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Exciton dispersion in molecular solids

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2015 J. Phys.: Condens. Matter 27 113204

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Journal of Physics: Condensed Matter

J. Phys.: Condens. Matter 27 (2015) 113204 (21pp) doi:10.1088/0953-8984/27/11/113204

Topical Review

Exciton dispersion in molecular solids

Pierluigi Cudazzo1,2, Francesco Sottile2,3, Angel Rubio1,2,4 andMatteo Gatti2,3,5

1 Nano-Bio Spectroscopy group, Universidad del Paıs Vasco, CFM CSIC-UPV/EHU-MPC and DIPC,E-20018 San Sebastian, Spain2 European Theoretical Spectroscopy Facility (ETSF)3 Laboratoire des Solides Irradies, Ecole Polytechnique, CNRS-CEA/DSM, F-91128 Palaiseau, France4 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany5 Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette, France

E-mail: [email protected]

Received 30 September 2014, revised 12 January 2015Accepted for publication 20 January 2015Published 4 March 2015

AbstractThe investigation of the exciton dispersion (i.e. the exciton energy dependence as a function ofthe momentum carried by the electron–hole pair) is a powerful approach to identify the excitoncharacter, ranging from the strongly localised Frenkel to the delocalised Wannier–Mottlimiting cases. We illustrate this possibility at the example of four prototypical molecularsolids (picene, pentacene, tetracene and coronene) on the basis of the parameter-free solutionof the many-body Bethe–Salpeter equation. We discuss the mixing between Frenkel andcharge-transfer excitons and the origin of their Davydov splitting in the framework ofmany-body perturbation theory and establish a link with model approaches based on molecularstates. Finally, we show how the interplay between the electronic band dispersion and theexchange electron–hole interaction plays a fundamental role in setting the nature of theexciton. This analysis has a general validity holding also for other systems in which theelectron wavefunctions are strongly localized, as in strongly correlated insulators.

Keywords: exciton dispersion, molecular solids, Bethe–Salpeter equation

(Some figures may appear in colour only in the online journal)

1. Introduction

Excitons are neutral elementary excitations of an electronicsystem in which electrons and holes are led to form boundstates by their Coulomb attraction. In insulating materialsthe creation of a bound exciton6 needs an energy that isless than the one required to excite independently electronsand holes (the difference is the exciton binding energy).Therefore, bound excitons, being the lowest-energy neutral

6 In other frameworks, e.g. for the interpretation of photoluminescenceexperiments in semiconductors [1], what are called here ‘bound excitons’ areusually referred to as ‘free excitons’, while ‘bound excitons’ are those that aretrapped by neutral impurities. Instead, in the present context bound excitonsare defined in contrast to resonant (or unbound) excitons, which are formed inthe electron–hole continuum and have an excitation energy that is larger thanthe fundamental gap (the latter are not explicitly considered here).

excitations, play a key role in the optical properties ofmaterials, as for light absorption or luminescence andin their technological applications, as for optoelectronics,photovoltaics, photocatalysis, etc.

It is often said [2, 3] that bound excitons come into twotypes: Frenkel or Wannier–Mott. While in Frenkel excitons[4, 5] the electron–hole pairs are strongly localised (i.e. withinthe same lattice unit cell), in Wannier–Mott excitons [6, 7] theelectron–hole pairs have a wider separation (i.e. the bound-state radius is much larger than the unit cell). Stronglybound Frenkel excitons are expected in wide-gap insulators,where the screening of the electron–hole attraction is weak,while weakly bound Wannier excitons are more probablein semiconductors. However, excitons in real materials aregenerally intermediate between the two limiting situationsdescribed by the models [8]. A notable example of excitons

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J. Phys.: Condens. Matter 27 (2015) 113204 Topical Review

of the intermediate type are charge-transfer excitons. Forexample, charge-transfer excitons can be found in ioniccrystals as a result of the excitation of valence electronslocalised around the anion into empty states localised aroundthe cation.

Due to the translation symmetry of the crystal, excitonsin a solid are characterised by the momentum q carried bythe electron–hole pair. Thus also the exciton energy dependson q, defining the exciton dispersion. The resulting ‘excitonband structure’ is linked to the way the exciton travels insidethe crystal (this can be called the ‘exciton kinematics’), inthe same manner as the electronic bands characterise how theelectrons (or the holes) can propagate in the solid. Frenkeland Wannier excitons give rise to two different mechanismsof electron–hole propagation. When the exciton is stronglylocalized as in the Frenkel model, it can travel by hoppingfrom site to site (mainly) through dipole–dipole interactions,whereas the center of mass of Wannier excitons propagates asa free particle (whose effective mass is given by the sum of theeffective masses of the electron and the hole). In more realisticterms, excitons can be thought to propagate as wavepackets,which interact and decay, as they can be scattered by latticevibrations, defects, impurites and other electronic excitations7.All these processes characterise the exciton lifetime and definethe ‘exciton dynamics’ [9–12], which is today at the heart ofa great deal of investigations and represents a challenge fortheory. In fact, even though excitons are not carrying netcharge (and thus are not contributing to electrical conduction),they still carry excitation energy that can be transformed andexploited by different means. The optimisation of this processis obviously of great relevance for technological applications.

