aggregates itself, rather thanfrom other species like single dye molecules.

This chapter is organized as follows. Section 4.2.1 presents the geometricalmodel that we use for the cylinders, Sec. 4.2.2 gives the Frenkel exciton Hamil-tonian and expressions for the absorption spectra. In Sec. 4.2.3 we discuss theparameters of the model, while Sec. 4.2.4 provides details about the fitting proce-dure. Section 4.3 is devoted to the numerical results and the comparison with theexperimental spectra, while we summarize and make several concluding remarks inSec. 4.4.

4.2 Theoretical model

4.2.1 Aggregate geometry

It is well-known that the tubular structure of the inner and outer cylinder inthe double-wall structure of the C8S3 aggregates can be obtained by wrapping

54



two-dimensional sheets of molecules around cylindrical surfaces of appropriate ra-dius [71, 97], similar to the way carbon nanotubes are formed out of graphenesheets. The number of absorption bands observed in the optical spectrum of theinner cylinder strongly suggest that the underlying molecular arrangement of thecylinder is based on a two molecule per unit cell geometry, giving rise to Davy-dov splitting of the characteristic parallel and perpendicular transition. Moreover,in planar geometries, amphiphilic cyanine dye molecules are generally understoodto self-assemble into either brick-layer or herringbone structures [138, 164–169].Therefore, a natural starting point to model the microscopic structure of the innerand outer wall is to roll a two molecule per unit cell lattice with an in-plane brick-layer or Herringbone arrangement of the molecules, onto cylinders of appropriateradius.

Common to all these structural models is that the simulation of their linearoptical response (not shown here) leads to intensities of the high-energy transitions,i.e. transitions IV and V in Fig. 4.1.1(a), which are significantly too small incomparison with the experimental data. Interestingly, of these structures onlythe Herringbone model could correctly reproduce the energy positions and thepolarizations of the four transition bands in the experimental absorption spectrumof the isolated inner wall cylinder. To increase the strength of the high-energytransitions IV and V , we used a Herringbone structure but, in addition, allowedthe molecules to be tilted out of the plane. This model, which we refer to as theExtended Herringbone model (abbreviated EHB), is described below.

The EHB geometry is constructed by starting with a simple planar (x− y)-lattice, shown in Fig. 4.2.1(a). Here each brick denotes a C8S3 molecule, whoselength and thickness are given by a and d, respectively. We assume that twoadjacent rows of bricks are not shifted with respect to each other. The transitiondipole of the molecule has magnitude µ and is directed along the y-direction, thatis, along the long axis of the bricks.

The EHB structure is now specified by performing two consecutive rotations ofthe C8S3 molecules. First, all C8S3 molecules are rotated counterclockwise overan angle φ around the y-axis, i.e. the axis which is parallel to the length of themolecule and passes through the center of the molecule, as depicted in Fig. 4.2.1(b).For future reference, we refer to this rotation by its complementary angle δ (i.e.,δ = π/2−φ). Second, all molecules are rotated around the x-axis by, alternatingly,an angle β and −β, as shown in Fig. 4.2.1(c). Note that the latter rotation givesrise to a lattice with two molecules per unit cell.


Figure 4.2.1: Scheme illustrating the Extended Herringbone (EHB) lattice. See themain text for details. Figures 4.2.1(b-d) published in Ref. [139], figures 4.2.1(a)and (e) adapted from Ref. [172], and figure 4.2.1(f) adapted from Ref. [97].

Clearly, after these rotations the C8S3 molecules have been tilted out of theoriginal planar lattice, and, as a consequence, also the molecular transition dipolevectors. Consequently, the distance ∆x between adjacent rows in the x-direction[Figs. 4.2.1(c) and 4.2.1(d)] is not directly determined by the dimensions of theC8S3 molecule, but rather follows from the interactions between the phenyl ringsof the C8S3 molecules (see Sec. 4.2.3). Hence, the EHB structure retains a highdegree of π-stacking, which is a feature common to many dye aggregates and playsa crucial role in the details of the microscopic arrangement within these aggregates.

The molecular structure of the tubular aggregate may now be obtained bywrapping the planar structure onto a cylindrical surface into the direction of achiral vector C [see Fig. 4.2.1(e)], whose beginning and end points should coincideafter the rolling [97]. In this way, the tube’s long axis is oriented perpendicular toC, while the length of C equals its circumference. The angle between the x-axisand C is the rolling angle, which we denote by θ [Fig. 4.2.1(e)]. After wrapping,we may use the common stack-of-rings representation to determine the positions

56



and transition dipole vectors of all molecules within the cylindrical coordinateframe [97]. Within this representation, the cylindrical aggregate is viewed as N1

equidistantly stacked rings [see Fig. 4.2.1(f)]; here, each ring has radius R and isoccupied by N2 unit cells that are positioned symmetrically on the ring. Each ringis rotated with respect to the previous one over a helical angle γ. The distancebetween adjacent rings is given by h. Using the notation n = (n1, n2) for the unitcell located at the nth

1 ring (n1 = 1, 2, ..., N1) at the nth2 helix (n2 = 1, 2, ..., N2),

and j = 1, 2 to denote the molecules within the unit cell, we obtain the followingexpression for the three-dimensional molecular position vectors rn,j in the standardCartesian coordinate system (given by unit vectors ex′′ , ey′′ and ez′′ ),

rn,j =R cos (n1γ + n2φ2 + δj,2∆φ) ex′′

+R sin (n1γ + n2φ2 + δj,2∆φ) ey′′ + (n1h+ δj,2∆z) ez′′ .(4.2.1)

The corresponding transition dipole vectors µn,j are given by,

µn,j =[µz,j cos (n1γ + n2φ2 + δj,2∆φ)

− (µx,j cos θ + µy,j sin θ) sin (n1γ + n2φ2 + δj,2∆φ)]ex′′

+[µz,j sin (n1γ + n2φ2 + δj,2∆φ)

+ (µx,j cos θ + µy,j sin θ) cos (n1γ + n2φ2 + δj,2∆φ)]ey′′

+[−µx,j sin θ + µy,j cos θ

]ez′′ .

(4.2.2)

Here we defined φ2 ≡ 2π/N2 and δj,2 is the Kronecker delta which results from thechoice to coincide the center of the unit cell with molecule j = 1. Furthermore, thetwo quantities ∆φ and ∆z define the relative position of the two molecules in theunit cell and are the components in the azimuthal direction and the axis direction,respectively. They are given by

∆z = ∆y cos θ −∆x sin θ, ∆φ =∆y sin θ +∆x cos θ

R. (4.2.3)

Finally, µx,j , µy,j and µz,j are the Cartesian components of the transition dipolevectors before the wrapping process. Expressing the rotations in the EHB modelby the following rotation matrix,

A =

cos δ ∓ sinβ sin δ − sin δ cosβ0 cosβ ∓ sinβ

sin δ ± cos δ sinβ cos δ cosβ

, (4.2.4)

allows us to directly determine µx,j , µy,j and µz,j through simple matrix multipli-cation.


4.2.2 Hamiltonian and absorption spectrum

We model the optically active electronic states of the C8S3 aggregates by a Frenkelexciton Hamiltonian. Taking into account energy transfer interactions betweenmolecules within a single tube (intra-tube) as well as interactions between moleculesbelonging to different tubes (inter-tube), the Hamiltonian reads

H =∑

X

∑

n,j

ωXn,j |n, j;X〉〈n, j;X|+

∑

X

∑′

n,m,j,j′

JXjj′ (n,m) |n, j;X〉〈m, j′;X|

+∑

X′ 6=X

∑

n,m,j,j′

JXX′

jj′ (n,m) |n, j;X〉〈m, j′;X ′| .

(4.2.5)

Here |n, j;X〉 denotes the state where only the jth molecule (j = 1, 2) in unit celln, located in cylinder X (either the inner or outer wall), is in its excited state,whereas all other molecules are in their ground state.

The first term in Eq. (4.2.5) describes the excitation energies of the molecules inboth cylinders. Here, energetic (static) disorder is taken into account by assumingrandom, uncorrelated values for ωX

n,j from a Gaussian distribution with mean ω0

and variance σ2. Thus, the quantity σ may be interpreted as a measure for thestrength of the disorder. The second term in Eq. (4.2.5) accounts for the excitationtransfer interaction between molecules situated on the same cylinder, giving riseto collective excitations (excitons) within each cylinder. Here the prime excludesthe terms with simultaneously n = m and j = j

′. The final term in Eq. (4.2.5)

represents the intermolecular coupling between the cylinders, which may possiblyresult in the occurrence of coherent exciton states shared by both. We assume allresonance interactions to be non-fluctuating quantities and calculate them usingthe extended dipole model [71, 138]. Here, the transition dipole of a molecule isthought of as two point charges with opposite sign, separated by a distance L.The transfer interactions between molecules are then found from the Coulombinteractions between the four charges involved.

For a particular disorder realization, numerical diagonalization of Eq. (4.2.5)yields the exciton eigenstates of the double-walled C8S3 aggregates. These may bewritten as

|q〉 =∑

X

∑

n,j

ϕq (n, j,X) |n, j;X〉 , (4.2.6)

where ϕq (n, j,X) denote the exciton wave function amplitudes of the state labeledby quantum number q. The corresponding eigenvalue gives the energy Eq of theexciton state with quantum number q.

The general expressions for the linear optical spectra in terms of the abovementioned eigenstates and energies can easily be derived using Fermi’s golden rule.

58



For explicitness, the absorption spectrum taken in an isotropic sample reads [97]

A(ω) =

⟨⟨∑

q

Oqδ(ω − Eq)

⟩⟩, (4.2.7)

where the oscillator strengths Oq are given by

Oq =∑

X,X′

∑

n,m

∑

j,j′

ϕq (n, j,X)ϕ∗q (m, j′, X ′)

⟨(µ

Xn,j · e

) (µ

Xm,j′ · e

)⟩. (4.2.8)

The double angular brackets 〈〈. . . 〉〉 in Eq. (4.2.7) denote an average over disorderrealizations, whereas the single brackets 〈. . . 〉 represent an orientational averagewith respect to the polarization vector e of the incoming light and, finally, the as-terisk ∗ indicates complex conjugation. Note that the isotropic absorption spectrumis basically a sum of delta functions located at the exciton energies and weightedby the oscillator strengths of the exciton states.

We now turn to discuss the situation in which both inter-wall resonance interac-tions and disorder are absent, i.e. ωX

n,j = ω0 for all n, j andX, so that we are in factdealing with single-walled homogeneous nanotubular aggregates. In this case, therotational symmetry around the cylinder axis imposes a Bloch form [exp (ik2n2φ2)]for the exciton wave function in the azimuthal direction (n2 direction) and, as aresult, there exist strong optical selection rules for homogeneous cylinders. Thatis, only exciton bands characterized by quantum wavenumbers k2 = 0 and k2 = ±1contain states that can carry oscillator strength [62, 97]. Moreover, even withinthese bands only very few states have significant oscillator strength, as may readilybe seen by imposing periodic boundary conditions in the cylinder axis direction, anapproximation justified by the cylinder’s large length. Specifically, when a Blochform for the exciton wave functions in the cylinder axis direction (n1 direction) isassumed, only states characterized by wave vectors k = (k1, k2) with k1 = 0 andk2 = 0, or k1 = γN1/ (2π) and k2 = ±1 contain oscillator strength [97,116].

Due to the fact that we deal with two molecules per unit cell, the above in-troduced Bloch exciton states do not fully diagonalize the Hamiltonian of thesingle-walled tubular aggregates. Rather, the resonance interactions between themolecules gives rise to (dimer-like) eigenstates that are symmetric and antisym-metric linear combinations of the Bloch exciton states with the same quantumvector k. The absorption spectrum of such nanotubular aggregates is expectedto be dominated by four distinct peaks. In particular, two peaks may be asso-ciated with Davidov-split states characterized by wavevector k = (0, 0) and arepolarized parallel to the cylinder axis. Furthermore, we find a total of four tran-sitions corresponding to the wave vector k = (γN1/ [2π] ,±1), all of which arepolarized perpendicular to the cylinder axis. As the states with quantum number


k2 = ±1 are degenerate, only two distinct peaks may be associated with thesefour (Davydov-split) transitions. As a result, we thus find spectra that contain upto four distinct peaks, which yields a simple explanation for the experimentallyobserved absorption spectrum of the isolated inner wall.

4.2.3 Parametrization

The EHB model contains several parameters. Values for most of them can bedetermined from experimental data, while the remaining parameters are treated asfitting parameters.

We begin with the relevant energetic parameters ω0, Q and L that characterizethe molecular transition frequencies and the resonance interactions. The valuesfor all these parameters have been taken identical to those in Ref. [71]; that is,we have ω0 = 18868 cm−1, corresponding to a single-molecule transition at 530nm. Furthermore, we take extended dipole charges Q = 0.34e (e the charge of anelectron) separated by a distance L = 0.7 nm, in good agreement with the valuesobtained for similar dye molecules [138]. This yields a single-molecule transitiondipole of µ = 11.4 Debye.

Besides the energetic parameters, we also have structural parameters that deter-mine the geometry of the aggregate. The parameters a and d define the dimensions(length and thickness, respectively) of the C8S3 molecules and have fixed valuesa = 2.0 nm and d = 0.4 nm [71, 138]. The radius of the outer wall nanotube wasdetermined experimentally from line scans taken on cryo-TEM images, yielding anouter wall radius of ROW = 6.5±0.5 nm [69,70]. The radius of the inner wall cylin-der was determined in Ref. [72], where the double-walled nanotube was first filledwith silver after which line scans of cryo-TEM images were taken to determine theradius of the silver nanowire. This procedure results in a lower boundary for thediameter of DIW = 6.4± 0.5 nm [72]. However, in this way the measured diameterdoes not reflect the center positions of the molecules, but of the sulfite groups. Tocompensate for this, we assume here a slightly higher value for the inner wall radiusRIW = 3.5±1. These values for the inner and outer wall radius are assumed fixed,although small deviations from these values (well within the experimental errors)can be induced by the discrete nature of the underlying lattice.

The geometrical parameter ∆x defines the distance between molecules in theunit cell in the x direction. It is mainly determined by the interactions between thephenyl rings of the C8S3 molecules. Its value is taken from Ref. [170] as ∆x = 0.55nm. Finally, the structural parameter ∆y denotes the distance between moleculesin the y direction. This parameter can be obtained by assuming that the moleculesstacked on top of each other are optimally packed. This gives ∆y = d/ sin (β),where β is restricted to values β > 11.3. The latter lower boundary condition

60



ensures that the stacked unit cells are connected (i.e., it follows from closed packingrequirements).

The remaining three geometrical parameters are a priori not known and shouldthus be treated as fitting parameters. They are the two rotation angles (β andδ) that create the two-dimensional structure of the herringbone model and thewrapping angle (θ) that determines how this lattice is rolled onto the cylindricalsurface.

4.2.4 Fitting procedure

In order to fit the experimental spectrum of the double-walled C8S3 aggregates[Fig. 4.1.1(a)], we first focus on the isolated inner cylinder spectrum [Fig. 4.1.1(b)].We stress here that any fit of this spectrum should, in principle, reproduce theenergy positions, polarization directions, and relative oscillator strengths of thefour absorption bands. The polarizations of the four inner wall transitions wereassigned based on previous linear dichroism experiments. Explicitly, transitions Aand C are polarized parallel to the cylinder axis, while transitions B and D havepolarizations perpendicular to this axis.

The fitting procedure for the inner wall spectrum is as follows. We first optimizethe energy positions of the four peaks. Here we focus mainly on the separationsbetween the various peaks (not their absolute positions) in the absorption spectra,thereby allowing for an additional overall energy shift to align the absolute positionof the simulated spectra with experiment (see also Sec. 4.3.1). This typically resultsin a number of different sets of possible fit parameters β, δ, and θ. From these, themodel parameters that best reproduced all the other quantities, i.e. the relativeoscillator strengths and absolute energy positions of the transiton bands, werechosen as the optimal fit parameters for the inner cylinder. We like to mentionthat searching the entire parameter space with three free parameters in order tofind the optimal fit for the experimental inner tube spectrum is quite a tremendoustask. This may, however, be greatly simplified by the observation that the positionsof the parallel transitions are almost independent of the rolling angle θ, due to thefact the cylinder radius is much larger than the molecular dimensions, so thatlocally the structure of the cylindrical surface is almost flat. Thus, we may firstuse the rotation angles β and δ to determine the (relative) positions of the paralleltransitions, afterwards the rolling angle θ may be determined from the positions ofthe perpendicular peaks relative to the parallel ones.

With the molecular arrangement of the inner cylinder completely specified, wecan as a next step fit the experimental spectrum of the double-walled C8S3 aggre-gates [Fig. 4.1.1(a)]. Obviously, the molecular packing within the outer wall tubecan not be determined in the same way as was done for the inner wall because

4.3 Numerical results and comparison to experiment 61

parameters inner cylinder outer cylinderR (nm) 3.551 6.465N2 6 2h (nm) 0.296 0.055γ 36.5 33.4

θ 53.7 53.4

β 23.6 23.1

δ 25.6 28.0

Table 4.3.1: Summary of model parameters for the inner and outer wall geometriesin the absence of disorder.

there is no spectrum for the isolated outer wall available. It seems, however, un-likely that the supramolecular structure of the outer wall will be entirely differentfrom that of the inner wall. Therefore, we model also the molecular geometry ofthe outer wall with the EHB structure. The values of the outer wall fit parame-ters are found by requiring an optimal agreement between the calculations of theabsorption spectrum of the double-walled system and the experimental absorptionspectrum.

4.3 Numerical results and comparison to experi-

ment

4.3.1 Inner wall absorption spectrum: homogeneous limit

In Fig. 4.3.1(a) we present the comparison between the experimental absorptionspectrum (dashed line) for the inner cylinder of the C8S3 aggregates and the simu-lated spectrum (solid line) in the absence of disorder. The widths in the model spec-trum result from convoluting the calculated absorption transitions with Lorentzianlineshapes of appropriate widths (see the caption of Fig. 4.3.1 for details) in orderto facilitate comparison with experiment. In addition, the theoretical spectrumhas been normalized in such a way that the heights of the highest wavelength ab-sorption band in the experimental and calculated spectrum are the same. In oursimulations the number of molecules in the inner wall is taken as N = 7992, cor-responding to a cylinder length of LIW = 197.0 nm. The optimal fit parameterswere found to be: β = 23.6, δ = 25.6 and θ = 53.7 (see Table 4.3.1).

Comparison of both spectra in Fig. 4.3.1(a) reveals that the general featuresof the experimental spectrum are very well reproduced by the EHB model. In

62



550 560 570 580 590 600 6100

0.2

0.4

0.6

0.8

1

λ (nm)

A(λ

)

Model

Experiment

Lorentzian lineshape A in model

Lorentzian lineshape B in model

Lorentzian lineshape C in model

Lorentzian lineshape D in model

550 560 570 580 590 600 6100

0.2

0.4

0.6

0.8

1

λ (nm)

A(λ

)

Model

Experiment(a) (b)

Figure 4.3.1: Comparison of the experimental (dashed line) and simulated (solidline) absorption spectrum of the inner wall tube. Spectra are normalized such thatthe heights of the highest wavelength absorption bands are identical in experimen-tal and calculated spectra. (a) Simulated spectrum in the absence of disorder. Fitparameters are β = 23.6, δ = 25.6, and θ = 53.7. The four transitions have beenbroadened by Lorentzian lineshapes (dotted lines) of widths (FWHMs): WA = 60cm−1, WB = 150 cm−1, WC = 100 cm−1, and WD = 360 cm−1. (b) Simulatedspectrum for disorder value σ = 250 cm−1 averaged over 50 disorder realizations.Fit parameters are β = 23.6, δ = 26.2 and θ = 53.7. Except for the high-est wavelength J-band absorption band, all transitions have been broadened byLorentzian lineshapes to facilitate comparison with the experiment. Experimentaland simulated homogeneous spectrum published in Ref. [139].

particular, we find that our model indeed gives rise to four clearly distinct absorp-tion peaks (labeled as A, B, C, and D in order of descending wavelength), withthe correct polarization properties. Specifically, A and C are polarized parallelto the cylinder axis, whereas peaks B and D are perpendicular polarized to thisaxis. Thus, our model provides strong evidence that all four peaks found in theexperiment are associated with exciton transitions.

A closer look at the simulated spectrum reveals that the absorption peaks A, B,C, and D are located at wavelengths λA = 598.8 nm, λB = 580.0 nm, λC = 571.3nm, and λD = 560.9 nm. It is clear that transitions A and B are located atthe same positions as the corresponding peaks in the experimental spectrum butthat transitions C and D have slightly overestimated wavelengths (in the order ofa few nm) in comparison to the experimental spectrum. This small discrepancydisappears if disorder is taken into account (i.e., see Fig. 4.3.1(b) and the discussionbelow). We stress that the overall simulated spectrum has been shifted to lower


wavelengths by approximately 15.3 nm in order to obtain the correct absoluteenergy values of the absorption peaks in experiment. Thus, our model correctlyreproduces the relative energies of the absorption peaks, but not their absolute

values. In general, upon aggregation there might be energetic shifts present whichare not included into our model. These can include solvent shifts and shifts dueto coupling to vibrational modes [13]. In this respect, we note that the overallenergetic shift found here is of comparable magnitude to what has been used inprevious studies on related cylindrical aggregates [71].

Thus far, we have examined the spectral position and polarization of the ab-sorption transitions. Another important property of these transitions are theiroscillator strengths (intensities), as they indicate the strength of the transition.Investigating Fig. 4.3.1(a), we see that the relative oscillator strengths of peaks A,B, and D as obtained from the model agree reasonably well with the experiment.In particular, transitions A and B have roughly the same oscillator strength asin experiment, whereas transition D has somewhat larger oscillator strength. Themain discrepancy regarding the intensities is related to transition C. The oscillatorstrength corresponding to this transition seems to be underestimated in our model(see also the caption of Fig. 4.3.1), although in experiment this peak also has thesmallest contribution to the spectrum. We like to stress here that, despite someminor discrepancies, in general our model can very well reproduce the qualitativefeatures of the experimentally observed inner wall spectrum.

To test the range of validity of the set of optimal geometrical angles for the innerwall, we first varied all three fit parameters separately, keeping the other two fixedat their optimal value and compared the resulting spectra with the experimentaldata. We found that the rotation angles could be varied up to β = 23.6 ± 0.1

and δ = 25.6 ± 0.5, while still retaining good agreement with the experimentallyobserved isolated inner wall spectrum. For the rolling angle we found an errormargin of roughly θ = 53.7 ± 3. Here we note that due to the discrete natureof the plane that is wrapped around the cylinder, this angle can only take oncertain discrete values, in contrast to the continuous nature of the rotation angles.Moreover, adjusting the rolling angle is accompanied by variations in the radius ofthe inner wall cylinder, RIW = 3.55± 0.3 nm, which remains well within the limitsset by the Cryo-TEM observations.