The detailed characterisation of the dispersion and thespatial localisation of excitons in real materials calls fortheoretical methods that are not restricted to the simple limitingcases of Frenkel or Wannier excitons. This is nowadayspossible thanks to powerful and accurate methods that havebeen developed in the last decades within the framework ofthe first-principle Green’s function theory [13–19]. Theseapproaches determine the solution of the Bethe–Salpeterequation (BSE) for the two-particle correlation function, aparticular kind of two-particle Green’s function describingthe correlated motion of the electron–hole pair [20]. TheBSE is an in principle exact equation for the determination ofneutral electronic excitations, yet resting on the validity of theBorn–Oppenheimer approximation and hence the possibilityof separating nuclear and electronic motions. Within thisframework the BSE does not make reference to any model(i.e. it is ab initio), thus being generally valid and does notmake use of any adjustable parameter, which potentially givesthe results a predictive power. In this context, conclusionsthat would be obtained from the models can be retrieved asparticular approximations to the general solution of the BSE.

7 In this context, an important concept is the exciton diffusion (or migration),also termed exciton delocalisation, which should not be confused with thespatial (de)localisation of the exciton that is discussed in the present topicalreview. While the former refers to the center-of-mass motion and for instance isused to define the exciton coherence lenght, the latter is linked to the electron–hole internal separation and is hence useful to tell a Frenkel from a Wannieror charge-transfer exciton.

This is the strategy that we will follow in this topicalreview, in which we will deal with excitons of molecularsolids. They are a textbook example of excitons showinglarge binding energies [21–26], which we have selected fortheir high pedagogical relevance. At the same time, excitonsin these materials are attracting a considerable interest fortechnological applications, as they play a crucial role inoptoelectronic devices, such as organic light-emitting diodes(OLED), in the photogeneration of carriers and for the energytransfer in organic solar cells. For example, in this classof materials there has been a recent explosion of studies onsinglet fission (see e.g. [27–38]), a multiple exciton generationprocess creating two triplet excitons from one singlet exciton,as it has the potential to dramatically increase the efficiency oforganic solar cells.

In this topical review we will focus in particular on thenature of the lowest-energy excitons in prototypical molecularsolids (tetracene, pentacene, picene and coronene), whichcontinues to be subject of debate. BSE calculations are usedto predict the exciton dispersion in these materials, which canbe directly compared for instance with electron-energy lossspectroscopy (EELS) experiments. Finally, from the generalBSE formalism we establish a link with tight-binding modelapproaches, unraveling the role played by localised Frenkeland delocalised charge-transfer excitons. The results havebeen in part recently published in previous works [39–41].Here we extend the description and the analysis of theexciton properties also to other selected prototypical molecularcrystals, putting them all in a broader perspective. We referto the excellent reviews and books on excitons, specificallyon molecular crystals (see e.g. two of the most recent ones[25, 26] and the references cited therein), in order to have a

general overview of the literature and common wisdoms in thefield, which we introduce in short in the next section. Theabundant available literature allows us to keep this generalbackground description within the pedagogical purposes ofthe present topical review. While in section 3 we summarizethe framework in which the Bethe–Salpeter equation is set andthe effective exciton hamiltonian that can be derived from it,in section 4 we briefly present the materials studied in the restof the article. Finally in sections 5 and 6 we establish thelink with model approaches that are very useful to understandand analyse the first-principles BSE results that are gathered insection 7. In section 8 we draw the conclusions of the presentinvestigation and outline possible perspectives.

2. Excitons in molecular solids

Molecular solids differ from ordinary solids as the elementaryunits are made of molecules instead of atoms. The interactionbetween different molecules is mediated by weak van derWaals forces and is hence much smaller than the covalentbonding between the atoms within the molecule. This impliesthat the electronic properties of the solid are mainly dictatedby the properties of the isolated molecules. The band widthsare generally small and the materials are insulators. Thisis the reason why excitations in these systems traditionallyhave been described using models based on molecular states.

2


(a) (b)

(c) (d)

Figure 1. Frenkel and charge transfer excitons in a molecular solids with two molecules of the same kind (A and B) per unit cell. In Frenkelexcitons, see panels (a) and (b), the electron and the hole are localised on the same molecule. In charge-transfer excitons, see panels (c) and(d), the electron and the hole are localised on different molecules (belonging to the same unit cell, as in the present figure, or to differentones). Frenkel excitons can be localised in either molecule of the unit cell (compare panels (a) and (b)). Contrary to donor-acceptorcomplexes, in this case in charge-transfer excitons an exchange of the hole position leads to correspondingly exchange also the electronposition (compare panels (c) and (d)).

In particular, in organic molecular solids the lowest-energyexcitations are due to transitions between bands that originatefrom π and π∗ molecular states.

In a first approximation the excited states of the crystal areconsidered as the situation in which one molecule is excited andthe others are in their ground state. In other terms, a Frenkelmodel with a molecule per site is adopted. Thus the excitonwavefunction is constructed from the wavefunctions of theisolated molecule and the interaction with the other moleculesin the solid is treated as a perturbation. In this framework, theproblem of describing absorption spectra of molecular crystalscan be schematically split in two steps. The first step amountsto the calculation of transitions between possible molecularconfigurations (obtained with different level of complexity ofquantum chemistry approaches). The second step deals withdipolar and non-dipolar intermolecular interaction. Moreover,one can assume that the direction of the transition dipolemoment in the molecule within the solid is the same as inthe isolate molecule. This assumption for crystals with severalmolecules of the same type per unit cell directly leads to theanalysis given by Davydov of the polarization splitting (named‘Davydov splitting’) of the absorption peaks (see section 5).

In addition to peaks deriving from pure electronicexcitations, also side-peaks corresponding to variouscombinations of electronic excitations with intramolecularvibrations (‘vibronic satellites’) can be taken into account

[42–44]. In fact, being organic solids made by light carbonatoms, the electron–phonon interaction is expected to beimportant. The interaction of excitons with phonons alsogoverns the nature of the exciton propagation. Coherentmotion occurs for weak exciton-phonon coupling, whileincoherent transport takes place for strong coupling. In thislimit, the interaction with the lattice not only influences themotion of the excitons, but it may also affect considerably theexciton state itself, leading to a ‘self-trapped exciton’ [45, 46],in which the lattice around the exciton is strongly deformed.The signature of exciton self-trapping can be found byconsidering the Stokes shift, which is the difference betweenabsorption and luminescence energies [47, 48].

Within those traditional approaches based on molecularstates the lowest-energy exciton is hence by definition anintramolecular Frenkel exciton [49, 50], in which the electronand the hole are confined in the same molecule (see figure 1).However, in some cases the energy separation between charge-transfer excitons, in which the electron and the hole arelocalised on different molecules (see figure 1) and the lowest-energy Frenkel exciton can be very small. In these situationsFrenkel and charge-transfer excitons strongly mix [26, 51–53].The extent of Frenkel or charge-transfer proportion of thelowest-energy exciton is still highly debated in several cases[32, 44, 54–57].

3


It is also important to note here that while in ionicsolids charge-transfer excitons induce a finite dipole momentbetween the anion and the cation, in molecular solids charge-transfer excitons often have a different meaning8. For example,when the unit cell is made by two molecules of the sametype, the valence and conduction wavefunctions are localisedon both inequivalent molecules in the unit cell. Therefore,an exchange of the position of the hole between the twomolecules correspondingly exchanges also the localizationof the electronic charge distribution (see figure 1). Thisimplies that the net dipole moment resulting from the sumof the different charge-transfer excitons is zero (which shouldbe mathematically the case in centrosymmetric crystals forsymmetry reasons). Therefore it has been suggested that theseexcited states could be better referred to as ‘charge-resonance’excitons9 [63].

Electroabsorption spectroscopy, in which a static electricfield is applied to the sample while its absorption spectrumis measured, has been instrumental in demonstrating thecontribution of charge-transfer excitons to the absorptionspectra of molecular solids [64–66]. The effect of the appliedelectric field is to shift the peaks in the spectra (i.e. to inducea ‘Stark shift’). In the case of charge-transfer excitons theshift in their energy levels is linear in the electric field (andproportional to the dipole moment of the single exciton).For Frenkel excitons, which do not have a permanent dipolemoment (assuming a point-like excitation), the shift in theirenergy levels is instead quadratic in the applied fields. Bytracking the corresponding variations in the absorption spectra(and their derivatives), it is in principle possible to identify thedifferent nature of the excitons. This assignment is howevercomplicated by the strong mixing between charge-transfer andFrenkel excitons [67–72]. More recently, the identificationof charge-transfer excitons was supported by electron-energyloss experiments (EELS) which measured the loss function asa function of momentum transfer [73–75]. The possibility toanalyse the measured spectra on the basis of first-principlesapproaches was lacking, which has been fulfilled by the recentworks [39, 40, 76] that are reviewed in the next sections.

In fact, only in recent years the description of excitoniceffects in these complex materials could be addressed using abinitio methods based on the solution of the BSE10. Pioneeringcalculations (see e.g. [78–87]) have discussed, for example,the effect of the crystal packing (also considering the effectof polymorphism and the application of external pressure),the exciton binding energy and the singlet-triplet splitting asa function of the molecular unit size. Generally, the resultsof those calculations were in good agreement with availableoptical absorption experiments.

8 Similarly to ionic solids, in quantum chemistry the charge-transfer charactercan also refer to intramolecular excitations occurring between different partsof large molecules (or between fragments close to the dissociation limit). Forthe theoretical treatment of this problem see e.g. [58–61].9 For the relation between charge transfer and excimers see [62].10 For a recent review of applications of BSE to organic moleculessee [77].