As a next step, we varied all three geometrical parameters of the inner wallsimultaneously around their optimal values to specify the region of the three-dimensional parameter space where unique sets of geometrical angles can be foundthat give good agreement with the experimental absorption spectra. We observedthat within the space limited by roughly one degree variation for the rotation an-gle β and several degrees of variation for the other two angles, multiple sets ofgeometrical parameters could be found that correctly reproduce the relative en-

64



ergy positions and polarization directions, and yielded reasonable intensities forthe four transition bands. From these possible structures, the optimal geometry(i.e., the one for which the spectrum is depicted in Fig. 4.3.1(a))was determined asthe structure that could give the correct intensities of the absorption peaks, whilemaintaining the overall energy shift minimal.

4.3.2 Inner wall absorption spectrum: inhomogeneous limit

Up till now, we discussed the spectrum of the inner tube aggregate without dis-order. In reality, however, local irregularities in the solvent host will generallygive rise to fluctuations in the solvent shifts in the transition energies and dipolesof the molecules within the aggregate [117]. Here we will model disorder of themolecular transition energies by choosing the excitation energies of all moleculesin the aggregate in a random, uncorrelated fashion from a Gaussian distributionwith mean value equal to ω0 = 18868 cm−1 (i.e., the averaged single moleculetransition energy) and standard deviation denoted by σ (see also Sec. 4.2.2). Thedisorder strength σ was determined from the width of the lowest energy J-band ofthe experimental spectrum, where we assume that this width completely derivesfrom inhomogenous (disorder-induced) broadening. The in this way determinedvalue provides an upper bound for σ, because, in reality, also homogeneous (life-time) broadening is likely to contribute to some extent to the width of the lowestenergy J-band, certainly at room temperature where intraband scattering typicallydominates the exciton lifetime [171].

Figure 4.3.1(b) shows the calculated inner wall spectrum for disorder σ = 250cm−1 together with the experimental spectrum. It is evident that the spectralfeatures resemble those already found for the homogeneous tubes. One slight im-provement here compared to the homogeneous case [Fig. 4.3.1(a)] is that absorptionbands C andD now appear at the same wavelength positions as in experiment. Thereason for this is that the highest wavelength peak A undergoes disorder-inducedredshifting, while the other transitions are rather insensitive to this phenomenon.This effectively decreases their wavelength position as we introduce an overall shiftof the spectrum such that transition A occurs at 598.8 nm). However, because tran-sition band B already occurred at the right position in the homogeneous spectrum,we slightly adjusted the value for δ to correct for this, which now equals δ = 26.2;the values of the other fitting parameters (β and θ) remained unchanged. Notethat the entire spectrum, taking the disorder-induced shift into account, has beenshifted by 16.4 nm to lower wavelengths, in order to match the experimental data.

From the model parameters that generated the fit in Fig. 4.3.1(b), we foundcylinders with radius RIW = 3.55 nm, N2 = 6 unit cells per ring (thus a totalof 12 molecules per ring), a ring-to-ring distance h = 0.30 nm, and a helicity


angle γ = 36.5. These values translate into 41 molecules on the cylinder surfaceper nanometer length, resulting in a molecular surface density which is somewhathigher than reported previously for related cylindrical aggregates [71].

The details of the molecular packing of the cylinders also allow us to determinethe extended dipolar resonance interactions between the molecules in the cylinder.The strongest negative interactions have magnitude Jmin ⋍ −1070 cm−1, whereasthe strongest positive interactions are equal to Jmax ⋍ 1576 cm−1. Surprisingly,we thus have negative as well as positive values for the dominant molecular inter-actions resulting in a mixture of both H and J-aggregation. The aggregate itself,however, is still classified as a J-aggregate, as is evident from the overall red shiftedspectrum compared to the monomer absorption. As a consequence of the differentsigns of the interactions, the energetically lowest superradiant state is no longersituated exactly at the bottom of the exciton band (as is the case when only nega-tive interactions are present), but rather lies somewhat higher up the band, whichmay severely reduce the fluorescence of the aggregates. For the optimal geometryof the inner wall (without disorder), we found that the energetically lowest super-radiant state is located roughly 172 cm−1 above the exciton band bottom. At roomtemperature, the superradiant state therefore remains significantly populated afterthermal equilibrium of an initial excitation, explaining the fact that the aggregateshows a strong fluorescence in experiment [70]. Also, because the disorder strength(σ = 250 cm−1) is larger than the energy difference between the superradiant stateand the exciton band bottom, energetically low lying exciton states may still ac-quire significant oscillator strength, thereby effectively increasing the amount offluorescence in comparison to the disorder-free situation.

To end this section, we like to mention that one of the significant effects ofdisorder is that it gives rise to localization of the exciton wave function on partsof the cylinder. Thus, the collective electronic excitations are not shared by all Nmolecules within the cylinder, but rather by some smaller number N∗. A typicalestimate of this localization number N∗ can be obtained by calculating the so-called participation ratio [33,34,124]. The results revealed that exciton states withenergies below the homogeneous superradiant state energy are typically shared byseveral hundreds of molecules for the disorder strength σ = 250 cm−1.

4.3.3 Full spectrum and effects of coupling

In the previous section we established a detailed microscopic model for the molecu-lar packing within the inner nanotube through comparison of the calculated opticalspectra with experiment. In this section we will also model the microscopic geom-etry of the outer wall by the EHB structure (see also Sec. 4.2.4) to determine theoptical response of the double-walled C8S3 aggregates. In particular, we focus on

66



the effects of the resonance interactions between the inner and outer wall by dis-tinguishing between the situation where all inter-tube molecular interactions aretaken into account and the case where they are completely neglected (i.e., the twocylinders are treated as two separate, uncoupled aggregates).

Figure 4.3.2 displays the calculated absorption spectra of the double-walled ag-gregates in the absence of disorder with and without the inclusion of the resonanceinteractions between the two walls. In the simulations, the values of the struc-tural parameters of the inner wall are taken identical to those determined from theisolated inner wall spectrum (see Fig. 4.3.1). The outer wall fit parameters, bothfor the absence and presence of inter-wall interactions, are given by βOW = 23.1,δOW = 28.0 and θOW = 53.5 (Table 4.3.1). These values are very close to thosefor the inner wall, which lends support to the model, as one does not expect strongdifferences in these structures because the cylinder radii are much larger than themolecular dimensions. Furthermore, the inner and outer wall are assumed to havethe same length, which gives NOW = 14260 molecules in the outer wall. We notethat the strongest inter-wall interactions are equal to 28 cm−1, which is of similarmagnitude to previous reported values [71].

It follows from Fig. 4.3.2 that both simulated absorption spectra (with and with-out inter-wall couplings) resemble the experimental spectrum of the double-walledaggregates reasonably well. In particular, both the peak positions and transitionstrengths of the high wavelength part of the spectrum (from 575 nm onward) areconsistent with the experimental data (i.e., peaks I−III). In this respect, we notethat in experiment the highest wavelength transition I of the double-walled systemis shifted by less then a nanometer compared to the inner wall spectrum. Thissmall shift is not observed in the calculated spectra. In the low wavelength regionof the spectrum (below roughly 570 nm) we observe larger discrepancies betweenexperiment and the calculations. Especially, the experimental results reveal twoseparate peaks IV and V around 568 nm and 559 nm, while the calculated spectrashow a single, rather broad absorption peak. The latter results from the lowestwavelength transition of both the inner and outer wall, which occur roughly at thesame position, and should thus be associated with peak V . Here it is interestingto notice the clear distinction between the calculations with and without inter-wallinteractions present. The reason that peak IV of the experiment is not visible inthe calculated spectra is most probably caused by the fact that the EHB modelunderestimates the intensity of this transition, as already shown in Fig. 4.3.1. Wealso performed simulations of the optical response of the double-walled aggregatesin the presence of disorder (not shown here). For σ = 250 cm−1, no significant im-provement was found in the thus obtained fit of the spectrum and also no significantchanges in the structural fit parameters were obtained.

As is evident from Fig. 4.3.2, in the high wavelength region the spectral features


540 550 560 570 580 590 600 6100

0.5

1

1.5

2

2.5

λ (nm)

A(λ

)

Model: uncoupled tubes

Experiment

Model: coupled tubes

III

IV

V

II

I

Figure 4.3.2: Comparison of the experimental (dashed line) and calculated absorp-tion spectra of the double-walled tubular aggregate in the absence of disorder. Thetwo simulated spectra shown are calculated with (dash-dot line) and without (solidline) resonance interactions between both walls taken into account. Spectra arenormalized such that the heights of absorption band II are identical in experimen-tal and calculated spectra. Fit parameters outer wall are β = 23.1, δ = 28.0,and θ = 53.5; fit parameters inner wall same as in Fig. 4.3.1. The calculatedstick spectra are broadened by Lorentzian lineshapes of various widths to facilitatecomparison with the experiment. Spectra are published in Ref. [139]

are not significantly affected by inter-wall couplings, that is, we observe no profoundpeak shifts and/or intensity changes. More specifically, we found upon electroni-cally coupling the walls for the highest wavelength superradiant state a shift of 2cm−1 and a change in intensity, given by the ratio of the oscillator strengths ofthe superradiant state before and after coupling of the tubes, of ∆I = 0.89. Forthe superradiant state at 589 nm we obtained no peak shift at all and only a smallintensity change of ∆I = 1.04. To corroborate on this further, we calculated forthese two states the probability to find the excitation on either one of the walls.In the absence of disorder, we found that the state at 599 nm is almost completelylocated on the inner wall (99%), while the superradiant state at 589 nm is primar-ily localized on the outer wall tube (91%). These findings agree with the aboveobservations regarding the absorption spectra and establish that in the absence of

disorder, even when accounting for inter-wall interactions, the two lowest energypeaks of the double-walled aggregate can indeed be associated with excitations oneither the inner or the outer wall.

68



Contrary to the high wavelength region, the low wavelength structure revealssome significant spectral changes upon inclusion of inter-wall interactions (seeFig. 4.3.2). Consequently, we expect that here the superradiant exciton statesare shared by both walls to certain extent. Calculation of the excitation proba-bilities in the absence of disorder for the superradiant states in the 580 nm region(absorption shoulder in Fig. 4.3.2) and the 550 nm region confirms this picture.Specifically, for the 580 nm region the probability to find the superradiant excitonson the inner wall equals roughly 40%, whereas in the 550 nm regime this probabilityis even close to 50%.

To end this section, we will discuss the excitation probabilities in more detail,focussing on the regions of the two absorption transitions near 599 nm and 589nm. Figure 4.3.3 presents histograms for the probability to find a certain excitonstate on the inner or outer wall, denoted respectively by ρ(PIW ) and ρ(POW ),both for the disorder free situation [Figs. 4.3.3(a) and 4.3.3(b)] and with disorderσ = 250 cm−1 included. Here all exciton states located near the maxima of thetwo absorption bands were taken into account (see caption Fig. 4.3.3 for details)rather than just the superradiant states.

In the absence of disorder, the histograms in Figs. 4.3.3(a) and 4.3.3(b) revealthat, contrary to the superradiant states, the dark exciton states (which do notcontribute to the spectra as they have no or very little oscillator strength) show alarger variety in their excitation probabilities. In particular, both near the 599 and589 nm regions there exist states which are shared (to any extent) by both tubes,although a certain preference for excitation on the inner wall (outer wall) in the599 nm (589 nm) region can still be observed. In principle, whether exciton statesassociated with the inner and outer walls strongly mix due to inter-wall interactions,thus resulting in new states that are shared by both walls, depends on the ratioof the coupling strength between the exciton states and their energy difference.Due to the high density of states of the cylinders (resulting from the large numberof molecules) and the overlap between the inner and outer wall exciton band inthe optical regime, even small values of the coupling between exciton states canlead to strong mixing of those states. This results in states where the excitation iscoherently shared by both walls. In this sense, the high wavelength superradiantk = (0, 0) states apparently obey strong selection rules that forbid the couplingwith energetically close exciton states belonging to the other tube, while theseselection rules relax for the higher energy superradiant exciton states.


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

PIW

ρ(P

IW)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

POW

ρ(P

OW

)

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

PIW

ρ(P

IW)

0.5 0.6 0.7 0.8 0.90

0.01

0.02

0.03

0.04

0.05

POW

ρ(P

OW

)

586.5 − 593.5 nm

σ = 250 cm−1

595.2 − 602.4 nm

σ = 250 cm−1

595.2 − 602.4 nm

σ = 0586.5 − 593.5 nm

σ = 0(a) (b)

(c) (d)

Figure 4.3.3: Probability histograms for finding an exciton state on either one ofthe two walls for two different energy intervals. [(a), (c)] Normalized number ofexciton states with probability PIW to be located on the inner wall tube ρ(PIW )without disorder σ = 0 (a) and for disorder σ = 250 cm−1 (c). All exciton stateswith wavelengths between 595.2 nm and 602.4 nm were taken into account. [(b),(d)] Normalized number of exciton states with probability POW to be located on theouter wall tube ρ(POW ) without disorder (b) and for disorder σ = 250 cm−1 (d).All exciton states with energies between 586.5 nm and 593.5 nm were considered.

In the presence of disorder (σ = 250 cm−1), the probability histograms inFigs. 4.3.3(c) and 4.3.3(d) clearly show that the exciton states are shared to alarger extent by both walls compared to the disorder-free situation [Figs. 4.3.3(a)and 4.3.3(b)]. More specifically, the density of excitations ρ(PIW ) near the 599nm region and ρ(PIW ) near the 589 nm region are both single peak distributionswith maxima close to PIW = 1/3 and POW = 2/3, respectively. The reason thatdisorder leads to exciton states which are typically to a larger degree delocalizedover both walls in comparison to the homogeneous states, is because exciton statesoriginally belonging to either the inner or the outer wall do not only couple to eachother as a result of intra-wall resonance interactions but may also strongly mixwith each other due to the disorder.

70



It is evident from Figs. 4.3.3(c) and 4.3.3(d) that the excitations are typicallyshared by both walls according to an inner to outer wall ratio of roughly 1 : 2. Inother words, all excitation have a probability of roughly 1/3 (2/3) to be located onthe inner wall (outer wall). This ratio can be understood as follows. The similarityof the inner and outer wall molecular structure results in nearly identical surfacedensities of the tubes such that the number of molecules (and thus also excitonstates) is roughly twice as large for the outer wall (we deal with cylinders with equallengths and radii that differ roughly by a factor of two). Moreover, the density ofstates of the tubes are very similar, such that for each exciton state originally (nointer-wall interactions) belonging to the inner wall, there are on average two excitonstates (close in energy) that belong to the outer wall. This leads (due to disorder-induced mixing of these states) to the observed 1 : 2 ratio. It also suggests thatthe excitation distributions are roughly independent of the actual energies of theexciton states taken into account, as long as the exciton bands of inner and outerwall overlap, which has indeed been confirmed. We note that in the distributionsshown in Fig. 4.3.3 no distinction was made between exciton states with significantoscillator strength and those that carry very little oscillator strength, althoughsimilar results were obtained when only optically dominant states were considered.

To end this section, we point out that the analysis of the interactions betweenthe inner and outer wall described here does not take into account dynamic disorder(temperature effects), resulting from the interaction of the excitons with vibrationsin the environment. In principle, the coherence between the walls may easily bedestroyed by thermal noise, owing to weak inter-wall interactions, while at the sametime the exciton coherence within each wall remains due to the strong intra-wallcouplings. These effects may be of importance in room temperature experiments.We will come back to this in more detail in the summary below and in the nextchapter.

4.4 Summary and concluding remarks

In this chapter, we studied the optical spectra of self-assembled tubular aggregatesof C8S3 dye molecules in a water/methanol solution (alcoholic route preparation).As observed from cryo-TEM, these aggregates possess a double-walled structurewith diameters of 3.5 nm and 6.5 nm for the inner and outer walls, respectively,and lengths that extend up to micrometers. Through oxidation experiments, theabsorption spectrum associated with the inner cylinder could be isolated from thefull spectrum, which clearly reveals four distinct absorption bands associated withthe inner cylinder [139]. We proposed a microscopic model with a two moleculesper unit cell Herringbone lattice structure, where the molecules are tilted out ofthe lattice that is eventually rolled onto a cylindrical surface. Employing a Frenkel

4.4 Summary and concluding remarks 71

exciton model to describe the excited states, we have demonstrated that this ex-tended Herringbone (EHB) structure can very well explain the polarizations, oscil-lator strengths and relative spectral positions of all four absorption bands of theexperimental inner wall spectrum. In particular, our results strongly suggest thatthe two lowest wavelength transitions, which were previously of unknown origin,should be associated with excitonic transitions of the aggregate. Moreover, usinga similar EHB structure to model the outer wall, we obtained a good fit to the ex-perimental spectrum for the double-walled aggregate system. Also, we performedsimulations including static disorder in the transition energies of the individualmolecules, from which the disorder strength (standard deviation of the Gaussiandisorder distribution) was found to be at most 250 cm−1.

A relevant issue for future applications in energy transport wires and artificialphotosynthesis complexes is to what extent the excitation transfer interactions be-tween the cylinders affect their optical response and energy transfer properties. Wehave shown by comparing the simulated optical spectra with and without inter-wall resonance interactions that the high wavelength part of the spectrum is notsignificantly changed due to the inter-wall interactions. In particular, almost nopeak shifts and only minor changes in the intensities of the low-energy absorp-tion bands were observed. The low wavelength side of the spectra, on the otherhand, showed more prominent spectral changes. Especially, as a result of inter-wallinteractions the lowest wavelength absorption band shifted by roughly 10 nm tolower wavelengths compared to the experimental spectrum. By calculating proba-bility distributions to find an excitation on one of the walls, we found that in thepresence of disorder almost all exciton states are shared by both cylinders, despitethe small inter-wall interactions in comparison to the intra-wall intermolecular in-teractions. The reason for this is as follows. The degree of delocalization of anexciton state over both cylinders depends on the ratio of the coupling betweenexciton states originally (without inter-wall interactions) associated with the innerand outer wall and their energy difference. Due to the high density of states of thecylinders (resulting from the large number of molecules) and the overlap betweenthe inner and outer wall exciton band in the optical regime, even fairly small cou-pling strengths between exciton states will typically lead to strong mixing of thosestates. This effect is even enhanced by disorder, which leads to a further mixing ofthe exciton states of the inner and outer wall.

As mentioned in the Introduction, previous pump-probe experiments on theclosely related C8S3 aggregates prepared in a pure water solution (direct routepreparation) strongly support the point of view that these double-walled aggre-gates consist of two distinct cylinders that are electronically (at most) weaklycoupled, resulting in separate optical responses of both cylinders and a Forster-likeincoherent energy transfer between them [71, 78]. Moreover, the oxidation experi-

72



ments and, most recently also two-dimensional electronic correlation spectroscopytechniques, performed on the alcoholic route prepared C8S3 aggregates give fur-ther proof for claim [139, 162]. Our results with inter-wall interactions, however,suggest the occurrence of coherent exciton states shared by both walls, resultingin a coherent energy transfer between the walls. A possible explanation for theseconflicting results may lie in the here neglected effects of dynamic disorder inducedby interactions of the excitons with vibrations in the environment. Because of theweak nature of the inter-wall resonance interactions, even a rather small amount ofdynamic disorder quickly tends to completely destroy the coherence between thewalls, while at the same time the exciton nature of the states within the walls isnot severely affected (due to the strong excitation transfer interactions within thecylinder). It would therefore be interesting to extend the model and include theeffects of exciton-vibration couplings and temperature to see to what extent the ex-citon coherence between the walls is destroyed. In the next chapter, we address thisissue in simplified model systems, in particular small one-dimensional aggregates.

Chapter 5

Vibronic effects and

destruction of exciton

coherence in optical spectra

of J-aggregates: a variational

polaron transformation

approach

Using a polaron transformation of the model Hamiltonian, we study the interplay ofexcitation-vibration couplings, resonance excitation transfer interactions, and tem-perature in the linear absorption spectra of molecular J-aggregates. Semi-analyticalexpressions for the spectra are derived and compared with results obtained fromdirect numerical diagonalization of the Hamiltonian in the two-particle basis setrepresentation. At zero temperature, we show that the polaron model reproducesboth the collective (exciton) and single-molecule (vibrational) optical response as-sociated with the appropriate standard perturbation limits. Specifically, for themolecular dimer excellent agreement with the spectra from the two-particle ap-proximation for the entire range of model parameters is obtained. With increasingaggregate size, the comparison starts to reveal some spectral discrepancies, espe-cially manifested in the higher energy vibrational side bands of the spectra, albeitthe main features of the spectra are still in reasonably good agreement. Upon in-

74

Vibronic effects and destruction of exciton coherence in optical spectra

of J-aggregates: a variational polaron transformation approach

creasing the temperature, the spectra show a transition from the collective to theindividual molecular features, which results from the thermal destruction of theexciton coherence.1

5.1 Introduction

Low-dimensional J-aggregates comprised of interacting molecules form a fascinatingtopic of research with a rich history [159], starting with the pioneering works of Jel-ley and Scheibe in the 1930s [1,2]. Owing to their unique spectroscopic and energytransport properties, which include very narrow absorption lines [1,2,22,23,159], en-hanced spontaneous emission [20, 21, 73], strong nonlinear susceptibilities [15–18],and highly efficient excitation energy transfer [11, 66, 133], J-aggregates have al-ready been utilized abundantly for applications, for instance, as spectral sensi-tizers in photographic processes [25, 26]. Moreover, a fundamental understand-ing of the nature of the excitations in molecular aggregates is of great scientificinterest to gain better insight into the principles behind the extraordinary lightabsorption properties and energy transport efficiency of natural light-harvestingcomplexes [27,28,40,68].

In general, the collective electronic excitations (Frenkel excitons) which re-sult from strong excitation transfer (resonance) interactions between the moleculesin the aggregate [3, 4], are altered by scattering on vibrational degrees of free-dom [33, 35]. Understanding such vibronic excitations has been an ongoing chal-lenge for several decades already [30–32,173–180]. In the simplest approach, a weakcoupling between the excitons and the vibrations is assumed, which is treated usingperturbation techniques. These methods have already proven successful in the pastto explain a variety of optical and dynamical phenomena observed for molecularaggregates [181].

Approaches based on weak exciton-vibration scattering are by their very natureinadequate to explain features associated with strong exciton-vibration coupling orhigh temperatures. Examples include the vibronic progression in the optical spec-tra of aggregates and the temperature-induced destruction of exciton coherences.In fact, the problem of the distortion of the single-molecule vibronic structureupon aggregation of the molecules has recently attracted much attention in one-dimensional helical supramolecular structures formed by monofunctionalized oligo-phenylenevinylene (MOPV4) chromophores [182, 183], self-assembled tubular ag-gregates of tetra(4-sulfonatophenyl)porphyrin (TPPS4) [95], and Fenna-Matthews-Olson (FMO) complexes of green sulphur bacteria [184]. In addition, studies of

1This chapter is based on E. A. Bloemsma, M. Silvis, A. Stradomska-Szymczak, and J.Knoester, in preparation.