3. The Bethe–Salpeter equation

The (complex) macroscopic dielectric function εM(q, ω) canbe calculated by solving the many-body Bethe–Salpeterequation (BSE) based on the GW approximation (GWA) [88]of the self-energy (for a detailed introduction to the theoreticalframework see [76, 89, 90]):

εM(q, ω) = 1 − 8π

q2

∑λ

∣∣∣ ∑kvc Akvcλ (q)〈φvk|e−iqr|φck+q〉

∣∣∣2

ω − Eλ(q) + iη,

(1)

where Aλ and Eλ are eigenstates and eigenvalues of theexcitonic Hamiltonian

Hex(q) =∑ck

Eck+qa†ck+qack+q −

∑vk

Evkb†vkbvk

+∑

vck,v′c′k′(v

vkck+qv′k′c′k′+q − W

vkck+qv′k′c′k′+q)a

†ck+qb

†vkbv′k′ac′k′+q, (2)

where a† (a) and b† (b) are creation (annihilation) operators forelectrons and holes, respectively. Here Hex(q) is written in theone-particle electron–hole (e-h) transition basis |vkck + q〉 =φvk(r1)φck+q(r2) (where k is in the first Brillouin zone and,in the Tamm–Dancoff approximation, v (c) runs over valence(conduction) bands). In equation (2), φvk are Kohn–Shamorbitals, Enk are the quasiparticle (QP) energies calculatedin the perturbative GW scheme11 [92, 93], v is a modifiedCoulomb interaction in which the long-range componentv(G = 0) is set to 0 in reciprocal space and W is the staticallyscreened Coulomb interaction, calculated in the random phaseapproximation (RPA). By using v instead of v in the BSE onewould calculate the two-particle correlation function L, whoseimaginary part is linked to the loss function.

The matrixelements of v and W enter the BSE kernelas repulsive exchange electron–hole interaction and attractivedirect electron–hole interaction, respectively:

vvkck+qv′k′c′k′+q = 2δM〈ck + q, v′k′|v(r1, r2)|vk, c′k′ + q〉

Wvkck+qv′k′c′k′+q = 〈ck + q, v′k′|W(r1, r2, ω = 0)|c′k′ + q, vk〉.

Here, in absence of spin–orbit coupling, δM = 1 for thesinglet channel and δM = 0 for the triplet (i.e. v is absentfor triplet states). The electron–hole exchange mediated byv is responsible for the crystal local-field effects [94, 95],linked to the induced Hartree potential in a inhomogeneousand polarizable electronic system. While the v term has theform of a dipole–dipole interaction, the direct charge–chargeinteraction W describes the electron–hole attraction, leadingto the formation of bound excitons. The difference between W

and the bare Coulomb interaction v is given by the screening,described by the static limit of the inverse of the microscopicdielectric function ε−1(r1, r2, ω = 0) (which within the GWapproximation is evaluated in the RPA):

W(r1, r2, ω = 0) =∫

dr′ε−1(r1, r′, ω = 0)v(|r′ − r2|).(3)

11 For the discussion of cases where the perturbative GW scheme breaks downwe refer e.g. to [91].

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In BSE calculations of optical properties of standard inorganicsemiconductors it is often possible to neglect local-field effectsin ε−1 assuming a homogeneous screening, i.e. that ε−1 isjust a function of the two-point separation |r1 − r2| [90].On the contrary, in molecular solids, where the screening ishighly anisotropic and inhomogeneous, we have verified thatthis assumption does not hold and it is necessary to fully takeinto account the microscopic nature of ε−1(r1, r2, ω = 0) (ina different context, see also [96]).

When the screening is small (typically in insulators andlow-dimensional materials), W remains strong giving rise toimportant excitonic effects (which conversely are generally notexpected in metallic systems, where the screening is large).

In the vanishing momentum transfer limit q → 0of ImεM(q, ω) one finds the absorption spectrum of thematerial [97]. In anisotropic materials, such as molecularcrystals, the spectra depend on the particular direction in whichthe q → 0 limit is taken. EELS spectra should be insteadcompared to the loss function −Imε−1

M (q, ω).Moreover, from the eigenvectors Aλ of the excitonic

Hamiltonian one also obtains the exciton wavefunction (i.e.a two-particle function):

�λ(rh, re) =∑kvc

Akvcλ (q = 0)φ∗

vk(rh)φck(re), (4)

where rh and re are the positions of the hole and theelectron, respectively. In absence of electron–hole interaction(i.e. neglecting the BSE kernel), the exciton wavefunctionbecomes the uncorrelated product of the single-particleorbitals: φ∗

vk(rh)φck(re). The effect of the interactionis therefore to strongly mix the independent electron–hole contributions (through the Aλ eigenvectors), whilesimultaneously modifying the transition energies from thedifferences Eck+q − Evk to the excitonic eigenvalues Eλ(q).If Eλ(q) is smaller than the smallest single-particle energydifference Eck+q −Evk, one can conclude that a bound excitonis present in the material.

Within the BSE framework it is nowadays possible totake into account the coupling between excitons and phononsfully ab initio, including the temperature-dependence of thespectra [98, 99]. It is in principle also possible to describeself-trapped excitons, through the calculation of excited-stateforces and the Stark shifts from the derivatives of the excitonenergies with respect to external perturbations (e.g. latticevibrations or electric fields) [100, 101]. However, in both casesto date there have not been applications of the BSE formalismto molecular crystals for their high computational cost [102].

4. Prototypical molecular solids

In the next sections we will consider the exciton propertiesof four prototypical molecular solids: tetracene, pentacene,picene and coronene. The building blocks of all thesecompounds are polycyclic aromatic hydrocarbon molecules(see figure 2). While acene molecules (such as tetraceneand pentacene) consist of a linear fusion of benzene rings,phenacene molecules (such as picene) are built up of benzene

Figure 2. Molecular units (left) and crystal structures (right) of thefour systems considered in the present work. A black arrow has beenadded in the case of picene to highlight the main molecular axis.

rings condensed in a W-shape. Coronene, instead, is made ofsix benzene rings joined in a circle.

The electronic properties of these molecules significantlydepend on the number of benzene rings forming them (fourfor tetracene, five for pentacene and picene, six for coronene)and on their edge structures (see e.g. [103, 104]), zigzagtype for the acenes and armchair type for the phenacenesand coronene. This finding is very general, holding also fortheir unsaturated counterparts, namely graphene nanoribbonsand nanoflakes (see e.g. [105–107]) and can be understoodalso in terms of Huckel molecular orbital theory and organicchemistry concepts such as Clar aromatic sextets and Kekulestructures [108, 109]. For example, in the acenes thehighest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO) are localized at theedges, whereas in the phenacenes they are delocalized all overthe molecules. While the stability and the HOMO-LUMOgaps of the acenes decrease rapidly with increasing numberof benzene rings, such rapid decrease is not observed for thephenacenes, which have larger gaps and are more stable thanthe corresponding acenes with the same number of benzenerings.

The acenes are one of the most investigated familiesof organic crystals for the possibility of a wide range ofpractical applications in optoelectronics, such as light-emitting

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diodes, thin-film transistors, sensors, electrodes for lithium ionbatteries, photovoltaic devices, just to mention few examples[110, 111]. Picene and coronene have attracted a large interestvery recently after the discovery of superconductivity withalkali-metal doping [112, 113].

In the solid phase all the investigated hydrocarbonmolecules crystallize in a peculiar structure, having aherringbone arrangement of two inequivalent molecules in theab plane and a layered structure of molecules stacked along thec axis (see figure 2). The crystal structure is either monoclinic(as for picene and coronene, space group no. 4) or triclinic (asfor tetracene and pentacene, space group no. 2).

In the solid the strong covalent intramolecular bondingdominates the weak van der Waals intermolecular interactions.For example, the planarity and conformational rigidity of themolecules due to their aromatic bonding are preserved also inthe crystal. The resulting structural and electronic propertiesof the crystals are thus highly anisotropic and can be directlyrelated to the properties of the isolated molecules.

The shallow energy potentials linked to the van derWaals bonding lead to a marked polymorphism, with differentphases obtained varying the growth conditions or changing thethickness of the thin films. For example, pentacene crystallizesin at least four different morphologies characterized by thedistance between the molecular units along the c direction[118, 125]: 14.1 Å [116] for single crystals and 14.4 Å[121, 123], 15.0 Å or 15.4 Å [126] for thin films. Moreover,the angles between the molecules change notably [127], i.e.by more than 20◦, from one structure [116] to the other[121, 123]. In the first part of table 1 we have gathered thelattice parameters used in the present work12.

In addition, the cell parameters naturally change as afunction of temperature. To have an idea of these variationsfor pentacene, which has been experimentally characterisedin details, in the second part of table 1 we also reportthe experimental crystal structure data obtained at differenttemperatures. Finally, in the last part of table 1 we comparethe crystal structures adopted in other BSE calculations foundin literature [81, 123].

Figure 3 shows the band structures of the four compoundscalculated in the local-density approximation (LDA) ofdensity-functional theory13. The band gap in the crystals isnaturally smaller than the HOMO-LUMO gap of the isolatedmolecules. Nevertheless, the behavior observed for theisolated molecules is reflected also in the crystal phase, with theband gap that is decreased by increasing the molecular sizes foracenes (e.g. tetracene versus pentacene) and increased movingfrom acenes to phenacenes (e.g. pentacene versus picene).

Each pair of bands in the solid mainly derives from amolecular level. In fact in each layer in the solid the molecularorbitals of each molecule split into bonding and antibondingones. There are exceptions to this general expectation, as forthe LUMO and LUMO+1 levels in picene that give rise to

12 For the other computational details we refer to the previously publishedarticles [39–41, 128, 129].13 LDA, which is known to underestimate band gaps, is used here as the zero-order reference for perturbative GW calculations that are supposed to give amore appropriate description of band structures and band gaps (see table 2).

highly hybridized bands [130]. Moreover, molecular levelsthat are degenerate in the molecule for symmetry reasons cansplit in the solid, as it happens for coronene that has a point-group symmetry D6h for the isolated molecule that is loweredto D2h in the crystal [131].