5.1 Introduction 75

double-layer tubular aggregates formed by amphiphilic cyanine dye molecules (seeRef. [61] for an excellent review) indicate that thermal destruction of the coher-ent nature of the exciton states may play an important role in these nanotubu-lar systems. In particular, both experimental findings and theoretical work haveshown that at room temperature the optical spectra of the C8S3 nanotubes canbe interpreted as the linear combination of the optical responses from two sepa-rate cylindrical aggregates, which strongly suggests that the two exciton manifoldsare (almost) completely uncoupled, with a Forster-like incoherent energy trans-fer between them [71, 78, 139]. The effect of interactions between two molecularassemblies, like the two cylindrical walls considered here, is however a complexproblem [29, 43, 83, 185]. In principle, two assemblies with overlapping excitonbands will give rise to many exciton states that are coherently shared betweenthem, even for very small interactions. The reason is that for most states in oneassembly, there will be one or more in the other one that has a very similar energy.However, if the coupling between the assemblies is weak, this coherence may easilybe broken by thermal noise (a dynamic environment), even if the coherence withineach assembly is maintained due to strong internal interactions.

Quantitative analysis of the aforementioned problems requires a method thattreats both the excitation transfer interactions between the assemblies and the in-teractions with the dynamic environment (vibration-excitation coupling) on equalfooting, i.e., non-perturbatively. Several numerical methods exist that go be-yond the standard perturbation approaches, among which are the density ma-trix renormalization group methods [186], hierarchial equations of motion tech-niques [187, 188], and multi-particle basis set approaches [178, 189–191]. Whilethese methods have clear advantages in the sense that they explore a larger part ofthe parameter space than the ordinary perturbation treatments, the correspondinghigh computational costs and complexity of these methods make them less suited.Also, the lack of analytical results makes these methods typically less insightfulthan perturbation treatments.

An alternative approach utilizes the polaron transformation [174,179,192,193].This approximate method is based upon a Lang-Firsov transformation of the modelHamiltonian [194], which leads to new exciton-vibration coupling terms that remainsmall over a larger range of parameters than the original vibronic interaction terms.Indeed, recently, it has been shown that this method can capture both the coherentenergy transfer limit and the Forster limit in a consistent way [195–197]. Cur-rently, however, the main focus lies on analyzing the energy transport phenomenawithin and between (multi)-chromophoric systems, while a detailed investigationof their spectroscopic features based on these polaron transformation methods hasremained relatively unexplored.

In this chapter, we study the signatures of exciton-vibration coupling, inter-

76



molecular interactions, and temperature (dynamic disorder) in the linear opticalabsorption spectra of J-aggregate model systems described by the Holstein Hamil-tonian [174]. We employ an augmented polaron transformation approach (abbrevi-ated as APTA), which exploits the symmetry of the model Hamiltonian [198,199].Specifically, it consists of two simultaneous transformations: a classical (full) po-laron transformation which is applied to the totally symmetric collective vibrationalmode, followed by a variational (partial) polaron transformation to minimize thecouplings between the electronic excitations and the remaining non-symmetric col-lective vibrational modes. The semi-analytical expressions for the low-temperatureabsorption spectra are compared with results obtained from direct numerical diago-nalization of the model Hamiltonian in the two-particle approximation (abbreviatedas TPA), where the latter serves as a benchmark to determine the range of validityof these expressions.

We establish that within this polaron framework the spectral properties asso-ciated with both of the standard perturbation limits are captured, while also inbetween these limits the spectra give reasonably good agreement with the resultsobtained from the TPA. In this respect, we find that the collective representationof the molecular vibrations combined with the variational nature of the polaron pa-rameter plays a key role for the accuracy of the results in this intermediate regime.In addition, we show that higher temperatures in general destroy the spatial coher-ence of the excitons and their related optical features, a phenomenon which may beof significant importance in the interpretation of room temperature experiments.

This chapter is organized as follows. In Sec. 5.2, we introduce the HolsteinHamiltonian model, present the augmented polaron transformation approach, andderive the expressions for the linear absorption spectrum of the molecular aggre-gate. In Sec. 5.3, we give the analytical results for the spectrum at zero temperatureand show explicitly how these contain both of the well-known perturbation limits.Section 5.4 is devoted to the numerical results; here, we show the comparison ofour method with the two-particle approximation, discuss the results in the light ofrelated polaron transformation methods, and present the temperature dependenceof the spectra. We conclude and make some final remarks in Sec. 5.5.

5.2 Theoretical framework 77

5.2 Theoretical framework

5.2.1 Model Hamiltonian

We consider the Holstein model to describe the optical properties of the molecularaggregate. It has the form (~ = 1) [174]

H =(E0 + λ2ω0

)∑

n

b†nbn +∑′

n,m

Jnmb†nbm +λω0

∑

n

(an + a†n

)b†nbn +ω0

∑

n

a†nan.

(5.2.1)Here, E0 and ω0 denote, respectively, the bare (adiabatic) molecular transitionenergy and vibrational mode frequency, Jnm is the matrix element of the excita-tion transfer interaction between molecules n and m (the prime in the summationexcludes the case n = m), and λ2 is the dimensionless Huang-Rhys factor, which de-termines the coupling strength between the vibrational and electronic excitations.As we will not account for electronic disorder, all molecular transition energies E0

are taken identical. Similarly, we do not consider disorder in the intermolecularinteractions Jnm. Furthermore, we assume that the frequency of each vibrationalmode ω0 is identical and that each mode couples to a single electronic excitationwith the same Huang-Rhys factor λ2.

The creation (annihilation) operators for electronic and vibrational excitationson molecule n are given by b†n (bn) and a†n (an), respectively. They satisfy the usual(anti-)commutation relations:

[bn, b

†m

]= δnm

(1− 2b†mbn

), b†nb

†n = bnbn = 0, and[

an, a†m

]= δnm. These relations reflect the bosonic nature of the vibrational oper-

ators, while the electronic operators are governed by Pauli commutation relations.We stress that each molecule can carry at most one electronic excitation, while thenumber of vibrational excitations is unlimited.

It is clear from Eq. (5.2.1) that the nature of the eigenstates of H is determinedby a competition between Jnm and λ. If λ = 0, a Bloch transformation diago-nalizes H (assuming periodic boundary conditions for the aggregate), giving riseto collective excitation waves (Frenkel excitons) and vibrations. The eigenstatesof the total Hamiltonian can then be expressed as Born-Oppenheimer products ofelectronic and vibrational wave functions for the whole aggregate. On the otherhand, if Jnm = 0, the eigenstates can be found by introducing displaced oscillatoroperators, yielding single-molecule excitation states. Thus, if one of these parame-ters (either Jnm or λ) is small compared to the other, one may attempt to find theeigenstates using a perturbation scheme.

If such a procedure fails, an alternative approach is to apply a full polaron trans-formation to Eq. (5.2.1) [194]. This transformation yields vibrational modes of themolecular excited state that are undisplaced compared to the ground state modes.Consequently, a new exciton-vibration interaction term appears after the transfor-

78



mation, which should be treated perturbatively. A generalization of this methodis the variational polaron transformation where the excited state vibrational modedisplacements are optimized using the variational polaron parameter [193]. Themain advantage of this method is that it typically yields exciton-vibration inter-actions terms which remain of perturbative nature for a wider range of systemparameters Jnm and λ compared to the full polaron transformation. It turns out,however, that these conventional polaron transformation approaches lead to inad-equate results for the optical response in the intermediate coupling regime (seeSec. 5.4.2), signifying the importance of the remaining exciton-vibration interac-tions in this regime. A natural way to overcome this is to develop approaches whichlead to further minimalization of these coupling terms. Below, we will demonstratethat this can be accomplished by rewriting the Hamiltonian [Eq. (5.2.1)] in thecollective vibrational mode representation, because it allows us to decouple in anexact way the totally symmetric vibrational mode from the electronic excitationsby means of a full polaron transformation. A variational polaron transformationis then applied to the other collective modes in order to minimize the remainingcoupling between the electronic excitations and these modes.

As mentioned above, the key point in our approach is to exploit the symmetryof the model. Thus, we introduce collective vibrational modes, defined by

a†q =1√N

∑

n

Φnqa†n, aq =

1√N

∑

n

Φnqan, (5.2.2)

where q denotes the vibrational quantum number and N is the total number ofmolecules in the aggregate. In what follows, we assume strictly real matrix elementsΦnq, which can be obtained as the symmetric and antisymmetric combinationsof the usual Bloch waves. Explicitly, for N odd we have: Φnq=0 = 1, Φnq =√2 cos (2πqn/N) if q = − (N − 1) /2, ...,−1, and Φnq =

√2 sin (2πqn/N) when

q = 1, ..., (N − 1) /2; for N even we find: Φnq=0 = 1, Φnq=N/2 = (−1)n+1

, Φnq =√2 cos (2πqn/N) if q = −N/2 + 1, ...,−1, and Φnq =

√2 sin (2πqn/N) when q =

1, ..., N/2− 1. In the collective vibrational mode representation H reads,

H =(E0 + λ2ω0

)∑

n

b†nbn +∑′

n,m

Jnmb†nbm +λω0√N

(a0 + a†0

)∑

n

b†nbn + ω0a†0a0

+λω0√N

∑

n,q 6=0

Φnq

(aq + a†q

)b†nbn + ω0

∑

q 6=0

a†qaq,

(5.2.3)

where we have explicitly split the vibrational part of H into two components: onepart describes the totally symmetric vibrational mode q = 0, while the other part


contains all the other modes. The reason for this is that the totally symmetricmode is unique in the sense that the excited state displacements are identical forall molecules. Due to this feature, the vibrations of this mode can be completelydecoupled from the electronic excitations by applying a full polaron transformation.

5.2.2 Polaron transformations

In general, the transformed Hamiltonian is defined by H = exp (G)H exp (−G),where G is the generator of the transformation. The full polaron transformationfor the symmetric vibrational mode is governed by the generator,

GF =λ√N

(a†0 − a0

)∑

n

b†nbn, (5.2.4)

which results, using Eq. (5.2.3), in the following form of H,

H =[E0 + λ2ω0

(1− 1

N

)]∑

n

b†nbn +∑′

n,m

Jnmb†nbm

+λω0√N

∑

n,q 6=0

Φnq

(aq + a†q

)b†nbn + ω0

∑

q

a†qaq.(5.2.5)

It should be noted that H is expressed in the original (untransformed) creationand annihilation operators for electronic excitations and vibrations, rather than interms of their transformed counterparts. Comparison of Eqs. (5.2.3) and (5.2.5)shows that after the transformation, the totally symmetric mode is completelydecoupled from the electronic excitations.

As explained above, the other collective vibrational modes can not, in general,be completely decoupled from the electronic excitations. Therefore, we proceed byapplying a variational polaron transformation to these modes. The generator ofthis transformation is given by

GV =ξ√N

∑

n

∑

q 6=0

Φnq

(a†q − aq

)b†nbn, (5.2.6)

where 0 ≤ ξ ≤ λ denotes the variational polaron parameter, of which the optimalvalue can be found from free energy minimalization arguments, as discussed at theend of this Section. We note that the case ξ = λ corresponds to a full polarontransformation, while ξ = 0 is associated with performing no transformation at all.After applying this second transformation to Eq. (5.2.5), we find

H = E0

∑

n

b†nbn+∑′

n,m

Jnmb†nbm+ω0

∑

q

a†qaq+ω0√N

(λ− ξ)∑

n,q 6=0

Φnq

(aq + a†q

)b†nbn,

(5.2.7)

80



where we have introduced the following renormalised molecular transition frequen-cies,

E0 = E0 + ω0

(1− 1

N

)(λ− ξ)

2, (5.2.8)

and renormalised interactions,

Jnm = Jnm exp

ξ√

N

∑

q 6=0

(Φnq − Φmq)(a†q − aq

) . (5.2.9)

Comparison of Eqs. (5.2.3) and (5.2.7) shows that, after applying both a full and avariational polaron transformation, the original linear exciton-vibration interactionterm is still present (except for the totally symmetric vibrational mode) albeit itsstrength is lowered. In addition, a new coupling term between the electronic andvibrational excitations appears in the intermolecular interactions, i.e., Eq. (5.2.9).

In order to find the excited states of H, we perform a perturbation expansion inboth exciton-vibration coupling terms using the standard projection superoperatorformalism [200]. The lowest perturbation order is given by the expectation valueof the exciton-vibration coupling terms with respect to the thermal equilibriumsituation of the vibrational modes q (mean field value). While the expectationvalue of the linear exciton-vibration coupling term is trivially equal to zero, the

mean field value of the intermolecular interactions, denoted⟨Jnm

⟩, is in general

different from zero. Its value can be calculated explicitly, yielding

⟨Jnm

⟩= Jnm exp

[−ξ2 coth

(ω0

2kBT

)], (5.2.10)

where kB is the Boltzmann constant and T the temperature. In the low-temperature

limit kBT ≪ ω0/2, the electronic intermolecular interactions reduce to⟨Jnm

⟩=

Jnm exp(−ξ2

), which depends on the value of the variational polaron parame-

ter: if ξ ≈ 0 (appropriate when |λω0| ≪ |Jnm|), we find⟨Jnm

⟩≈ Jnm, while

in the opposite case ξ ≈ λ (which holds if |λω0| ≫ |Jnm|), we have⟨Jnm

⟩≈

Jnm exp(−λ2

)< Jnm. Thus, due to exciton-vibration coupling the interactions

between the molecules are effectively lowered, resulting in a more narrow excitonband. A similar effect also occurs due to temperature: increasing the temperature

leads to a decrease of⟨Jnm

⟩, which in the high temperature limit kBT ≫ ω0/2,

approaches⟨Jnm

⟩≈ 0. This phenomenon is typically referred to as temperature-

induced narrowing of the exciton band, although strictly speaking it originates fromthe presence of vibrations.


By both adding and subtracting⟨Jnm

⟩from the right-hand side of Eq. (5.2.7),

we may, after rearranging the terms, cast it into the form H = H0 + HI ; the freeHamiltonian H0 reads

H0 = E0

∑

n

b†nbn +∑′

n,m

⟨Jnm

⟩b†nbm + ω0

∑

k

a†qaq, (5.2.11)

while the interaction Hamiltonian HI has the form,

HI =∑′

n,m

(Jnm −

⟨Jnm

⟩)b†nbm +

ω0√N

(λ− ξ)∑

n,q 6=0

Φnq

(aq + a†q

)b†nbn. (5.2.12)

Equations 5.2.11 and 5.2.12 constitute the final result for the transformed Hol-stein Hamiltonian. Here H0 describes the average effect of the exciton-vibrationcoupling on the excited state properties of the original Holstein Hamiltonian, whileHI contains the residual exciton-vibration interaction terms. The latter dependsexplicitly on the value of ξ: for ξ ≈ 0, the intermolecular interaction term (firstright-hand side term of Eq. (5.2.12)) disappears, while for ξ ≈ λ, the linear exciton-vibration coupling term (second right-hand side term of Eq. (5.2.12)) vanishes.These two extremes roughly coincide with, respectively, the weak exciton-vibrationcoupling and weak intermolecular interactions limit of H in Eq. (5.2.3).

To end this section, we note that up to now the variational polaron parameterξ is still undetermined. As mentioned, setting ξ = 0 is similar to performingno transformation, while taking ξ = λ corresponds to applying a full polarontransformation. The main idea behind the variational polaron transformation isthat the value of ξ can be adjusted such that HI remains small for a wide range ofparameters λ, Jnm, and T , and consequently may be treated perturbatively. Thisvalue of ξ may be obtained from the Bogoliubov upper bound on the free energy,which is given by [192,193]

AB = −kBT ln

[Tr

exp

(−H0

kBT

)], (5.2.13)

where Tr... denotes the trace. The optimized value of ξ is that for which the freeenergy AB is minimal, i.e., it can be found from the solution of the self-consistencyequation dAB/dξ = 0. In general, this equation can not be solved analytically andone has to resort to numerical methods in order to find ξ.

82



5.2.3 Linear absorption spectrum

The absorption spectrum A (E) of molecular aggregates can be obtained fromFermi’s Golden Rule. Neglecting numerical prefactors, it has the following form

A (E) =∑

i,f

Pi |〈f |M · e |i〉|2 δ (E − ωfi) . (5.2.14)

Here i and f label the initial (before absorption) and final (after absorption) states,respectively, Pi denotes the probability that initially the system resides in statei, and ωfi ≡ ωf − ωi is the energy difference between final and initial states.Furthermore, e gives the polarization of the incident light beam, while M is thetotal dipole operator of the aggregate, which is taken as a sum of single-moleculedipole operators. It is clear from Eq. (5.2.14) that the calculation of A (E) requiresknowledge of the initial and final states. These can not, in principle, be obtained ina simple way from Eq. (5.2.1), because the exciton-vibration coupling term mixesthe vibrational and electronic parts of the eigenstates. Below we will show howA (E) can be obtained in a straightforward manner from H.

In Sec. 5.2.2, we showed that the transformed Hamiltonian can be written intothe form H = H0 + HI and discussed that the variational polaron parameter ξcan be used to minimize the transformed exciton-vibration interaction term HI .Henceforth, to find the eigenstates of H, we perform a perturbation expansion inHI thereby neglecting any terms of higher order than the lowest one. In this ap-

proximation, we have H = H0 (by construction, the mean field value⟨HI

⟩= 0);

therefore, the eigenstates of H are identical to those of H0. Because electronic exci-tations and vibrations are completely decoupled in H0 (Eq. 5.2.11), the eigenstatesfollow directly as products of the individual electronic and vibrational excited statesof the aggregate. This allows us to evaluate them separately, as detailed below.

We first consider the electronic part of the eigenstates. They can be obtainedby diagonalizing the electronic part of H0, i.e., the first two terms on the right-handside of Eq. (5.2.11). This yields exciton states of the general form

|k〉 =∑

n

Ψnkb†n |g〉 , (5.2.15)

with corresponding energies denoted by Ek. Here k denotes the excitation quantumnumber and |g〉 is the electronic ground state of the aggregate (i.e., the state whereall molecules reside in their respective ground states). Explicitly, the exciton statecoefficients Ψnk follow from the usual Bloch waves (we assume periodic boundaryconditions for the aggregate and neglect electronic and intermolecular interactiondisorder, see Eq. 5.2.1), while the exciton energies obey the typical cosine behavior.


Next, we address the vibrational part of H0, i.e., the last term on the right-handside of Eq. (5.2.11). Expressed in terms of the ladder operators, they are given by

|Nq〉 =(a†q)Nq

√Nq!

|0q〉 , (5.2.16)

where Nq is the number of vibrational quanta of mode q and |0q〉 denotes the lowest(ground) vibrational state of this mode, i.e., we have aq |0q〉 = 0. The energy ofthe eigenstate |Nq〉 is given by ENq

= Nqω0.The absorption spectrum of the molecular aggregate can now be derived from

Eq. (5.2.14). Assuming that the aggregate starts in the collective electronic groundstate, the initial state in Eq. (5.2.14) is given by |i〉 = |g〉∏q |Nq〉. The final stateis expressed likewise as |f〉 = |k〉∏q |Mq〉, where the aggregate is assumed to beraised to an electronically excited state |k〉 right after absorption. Because the

initial and final states are obtained from the transformed Hamiltonian H0, we needto transform A (E) as well. Replacing the total dipole operator M by its transform

M, we obtain the following general expression for A (E)

A (E) =µ2∑

k,Nq,MqPNq

×

∣∣∣∣∣∣〈M0|e

λ√N(a†

0−a0)|N0〉∑

n

Ψnk

∏

q 6=0

〈Mq|eξ√N

Φnq(a†q−aq)|Nq〉

∣∣∣∣∣∣

2

× δ(E −

(Ek + EMq − ENq

)).

(5.2.17)

Here Nq (Mq) denotes any possible configuration of vibrational quanta of thevibrational modes for the initial (final) state, ENq (EMq) is the energy cor-responding to these configurations and PNq gives the probability that a certainconfiguration Nq occurs initially. Furthermore, we took for simplicity the single-molecule transition dipoles equal in magnitude, i.e. |µn| = µ, and oriented alonge.

It follows from Eq. (5.2.17) that, in principle, transitions between all vibrationalstates in initial and final states are allowed; the strengths of these transitionsare determined by vibrational overlap integrals UNM ≡ 〈M | exp

(s(a† − a

))|N〉,

explicitly given by

UNM = exp

(−s2

2

) N∑

p=0

sM−N+p

(M −N + p)!

(−s)p

p!

√N !M !

(N − p)!. (5.2.18)

Beside the vibrational overlap integrals, PNq also has to be determined. Assuming

84



that the vibrational states are in thermal equilibrium, PNq has the form,

PNq =exp

(−ENq/kT

)∑

Nq exp(−ENq/kT

) , (5.2.19)

where the denominator is introduced to guarantee that∑

Nq PNq = 1.

5.3 Zero-temperature absorption spectrum

5.3.1 General expressions

At T = 0, Eq. (5.2.17) is considerably simplified because initially (before absorp-tion) only the vibrational ground states are occupied [see Eq. (5.2.19)]. Specifically,the initial state is given by |i〉 = |g〉∏q |Nq = 0〉. As a result, the only important

vibrational overlap integrals are U0M ≡⟨M | exp

[s(a† − a

)]|0⟩. These can be cal-

culated using Eq. (5.2.18), yielding U0M = exp(−s2/2

)sM (M !)

−1/2. This then

leads to the following expression for A (E),

A (E) =µ2 exp

(−λ2

N

)exp

(−ξ2

N(N − 1)

) ∑

k,Mq

1∏q Mq!

(λ2

N

)M0

×

∣∣∣∣∣∣

∑

n

Ψnk

∏

q 6=0

(ξΦnq√

N

)Mq

∣∣∣∣∣∣

2

δ

(E −

(Ek + ω0

∑

q

Mq

)),

(5.3.1)

where we made use of the relation∑

q 6=0 Φ2nq = N − 1 (completeness).

Equation (5.3.1) may be simplified even further by realizing that the vibronictransitions associated with each of the exciton states can be classified into mani-folds according to the number of vibrational excitation quanta involved, as thesetransitions have the same energy. The energetically lowest manifold consists of thetransitions to the vibrational ground state, which is the state where all modes arein their respective vibrational ground states (Mq = 0 for all q). This manifold istypically referred to as the 0 − 0 vibronic band (or 0 − 0 vibronic line). The sec-ond lowest manifold contains transitions to N states that carry a single vibrationalquantum. This manifold is termed accordingly as the 0− 1 vibronic band and hasenergy equal to E = Ek+ω0 (where Ek is the energy of the kth exciton state). Thenext manifold, with energy equal to E = Ek + 2ω0, consists of all possible statesthat have two vibrational quanta, etc. The transition strength of the manifoldwith M vibrational excited quanta together with the simultaneous excitation of

5.3 Zero-temperature absorption spectrum 85

the exciton state k, denoted Ik0−M , is given by

Ik0−M =µ2 exp

(−λ2

N

)exp

(−ξ2

N(N − 1)

)

× 1

M !

[(λ2 + ξ2 (N − 1)

N

)M

+

(∑′

n,m

ΨnkΨmk

)(λ2 − ξ2

N

)M],

(5.3.2)

where the prime in the summation excludes the case n = m; the energies corre-sponding to these transitions are given by E = Ek +ω0M . In deriving Eq. (5.3.2),we used the relation

∑q 6=0 ΦnqΦmq = −1. We note that the spectrum in terms of

Ik0−M is given by

A (E) =∑

k,M

Ik0−Mδ (E − (Ek +Mω0)) . (5.3.3)

5.3.2 Molecular dimer

To gain more physical insight into Eqs. (5.3.1), (5.3.2), and (5.3.3) we will discussin this section the most simple example of an aggregate, namely the dimer, andshow how these expressions give rise to the well-known zero-temperature absorp-tion spectra in the standard perturbation limits of the Holstein Hamiltonian. Forexplicitness, our dimer model consists of two molecules, labeled 1 and 2, with equaltransition energies E0 (i.e., no electronic disorder) and coupled via a resonant en-ergy transfer interaction, denoted by J (we take J < 0). The transition dipolesare oriented parallel to each other with equal magnitude µ and each electronicexcitation is coupled to a vibrational mode of frequency ω0 with strength λ.

Following the procedure outlined in Sec. 5.2.2, H0 for the dimer is given by

H0 =(E0 +

ω0

2 (λ− ξ)2)[

|1〉〈1|+ |2〉〈2|]+ J exp

(−ξ2

) [|1〉〈2|+ |2〉〈1|

]

+ ω0

(a†+a+ + a†−a−

),

(5.3.4)

where we introduced electronic excitation state vectors |n〉 = b†n |g〉 (n = 1, 2) andincluded the explicit expressions for the renormalised molecular transition energyand renormalised interaction using Eqs. (5.2.8) and (5.2.10), respectively. Thecreation and annihilation operators for the collective symmetric (+) and antisym-metric (−) vibrational modes are expressed in terms of the original modes (labeled

1 and 2) as a†± = (a†1±a†2)/√2 and a± = (a1±a2)/

√2, respectively. The electronic

part of H0 is a 2 × 2 matrix that can be diagonalized analytically, yielding the

86



symmetric and antisymmetric exciton states |±〉 and corresponding energies

|±〉 = 1√2

[|1〉 ± |2〉

], E± =

(E0 +

ω0

2(λ− ξ)

2)± J exp

(−ξ2

). (5.3.5)

The absorption spectrum of the molecular dimer is, using Eqs. (5.3.1) and (5.3.5),expressed as,

A (E) =µ2

2exp

(−λ2 + ξ2

2

) ∑

M+,M−

1

M+!M−!

(λ2

2

)M+

[(ξ√2

)M−

±(−ξ√

2

)M−]2

× δ (E − (E± + ω0M+ + ω0M−)) ,(5.3.6)

where the ± sign refers to the symmetric and antisymmetric exciton state |±〉,respectively. Accordingly, using Eq. (5.3.2), the transition strengths of the M th

vibrational manifold associated with excitation of one of the exciton states |±〉,denoted I±0−M , is given by

I±0−M = µ2 exp

(−λ2 + ξ2

2

)1

M !

[(λ2 + ξ2

2

)M

±(λ2 − ξ2

2

)M]. (5.3.7)

Here M = M++M− denotes the total number of vibrational quanta. It is immedi-ately evident from Eq. (5.3.7) that the total oscillator strength O, i.e., the sum of allthe transition strengths, is a conserved quantity given by O =

∑M (I+0−M+I−0−m) =

2µ2.It follows from Eqs. (5.3.6) and (5.3.7) that the spectra strongly depend on the

variational polaron parameter ξ. As explained in Sec. 5.2.2, ξ is determined byminimizing the free energy [see Eq. (5.2.13)]. For the dimer at T = 0, this leads tothe following self-consistency equation [197],

ξ = λ

[1 +

2 |J |ω0

exp(−ξ2

)]−1

, (5.3.8)

from which it is immediately clear that 0 ≤ ξ ≤ λ. In principle, the values of ξcan only be obtained by numerically solving Eq. (5.3.8). However, in the standardperturbation limits of weak and strong exciton-vibration coupling they can beapproximated, respectively, by ξ ≈ 0 and ξ ≈ λ. Below, we examine these extremelimits and show that Eq. (5.3.6) correctly reproduces the optical response of thedimer in these situations.

i) |J |/ω0 ≪ |λ|. In this limit, the exciton-vibration coupling dominates theintermolecular interaction such that the optical response is expected to be deter-mined by the single-molecule state properties rather than by delocalized exciton

5.3 Zero-temperature absorption spectrum 87

states. Consequently, the variational polaron parameter is roughly equal to ξ ≈ λ,associated with a full polaron transformation. Setting ξ = λ in Eq. (5.3.6), weobtain the following approximate solution for A (E),

A (E) ≈ 2µ2 exp(−λ2

) ∞∑

M=0

λ2M

M !δ (E − E0 −Mω0) . (5.3.9)

where we used E± ≈ E0 (|J |/ω0 ≪ 1) and denoted the total number of vibrationalquanta by M . It follows directly from Eq. (5.3.9) that A (E) is simply given bythe sum of the two single-molecule spectra. It consists of several active transitionswith energy separation between adjacent peaks given by ω0, while the transitionstrengths depend both on λ and M . Here we can distinguish between the twolimits: weak (|λ| ≪ 1) and strong (|λ| ≫ 1) exciton-vibration coupling. In theformer limit, A (E) is dominated by the 0−0 transition (M = 0), located at E0 andstrength roughly equal to µ2. In the opposite limit, |λ| ≫ 1, the spectrum consistsof multiple transition lines with strengths that obey the Poisson distribution. Thetransition with M ≈ λ2 has the highest intensity and is positioned at E ≈ E0 +λ2ω0.

ii) 0 ≈ |λ| ≪ |J |/ω0. In this case, the intermolecular interaction dominates thevibronic coupling strength, resulting in a collective optical response of the dimer,and we have ξ ≈ 0, consistent with performing no transformation as expected forthis limit. Keeping only the lowest order term in λ in Eq. (5.3.6), A (E) reduces to

A (E) ≈ 2µ2δ (E − E+) , (5.3.10)

where E+ ≈ E0+J . Equation (5.3.10) shows the characteristic optical response ofa molecular dimer in the absence of exciton-vibration coupling: a single absorptionpeak that carries twice the single molecular oscillater strength and is shifted towardsthe red (J < 0) or blue (J > 0) side of the energy spectrum compared to the singlemolecule transition energy.

iii) 1 ≪ |λ| ≪ |J |/ω0. Similar to the previous limit, the optical response isexpected to be dominated by the formation of excitons and we may set ξ ≈ 0again. Contrary to the previous case, however, is that higher order terms in λ cannot be, a priori, discarded. In this case, A (E) is approximately given by

A (E) ≈ 2µ2 exp

(−λ2

2

)∑

M+

1

M+!

(λ2

2

)M+

δ (E − E+ −M+ω0) , (5.3.11)

with E+ ≈ E0+J . Thus, we see from Eq. (5.3.11) that the symmetric exciton statecarries all the oscillator strength while the antisymmetric exciton state is essentially

88



an optically dark state, similar to Eq. (5.3.10). However, due to the strong exciton-vibration coupling the oscillator strength is redistributed over various vibrationaltransitions, giving rise to a vibrational progression structure, similar to that forsingle molecules [Eq. (5.3.9)]. We note that in general the 0− 0 transition carriessomewhat more than twice the single-molecule 0− 0 oscillator strength, while thedistribution of oscillator strengths of the higher vibrational replicas also differsslightly from the sum of single-molecule transitions. These effects are related tothe collective nature of the eigenstates of the dimer, resulting in a reduced effectiveexciton-vibration coupling strength λ∗ = λ/

√2.

5.4 Numerical results and discussion

5.4.1 Comparison with the two-particle approximation

So far we have demonstrated that the APTA successfully reproduces the opticalresponse in the perturbation limits of the Holstein Hamiltonian. An intriguingquestion remains: is this method also applicable in the intermediate regime, whichis inaccessible by means of the usual perturbation approaches? To address thisissue, we compare the absorption spectra obtained using the APTA with thoseresulting from the two-particle approximation (TPA). We stress that in case of amolecular dimer, this method gives numerically exact results which makes it anideal method to test the validity of the APTA. Within the TPA, the excited statesof the Hamiltonian in Eq. (5.2.1) are expressed in terms of a basis set of n-particlestates. Here, an n-particle state consists of a vibronically (i.e., both vibrationallyand electronically) excited molecule and (n − 1) molecules that are excited vibra-tionally (but not electronically). Numerical diagonalization of Eq. (5.2.1) expressedin the two-particle basis state representation yields the optical spectra in a straight-forward manner [178,190,191].

Figure 5.4.1 presents the calculated zero-temperature absorption spectra A(E)of the molecular dimer for various values of J/ω0 using the APTA (solid lines), i.e.,based on Eq. (5.3.6), together with the numerical results obtained from the TPA(dashed lines). In all calculations we take λ = 1. It is clear from Fig. 5.4.1 thatthe spectra from both methods are almost identical to each other for a large rangeof values J/ω0. Thus, the APTA captures not only the standard perturbationregimes, but in fact also can successfully reproduce the optical response of thedimer in the intermediate regime. This will now be discussed in more detail below.

Figure 5.4.1(a) shows A(E) for |J |/ω0 = 0; here, as expected, A(E) simplyconsists of the sum of the single-molecule spectra, clearly showing the characteristicvibrational progression of the 0−M transitions for single molecules [i.e., Eq. (5.3.9)].Note that the transition strengths of the 0 − 0 and 0 − 1 transition are identical

5.4 Numerical results and discussion 89

−2 0 2 4 60

0.2

0.4

0.6

0.8

1

A(E

)

J/ω0=0

−2 0 2 4 60

0.2

0.4

0.6

0.8

1

J/ω0 = −0.25λ

−2 0 2 4 60

0.2

0.4

0.6

0.8

1

J/ω0= −0.75λ

−2 0 2 4 60

0.2

0.4

0.6

0.8

1

E

J/ω0 = −λ

A(E

)

−2 0 2 4 60

0.2

0.4

0.6

0.8

1

E

J/ω0 = −1.25λ

−5 0 5 100

0.2

0.4

0.6

0.8

1

E

J/ω0 = −2λ

(b) (c)

(d) (e) (f)

(a)

Figure 5.4.1: Calculated zero-temperature absorption spectra for a homogeneousmolecular dimer from the augmented polaron transformation approach (APTA,solid lines) and within the two-particle approximation (TPA, dashed lines). In allcalculations we set E0 = 0, µ = 1, ω0 = 1 and take λ = 1. The transitions havebeen broadened for clarity by Gaussian functions with width given by Γ = 0.28ω0.The intermolecular interaction J/ω0 increases from left to right and from top tobottom.

for λ = 1. Increasing the intermolecular interaction to J/ω0 = −0.25λ, we haveA(E) as in Fig. 5.4.1(b). We observe that the 0 − 0 transition strength increasesslightly, while the replicas tend to loose some of their strengths. Besides these smalldiscrepancies, A(E) still resembles the single-molecule spectrum to a large extent,as expected for the weak intermolecular interaction regime.

In Figs. 5.4.1(c-e), we display A(E) for the intermediate coupling regime J/ω0 ≈−λ. The spectra clearly start to show the optical signatures of exciton formation;that is, the absorption peaks are shifted to lower energies compared to the singlemolecule case and a clear increase in the intensity of the 0 − 0 transition (lowestenergy peak) is observed. We point out that in this intermediate regime the spec-tra resulting from the APTA and those obtained from the TPA show somewhat

90



−2 0 2 40

0.5

1

1.5

2

2.5

A(E

)

TPA

OPA

APTA

−2 0 2 40

0.5

1

1.5

2

2.5TPA

OPA

APTA

−4 −2 0 20

0.5

1

1.5

2

2.5TPA

OPA

APTA

−2 0 2 40

0.5

1

1.5

2

2.5

E

A(E

)

TPA

OPA

APTA

−2 0 2 40

0.5

1

1.5

2

2.5

E

TPA

OPA

APTA

−4 −2 0 20

0.5

1

1.5

2

2.5

E

TPA

OPA

APTA

(a)

(d) (e)

(b) (c)

(f)

N = 8, J / ω0 = −0.25λ N = 8, J / ω

0= −λ N = 8, J / ω

0 = −2λ

N = 32, J / ω0 = −0.25λ N = 32, J / ω

0 = −λ N = 32, J / ω

0 = −2λ

Figure 5.4.2: Calculated zero-temperature absorption spectra for aggregates (usingperiodic boundary conditions) with N = 8 (top row) and N = 32 (bottom row)molecules and parameters similar to those in Fig. 5.4.1. Each panel shows theresults based on the augmented polaron transformation approach (APTA, bluesolid line), one-particle approximation (OPA, black dash-dot line) and the two-particle approximation (TPA, red dashed line). The intermolecular interactionJ/ω0 increases from left to right.

larger discrepancies than in the weak and strong coupling limits [Figs. 5.4.1(b)and 5.4.1(f)], mostly featured in the high energy region of the spectra E ≥ 2[Figs 5.4.1(d) and 5.4.1(e)]. This part of the spectrum is mostly dominated byvibronic transitions associated with the antisymmetric exciton state, while the lowenergy part is mainly determined by vibronic transitions belonging to the sym-metric exciton state. We stress, however, that the overall qualitative agreementbetween the spectra is still very good in the intermediate coupling regime.

Finally, in Fig. 5.4.1(f) we present A(E) for J/ω0 = −2λ, which is close tothe weak exciton-vibration coupling regime. In this case, the spectrum is almostentirely determined, except for the weak absorption band seen for energies E ≈ 3,by the vibronic transitions connected to the symmetric exciton state, in accordance


with Eq. (5.3.11). These transitions are in perfect agreement with the TPA results.

We now turn our attention to the absorption spectra for larger aggregates. Tothis end, we consider linear aggregates with nearest-neighbor electronic interac-tions only and assume periodic boundary conditions. In Fig. 5.4.2, we present thecalculated zero-temperature spectra for N = 8 (top row) and N = 32 (bottom row)molecules based on the APTA. These results (shown as blue solid lines) are com-pared with both the one (black dash-dot lines) and two (red dashed lines) particleapproximation (OPA and TPA, respectively). We point out that, in contrast to thedimer, the TPA method no longer provides exact results. However, comparing theresults from the OPA and the TPA does give some insight into the validity of theTPA method. In all calculations we took λ = 1. Similar to the case of the dimer,again, all the results clearly show the evolution of the spectra from the single-molecule spectrum at small values of J/ω0 [Figs. 5.4.2(a) and 5.4.2(d)] towards thespectrum with the characteristic exciton features [Figs. 5.4.2(c) and 5.4.2(f)] forlarge values of J/ω0.

We first consider the weak intermolecular interaction regime (J/ω0 = −0.25λ),depicted in Figs. 5.4.2(a) and 5.4.2(d). Here, the spectra obtained from the OPAand the TPA are almost identical, indicating that the TPA method forms a goodframe of reference in this regime. Comparison of the spectra from the TPA andthe APTA reveals that the number of transitions as well as their energy positionscoincide reasonably well. However, the higher energy absorption bands, arisingfrom 0−1, 0−2 transitions etc., have a somewhat broader lineshape in the APTA-based spectra compared to the TPA results. The reason for this is that these bandsconsist of vibronic transitions associated with every exciton state. Because theexciton states have slightly different energies, this gives rise to an overall broadeningof these bands. Despite this, the results obtained using the APTA are still inreasonable agreement with the TPA spectra.

Next, we increase the intermolecular interactions to J/ω = −λ. The resultingspectra for N = 8 and N = 32 molecules in this intermediate coupling regime aredisplayed in Fig. 5.4.2(b) and 5.4.2(e), respectively. Similar to the dimer, we startto see exciton features where the intensity of the 0− 0 transition is increased andshifted to lower energy compared to the single-molecule spectrum. The calculatedoscillator strength and spectral position of this transition roughly coincide for allthree methods. On the other hand, the higher energy structure in the spectra,associated with vibronic replicas, shows several clear differences which are mostpronounced in the N = 32 case. In particular, in the APTA spectra the remainingoscillator strength is distributed over the entire exciton band, resulting in a longabsorption tail [see Fig. 5.4.2(e)]. This stems from the fact that all non-symmetricvibrational modes are treated with the same variational parameter, such that allnon-symmetric electronic eigenstates have an identical vibronic structure (M > 0).

92



In contrast, the TPA method does not give rise to a broad vibrational structurebut rather yields distinct 0− 1 and 0− 2 peaks in the spectra. A possible solutionto overcome this discrepancy is to describe each non-symmetric vibrational modeby its own variational parameter. In this respect, we point out that the OPA- andTPA-based spectra also deviate significantly from each other (for higher vibronicreplicas), such that the validity of the TPA results is not clear in this region.Testing it would demand performing calculations in three-particle approximation,which is beyond the scope of this chapter.

In Figs. 5.4.2(c) and 5.4.2(f) we present the spectra in the strong intermolecularinteraction regime (J/ω0 = −5λ). The spectra from the APTA and the TPA arefound to be in good agreement with each other. The spectra in this regime consistof the vibronic transitions (mainly 0 − 0 and 0 − 1) associated with the totallysymmetric, lowest energy, exciton state. In particular, we found that the ratiobetween the two lowest transitions is roughly equal to N/λ2, in agreement witha weak exciton-vibration perturbation analysis. Note also the similarity betweenthe OPA and the TPA, which suggests that the TPA method provides trustworthyresults in this region.

5.4.2 Comparison with related polaron transformations

One of the key elements of the APTA, as already explained in Sec. 5.2, is torepresent the molecular vibrational modes in terms of a set of collective modes[Eq. (5.2.2)]. This allows one to completely decouple the symmetric vibrationalmode from the electronic excitations using a full polaron transformation, while theother modes are partially decoupled by means of a variational polaron transforma-tion. In this section, we compare our results of the dimer with those obtained usinga different approach in which the (variational) polaron transformation is applieddirectly to Eq. (5.2.1), i.e., without introducing collective modes. We distinguishbetween two different methods: (i) a full polaron transformation applied to bothmodes (FPT method), and (ii) utilizing a variational polaron transformation forboth modes (VPT method).

Figure 5.4.3 presents the spectra calculated based on the various methods (dot-dash lines: FPT, dashed lines: VPT, and solid lines: APTA). The results clearlyreveal that in the single-molecule limit J/ω0 ≪ λ2 = 1 the three methods give verysimilar spectra [Fig. 5.4.3(a)]. The reason is that in this limit we have ξ ≈ λ, whichmeans that the transformations performed in the VPT and APTA method are veryclose to full polaron transformations, as performed in the FPT method. With in-creasing values of J/ω0 (keeping λ = 1), we see from Figs. 5.4.3(b) and 5.4.3(c) thatthe resulting spectra differ significantly for the different methods. In particular,for J/ω0 ≫ λ = 1 [Fig. 5.4.3(c)], the FPT method yields an absorption structure


−2 0 2 40

0.1

0.2

0.3

0.4

0.5

0.6

A(E

)

λ2 ω0=1,J = −0.25ω

0

FPT

VPT

APTA

−2 0 2 40

0.2

0.4

0.6

0.8

1

λ2 ω0=1, J = − ω

0

FPT

VPT

APTA

−6 −4 −2 0 2 4 60

0.2

0.4

0.6

0.8

1

1.2

1.4

A(E

)

λ2 ω0 = 1, J = −5ω

0

E−8 −6 −4 −2 0 2 4 60

0.5

1

1.5

E

λ2 ω0 = 0.1, J = −5ω

0

FPT

VPT

APTA

FPT

VPT

APTA

(a) (b)

(d)(c)

Figure 5.4.3: Comparison of zero-temperature dimer absorption spectra obtainedfrom the full polaron transformation method (FPT, dot-dash lines), variational po-laron transformation method (VPT, dashed lines), and augmented polaron trans-formation (APTA, solid lines). In all calculations we set E0 = 0, µ = 1, and ω0 = 1.The transitions have been broadened for clarity by Gaussian functions with widthgiven by Γ = 0.28ω0.

associated with excitation of the symmetric exciton state which still looks quitesimilar in shape to that of the APTA method, although the energy positions of thetransitions differ. The intensities of the higher vibrational replicas associated withexcitation of the antisymmetric exciton state, however, are highly overestimated inthe FPT method compared to the APTA. Specifically, the intensities of the highervibrational replicas connected with the symmetric and antisymmetric exciton stateare exactly the same within the FPT method. On the other hand, in the VPTmethod the absorption spectrum in this limit [Fig. 5.4.3(c)] consists of only the 0-0vibronic transition associated with the symmetric exciton state, i.e., all vibrationalstructure is lost. The reason for this is that in this limit the variational polaronparameter is roughly equal to ξ ≈ 0.

The above arguments suggest that in the limit of strong intermolecular inter-

94



actions J/ω0 ≫ λ and weak exciton-vibration coupling λ ≪ 1 the three methodsshould again give very similar results, as the effects of the higher vibrational statesmay to good approximation be ignored in this situation. This is illustrated inFig. 5.4.3(d), where the absorption spectra are plotted for λ2 = 0.1. Despite afew small discrepancies, all three methods indeed give a very similar absorptionspectrum, consisting mainly of the single 0− 0 transition of the symmetric excitonstate.

Thus, we found that in the limits J/ω0 ≪ λ and 1 ≪ λ ≪ J/ω0, the opticalresponse of the dimer is very similar for all polaron transformation methods, whileoutside these regions only the APTA yields correct absorption spectra of the dimer,as was already established through the comparison with the TPA (see Fig. 5.4.1).These results clearly demonstrate that the collective mode representation of thevibrations together with the variational nature of the polaron parameter for theantisymmetric vibrational mode are essential to correctly describe the optical re-sponse in the intermediate regime (i.e., in between the perturbation limits).

5.4.3 Thermal effects

Up till now we discussed the absorption spectra at zero temperature. In this sectionwe address the effects of temperature on the optical spectra, again focusing on themolecular dimer.