The bands are rather flat especially along the stackingdirection of the molecules in the crystal. In contrast, the largestband dispersions are found in directions within the herringboneplane, in correspondence to larger intermolecular interactions.In general, in molecular solids the bandwidths are rather smallsince van der Waals interactions are weak. The intermolecularinteractions strongly depend on the relative orientations of themolecules: changing the herringbone angle significantly altersthe intermolecular interactions and thus the band dispersion inthe herringbone plane. Therefore also the calculated electronicproperties are highly sensitive to the differences of the crystalstructures of the various polymorphs [78, 81, 127]. Alsokeeping in mind this important observation, calculated LDAband structures in figure 3 compare well with other similarcalculations [54, 78, 81, 127, 130, 131].

5. From the molecule to the solid

The traditional paradigm for molecular crystals is based on thefact that starting from the electronic excitations of the isolatedmolecule it is possible to directly obtain a qualitative pictureof the electronic excitations of the crystal as well. In thefollowing we will illustrate this intuitive connection focusingon the particular example of the neutral excitations of the aceneand phenacene molecules, but the analysis is of course generaland can be extended also to coronene and other classes ofmolecular solids.

Phenacenes (as picene) and acenes (as tetracene andpentacene) belong to the C2v and D2h point group, respectively.Thus, by symmetry the only allowed transitions in the dipoleapproximation are π → π∗ and σ → σ ∗ for polarizationdirections on the plane of the molecule and π → σ ∗ andσ → π∗ for directions perpendicular to the molecular plane.In the condensed phase, due to the small overlap betweenwavefunctions localized on different sites, this picture isapproximately still valid. The a and b axes of the cyrstalhave a large component normal to the plane of the molecule,while c crystal axis is nearly parallel to the molecular mainaxis (see figure 2). Therefore, in the low-energy region, whereonly π → π∗ transitions are active, the oscillator strengthsfor light polarized in the ab plane have low intensities and thespectrum is dominated by transitions with polarization alongc. However, in the acenes the HOMO-LUMO transition fora polarization parallel to the main axis of the molecule isforbidden by symmetry. So also in the solid the absorptiononset in pentacene and tetracene has only contributions arisingfrom transitions with polarization belonging to the ab plane,contrary to picene where the HOMO-LUMO transition isallowed.

In the Frenkel picture each neutral excitation of theisolated molecule gives rise in the condensed phase to anumber of excitations equal to the number of molecules inthe unit cell. In the present case, due to the presence of

6


Table 1. (First part) Experimental crystal structures (lengths of the primitive unit cell vectors and angles between them) that have beenadopted in the present work. In the last column the original experimental reference is given. When available, the temperature of themeasurement is reported. (Second part) Experimental crystal structures as a function of temperature. (Third part) Lattice parameters used asinput for other BSE calculations. Reference [81] uses the structure from [123] within the more modern Niggli cell convention [124] wherethe spacing in the first axis direction is the smallest (compare the two last rows).

System a (Å) b (Å) c (Å) α (◦) β (◦) γ (◦) T (K) References

Picene 8.480 6.154 13.515 90 90.46 90 [114]Pentacene 6.277 7.663 14.363 76.893 88.182 84.326 100 [115]Tetracene 6.0565 7.8376 13.0104 77.127 72.118 85.792 155 [116]Coronene 16.094 4.690 10.049 90 110.79 90 [117]

Pentacene 6.275 7.791 14.556 76.434 87.585 84.707 293 [115]Pentacene 6.239 7.636 14.330 76.978 88.136 84.415 90 [118]Pentacene 6.266 7.775 14.530 76.475 87.682 84.684 300 [118]

Tetracene [80] 7.98 6.14 13.57 101.3 113.2 87.5 [119]Pentacene [78] 6.06 7.90 14.88 96.74 100.54 94.2 [120]Pentacene [78] 6.253 7.786 14.511 76.65 87.50 84.6 [120]Pentacene [80] 7.93 6.14 16.03 101.9 112.6 85.8 [121]Pentacene [81] 5.92 7.54 15.63 81.5 87.2 89.9 [122]Pentacene [81] 6.06 7.90 15.01 81.6 77.2 85.8 [123]Pentacene [54] 7.90 6.06 16.01 101.9 112.6 85.6 [123]

Z Γ X V U Γ V U R X V-3

-2

-1

0

1

2

3

4

Ene

rgy

(eV

)

εF

Pentacene

Γ Z T R V X Γ Y V-3

-2

-1

0

1

2

3

4

Ene

rgy

(eV

)

εF

Tetracene

Γ B D Z C Y Γ-3

-2

-1

0

1

2

3

4

Ene

rgy

(eV

)

Picene

Γ Z C Y Γ X D E A X E C Γ-3

-2

-1

0

1

2

3

4

Ene

rgy

(eV

)

εF

Coronene

Figure 3. LDA band structures. The zero of the energy axis is set at the energy of the top of the occupied bands. k-point definitions can befound in the Bilbao Crystallographic Server [132].

two inequivalent molecules (A and B) in the unit cell, foreach molecular excitation two different configurations arepossible: one where the excitation is localized on moleculeA and the other one where it is localized on molecule B (seefigure 1). The resulting excitonic states �±

λ are hence thesymmetric (+) and antisymmetric (−) combinations of thetwo configurations (beyond the Frenkel picture, this is validalso for charge-transfer excitations that of course are absent in

the isolated molecules). In absence of interaction symmetricand antisymmetric excitons are degenerate. However, thisdegeneracy is removed when the Coulomb interaction betweenexcitations localized on different molecules is taken intoaccount, giving rise to the splitting between symmetric andantisymmetric excitons that is called Davydov splitting (DS).In general, the formally corrected description of the Davydovsplitting requires an analysis based on the group theory.