For temperatures T 6= 0, in principle any vibrational state can be occupiedinitially; the probability of this follows from the thermal equilibrium distributionfunction in Eq. (5.2.19). As a consequence, for non-zero temperatures more tran-sitions are optically accessible, spanned over a wider range of energies comparedto the T = 0 limit. If the linewidths of the transitions are large compared to theirtypical energy separation, this results in temperature induced broadening of thelineshape. Moreover, as explained in Sec. 5.2.2, temperature also gives rise to reduc-tion of the electronic intermolecular interactions, as can be seen from Eq. (5.2.10).Thus, we expect that with increasing temperature the spectral features associatedwith the coherent nature of excitons (for appropriate values of J/ω0 and λ) dimin-ish and, correspondingly, the optical response evolves towards the single-moleculespectra.

It is noteworthy that the value of the variational polaron parameter ξ itself alsodepends on temperature. For the dimer, the value of ξ follows, using Eq. (5.2.13),from the following self-consistency equation [197],

ξ = λ

[1 +

2|〈J〉|ω0

coth

(ω0

2kBT

)tanh

(|〈J〉|kBT

)]−1

, (5.4.1)

where 〈J〉 is the renormalized intermolecular interaction defined in Eq. (5.2.10).


−6 −4 −2 0 2 4 6 80

0.2

0.4

0.6

0.8

E

A(E

)

N = 1

N = 2

−6 −4 −2 0 2 4 6 80

0.2

0.4

0.6

0.8

E

A(E

)

N = 1

N = 2

−6 −4 −2 0 2 4 6 80

0.2

0.4

0.6

0.8

E

A(E

)

N = 1

N = 2

−6 −4 −2 0 2 4 6 80

0.2

0.4

0.6

0.8

E

A(E

)

N = 1

N = 2

(a) (b)

(d)(c)

kBT/ω

0 = 2k

BT/ω

0 = 1.5

kBT/ω

0 = 0.5 k

BT/ω

0 = 1

Figure 5.4.4: Calculated temperature-dependent absorption spectra for the molec-ular dimer (solid lines) for system parameters J/ω0 = −λ. For reference, the sum ofthe two temperature-dependent single-molecule spectra (dashed lines) are plottedin each panel. In all calculations we set E0 = 0, µ = 1, ω0 = 1. The transitions havebeen broadened for clarity by Gaussian functions with width given by Γ = 0.28ω0.The temperature increases from left to right and from top to bottom.

For T = 0, the above relation reduces to Eq. (5.3.8), while in the high temperaturelimit (T → ∞) we find ξ ≈ λ. Similar to Eq. (5.3.8), the values of ξ should, ingeneral, be found by numerically solving the above self-consistency equation. Herewe point out that with increasing temperatures, the optimal values of ξ obtainedfrom Eq. (5.4.1) not necessarily increase towards λ in a regular fashion, as physicallyexpected, but rather show a sudden jump towards ξ = λ (see Ref. [192] for adetailed discussion). This also has obvious consequences for the spectral changesof the dimer with increasing temperature, as we will see below.

In Fig. 5.4.4 we show the calculated spectra of the homogeneous dimer for var-ious values of the temperature, taking system parameters J/ω0 = −λ. The spec-tra clearly reveal the destruction of the optical features associated with coherentelectronic excitations with increasing temperature, as already anticipated above.

96



For low temperatures, as plotted in Fig. 5.4.4(a) (kBT/ω0 = 0.5), the absorptionspectrum is not significantly changed compared to the zero temperature case [seeFig. 5.4.1(d)]. The most pronounced effect of temperature here is the formationof an additional small low energy transition, which can be associated with 1 − 0vibronic transitions of the symmetric exciton state. Increasing the temperatureto kBT/ω0 = 1, we see from Fig. 5.4.4(b) that the intensity of this transition isincreased while at the same time a new low energy transition appears (associatedwith 2 − 0 transitions). While these thermal effects lead to minor redistributionsof the oscillator strengths over the various transitions, the excitonic character isstill preserved, as can clearly be seen through comparison with the single-moleculespectra displayed in Figs. 5.4.4(a) and 5.4.4(b).

Further increasing the temperature to kBT/ω0 = 1.5 gives rise to a drasticchange in the optical response, as seen in Fig. 5.4.4(c). In fact, at this temper-ature the spectrum of the dimer is identical to the sum of the single-moleculespectra, reflecting the destruction of intermolecular coherence when kBT exceedsω0. Mathematically, the reason for the drastic change is that the value of thevariational polaron parameter makes a sudden jump towards the value ξ = λ, astouched upon already below Eq. (5.4.1), resulting in single-molecule features. Forthe system parameters used here (J/ω0 = −λ), we found that this discontinuity(sudden jump) occurs around kBT/ω0 ≈ 1.2. Finally, if we increase the tempera-ture even more to kBT/ω0 = 2 we observe that the spectrum of the dimer remainssimilar to the sum of the single-molecule spectra, as expected. The only differencebetween the spectra in Figs. 5.4.4(c) and 5.4.4(d) is that for higher temperaturesgenerally more vibrational transitions are optically active, which occur both at thelow and high energy sides of the spectrum.

5.5 Summary and concluding remarks

We have investigated the linear optical response of molecular J-aggregates by ap-plying an augmented polaron transformation approach (APTA) to the underlyingHolstein Hamiltonian. This method is based on the symmetry of the model andconsists of two polaron transformations: (i) a full transformation to completely de-couple the symmetric collective vibrational mode from the electronic excitations,and (ii) a variational (partial) transformation to minimize the coupling betweenthe electronic excitations and the remaining (non-symmetric) vibrational modes.As a result, the expressions for the absorption spectrum are expected to be validfor a wide range of exciton-vibration couplings, intermolecular interactions, andtemperatures, beyond the standard perturbation limits of weak (strong) exciton-vibration (excitation transfer) interactions. To establish this, we have comparedour results with those obtained from direct numerical diagonalization of the model

5.5 Summary and concluding remarks 97

Hamiltonian in the two-particle basis set approximation (TPA).

At zero temperature, we have shown that our approach captures both the per-turbation limits of weak exciton-vibration coupling, giving rise to a collective (exci-ton) optical response, and weak intermolecular interaction, resulting in absorptionspectra with mostly single-molecule features. In between these limits, we found thatour results showed reasonably good agreement with those obtained from the TPA,which strongly suggests that the APTA can also accurately describe the spectralcombination of single-molecule and exciton properties associated with this interme-diate regime, which in general is not accessible by means of standard perturbationtechniques.

In particular, for the molecular dimer the zero-temperature spectra obtainedfrom both methods coincided almost perfectly with each other for the entire range ofmodel parameters. With increasing number of molecules in the aggregate, however,the results obtained from the APTA and the TPA method started to reveal somespectral discrepancies for the intermediate regime, which were mostly manifestedin the higher energy vibronic absorption bands. The intensity and energy positionof the optically relevant vibrationless transition, on the other hand, still showedgood agreement between both methods. These findings reflect that using a singlepolaron transformation for all the non-symmetric vibrational modes is likely anoversimplification for aggregates consisting of many molecules.

We have also compared the zero-temperature spectra of the dimer with thoseobtained from related polaron techniques, where either a full or variational trans-formation is applied to both vibrational modes in the Hamiltonian. While allthree approaches gave similar results for the weak exciton-vibration coupling andweak intermolecular interaction limits, only the augmented polaron scheme couldsuccessfully explain the optical response of the dimer in the intermediate regime.These results illustrate the key importance of introducing a collective mode rep-resentation of the molecular vibrations together with a polaron transformation ofvariational nature to minimize the remaining vibronic interactions.

With increasing temperature, the absorption spectra evolved from having col-lective optical properties, owing to the coherent nature of the exciton states, tothe single-molecule features, associated with spatially incoherent electronic exci-tations (i.e., single-molecule excitations). These spectral changes originate fromreduced intermolecular interaction strengths at higher temperatures. This ther-mal breakdown of the exciton coherence is of importance to understand the effectsof interactions between multi-chromophoric assemblies, which is significant in thelight of recent findings on double-layer nanotubes. We note that within our APTAmethod, the spectral changes do not occur in a smooth, regular fashion with in-creasing temperature, as would physically be expected. Rather, the spectra reveal asudden drastic change of their optical features, resulting from a discontinuity in the

98



variational parameter. It will be worthwhile, therefore, to explore alternative waysto optimize this parameter which allows one to circumvent such discontinuities.

To end, we point out that all of the results shown here were calculated on astandard, commercially available computer and took at most a calculation time inthe order of minutes. This clearly demonstrates the low computational costs andimplementation simplicity of the presented method. In that respect, our approachcan also be extended to incorporate more complex models of the environmentof the aggregate, including for example spectral densities, possible environmentalcorrelations and non-equilibrium situations. Finally, the perturbation nature of ourmethod allows for the evaluation of (semi-)analytical expressions for the spectra,which often can provide deeper insight into the physics than numerical methods.

Chapter 6

Photon emission statistics

and photon tracking in

single-molecule spectroscopy

of molecular aggregates:

dimers and trimers

Based on the generating function formalism, we investigate broadband photonstatistics of emission for single dimers and trimers driven by a continuous monochro-matic laser field. In particular, we study the first and second moments of theemission statistics, which are the fluorescence excitation lineshape and Mandel’sQ parameter. Numerical results for this lineshape and the Q parameter versuslaser frequency in the limit of long measurement times are obtained. We showthat in the limit of small Rabi frequencies and laser frequencies close to resonancewith one of the one-exciton states, the results for the line shape and Q parameterreduce to those of a two-level monomer. For laser frequencies halfway the transi-tion frequency of a two-exciton state, the photon bunching effect associated withtwo-photon absorption processes is observed. This super-Poissonian peak is char-acterized in terms of the ratio between the two-photon absorption lineshape andthe underlying two-level monomer lineshapes. Upon increasing the Rabi frequency,the Q parameter shows a transition from super- to sub- to super-Poissonian statis-tics. Results of broadband photon statistics are also discussed in the context of a

100

Photon emission statistics and photon tracking in single-molecule

spectroscopy of molecular aggregates: dimers and trimers

transition (frequency) resolved photon detection scheme, photon tracking, whichprovides a greater insight in the different physical processes that occur in the multi-level systems.1

6.1 Introduction

The optical properties and dynamics of excitons in low-dimensional aggregates ofinteracting molecules form a rich research area, spanning a wide range of systemscomposed of (bio)organic molecules [160]. These include, for instance, the class ofJ-aggregates of synthetic cyanine dye molecules [1, 2, 26, 62, 65], with applicationsas photosensitizers, but also the large family of natural light-harvesting complexesof chlorophyll molecules that occur in bacteria and higher plants [27, 28], as wellas crystals and thin layers of conjugated oligomers and polymers used in currentorganic optoelectronic devices [201].

Generally, low-dimensional molecular aggregates are complex systems, whoseexcitonic properties result from an interplay of intermolecular excitation transfer(resonance) interactions, static disorder caused by interactions with frozen (slow)degrees of freedom in the environment of the aggregate, as well as dynamic dis-order, caused by faster degrees of freedom interacting with the excitons. Opticalspectroscopy offers a large toolbox to study this interplay. The vast majority ofoptical spectroscopies applied to these complex systems consist of bulk measure-ments, where since the late 1980’s in particular nonlinear optical techniques, suchas photon echoes [20, 21, 34, 73], pump-probe [74–77], and, most recently, two-dimensional correlation spectroscopy [28, 55, 56, 80–82], have opened new ways tounravel details hidden in linear spectroscopy. In parallel to these developments, alsosingle-molecule spectroscopy (SMS), introduced around 1990 [84, 85], has provena powerful experimental technique to study complex molecular systems [86–90].SMS allows one to directly address single systems of interest in a -possibly- hetero-geneous ensemble, thereby avoiding the ensemble averaging that necessarily takesplace in bulk experiments and allowing one to directly look under the inhomo-geneous lineshapes measured in ensemble spectra. SMS experiments performedon single bacteriochlorophyll aggregates [40], terylene dimers [91], tetraphenoxy-perylene diimide trimers [92], and single aggregates of amphi-pseudoisocyanine [93]have indeed revealed that interesting details on collective optical transitions arehidden underneath the ensemble average.

The analysis of the discrete data stream of spontaneously emitted photons ob-tained in SMS experiments has generated an interesting theoretical field of research

1This chapter has been published as E. A. Bloemsma and J. Knoester, J. Chem. Phys. 136,224507 (2012).

6.1 Introduction 101

(see Refs. [202–204] and references therein). Here, the main effort has been toanalyze the photon stream generated by a single molecule interacting with its en-vironment. In general, photon emission streams obey a probability distributionPn (ωL, T ) that describes the possibility of detecting a number of photons n in acertain time interval T for a particular laser frequency ωL. Both in theoreticaland experimental work, the focus lies on determining and interpreting the firstand second moments of this distribution. More specific, the first moment, denotedI (ωL, T ), is defined as the average number of photons emitted (broadband detec-tion) from the system per unit time for particular frequencies of the incoming light.For measurement times longer than any dynamical timescale present in the system,the lineshape I (ωL, T ) is identical to the (fluorescence) excitation spectrum. Thesecond moment of the distribution is conveniently represented by the Mandel pa-rameter [205], denoted Q (ωL, T ), which provides a measure for the variance of thenumber of emitted photons. Here negative (positive) values of Q imply that thevariance of the distribution is decreased (increased) relative to the Poisson distri-bution. Such statistics is typically referred to as sub-Poissonian (super-Poissonian)and is indicative of photon antibunching (bunching) behavior. For Q equal to zero,the case of purely Poissonian statistics is recovered. We note that Mandel’s Qparameter is closely related to the two-point correlation function, or fluorescenceintensity correlation function, g(2)(t) that describes the correlation between arrivaltimes of emitted photons [203].

The different approaches used to model the photon emission data streams [203,204, 206–208] have provided us with a good understanding of the mechanisms atwork that account for the fluctuations in the photon counts. Examples include theeffect of blinking caused by a long-lived triplet state in the molecule [209–211], thephenomenon of photon antibunching inherent to the quantum nature of radiationand observed in the fluorescence of a single two-level molecule in (near)-resonancewith the laser field [205,207,212,213], and photon bunching which may result, forinstance, from spectral diffusion [203, 206, 208, 214–217]. However, thus far onlyvery few theoretical studies have been devoted to the analysis of photon statisticsfor assemblies of coupled molecules. Jang and Silbey [218, 219] derived a theoreti-cal framework for the single-molecule lineshapes of multichromophoric systems andapplied it to a model of the B850 ring in the light-harvesting complex 2 of purplebacteria. Sanda and Mukamel [220] calculated the two-point fluorescence intensitycorrelation function g(2)(t) and the time-dependence of Mandel’s Q parameter Q(t)for a strongly pumped dimer undergoing Gaussian-Markovian frequency fluctua-tions.

In this chapter we present the framework for calculating photon emission statis-tics of an aggregate described by the Frenkel exciton model [11, 13] interactingwith a classical continuous monochromatic laser field. To extract photon statis-

102



tics, the Generalized Bloch Equations (GBE) formalism developed by Zheng andBrown [207,208] is adapted. Within this method, the ordinary optical Bloch equa-tions governing the dynamics of the system and its interaction with the laser fieldare rewritten using generating functions [221] from which statistical moments ofthe photon counting process follow naturally. The equations for the generatingfunctions are exact within the rotating wave approximation and the limits set bythe Hamiltonian. This method has already been applied sucessfully to single two-level chromophores [207, 208, 214–217, 222, 223] and their extension to multilevelquantum systems [224,225], and has also been used by Sanda and Mukamel [220].Using numerical methods, we apply the general equations to investigate the flu-orescence excitation lineshape (we will refer to this, for short, as the lineshape)and Mandel’s Q parameter as a function of laser frequency in the limit of longmeasurement times, i.e., much longer than any dynamical timescale present, forboth dimers and trimers of interacting molecules.

Our analysis for the dimer shows that laser frequencies close to resonance withone of the one-exciton transitions lead to sub-Poissonian statistics as a result ofphoton antibunching. This is expected, because in this frequency region the dimercan to a good approximation be regarded as an effective two-level system. For laserfrequencies halfway the transition frequencies of the individual two-level monomers,however, it turns out that the photon statistics is more complicated. Hettich etal. [91] measured both the fluorescence excitation spectrum and autocorrelationfunction g(2)(t) for single pairs of strongly interacting terylene molecules embeddedin a para-terphenyl crystal. For intense laser illumination a new peak was foundin the excitation spectrum, halfway the two one-exciton transition frequencies ofthe dimer. This peak arises from the significant enhancement of the resonant two-photon absorption and corresponding two-photon emission process under intenselaser fields. The corresponding autocorrelation function g(2)(t) showed that thestatistics was super-Poissonian, revealing the signature of photon bunching. Wewill show that upon increasing the laser intensity, the photon statistics in thisregion undergoes a transition from super- to sub-Poissonian and back again tosuper-Poissonian. For larger systems, in particular for the trimer, similar effectsoccur.

The photon counting measurements from SMS and their statistical analysisand interpretation have focused mainly on broadband photon detection schemes,in which the color of the emitted photons has been ignored. Two of the exceptionsare the study by Gopich and Szabo [226] concerning the distribution of the numberof donor and acceptor photons from single molecule Forster resonance energy trans-fer measurements and the work of Bel, Zheng, and Brown [224], in which statisticalmoments were calculated for multi-level quantum systems, based on an extensionof the generating function method. Using extended generating functions, we de-


termine a general scheme for calculating statistical moments of transition resolved(frequency resolved) photons emitted from a molecular aggregate. This photontracking method provides us with a better insight in the physical processes thatoccur under illumination.

This chapter is organized as follows. In Sec. 6.2.1 we present the Frenkel ex-citon model Hamiltonian for an aggregate of interacting molecules. Section 6.2.2describes the conversion of the resulting optical Bloch equations to the set of gen-eralized Bloch Equations and reviews the extraction of broadband photon emissionstatistics from this formalism. Section 6.2.3 is devoted to the description of thephoton tracking method. In Sec. 6.3 we present numerical results for the fluores-cence excitation lineshape and Q parameter of several specific dimer and trimersystems and discuss these results in the context of the photon tracking methodand the well-known results for the single two-level chromophore [205,217]. Finally,we present our conclusions in Sec. 6.4. Several technical details are given in anappendix.

6.2 Theoretical framework

6.2.1 Model Hamiltonian

The optical response of molecular aggregates is well described by the Frenkel exci-ton model. In this model, the monomers are treated as effective two-level systemsthat interact with each other via strong resonance dipole-dipole couplings. Allow-ing for site dependent transition frequencies and interactions (i.e, disorder), thecorresponding Hamiltonian within the Heitler-London approximation [11,13] reads(~ = 1)

Hagg =∑

n

ωnb†nbn +

∑

n,m

Jnmb†nbm. (6.2.1)

Here, ωn denotes the transition frequency of the nth monomer, Jnm is the matrixelement of the resonant transfer interaction between monomers n and m, and b†n(bn) is the Pauli creation (annihilation) operator of an excitation on monomer n.The operators satisfy the (anti-)commutation relations

[bn, b†m] = δnm(1− 2b†mbn), b†nb

†n = bnbn = 0. (6.2.2)

These relations express the fact that the two-level monomer can carry at most oneexcitation.

For an aggregate of N molecules, diagonalizing the Hamiltonian yields 2N eigen-states, denoted |i〉 (i = 1, 2, ..., 2N ), and their energies Ei. Because the Hamilto-nian conserves the total number of excitons, these eigenstates can be classified into

104



manifolds of multi-exciton states according to the number of excitation quanta theyshare. Therefore, in the exciton state basis |i〉 the aggregate can be treated as a2N energy-level diagram consisting of (N +1) manifolds. Here the lowest manifoldcontains only the ground state of the aggregate, which in the Heitler-London ap-proximation is the state with all molecules in their ground state. The correspondingenergy is set E1 = 0. The next lowest manifold contains the N one-exciton states,which share a single excitation quantum, the third lowest manifold the N(N−1)/2two-exciton states that share two excitations, etc.

The interaction of a classical continuous wave laser field of frequency ωL andelectric field amplitude E0 with the optical transitions of the aggregates is in thedipole approximation (aggregate small compared to an optical wavelength) givenby

Hint(t) = −M ·E0 cosωLt, (6.2.3)

where M denotes the transition dipole operator of the aggregate, which is givenby the sum of single monomer dipole operators, M =

∑Nn=1 µn(b

†n + bn), with µn

indicating the transition dipole matrix element (assumed real) between the ground

and excited state of monomer n. As M is a sum of single monomer operators,in the exciton eigenstate basis the only possible non-zero matrix elements of Mare those corresponding to transitions between states that lie in adjacent excitonmanifolds. The Rabi frequencies corresponding to these transitions are defined as

Ωij ≡ −Mij ·E0

2, (6.2.4)

with Mij =⟨i∣∣∣M∣∣∣ j⟩.

The dynamics of the aggregate interacting with a classical laser field can bedescribed in terms of the density operator ρ(t), whose evolution is determined bythe Liouville-von Neumann equation [227]. To account for spontaneous emissionevents, we add phenomological damping terms to this equation leading to thefollowing 22N equations for the (multi-)exciton populations ρii(t) and coherencesρij(t)(i 6= j)

(dρ(t)

dt

)

ii

= −ı[H(t), ρ(t)

]ii+∑′

j

Γijρjj(t)−∑′′

j

Γjiρii(t) (6.2.5a)

(dρ(t)

dt

)

ij

= −ı[H(t), ρ(t)

]ij− 1

2(Γi + Γj)ρij(t), (6.2.5b)

with ı denoting the imaginary unit. Here, H(t) = Hagg + Hint(t) is the total

Hamiltonian for an aggregate interacting with a classical laser field,∑′

j (∑′′

j )


denotes the sum over j for which Ej > Ei (Ej < Ei), Γij gives the rate at whichpopulation decays from level |j〉 to |i〉 due to a spontaneous emission event, and Γi

denotes the total decay rate of population out of level |i〉, i.e. Γi =∑′′

j Γji. Thedecay rate Γij can be described by Einstein’s A coefficient

Γij =1

3πǫ

(Ej − Ei

c

)3

|Mij |2 , (6.2.6)

with ǫ the permittivity of the surrounding medium and c the vacuum speed of light.

6.2.2 Generating function formalism and photon statistics

Although the set of 22N optical Bloch equations as given by Eqs. (6.2.5) suffices todetermine all dynamics of the aggregate, statistical moments of the spontaneousphoton emission process can not be obtained directly from these equations. Aspointed out previously [208,221], information about these statistics is contained inthe generating function

Gij(s, t) =

∞∑

n=0

σ(n)ij (t)sn. (6.2.7)

Here, s is an auxiliary variable and the σ(n)ij (t) denote the generalized populations

(i = j) or coherences (i 6= j), which are defined by ρij(t) =∑

n σ(n)ij (t), where

n denotes the number of photons spontaneously emitted prior to time t. There-

fore, σ(n)ij (t) can be interpreted as the density operator of systems which have

spontaneously emitted n photons prior to time t. Using arguments given else-where [207, 222], the equations of motion for Gij(s, t) follow from Eqs. (6.2.5) as

(dG(s, t)

dt

)

ii

= −ı[H(t), G(s, t)

]ii+ s

∑′

j

ΓijGjj(s, t)−∑′′

j

ΓjiGii(s, t)

(6.2.8a)(dG(s, t)

dt

)

ij

= −ı[H(t), G(s, t)

]ij− 1

2(Γi + Γj)Gij(s, t). (6.2.8b)

These equations differ from those for the total density operator [Eqs. (6.2.5)] onlyin the fact that terms describing population increase of a level due to radiativedecay out of a higher energy level are accompanied by an extra factor s. It is theauxilary variable s that allows us to extract photon statistics from these equations,as will be shown below.

106



To obtain the solution to the set of equations for Gij(s, t), we invoke the rotatingwave approximation (RWA) [228] and introduce slowly changing variables to ensurethat the resulting equations have only time independent and real coefficients. Thesevariables are of the form

C±ij (s, t) =

ı−12± 1

2

2

(Gij(s, t)e

−ıωijt ±Gji(s, t)eıωijt

)(6.2.9a)

P±kl (s, t) =

1

2

(Gkk(s, t)±Gll(s, t)

). (6.2.9b)

Here, i and j run from 1, ..., 22N with i < j, k, l = 1, 2, 3, 4, ..., 2N−1, 2N,and ωij is defined as zero if i and j represent levels that lie in the same excitonband, ωL if they represent levels in adjacent bands, 2ωL for levels that are twoexciton bands apart, etc. The resulting set of equations, often referred to as theset of Generalized Bloch equations, can be cast in the form

X(s, t) = M(s)X(s, t), (6.2.10)

where X(s, t) = (C+ij (s, t);C

−ij (s, t);P

−ij (s, t);P

+ij (s, t))

T is the column vector con-taining all the slowly changing variables, the dot reflects the time derivative, andM(s) denotes a 22N × 22N time independent matrix. The solution to Eq. (6.2.10)is given by

X(s, t) = eM(s)tX0(s, t0). (6.2.11)

Throughout this chapter, the initial condition is chosen to reflect the aggregate’sground state, i.e., X0(s, t0 = 0) = (0; 0; 1/2, 0, ..., 0; 1/2, 0, ..., 0).

Once the solution to the Generalized Bloch equations is known, information onphoton statistics is obtained through the following equality [208]

2∑

k,lP+kl(s, t) =

∞∑

n=0

2N∑

i=1

σnii(t)s

n =

∞∑

n=0

Pn(t)sn. (6.2.12)

Here, Pn(t) is identified as the probability that n photons have been emitted in thetime interval [0, t]. The two quantities of interest in this chapter are the fluorescence

excitation lineshape I(ωL) ≡ limt→∞〈n(t)〉

t (hereafter referred to as the lineshape)

and the Mandel parameterQ(ωL, t) ≡ 〈n2(t)〉−〈n(t)〉2〈n(t)〉 −1 [213], where< ... > denotes

the average over the spontaneous photon emission process. Using Eq. (6.2.12),I(ωL) may be expressed as

I(ωL) = limt→∞

2 ∂∂s

∑k,l P

+kl(s, t)|s=1

t, (6.2.13)


and Q(ωL, t) as

Q(ωL, t) =2 ∂2

∂s2

∑k,l P

+kl(s, t)|s=1 −

(2 ∂∂s

∑k,l P

+kl(s, t)|s=1

)2

2 ∂∂s

∑k,l P

+kl(s, t)|s=1

. (6.2.14)

The Q parameter is defined so that Q = 0 corresponds to Poissonian statistics,whereas the values Q < 0 (Q > 0) are referred to as sub(super)-Poissonian statis-tics. Sub-Poissonian behavior is associated with the effect of photon antibunch-ing, whereas super-Poissonian statistics is related to photon bunching. In general,Mandel’s Q parameter is a function of both the measurement time t and the laserfrequency ωL; throughout this chapter we will focus only on the limit t → ∞ oflong measurement times.

6.2.3 Photon tracking

The formalism derived in Secs. 6.2.1 and 6.2.2 determines broadband photon statis-tics, i.e., statistics of all photons spontaneously emitted from the aggregate, inde-pendent of their frequencies. With minor modifications, it is possible to calculatestatistics of photons originating from a specific transition in the aggregate.

Let n = (n1, n2, ..., nκ) be a vector, where each element ni gives the totalnumber of photons emitted due to a single radiatively allowed transition and let s =(s1, s2, ...sκ) be the vector of auxilary variables that correspond to these transitions.In order to find the statistics, we introduce the generating function

Gij(s, t) =∑

n1,n2,...,nκ

σ(n)ij (t)

[ κ∏

l=1

snl

l

]. (6.2.15)

Here, each element ni of the summation runs from zero to infinity. The main differ-ence between the generating function for broadband photon statistics [Eq. (6.2.7)]

and the one given here, is that σ(n)ij (t) explicitly depends on the number of photons

ni emitted within each allowed transition, whereas σ(n)ij (t) [Eq. (6.2.7)] depends

solely on the total number of emitted photons n (n =∑

i ni).

The set of equations for the generating functions Gij(s, t) is given by(dG(s, t)

dt

)

ii

= −ı[H(t), G(s, t)

]ii+ sji

∑′

j

ΓijGjj(s, t)−∑′′

j

ΓjiGii(s, t)

(6.2.16a)(dG(s, t)

dt

)

ij

= −ı[H(t), G(s, t)

]ij− 1

2(Γi + Γj)Gij(s, t). (6.2.16b)

108



Here, sji is the auxiliary variable related to the specific radiative transition |j〉→ |i〉. The solution to Eq. (6.2.16) is obtained in the same way as before, thatis, one introduces a set of slowly changing variables similar to Eqs. (6.2.9) andapplies the RWA. The statistics of photons due to a specific allowed transition isthen obtained from the derivative of the solution to these equations with respectto the auxiliary variable associated with that transition. For instance, the averagenumber of photons emitted as a result of the transition |j〉 → |i〉 reads

〈nji(t)〉 = 2∂

∂sji

∑

k,lP+kl(s, t)|s=(1,1,...,1). (6.2.17)

6.3 Numerical analysis

This section is devoted to the numerical analysis of the lineshape and Q parameterin the long measurement time limit for dimers and trimers. The dimer consistsof two two-level monomers with transition frequencies ω1 and ω2, that interactthrough an intermolecular excitation transfer interaction J , leading to collectiveoptical transitions. For explicitness, we will choose J to be positive. Appendix 6.5provides the exciton eigenstates and energies together with all system parametersΩij and Γij , expressed in their single monomer quantities Ω0 ≡ −µ·E0

2 and Γ0 ≡µ2

3πǫ

(ω0

c

)3(ω0 ≡ ω1+ω2

2 ), respectively. Also the set of Generalized Bloch equationsfor the dimer as derived from Eqs. (6.2.8) and (6.2.9) is listed there. As an exampleof a larger system, we will discuss photon statistics of a chain consisting of threeidentical two-level monomers with dipole-dipole interactions.

6.3.1 Homogeneous dimer

We first consider the special case of a completely homogeneous dimer where bothmonomers have the same transition frequency ω0. Furthermore, we assume thetransition dipole vectors µ1 and µ2 to be parallel. It then follows directly fromEqs. (6.5.3) and (6.5.4) that Γg− = Γ−e = Ωg− = Ω−e = 0, so that the anti-symmetric one-exciton state |−〉 can not absorb or emit photons; thus, the systemcan be represented as an effective three-level system characterized by its groundstate |g〉, the symmetric one-exciton state |+〉 and the two-exciton state |e〉, withRabi frequencies Ωg+ = Ω+e ≡ Ω+ and spontaneous decay rates Γg+,Γ+e (seeFig. 6.3.1).

6.3 Numerical analysis 109

Figure 6.3.1: Level diagram of the dimer. In the special case of a homogeneousdimer, the dotted anti-symmetric state |−〉 is optically dark and the dimer can beregarded as an effective three-level system.

Small Rabi frequency limit

Figure 6.3.2 presents the numerical results for I(ωL) and Q(ωL) in the limit of smallRabi frequencies, |Ω+| ≪ Γ+g,Γe+. As observed, I(ωL) consists of a single peakcentered around the one-exciton transition frequency E+ = ω0 + J . This peak isexplained as follows. There are two contributions to I(ωL): spontaneous emissionout of the |+〉 state and out of the two-exciton state |e〉. In the limit consideredhere, where |Ω+| ≪ Γ+g, the excitation probability of the two-exciton state isnegligible and its contribution to the emission spectrum may be ignored. Near thepeak at ωL ≈ E+, the off-resonance nature of two-exciton creation, either from the|+〉 state by one-photon excitation or from the ground state |g〉 via two-photonexcitation, further reduces the possible contribution from the |e〉 state. Using thephoton tracking method introduced in Sec. 6.2.3, we calculated this contribution,taking R = 〈ne+〉 / (〈ne+〉+ 〈n+g〉) as a characteristic measure. For (ωL−ω0) = Jand Ω0 = −1 · 10−3J , this gives R ≈ 2.7 · 10−7, which shows that near the one-exciton resonance the contribution to I(ωL) of photons emitted as a result of theexcitation of the |e〉 state is indeed negligibly small. Therefore, when using a laserwith low intensity and tuned close to the one-exciton resonance, the dimer caneffectively be treated as a two-level monomer. This may be corroborated furtherby considering the lineshape for the two-level system [217]

ITLS(ωL) =ΓΩ2

Γ2 + 2Ω2 + 4∆2. (6.3.1)

Here, the symbols have their usual meaning, Γ is the spontaneous decay rate,

Ω ≡ −µ|E0|~

is the Rabi frequency, and ∆ indicates the detuning of the laser awayfrom the resonance frequency of the two-level system. Identifying these parameters

110



−2 −1 0 1 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x 10−4

(ωL−ω

0)/|J|

I(ω

L)

−2 −1 0 1 2−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

(ωL−ω

0)/|J|

Q(ω

L)

−0.2 0 0.2

(a) (b)

Figure 6.3.2: I(ωL) and Q(ωL) versus (ωL−ω0)/J in the small Rabi frequency limitfor the homogeneous dimer. Chosen parameters are: ω0 = 10J , Γ0 = 2 · 10−2J ,Ω0 = −1 · 10−3J . Calculated parameters based on Eqs. (6.5.3) and (6.5.4) are:Γg+ = 5.3 · 10−2J , Γ+e = 2.9 · 10−2J , Ω+ = −1.4 · 10−3J . Inset: our numericalresults for the super-Poissonian peak (solid line) compared to the results derivedfrom Eq. (6.3.3).

with the values appropriate for the |g〉 → |+〉 transition in the dimer, we indeedobtain a lineshape which can not be distinguished from the peak in Fig. 6.3.2(a).

We now turn to discussing the results for Q(ωL) in Fig. 6.3.2(b). Based onthe foregoing discussion, for ωL ≈ ω0 + J the behavior of Q(ωL) is expected toreproduce that of a single two-level monomer [217]

QTLS(ωL) = − 2Ω2(3Γ2 − 4∆2

)

(Γ2 + 2Ω2 + 4∆2)2 , (6.3.2)

with the appropriate parameters. Indeed, the numerical results for Q (ωL) are inperfect agreement with Eq. (6.3.2). Therefore, for ωL ≈ E+, the photon statisticsis sub-Poissonian. We point out that in general QTLS (ωL) has a second-derivativelike structure in the sense that its value is negative for resonant laser frequencies(∆ = 0), but turns positive (when ∆2 > (4/3) Γ2) before approaching zero inthe limit ∆ → ∞. This second-derivative structure can, for example, clearly beobserved in Fig. 6.3.3(b).

In contrast to the above, for frequencies ωL ≈ ω0 the observed behavior ofQ(ωL)in Fig. 6.3.2(b) can not be explained from a simple two-level system picture. Thepositive peak of Q(ωL), i.e., super-Poissonian statistics, indicates the presence of


photon bunching. Physically, this effect is a consequence of the |e〉 state and isexplained as follows. Of the two possible processes mentioned above to populatethe |e〉 state, at ωL ≈ ω0 only the two-photon absorption process from the groundstate is resonant. Therefore, this process is expected to dominate the creation ofthe |e〉 state. From the |e〉 state, the system decays to the |+〉 state through thespontaneous emission of a single photon. In the |+〉 state, the system can furtherdecay to the |g〉 state by the emission of another photon or return to the |e〉 stateby absorbing another photon from the laser beam. As the latter process is notresonant and |Ω+| ≪ Γg+,Γ+e, after populating the |e〉 state the system will decayto the |g〉 state by the emission of two photons rapidly after each other compared

to |Ω+|−1. This effect leads to photon bunching.

Using photon tracking we calculated R ≈ 5.1 · 10−3 for ωL = ω0. This showsthat even for laser frequencies in resonance with the two-photon absorption process,the |g〉 ⇐⇒ |+〉 cycle is the dominant process to occur for the Rabi frequenciesconsidered here. However, the contribution of this two-level system process to

Q(ωL) is very small, Q(ωL) ∝ Ω2+

J2 ≪ 1, as seen from Eq. (6.3.2). Therefore, the|g〉 ⇐⇒ |e〉 cycle is the process that determines the behavior of Q(ωL) for ωL ≈ ω0,which leads to the observed super-Poissonian statistics. It should be stressed thatI(ωL) does not show clear signatures of the two-photon absorption process. Thus,Q(ωL) provides complementary information on the role of two-exciton states thatcan not be deduced from I(ωL).

The latter may be demonstrated further by considering the super-Poissonianbehavior in more detail [inset Fig. 6.3.2(b)]. To this end, we note that studiesof single molecules undergoing a stochastic spectral diffusion (Kubo-Anderson)process have demonstrated that (in certain limits) super-Poissonian statistics maybe expressed in terms of a ratio of lineshapes involved [206,214]. In analogy to this,we found that if ωL ≈ ω0 and |Ω+| ≪ Γg+,Γ+e ≪ J , the observed super-Poissonianpeak is indistinguishable from the expression

Q(ωL) =I2(ωL)

2I+(ωL). (6.3.3)

Here, I+(ωL) is the two-level monomer lineshape [Eq. (6.3.1)] corresponding tothe |g〉 → |+〉 transition in the dimer and I2(ωL) is the two-photon absorptionlineshape, which, in second-order perturbation theory, can be expressed as [229]

I2(ωL) =

∣∣∣∣∣∑

k

ΩkeΩgk

ωk − ωL

∣∣∣∣∣

2Γe

Γ2e + 4(ωe − 2ωL)2

. (6.3.4)

The summation extends over all intermediate states |k〉 (in our case |k〉 = |+〉only), ωk is the transition frequency between the ground state and |k〉, and Γ−1

e is

112



the lifetime of the two-exciton state |e〉. Thus, the super-Poissonian peak of thehomogeneous dimer is found to be approximately Lorentzian with FWHM Γ+e andmaximum value Qmax ≈ 2Ω2

+/(Γg+Γ+e).The above discussion shows that in the low intensity limit, the Q parameter

in principle can be used as a tool to address the role of the two-exciton state inthe dimer. We point out that such information is accessible in an experimentalsituation, provided that enough photon emission events are recorded to accuratelymeasure Qmax. Because the number of photon events per unit time resulting fromexcitation of the two-exciton state is obviously small for low laser intensities, thisrequires in general long measurement times. These times, however, are limited bythe fact that after a number of excitation cycles the molecules are destroyed bythe light to which they are exposed (photobleaching). Therefore, to experimentallygain information on the two-exciton state at low laser intensities requires the useof molecules that are not too sensitive to photobleaching (i.e., can go through alarge number of excitation cycles).

Intermediate Rabi frequencies

To further study the effects of the two-exciton state on photon statistics, the sin-gle monomer Rabi frequency Ω0 was increased, leading to faster Rabi oscillations|Ω+| ≈ Γg+,Γ+e. Figure 6.3.3 displays I(ωL) and Q(ωL) for the homogeneousdimer for three different values of Ω0. For laser frequencies ωL ≈ ω0 + J , I(ωL)and Q(ωL) show characteristics also found in the small Rabi frequency limit. Theessential differences with the low-intensity limit is that the widths of spectral fea-tures in I(ωL) and Q(ωL) clearly undergo power broadening; in addition, theirmaxima and minima also appear to be shifted with respect to the two-level systemapproximation introduced in Sec. 6.3.1. Numerically, this shift may be estimatedto be ∆s ≈ Ω2

+/2J . This shift finds its origin in a shift of the one-exciton state |+〉, due to the effective coupling between the |+〉 and the |e〉 states induced by thestrong laser fields.

For ωL ≈ ω0 both I(ωL) and Q(ωL) fundamentally differ from the results foundin the small Rabi frequency limit. I(ωL) shows a peak that increases with increasingfield intensity Ω0, which was not seen for small Rabi frequencies. This peak isa direct manifestation of the significant enhancement of the resonant two-photonabsorption process upon increasing laser intensities [227]. Indeed, the appearance ofsuch a peak under intense laser illumination has also been observed experimentallyfor two strongly coupled terylene molecules by Hettich et al. in Ref. [91]. Using thephoton tracking method we calculated the average number of emitted photons perunit time for the different possible transitions, taking ωL = ω0 and Ω0 = −0.08.This yielded 〈n+g〉 ≈ 7.7 ·10−3 and 〈ne+〉 ≈ 7.3 ·10−3, which gives R ≈ 0.49. Thesedata suggest that not only the |e〉 state is easily populated due to a two-photon


−2 −1 0 1 20

0.005

0.01

0.015

0.02

0.025

0.03

(ωL−ω

0)/|J|

I(ω

L)

Ω0 = −0.04|J|

Ω0 = −0.08|J|

Ω0 = −0.12|J|

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

(ωL−ω

0)/|J|

Q(ω

L)

−0.2 0 0.2

0

(a) (b)

Figure 6.3.3: I(ωL) and Q(ωL) for the homogeneous dimer as a function of (ωL −ω0)/J for intermediate Rabi frequencies, Ω0 = −4 · 10−2J,−8 · 10−2J, and −1.2 ·10−1J . Parameters are the same as those of Fig. 6.3.2, leading to Ω+ = −5.7 ·10−2J,−1.1 · 10−1J, and −1.7 · 10−1J , respectively. Inset: transition from super-to sub-Poissonian behavior in more detail.

absorption process, but also that the system decays from the |e〉 state via the |+〉state to the |g〉 state by the emission of two photons rather then re-excite from the|+〉 state to the |e〉 state by absorbing a photon. Thus, the |g〉 ⇐⇒ |e〉 cycle is thedominant process to occur in the dimer at intermediate Rabi frequencies for laserfrequencies ωL ≈ ω0.

For Q(ωL) an interesting phenomenon occurs when ωL ≈ ω0. It is seen fromFig. 6.3.3(b) that Q(ωL) has a second-derivative like shape, quite similar to thatfor a two-level monomer but clearly distinct from the Lorentzian shape observedfor small Rabi frequencies. Moreover, its minimum value (occurring at ωL = ω0)indicates that exactly at the two-photon resonance, photon statistics can either besuper- or sub-Poissonian, depending on the strength of the applied laser field Ω0,in contrary to the purely super-Poissonian statistics obtained for slow Rabi oscil-lations. We note that in the experiment by Hettich et al. it was found that underintense laser illumination photon statistics connected to the simultaneous excita-tion of both strongly coupled terylene molecules in a dimer is super-Poissonian [91].Our results, however, imply that such statistics sensitively depend on the strengthof the applied laser field and that in general it can give rise both to super- andsub-Poissonian behavior.

114



The observation of sub-Poissonian statistics, i.e., the fact that the emitted pho-tons are correlated in their arrival times, can be explained as follows. Once thedimer is excited into the |e〉 state, it emits two photons (not correlated) and col-lapses into the ground state. Before the dimer can emit another photon, it firstneeds to be reexcited to the |e〉 state through two-photon absorption (dominantprocess), which takes a finite amount of time. Hence, emission events of pairs ofphotons are stretched on the time axis, which leads to sub-Poissonian statistics.This effect disappears if the Rabi frequencies become too large or too small com-pared to the spontaneous decay rates. The above reasoning then determines aregion of possible values for the strength of the applied laser field for which sub-Poissonian statistics can be observed. For the model parameters used to generateFig. 6.3.3 we find numerically that this region is given by 0.05J / |Ω0| / 0.12J .

6.3.2 Inhomogeneous dimer

The case of a completely homogeneous dimer will not occur in practice, becausethe host matrix of dimers and larger aggregates usually exhibits structural disorder(liquid solvents, glasses and protein matrices are frequently occurring hosts), whichleads to differences in the transition frequencies of the individual molecules. Here,we will assume these frequencies to be constant on the time scale of the experiment(static disorder) and refer to them as ω1 and ω2. The quantity σ ≡

∣∣ω1−ω2

2

∣∣ thenserves as the characteristic measure for the amount of inhomogeneity within thedimer and ω0 ≡ ω1+ω2

2 denotes the average transition frequency. The eigenstates,emission constants, and Rabi frequencies for general ω1 and ω2 are given in theAppendix. The aim of this section is to analyze the effect of the inhomogeneity onI(ωL) and Q(ωL).

Figures 6.3.4a and 6.3.4c show I(ωL) and Q(ωL), respectively, in the limitof small inhomogeneity, σ ≪ J , and small Rabi frequencies, |Ωij | ≪ Γij (weassumed equal transition dipoles µ1 = µ2). I(ωL) then consists of two Lorentzians,centered at the one-exciton transition frequencies E±. We have verified, using theparameter values corresponding to the |g〉 → |±〉 transitions, that the observedLorentzian lineshapes are in agreement with Eq. (6.3.1). For Q(ωL), we observeclose to the one-exciton transition frequencies the second-derivative structures withsub-Poissonian behavior for resonant laser frequencies, as dictated by Eq. (6.3.2),while at ωL ≈ ω0 we find a Lorentzian shaped super-Poissonian peak caused bythe presence of the two-exciton state. As the inhomogeneity induces a non-zerodipole moment between the ground state and the one-exciton state |−〉 (which isdipole-forbidden in the homogeneous dimer), the super-Poissonian peak is better


−1,5 −1 −0,5 0 0,5 1 1,50

1

2

3

4

5

6x 10

−4

(ωL−ω

0)/|J|

I(ω

L)

−1 −0.5 0 0.5 1 1.5−0.1

−0.05

0

0.05

0.1

(ωL−ω

0)/|J|

Q(ω

L)

−10 −5 0 5 100

0.2

0.4

0.6

0.8

1x 10

−3

(ωL−ω

0)/|J|

I(ω

L)

−10 −5 0 5 10−0.4

−0.2

0

0.2

0.4

(ωL−ω

0)/|J|

Q(ω

L)

−0.2 −0.1 0 0.1 0.20

0.0002

0.0004

−1.01 −10

1

2

3

4x 10

−5 (a)

(c) (d)

(b)

Figure 6.3.4: Plots of I(ωL) and Q(ωL) versus (ωL − ω0)/J for the dimer in bothlimits of inhomogeneity. Panels (a) and (c) present the data for small inhomogene-ity, with parameter choices ω1 = 10J , ω2 = 10.2J (i.e., σ = 0.1J), Γ0 = 2 · 10−2J ,and Ω0 = −2 · 10−3J . Panels (b) and (d) present data for large inhomogeneity,with parameter choices ω1 = 10J , ω2 = 30J (i.e., σ = 10J), Γ0 = 2 · 10−2J , andΩ0 = −2 · 10−3J . Inset (a): details of the |g〉 → |−〉 transition lineshape. Inset(d): observation of the small super-Poissonian peak for ωL ≈ ω0

approximated by a generalization of Eq. (6.3.3),

Q(ωL) =I2(ωL)

2 [I+(ωL) + I−(ωL)], ωL ≈ ω0. (6.3.5)

We have confirmed numerically that the super-Poissonian peak in Fig. 6.3.4(c) isin perfect agreement with Eq. (6.3.5).