7


However, in the case of phenacene and acene molecularcrystals it can be interpreted also using a rather simple picture,as discussed by Davydov in [21]. The dipole matrix elementsfor the exciton �±

λ can be generically written in terms of thedipole matrix elements of the isolated molecule, which we callx and y for the HOMO-LUMO transition for a polarizationdirection parallel to the short (x) and the long (y) axis of themolecule, respectively, (see figure 4):

〈�±λ |r|0〉 = (αA ± αB)x + (βA ± βB)y (5)

where αA and βA are the projections of r along the x and y

axes of the molecule A (and, analogously, αB and βB for thesecond molecule B in the unit cell). In the present case, dueto the symmetry of the acene molecules, the y dipole is zero.Therefore, the dipole matrix element defined in equation (5)simplifies into p± = 〈�±

λ |r|0〉 = αAx±αB x = pA ±pB . Thissimplification occurs also for the crystal structure of picene.In fact, in that case, even if the y dipole is different from zero,|βA| and |βB | ≈ 0 for both a and b directions, since y is nearlyperpendicular to both axes.

We thus see that the dipoles associated to the two Davydovcomponents are approximately perpendicular each other (infact, p+ = pA + pB is perpendicular to p− = pA − pB).Moreover, the angle between b and the x axis is approximatelythe same for the two molecules, i.e. for the b direction|αA| � |αB |. Thus one of the two Davydov components hasits dipole moment mainly oriented along b (i.e. it is visiblealong b). Since a and b are also approximately perpendiculareach other, the other Davydov component is polarized mainlyalong a. We also understand that in all the directions thatare intermediate between a and b, the weight of the twoDavydov components changes according to the projection ofthe polarization direction with respect to the directions of thetwo dipoles p+ and p− (see [41] for a practical example).

Just on the basis of geometrical considerations, we havefound that for the dipole matrix elements (5) oriented along a

or b one has |αA| � |αB | and |βA| � |βB | � 0 and thus we havebeen able to conclude that each Davydov component is mainlyvisible along one of the two axes only. In order to identifywhich component (the symmetric or the antisymmetric) isvisible along a given axis (a) or (b), one has still to say whetheralong that axis αA � +αB or αA � −αB . In other words, onehas to identify the orientation of the molecular dipole (so farwe have just examined its direction).

In the case of picene, the identification of the mutualorientation of the two molecules in the unit cell is rather easyon the basis of the structural properties alone. In fact we findthat in picene αA � −αB for dipole moment along a andαA � αB along b. Thus the antisymmetric state is visible onlyalong the a axis, while the symmetric one is visible along b.On the contrary, for the acenes, being symmetric with respectto both x and y axes, the identification of the symmetric andantisymmetric components is not possible from the structuralproperties alone, without choosing an additional (arbitrary)convention (of course, the two Davydov components can bestill unambiguously identified for example by the symmetryproperties of the excitonic wavefunctions or from their energylevels, see section 6).

We illustrate the difference between picene and pentacenein figure 4, where, together with the definition of the respectivedipolar axes x and y, for both picene and pentacene weshow two pairs of molecules. In the second pair, the secondmolecule is rotated by 180◦ around the y direction, thereforethe orientation of its x dipole is opposite than in the first pair.However, while for picene the different mutual orientationof the dipoles corresponds also to a different geometricalorientation, in pentacene, for its symmetry properties, it is notpossible to distinguish the two geometries associated to thedifferent dipoles.

In summary, we have described how just fromconsiderations about the geometry of the crystal structure andthe mutual orientations of the molecules, together with thesymmetry properties of the isolated molecule, it is possible topredict the anisotropy of the spectra. In the next section we willdiscuss how it is also possible to intuitively infer the excitationenergies, distinguishing between Frenkel and charge-transferexcitons.

6. Localised wavefunctions: a simplified model

To better understand the excitonic effects in molecular solids,we rewrite the excitonic hamiltonian equation (2) in a basis ofwavefunctions localized on the molecular units, which in a firstapproximation are the wavefunctions of the isolated molecules:

Hex =∑Ri,Sj

heRi,Sj a

†RiaSj −

∑Ri,Sj

hhRi,Sj b

†RibSj

+∑

Ri,Sj,Pl,Qm

(vSj,Pl

Qm,Ri − WSj,Pl

QmRi )a†Rib

†QmbSj aPl , (6)

with:

vSj,Pl

Qm,Ri = 2δM〈cRi, vSj |v|vQm, cPl〉W

Sj,Pl

QmRi = 〈cRi, vSj |W |cPl, vQm〉.Here the boldface and italic letters indicate the lattice vectorsand the molecular unit in the primitive cell, respectively. he

is the single-particle hamiltonian in the GWA describing themotion of independent electrons: he

Ri,Sj = EcδRi,Sj + t cRi,Sj ,where Ec is the electron level and t c is the electron hoppingintegral. hh is defined correspondingly for holes. E =Ec −Ev is the band gap. From now on we drop the band indexsince we are interested in the lowest excitation that involvesmainly the HOMO-LUMO states of the isolated molecule.