Figures 6.3.4b and 6.3.4d present I(ωL) and Q(ωL), respectively, in the oppositelimit σ ≫ J of large inhomogeneity. In this limit, the one-exciton states reduce tothe excited states of the uncoupled molecules and any collective optical propertiesof the dimer are expected to vanish. The observed I(ωL) indeed confirms this idea,as it consists of two Lorentzians centered roughly at the transition frequencies of the

116



10−3

10−2

10−1

100

101

102

10−7

10−6

10−5

10−4

10−3

10−2

10−1

σ

Qm

ax

10−2

100

102

10−6

10−4

10−2

100

Figure 6.3.5: Maximum value of the observed super-Poissonian peak Qmax as afunction of the disorder parameter σ. Numerical results (squares) are comparedwith those obtained from Eq. (6.3.5) (solid line). Chosen parameters are: ω1 = 10J ,Γ0 = 2 ·10−2J , Ω0 = −1 ·10−3J . Inset: Same as the main plot, but now the energydependence of the spontaneous decay rates Γij is neglected.

single molecules and in perfect agreement with Eq. (6.3.1) for the single monomerparameters. Note that the difference in linewidth and height of the two Lorentziansis a direct consequence of the transition energy dependence of the spontaneous de-cay rates Γg− and Γg+ [Eq. (6.2.6)]. In accordance, Q(ωL) shows the characteristicsecond-derivative structure, where statistics is sub-Poissonian for laser frequenciesresonant with one of the monomer transition frequencies. The inset of Fig 6.3.4(d)shows Q(ωL) for ωL ≈ ω0. The super-Poissonian peak, characteristic for the in-fluence of the two-exciton state, is still observed, although its maximum value isdecreased by two orders of magnitude with respect to the small inhomogeneitylimit.

In Fig. 6.3.5, we further analyze the maximum value Qmax of the observedsuper-Poissonian peak as a function of the disorder parameter σ. The two graphscorrespond to the cases where the energy dependence of the spontaneous decayrates is taken into account (main plot) and where it is neglected (inset). Forboth graphs and all disorder strengths, we find that the numerically calculateddata (squares) is in excellent agreement with the results obtained from Eq. (6.3.5)(solid line). The distinction between the two different regimes of small and largeinhomogeneity is clearly seen in Fig. 6.3.5. In the limit σ ≪ J , Qmax is roughly


independent of disorder. This is expected, as to first order in J the eigenstatesof the dimer are identical to those of the homogeneous dimer. In the oppositelimit of σ ≫ J , it follows from Eq. (6.3.5) that, when the dependence of Γij

on the transition energy is neglected, Qmax depends on the disorder strength σ

through the power law Qmax = 2(

Ω0

Γ0

)2(σ/J)

−2. If the energy dependence of the

decay rates is taken into account, no strict power law behavior could be derivedfrom Eq. (6.3.5). By fitting our numerical results, however, we found that alsoin this case Qmax for large inhomogeneity decreases with σ according to a powerlaw: Qmax ∝ (σ/J)

−2.65. Physically, the observed decrease of super-Poissonian

statistics with increasing inhomogeneity (in both models for the decay rates) is adirect consequence of the disorder induced localization of the exciton states. Thislocalization implies that the collective optical properties of the dimer, such as theoccurrence of super-Poissonian statistics at ωL ≈ ω0, vanish.

To end this section we point out that for laser intensities beyond the small Rabifrequency limit, the effects of disorder on the second order statistics associated withthe two-exciton states are qualitatively very similar to those described above. Morespecific, for small disorder values (σ ≪ J) and laser frequencies ωL = ω0 we maystill observe, depending on the intensity of the light, both sub- and super-Poissonianstatistics (as found in Fig. 6.3.3(b)), because the disorder is too small to destroy thecollective nature of the exciton states. In the opposite limit, σ ≫ J , the moleculesare effectively almost completely decoupled and, as a result, transitions betweensuper- and sub-Poissonian statistics (at ωL = ω0) will disappear. The statistics inthis limit is super-Poissonian, whose magnitude diminishes (grows) with increasing(decreasing) value of the ratio of disorder strength and laser intensity, i.e., σ2/Ω2

0.

6.3.3 Linear homogeneous trimer

We consider a one-dimensional chain consisting of three two-level monomers withequal transition frequencies ωn = ω0 (n = 1, 2, 3) and equal transition dipoles(µ1 = µ2 = µ3), separated by equal distances. Restricting ourselves to nearest-neighbor interactions J (> 0) only and open boundary conditions, diagonalisingthe 3 × 3 Hamiltonian matrix [Eq. (6.2.1)] yields the one-exciton states, denoted|1; ρ〉 (ρ = 1, 2, 3), and energies E1;ρ

|1; 1〉 =1

2|1〉 − 1√

2|2〉+ 1

2|3〉 E1;1 = ω0 −

√2J

|1; 2〉 =1√2|1〉 − 1√

2|3〉 E1;2 = ω0

|1; 3〉 =1

2|1〉+ 1√

2|2〉+ 1

2|3〉 E1;3 = ω0 +

√2J. (6.3.6)

118



Here, |i〉 = b†i |g〉. The two-exciton states |2;σ〉 (σ = 1, 2, 3) are found as the Slaterdeterminants of two different one-exciton states |1; ρ〉 and |1; ρ′〉 with eigenenergiesE2;σ = E1;ρ +E1;ρ′ . Furthermore, the three-exciton state |e〉 is the state where allthree monomers are excited (Ee = 3ω0). Using Eq. (6.3.6), it is a straightforwardexercise to determine all Rabi frequencies and spontaneous decay rates of allowedtransitions in the trimer, depicted schematically in Fig. 6.3.6.

Figure 6.3.6: Level diagram of the linear homogeneous trimer with all moleculeshaving equal transition dipole vectors. The arrows correspond to the opticallyallowed transitions.

Figure 6.3.7 shows the numerical results for I(ωL) and Q(ωL) in the limit ofsmall Rabi frequency. As is observed, I(ωL) consists of two peaks centered aroundthe one-exciton transition frequencies E1;1 and E1;3. Q(ωL), correspondingly, showsthe second-derivative like structure characteristic for two-level monomers, wherestatistics is sub-Poissonian for resonant laser frequencies. Using the parameterscorresponding to the |g〉 → |1; 1〉 , |1; 3〉 transitions, we confirmed that indeed thetrimer behaves as an effective two-level monomer in these frequency regions. Notethat the |g〉 → |1; 2〉 transition is dipole forbidden, which explains the absence ofsub-Poissonian statistics for ωL ≈ ω0.

Furthermore, in Fig. 6.3.7(b) two super-Poissonian peaks are found at laserfrequencies ωL ≈ ω0 ± J/

√2, i.e., halfway the transition frequencies of the two-

exciton states |2; 1〉 and |2; 3〉. As in the case of the dimer, we expect these peaksto result from the direct excitation of the two-exciton states from the ground stateby means of a two-photon absorption process followed by the deexcitation to theground state through the emission of two photons. Thus, the peaks should be inperfect agreement with Eq. (6.3.5), which was indeed numerically confirmed. We


−2 −1 0 1 20

1

x 10−4

(ωL−ω

0)/|J|

I(ω

L)

−2 −1 0 1 2−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

(ωL−ω

0)/|J|

Q(ω

L)

(a) (b)

Figure 6.3.7: I(ωL) and Q(ωL) versus (ωL − ω0)/J for the homogeneous trimer inthe limit of small Rabi frequency. Chosen parameters are: ω0 = 10J , Γ0 = 2·10−2J ,and Ω0 = −1 · 10−3J .

note that similar behavior is not observed for laser frequencies ωL ≈ ω0 halfwaythe transition frequency of the |2; 2〉 state, because this state can not be createdby any (multi-)photon excitation or emission process.

Figure 6.3.8 displays the numerical results for I(ωL) and Q(ωL) for increasinglyfast Rabi oscillations, revealing several characteristics of the influence of multi-exciton states on the photon statistics. For ωL ≈ ω0±J/

√2, I(ωL) in Fig. 6.3.8(a)

shows the appearance of two new peaks compared to the low Rabi frequency limit.Similar to the dimer case, they result from the increasing occurrence (for increasinglaser intensities) of two-photon absorption processes that populate the two-excitonstates |2; 1〉 and |2; 3〉 and the corresponding decay to the |g〉 state via one of theone-exciton states. For ωL ≈ ω0, a third new peak can be observed in the spectrum,which has no analogue in the dimer case. It originates from the resonance natureof three-exciton creation via a three-photon absorption process, which, at largerRabi frequencies, significantly populates the |e〉 state .

As seen in Fig. 6.3.8(b), for ωL ≈ ω0 ± J/√2, Q(ωL) shows a transition from

the Lorentzian super-Poissonian peak shape (as observed in the small Rabi fre-quency limit) to the second-derivative like structure. This transition was alreadyencountered for the homogeneous dimer upon increasing laser intensities and gaverise to the sub-Poissonian statistics seen there. Here, for laser frequencies halfwaythe transition frequency of the |2; 1〉 state, sub-Poissonian statistics is again ob-served, although the minimum values of Q(ωL) lie closer to zero than in the dimer

120



−1 −0.5 0 0.5 10

0.005

0.01

0.015

0.02

0.025

0.03

(ωL−ω

0)/|J|

I(ω

L)

Ω0 = −0.1|J|

Ω0 = −0.05|J|

−1 −0.5 0 0.5 1−5

0

5

10

(ωL−ω

0)/|J|

Q(ω

L)

−0.8 −0.7 −0.6−0.5

0

0.5

1(a) (b)

Figure 6.3.8: I(ωL) and Q(ωL) for the trimer plotted against (ωL − ω0)/J forΩ0 = −0.05J and −0.1J . Chosen parameters are the same as those of Fig. 6.3.7.The range for ωL is chosen to reflect the characteristics of multi-exciton influences inmore detail. Inset: detailed behavior of the frequency regime near ωL = ω0−J/

√2,

which is the transition frequency between the ground state and the |1; 1〉 one-exciton state.

case, indicating that the sub-Poissonian statistics resulting from the excitation oftwo-exciton states is less pronounced for the trimer. In fact, for laser frequencieshalfway the transition frequency of the |2; 3〉 state, we do observe the transition tothe second-derivative like structure, but statistics remains super-Poissonian for alllaser intensities.

These findings illustrate that for larger aggregates signs of multi-exciton statesmay still be observed in Q(ωL), although interesting characteristics connected withthese states, for example the super- to sub-Poissonian transition for resonant laserfrequencies, tend to disappear. This results from the rapidly increasing numberof (resonant) excitation and decay pathways for larger aggregates, which, in theend, destroys the (anti-)correlations in the photon emission process arising fromthe multi-exciton states.

For laser frequencies resonant with the three-photon excitation process, i.e.ωL ≈ ω0, we observe in Fig. 6.3.8(b) a super-Poissonian single peak structure,which can be understood as follows. Once the |e〉 state is populated, the system willrather decay via the emission of two consecutive photons to one of the one-excitonstates than recreate the |e〉 state through one-photon excitation from the two-exciton state, as the latter process is not resonant with the laser frequency. In the

6.4 Conclusions 121

one-exciton state, both the reexcitation (one-photon absorption) to the two-excitonstate (resonant process) and the decay towards the ground state occur; whetherone of these pathways dominates the other depends sensitively on the values forthe emission constants and Rabi frequencies involved. In the end, however, in bothcases the decay from the |e〉 state gives rise to photon bunching (either from twoor from three photons) leading to the observed super-Poissonian statistics.

To corroborate on this further, we expect that this super-Poissonian peak isrelated to the three-photon absorption lineshape, similar to the way in which thetwo-photon absorption lineshape was connected to super-Poissonian behavior ofthe two-exciton state in the dimer [i.e., see Eqs. (6.3.3) and (6.3.4)]. As a result,the maximum value Qmax (at ωL = ω0) of the super-Poissonian peak should scalewith increasing laser intensity Ω0 according to a power law with cubic exponent.Numerical calculation of Qmax for different values of the laser intensity (betweenΩ0 = −5× 10−3J and Ω0 = −5 × 10−2J) revealed that indeed the dependence ofQmax on laser intensity obeys a power law. The corresponding exponent was foundto be 3.25, which is in good agreement with the expected value of three.

It is interesting to notice that for the Rabi frequencies considered in Fig. 6.3.8(b),no transition towards the second-derivative like structure is (yet) observed in thefrequency region ωL ≈ ω0, as opposed to the statistics resulting from the two-exciton states. This derives from the fact that higher laser intensities are requiredto excite the three-exciton state than two-exciton states. This is already apparentfrom Fig. 6.3.7(b), where at low laser intensities no signal was found in Q(ωL)at frequencies resonant with the three-photon absorption process, in contrast tothe statistics connected with the two-exciton states, which clearly show up at theintensities used there. If the Rabi frequencies are increased even further, we findthat also the statistics connected with three-exciton creation undergoes the tran-sition towards a second-derivative like structure, although possible sub-Poissonianbehavior at ωL = ω0 was not observed.

6.4 Conclusions

Using the generating function formalism, we have studied broadband photon emis-sion statistics for small molecular aggregates (Frenkel exciton systems) driven bya monochromatic laser field. This method allows one to extract statistical mo-ments from the set of Generalized Bloch Equations (GBE). Numerically, we foundit convenient to invoke the rotating wave approximation as the resulting linear dif-ferential equations only have time-independent coefficients. To analyze and explainthe results of broadband photon statistics more carefully, we introduced a photontracking method. This method, based on extending the generating function, allowsus to distinguish between photons that originate from different transitions.

122



The statistical moments of the photon emission process in the limit of longmeasurement times were obtained numerically in terms of the lineshape and Man-del’s Q parameter for the dimer and the linear homogeneous trimer. Especially,the Q parameter provided interesting information on the role of multi-excitonicstates in these systems, even in the limit of low laser intensity. For laser frequen-cies close to resonance with the transition frequency of a one-exciton state, wefound that photon statistics can to a good approximation be reduced to the two-level monomer statistics, although slight deviations occurred for increasing laserintensities. Furthermore, for laser frequencies halfway the transition frequency ofa two-exciton state and using low laser intensity, a Lorentzian super-Poissonianpeak was observed in the Q parameter. This phenomenon is related to populatingthe two-exciton state by a resonant two-photon absorption process and the cor-responding decay back to the ground state by the rapid emission of two photons(bunching). Interestingly, this peak was found to be in excellent agreement with theratio between the two-photon absorption lineshape and the two-level monomer line-shapes. Therefore, information on the two-exciton states in the system can be de-termined from Q. For increasing laser intensities, a transition from the Lorentziansuper-Poissonian peak to a second-derivative like structure was observed for theselaser frequencies. In several cases, this second-derivative structure showed a valleyof sub-Poissonian statistics in between two super-Poissonian peaks, a character-istic usually only observed for monomers. We point out that Ω0/J is used as ameasure for the excitation intensity throughout this chapter. To connect to ex-periment, we may express this measure of excitation intensity in absolute units,such as a power per unit area, using the Poynting vector S. Assuming a typicalvalue for the dipole moment of µ = 10 Debye and a typical interaction strength ofJ = 600 cm−1, we found the following relation between the power flux and Ω0/J :S ≈ 6.8× 102

(Ω2

0/J2)W µm−2. Thus, the region of low laser intensities, defined

in this chapter as Ω0 ≤ 10−3J , translates into S / 0.7 mW µm−2 for the powerper unit area. High laser intensities of Ω0 = 0.1J correspond to S ≈ 7 W µm−2.

By studying both the dimer and trimer system, we found that photon statisticsof the trimer, although more complex than the dimer, shows the same essential fea-tures. In particular, for intermediate laser intensities and exciting with a frequencyhalfway that of the two-exciton states, we observed the second derivative struc-ture already encountered in the dimer case, although the valley of sub-Poissonianstatistics was less pronounced for the trimer. This finding illustrates that for largeraggregates, detailed characteristics of the emission process associated with multi-exciton states are expected to disappear in the Q parameter. This is a consequenceof the increasing number of excitation and decay pathways for larger aggregates,which reduces (anti-)correlations in the photon arrival times. In addition, explor-ing the role of higher multi-exciton states in large aggregates acquires strong(er)

6.5 Appendix: eigenstates and Generalized Bloch Equations for the

dimer 123

laser illumination. On the other hand, higher order correlations, i.e. higher ordermoments of Pn (ωL, T ), may give strong signals even for larger aggregates. Thisposes an interesting challenge for future research, as such correlations are in generalhard to calculate for large aggregates.

6.5 Appendix: eigenstates and Generalized Bloch

Equations for the dimer

Consider two two-level monomers with transition frequencies ω1 and ω2, respec-tively, which interact through a resonance dipole-dipole coupling J . For explicit-ness, we chose J to be positive. Using the Frenkel exciton Hamiltonian of Eq. (6.2.1)we find the one-exciton eigenstates

|−〉 =1√

1 + η2

[|1〉 − η |2〉

]

|+〉 =1√

1 + η2

[η |1〉+ |2〉

]. (6.5.1)

Here, |i〉 = b†i |g〉 denotes the state in which only monomer i is excited and η ≡E+−ω2

J , where E+ is the energy corresponding to the |+〉 state. The Heitler-Londonapproximation ensures that the dimer ground state |g〉 is the state where bothmonomers are in their ground state with energy Eg = 0 and that the two-excitonstate |e〉 is the state where both monomers are excited with corresponding energyEe = ω1 +ω2. The eigenenergies corresponding to the one-exciton states are givenby

E± =1

2

(ω1 + ω2 ±

√(ω1 − ω2)2 + 4J2

). (6.5.2)

There are two limiting cases for which Eqs. (6.5.1) and (6.5.2) reduce to asimpler form. In the limit of small inhomogeneity, σ ≡

∣∣ω1−ω2

2

∣∣ ≪ J , the one-exciton states reduce to those of the homogeneous dimer, |±〉 = 1√

2(|1〉 ± |2〉) with

corresponding energies E± = ω0 ± J . In the opposite limit of large inhomogene-ity, σ ≫ J , the one-exciton states reduce to the excited states of the uncoupledmolecules |i〉 = b†i |g〉 with energies ω1 and ω2.

To calculate the Rabi frequencies corresponding to the possible transitions, itis assumed for simplicity that the transition dipoles of the monomers have equalmagnitude and orientation (µ1 = µ2 = µ). Using Eq. (6.2.4), the Rabi frequenciesare given by

Ωg± = Ω±e =Ω0√1 + η2

(1± η) . (6.5.3)

124



Here, Ω0 ≡ −µ·E0

2 is the Rabi frequency for a single molecule. The spontaneousdecay rates for the possible transitions follow directly from Eq. (6.2.6) as

Γ±g = Γ0

(E±ω0

)31

1 + η2(1± η)

2

Γe± = Γ0

(2− E±

ω0

)31

1 + η2(1± η)

2. (6.5.4)

Here, ω0 ≡ ω1+ω2

2 is the mean of the two monomer transition frequencies and Γ0 ≡µ2

3πǫ

(ω0

c

)3is the spontaneous decay rate for a monomer with this mean frequency.

Thus, Γ0 may be regarded as a typical monomer spontaneous emission rate.The Hamiltonian H(t) = Hagg + Hint(t) for the inhomogeneous dimer interact-

ing with a continuous wave laserfield in the exciton basis |g〉 , |±〉 , |e〉 now canbe expressed as

H(t) =

0 0 0 00 E− 0 00 0 E+ 00 0 0 ω1 + ω2

+ cosωLt

0 Ωg− Ωg+ 0Ωg− 0 0 Ω−e

Ωg+ 0 0 Ω+e

0 Ω−e Ω+e 0

.

(6.5.5)Here, the first term represents the free inhomogeneous dimer, whereas the secondterm represents its interaction with the laser field. Using Eqs. (6.2.8) and (6.2.9)we obtain, within the rotating wave approximation, the following set of GeneralizedBloch equations

C+g− = −Γ−g

2C+

g− + [ωL − E−]C−g− − Ω−eC

−ge − Ωg+C

−−+, (6.5.6)

C+g+ = −Γ+g

2C+

g+ + [ωL − E+]C−g+ − Ω+eC

−ge +Ωg−C

−−+, (6.5.7)

C+−e =− Γ−g + Γe− + Γe+

2C+

−e + [ωL − (ω1 + ω2 − E−)]C−−e

+Ωg−C−ge − Ω+eC

−−+,

(6.5.8)

C++e = −Γ+g + Γe− + Γe+

2C+

+e + [ωL − (ω1 + ω2 − E+)]C−+e

+Ωg+C−ge +Ω−eC

−−+,

(6.5.9)

6.5 Appendix: eigenstates and Generalized Bloch Equations for the

dimer 125

C+ge =− Γe− + Γe+

2C+

ge + [2ωL − (ω1 + ω2)]C−ge

− Ω−eC−g− − Ω+eC

−g+ +Ωg−C

−−e +Ωg+C

−+e,

(6.5.10)

C+−+ =− Γ+g + Γ−g

2C+

−+ − [E+ − E−]C−−+

+Ωg+C−g− +Ωg−C

−g+ − Ω+eC

−−e − Ω−eC

−+e,

(6.5.11)

C−g− = −Γ−g

2C−

g− − [ωL − E−]C−g− +Ω−eC

+ge − Ωg+C

+−+ − 2Ωg−P

−−g, (6.5.12)

C−g+ =− Γ+g

2C−

g+ − [ωL − E+]C+g+ +Ω+eC

+ge − Ωg−C

+−+

+Ωg+P−e+ − Ωg+P

−−g − Ωg+P

+e+ +Ωg+P

+g−,

(6.5.13)

C−−e =− Γ−g + Γe− + Γe+

2C−

−e − [ωL − (ω1 + ω2 − E−)]C+−e − Ωg−C

+ge

+Ω+eC+−+ − Ω−eP

−e+ +Ω−eP

−−g − Ω−eP

+e+ +Ω−eP

+g−,

(6.5.14)

C−+e =− Γ+g + Γe− + Γe+

2C−

+e − [ωL − (ω1 + ω2 − E+)]C++e

− Ωg+C+ge +Ω−eC

+−+ − 2Ω+eP

−e+,

(6.5.15)

C−ge =− Γe− + Γe+

2C−

ge − [2ωL − (ω1 + ω2)]C+ge

+Ω−eC+g− +Ω+eC

+g+ − Ωg−C

+−e − Ωg+C

++e,

(6.5.16)

C−−+ =− Γ+g + Γ−g

2C−

−+ + [E+ − E−]C+−+

+Ωg+C+g− − Ωg−C

+g+ +Ω+eC

+−e − Ω−eC

++e,

(6.5.17)

P−e+ =− Ωg+C

−g+ +Ω−eC

−−e + 2Ω+eC

−+e −

1

2Γe+(1 + s)(P+

e+ + P−e+)

− 1

2Γe−(P

+e+ + P−

e+) +1

2Γ+g(P

+e+ − P−

e+),

(6.5.18)

P−−g =2Ωg−C

−g− +Ωg+C

−g+ − Ω−eC

−−e +

1

2sΓe−(P

+e+ + P−

e+)

− 1

2sΓ+g(P

+e+ − P−

e+)−1

2Γ−g(1 + s)(P−

−g + P+g−),

(6.5.19)

P+e+ =+Ωg+C

−g+ +Ω−eC

−−e −

1

2Γe+(1− s)(P+

e+ + P−e+)

− 1

2Γe−(P

+e+ + P−

e+)−1

2Γ+g(P

+e+ − P−

e+),

(6.5.20)

126



P+g− =− Ωg+C

−g+ − Ω−eC

−−e +

1

2sΓe−(P

+e+ + P−

e+)

+1

2sΓ+g(P

+e+ − P−

e+)−1

2Γ−g(1− s)(P−

−g + P+g−).

(6.5.21)

Samenvatting

In dit proefschrift worden de optische eigenschappen van zogeheten moleculaireaggregaten beschreven. Onder moleculaire aggregaten verstaan we clusters vanmoleculen die in een bepaalde geometrie bijeen gehouden worden door, bijvoor-beeld, elektrostatische krachten en/of hydrofobe interacties. De moleculen waaruitdeze aggregaten bestaan worden gekenmerkt door enerzijds de sterke wisselwerkingdie ze hebben met zichtbaar licht en anderzijds door de sterke interacties die demoleculen onderling hebben. Tengevolge van deze laatste eigenschap is de optischerespons van een aggregaat verschillend van dat van de losse moleculen waaruit hetopgebouwd is. In het bijzonder wordt er onderscheid gemaakt tussen zogenaamdeJ-aggregaten en H-aggregaten. De eerste groep wordt gekenmerkt door een ab-sorptieband die verschoven is naar lagere energie (roodverschuiving) in vergelijkingmet het moleculaire absorptiespectrum, terwijl voor de tweede groep aggregatende absorptieband juist verschoven is naar hogere energie (blauwverschuiving).

Moleculaire aggregaten zijn in overvloed aanwezig in de fotosynthetische sys-temen van vele planten en bacterien. In deze complexen zorgen moleculaire ag-gregaten voor de opvang (absorptie) van zonlicht en het transport van de resul-terende excitatie-energie naar reactiecentra, waar uiteindelijk de chemische reactiesplaatsvinden waardoor deze organismen hun biologische functies kunnen uitvoeren.Een bijzonder voorbeeld van een organisme dat overleeft door middel van fotosyn-these is de groep van groene bacterien. Bepaalde soorten van deze bacterie zijngevonden in omgevingen waar de lichtintensiteit zeer laag is, zoals bijvoorbeeld inde Zwarte Zee op dieptes rond de 100 meter. Het proces van lichtabsorptie enenergietransport in deze bacterien is buitengewoon efficient; meer dan 95% van deopgevangen fotonen bereikt uiteindelijk een van de reactiecentra. De antennesys-temen van deze groene bacterien die verantwoordelijk zijn voor de absorptie en hettransport van de zonne-energie, de zogeheten chlorosomen, bestaan uit cylinder-vormige aggregaten die opgebouwd zijn uit duizenden chlorofyl moleculen.

De fascinerende optische eigenschappen en het zeer efficiente energietransportvan natuurlijke fotosynthetische complexen vormen een belangrijke drijfveer achter

128 Samenvatting

veel van het onderzoek naar moleculaire aggregaten. Vanuit fundamenteel oog-punt gaat de interesse uit naar een beter begrip van de collectieve excitaties diein moleculaire aggregaten voorkomen. Belangrijke onderwerpen die in dit proef-schrift specifiek aan bod komen zijn de invloeden van de geometrie en moleculairestructuur van het aggregaat zelf op de optische eigenschappen van deze collectieveexcitaties en de effecten die de omgeving waarin het aggregaat zich bevindt hierophebben. Naast het beter begrijpen van de natuurkundige principes die ten grond-slag liggen aan de werking van natuurlijke fotosynthetische systemen, vormen dezecomplexen ook een blauwdruk voor het ontwerpen van synthetische antennesyste-men met gelijkwaardige of misschien zelfs betere efficientie van lichtabsorptie enenergietransport. Een van de meest veelbelovende kandidaten bestaat uit kool-stofcyanine moleculen waaraan hydrofobe en hydrofiele zijgroepen vastzitten. Ineen waterige oplossing vormen deze moleculen namelijk cylindrische aggregaten,waarvan de geometrie nauw verwant is aan de natuurlijke chlorosoom-antennes dievoorkomen in groene bacterien.

De specifieke optische eigenschappen van moleculaire aggregaten zijn het di-recte gevolg van collectieve excitaties. Hiermee wordt bedoeld dat wanneer er lichtgeschenen wordt op zulke systemen er niet een enkel molecuul gexciteerd wordt,maar dat de excitatie collectief gedeeld wordt door de moleculen. Om dit soortfenomenen kwantummechanisch te kunnen beschrijven, beschouwen we over hetalgemeen een molecuul als een twee-niveau systeem, bestaande uit een grondtoe-stand en een aangeslagen toestand. De wisselwerking tussen de overgangsdipolenvan de moleculen zorgt er voor dat een excitatie kan verspringen van molecuul naarmolecuul, waardoor er een excitatiegolf in het aggregaat onstaat. Deze collectieveexcitatie wordt ook wel een Frenkel exciton genoemd. Wanneer de moleculen in hetaggregaat identiek zijn, zullen tengevolge hiervan de excitontoestanden verspreidzijn over alle moleculen. In dit geval spreken we ook wel van compleet gedelo-caliseerde excitonen. Echter, in werkelijkheid zal dit niet voorkomen aangeziende moleculen wisselwerkingen hebben met hun directe omgeving. Omdat dezeomgeving lokaal zal verschillen van molecuul tot molecuul, zullen ook de excitatie-energieen en overgangsdipolen verschillend zijn voor elk molecuul afzonderlijk. Eendergelijk effect wordt wanorde genoemd en leidt tot excitontoestanden die gedeeldworden door een beperkt aantal moleculen, vaak veel minder dan het totaal aan-tal moleculen in het aggregaat. De mate waarin de excitaties verspreid zijn overeen aggregaat wordt dus bepaald door de wisselwerking tussen de intermoleculairekoppelingssterktes enerzijds (die zorgen voor delocalisatie van de excitaties) ende grootte van de aanwezige wanorde anderzijds (resulterend in localisatie van deexcitaties).

Met behulp van bovenstaande beschouwingen kunnen we nu een aantal ken-merkende optische eigenschappen van moleculaire J-aggregaten kwalitatief beschri-

Samenvatting 129

jven, zoals de roodverschuiving en de versmalling van de absorptieband ten opzichtevan het moleculaire spectrum, en de bijzonder snelle spontane emissie in vergeli-jking met individuele moleculen. Voor een lineair J-aggregaat bestaande uit Nmoleculen zijn er ook N mogelijke excitontoestanden. Als we wanorde niet meen-emen en ervan uitgaan dat het aggregaat opgebouwd is uit veel moleculen (metmoleculaire overgangsdipolen die parallel zijn) zal over het algemeen alleen de ex-citontoestand met de laagste energie het licht absorberen. Aangezien de excitatie-energie van deze excitontoestand lager is dan de excitatie-energie van de individuelemoleculen, leidt dit tot een absorptiespectrum wat verschoven is naar lagere en-ergie (roodverschuiving) ten opzichte van het moleculaire spectrum. De snelheidvan spontane emissie en de absorptie-intensiteit van deze toestand is ongeveer Nmaal groter dan dat van een individueel molecuul als gevolg van de collectievenatuur van deze excitatie. Deze bijzondere excitatietoestand wordt vaak ook welde superstralende excitontoestand genoemd. De breedte van de absorptiepiek vaneen aggregaat wordt bepaald door de grootte van de wanorde. Omdat in een col-lectieve excitontoestand de fluctuaties van de moleculen ten gevolge van wanordetot op zekere hoogte uitmiddelen, zal een excitontoestand effectief een kleinerewanorde ervaren dan de individuele moleculen. Hierdoor is de absorptiepiek vanhet aggregaat vaak veel smaller dan de absorptiepiek van het moleculaire spec-trum. Dit fenomeen wordt ook wel aangeduid als ”exchange narrowing” en speelteen belangrijke rol in de optische eigenschappen van moleculaire aggregaten.

De hierboven beschreven optische eigenschappen van J-aggregaten zijn nietalleen van toepassing op aggregaten met een lineaire geometrie, maar gelden ookvoor cylindervormige J-aggregaten. Een belangrijk verschil ten opzichte van lin-eaire aggregaten is dat het absorptiespectrum van cylindrische aggregaten vaakniet een, maar meerdere absorptiebanden laat zien. Dit is een consequentie vanhet feit dat de overgangsdipolen van de moleculen de symmetrie van de cylindervolgen en dus niet allemaal in dezelfde richting staan, zoals gebruikelijk wel wordtaangenomen in het geval van lineaire aggregaten. Als gevolg hiervan zijn er overhet algemeen meerdere superstralende excitontoestanden met verschillende polar-isatierichtingen. Voor een moleculaire structuur van een cylindervormig aggregaatbestaande uit een molecuul per eenheidscel (gebruikt in hoofdstukken 2 en 3) zijner in totaal drie superstralende toestanden. Een van deze excitontoestanden heefteen overgangsdipool die in de richting van de cylinderas wijst, terwijl de anderetwee toestanden loodrecht gepolariseerd zijn op deze as. Deze laatste twee toes-tanden hebben dezelfde energie en kunnen daardoor niet onderscheiden worden inhet absorptie spectrum, terwijl de parallel gepolariseerde toestand in het algemeeneen andere excitatie-energie zal hebben. Deze zogeheten optische selectieregels diehet aantal superstralende toestanden en hun polarisatie richtingen aangeven zijneen directe consequentie van het feit dat de excitonen compleet verspreid (geen

130 Samenvatting

wanorde) zijn over alle moleculen.

Een van de onderwerpen van het onderzoek in dit proefschrift (beschreven inhoofdstukken 2 en 3) behelst de effecten van wanorde op de optische eigenschap-pen van cylindrische J-aggregaten. Specifiek hebben we hier gekeken naar wanordein de excitatie-energieen van de moleculen, waarbij de excitatie-energie van iedermolecuul willekeurig gekozen is volgens een Gaussische distributie. De breedtevan deze distributie vormt een maat voor de grootte van de wanorde. In hoofd-stuk 2 laten we zien dat de reeds beschreven optische selectieregels niet langergeldig zijn wanneer wanorde geıntroduceerd wordt, doordat de excitontoestandengelocaliseerd zijn. Dit resulteert in een distributie van polarisatierichtingen van deoptisch relevante excitontoestanden. De mate waarin de selectieregels geschondenworden, wordt bepaald door de verhouding tussen de localisatielengte van het ex-citon (die bepaald wordt door de sterkte van de wanorde) en de omtrek van decylinder. Dit betekent dat de selectieregels sterker gebroken worden naarmate destraal van de cylinder toeneemt. Hoofdstuk 3 behandelt vervolgens het effect vanwanorde op de breedte van de energetisch laagste absorptiepiek. Over het alge-meen geldt dat de lijnbreedte groter wordt naarmate de wanorde ook groter wordt,als een gevolg van het feit dat de localisatielengte van de excitonen kleiner is bijgrotere wanorde. De mate waarin dit gebeurt by cylindrische aggregaten verschiltechter drastisch van dat voor aggregaten met een lineaire geometrie. De reden hier-voor is dat excitonlocalisatie in cylindrische aggregaten sterk onderdrukt wordt tengevolge van de tweedimensionale structuur van de cylinder. Dit fenomeen vormteen mogelijke verklaring waarom de cylindervormige antennecomplexen in groenebacterien zo efficient zijn. De geometrie van de cylinder zorgt er in dit geval voordat excitatie-energie zich makkelijk door het antennecomplex kan verspreiden, zelfsin de aanwezigheid van inhomogeniteiten.

In hoofstuk 4 zijn de resultaten gepresenteerd van een samenwerkingsverbandmet experimentele groepen om de optische eigenschappen en moleculaire structuurvan zelf-geassembleerde dubbelwandige cylindrische aggregaten bestaande uit eenbepaald type amfifiele koolstofcyanine moleculen te ontrafelen. Zoals eerder opge-merkt vertonen deze synthetische aggregaten wat betreft hun geometrie grote geli-jkenis met de chlorosomen van groene bacterien waardoor ze aantrekkelijke model-systemen vormen voor toepassingen in kunstmatige antennecomplexen. Door mid-del van vergelijking met de experimentele spectra hebben we een model opgesteldvoor de moleculaire structuur van deze aggregaten. Dit model is gebaseerd ophet oprollen van tweedimensionale vlakken met een visgraatstructuur, waarbij elkeeenheidscel bestaat uit twee moleculen. Deze samenwerking tussen experiment entheorie heeft niet alleen geleid tot een gedetailleerde beschrijving van de struc-tuur op moleculair niveau maar heeft ook nieuwe inzichten gebracht over de rolvan interacties tussen de binnenste en de buitenste cylinder in deze buisvormige

Samenvatting 131

aggregaten.Hoofdstuk 5 behandelt het effect van de wisselwerking tussen de collectieve

excitaties van een lineair aggregaat en vibraties in de omgeving. Wanneer de kop-pelingen tussen de moleculen in het aggregaat veel groter zijn dan de interactiesmet de vibrationele toestanden in de omgeving kan de optische respons over hetalgemeen goed beschreven worden met behulp van storingsrekening. Echter, dezemethode is in het algemeen niet toepasbaar wanneer de intermoleculaire koppelin-gen van dezelfde orde van grootte zijn als de wisselwerkingen met de vibraties. Eenvoorbeeld waarvoor deze situatie zich aan kan dienen is bij de hierboven beschrevendubbelwandige aggregaten, waar de interacties tussen de binnenste en buitenstewand mogelijk van dezelfde grootte zijn als de koppelingen met de vibraties. Wehebben in dit hoofdstuk een model opgesteld om optische spectra van lineaire aggre-gaten te berekenen, waarbij de koppeling met de vibraties expliciet is meegenomen.Door middel van een polarontransformatie hebben we algemene semi-analytischeuitdrukkingen voor de spectra bepaald voor willekeurige grootte van de vibra-tionele en moleculaire koppelingen. De geldigheid van deze spectrale expressieswordt bestudeerd door vergelijking met een alternatieve numerieke methode, dezogeheten twee-deeltjes benadering. Onze resultaten laten zien dat vergeleken metstandaard storingstheoriemethodes dit polarontransformatiemodel de optische re-spons voor een grotere spanwijdte van vibrationele en moleculaire interactie sterktesgoed beschrijft, terwijl de complexiteit en computationele kosten vergelijkbaar zijnmet die van storingstheorieen.

Het laatste hoofdstuk van dit proefschrift (hoofdstuk 6) beschrijft de analyse vanhet foton-emissieproces van kleine moleculaire aggregaten, in het bijzonder hebbenwe hier gekeken naar dimeren en trimeren. Dit emissieproces is experimenteelmeetbaar met behulp van ”single-molecule” spectroscopie (SMS) technieken. Do-ordat in SMS gemeten wordt aan individuele aggregaten in plaats van ensembleslevert dit unieke informatie op over de eigenschappen van de collectieve exciton-toestanden van afzonderlijke aggregaten en de mogelijke heterogeniteit tussen deverschillende aggregaten. Met behulp van een theoretisch model hebben we destatistiek van het foton-emissieproces berekend. Onze resultaten tonen inzicht in derol van meervoudig-geexciteerde toestanden in moleculaire aggregaten, zelfs bij lagelichtintensiteit, die normaal gesproken verborgen blijven in ensemble spectroscopie-experimenten.

132 Samenvatting

Dankwoord

Allereerst wil ik graag mijn promotor, Jasper Knoester, bedanken voor de kans diehij mij gegeven heeft om mijn promotieonderzoek in zijn groep te doen. Vanaf hetmoment dat ik in je groep terecht ben gekomen, heb ik met veel plezier onderzoekgedaan. Je hebt me altijd gesteund en begeleid tijdens mijn onderzoek, maar meook de vrijheid gegeven om de wetenschappelijke vragen te beantwoorden die ikzelf interessant vond. Je feilloze natuurkundige intuıtie voor aggregaten, je talentom altijd direct de juiste vragen te stellen en je kritische opmerkingen bij hetschrijven van wetenschappelijke artikelen zijn ontzettend waardevol geweest voorde totstandkoming en kwaliteit van dit proefschrift. Ik wil je bij deze hiervoor danook hartelijk bedanken. Ten tweede wil ik ook graag Victor Malyshev bedanken.Voornamelijk in de laatste cruciale fase van mijn promotietijd kon ik altijd bij jeterecht. Ik heb met heel veel plezier met je samengewerkt, waarbij je inzicht enkennis van excitonlocalisatie van groot belang zijn geweest voor dit proefschrift.

I would like to thank the reading committee, consisting of Caspar van der Wal,Rienk van Grondelle and Peter Reineker, for their careful reading of the manuscriptand their positive assessment.

Verder wil ik mijn twee paranimfen Mannold en Bas bedanken. Mannold, ik hebhet met jou, zowel op de universiteit als daarbuiten, altijd ontzettend naar mijn zingehad, vooral natuurlijk ook in het jaar dat we huisgenoten zijn geweest. Bas, ikheb altijd enorm genoten van onze fijne samenwerking aan cylindrische aggregaten,waarvan veel ook in dit proefschrift terug te vinden is, en van de gezellige tijd diewe doorbrachten buiten de universiteit om. Tevens wil ik bij deze ook Klaas Berndbedanken voor alle hulp met het maken van de omslag van dit proefschrift.

Natuurlijk wil ik iedereen van onze groep van harte bedanken voor de gezelligegroepsuitjes, de nuttige groepsbesprekingen en de leuke borrels. In het bijzonder wilik Anna en Maurits bedanken voor het leerzame en leuke project over polaronenwaar we een jaar lang samen aan gewerkt hebben. Verder wil ik ook Thomasbedanken voor de tijd en energie die hij steekt in het regelen van allerhande zakendie te maken hebben met de groep. Natuurlijk gaat mijn dank ook uit naar mijn

134 Dankwoord

kamergenoten van de afgelopen jaren, Chiel en Oleksander. Tenslotte wil ik graagIris de Roo-Kwant en Annelien Blanksma van het secretariaat bedanken voor allehulp met administratieve zaken.

One of the chapters in this thesis describes the results of a collaboration withseveral experimental groups. I would like to thank everybody involved for thefruitful collaboration and enjoyable experience. My special thanks to Dorthe Eisele.Your never-ceasing enthusiasm and expertise of cylindrical J-aggregates have beena great source of inspiration for me. I have learned a lot from our many discussions,which in the end resulted in a, in my opinion, beautiful paper.

I would like to thank Robert Silbey for the pleasant discussions that we hadabout single-molecule spectroscopy during his visit to Groningen in my first year.You were truly one of the nicest people I met. Science has lost one of its greatminds too early, may you rest in peace.

Buiten het leven als onderzoeker moet er natuurlijk ook ruimte zijn voor socialeactiviteiten die zorgen voor de broodnodige ontspanning. Bij deze wil ik graagten eerste alle Franckenleden bedanken die ik heb leren kennen gedurende mijnstudie en ook later tijdens mijn promotie. Dankzij jullie heb ik een fantastischetijd gehad hier in Groningen. Speciaal gaat mijn dank uit naar Paul, Mannold,Arno, Robin, Jorik, Bas, Erik van der H., Bijl, Roel, Sander ter V., Jorn, Henkie,Wendy, Maaike, Sandra, Onur, KB, Remko, Detsi, Thijs, Hedde, Mark, Rudy,Jakko, Peynacker, Christiaan, Tim, Tom de B., Jasper van den B., Tom B., Bosch,Boerma en Reeuwerd. Verder wil ik ook graag alle mensen bedanken met wieik menig gezellig uurtje op de tennisbaan heb doorgebracht, in het bijzonder demensen met wie ik in de afgelopen jaren competitie heb gespeeld: Jeroen, Jelmer,Michel, Joost, Jarno, Pen, Marc, Cecile, Ilona, Annet, Morten en Piso.

Graag bedank ik ook de familie van Anouk, speciaal haar ouders Sjaak en Inge,en haar twee zusjes Iris en Merel. Ik waardeer het enorm dat jullie mij altijd metopen armen hebben ontvangen, waardoor ik me vanaf het eerste moment bij jullieheb thuis gevoeld.

Natuurlijk wil ik ook mijn familie, in het bijzonder mijn ouders en zus, bedankenvoor de steun en het vertrouwen die ik vanaf het eerste moment van ze heb gekregen,en nog steeds krijg. Ik ben ontzettend dankbaar voor het feit dat jullie mij altijdhebben geholpen. Zonder jullie had ik nooit zoveel kunnen bereiken en was ik nietdiegene geweest die ik nu ben.

Als laatste wil ik mijn lieve vriendin Anouk bedanken. Jij bent altijd mijn steunen toeverlaat geweest gedurende mijn promotietijd. Ik hoop dat ik je in de komendejaren net zoveel steun kan geven bij je promotieonderzoek als jij mij gegeven hebtin de afgelopen jaren. We hebben samen al een fantastische tijd achter de rug,maar ik hoop dat dit pas het begin is van onze mooie reis samen. Ik hou van je,moppie!

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University of Groningen Exciton localization and optical … · absorption spectrum. In contrast, to calculate the nonlinear optical features [14–18] it is necessary to also invoke