The excitonic wavefunction can be written as asuperposition of electron–hole pairs localized on differentsites [133]:

|�ex〉 =∑

R

eiq·R ∑S

∑ij

CqS,ij a

†Rib

†R+Sj |0〉, (7)

where the coefficient CqS,ij are given by solution of the secular

problem:

〈0|aRibR+Sj e−iq·RHex|�ex〉 = Eex(q)CqS,ij . (8)

If we neglect the overlap between wavefunctions localized ondifferent molecules, matrixelements of v are thus zero unless

8


(a) (b)

Figure 4. In the case of picene, see panel (a), the inversion of the orientation of the dipole axis x of one of the two molecules corresponds toa different mutual geometrical orientation, as it can be understood by comparing the two pairs. This is not the case for pentacene, see panel(b) the inversion of the dipole axis x leaves unaltered the mutual geometrical orientation of the two molecules.

the condition Ri = Qm and Sj = Pl is verified, that iselectron–hole pairs are on the same site. At the same time,matrixelements of W are not zero only when Ri = Pl andSj = Qm, which means that an electron (or a hole) cannotscatter on a different site. The hamiltonian equation (6) in thisway becomes:

Hex =∑Ri,Sj

heRi,Sj a

†RiaSj −

∑Ri,Sj

hhRi,Sj b

†RibSj

+∑Ri,Sj

(2δMvSj,Sj

Ri,Ri − δRi,SjWSj,Sj

RiRi )a†Rib

†RibSj aSj

+∑Ri,Sj

(1 − δRi,Sj )WSj,Ri

SjRi a†Rib

†Sj bSj aRi . (9)

Here the third term describes the interaction between anelectron and a hole localized on the same site. The fourthterm instead describes the interaction between an electron anda hole on different sites. These two terms are coupled by thehopping terms (i.e. the first two), which are responsible forscattering processes of an electron (or a hole) from site to site.

We thus realise that, when the wavefunctions are localisedand we can neglect the overlap between different sites, Hex hasa simple block form:

Hex =(

H FRex H CT−FR

ex

H CT−FR∗ex H CT

ex

). (10)

The first diagonal block H FRex gives rise to a Frenkel (FR)

exciton with both electron and hole localised on the same site:

|�FRex 〉 =

∑Ri

eiq·RCqS=0,iia

†Rib

†Ri |0〉. (11)

The second diagonal block H CTex , instead, produces a charge-

transfer (CT) exciton with electron and hole localised ondifferent sites:

|�CTex 〉 =

∑Ri,Sj

(1 − δRi,R+Sj )eiq·RC

qS,ij a

†Rib

†R+Sj |0〉. (12)

The off-diagonal coupling term H CT−FRex originates from

hopping processes of independent electron–hole pairs and isrelated to the band dispersion. It is responsible for mixing theFR and CT contributions.

6.1. Flat bands

Neglecting the hopping integrals in he and hh is equivalent toassume that the electronic bands in the solid are completely flat,since he and hh give rise to the finite dispersion of the bands.In this way, the excitonic hamiltonian (10) decouples in thetwo independent blocks H FR

ex and H CTex , since H CT−FR

ex = 0.It is important here to stress that only within this

hypothetical situation pure FR and pure CT excitons can beobtained. In any real material, on the contrary, the banddispersion is never zero, implying that in any realistic situationthere is always a mixing of FR and CT excitons (see nextsection).

Considering for the moment only singlet excitations, theinteracting parts Kex = 2v − W of H FR

ex and H CTex , which

are responsible for the formation of excitons, can be writtenexplicitly as:

KFRex =

∑Ri,Sj

(2vSj,Sj

Ri,Ri − δRi,SjWSj,Sj

Ri,Ri )a†Rib

†RibSj aSj

KCTex =

∑Ri,Sj

(1 − δRi,Sj )WSj,Ri

Sj,Ri a†Rib

†Sj bSj aRi .

In particular, for a system with two non-equivalent moleculesin the unit cell, the secular problem of equation (8) for the FRhamiltonian simplifies to the diagonalization of a 2 × 2 matrixwith eigenvalues:

EFR±ex (q) = E + I(q) − W ± |J (q)|, (13)

where W = WRi,RiRi,Ri is the on-site screened Coulomb

interaction and I and J are given by:

I(q) = 2vRi,RiRi,Ri +

∑R′

2vR′i,R′iRi,Ri eiq·R′

J (q) =∑

R′2v

R′j,R′jRi,Ri;(i =j)e

iq·R′(14)

J and the last term in I are the excitation transfer interactions[3] (for simplicity we assume here that v

R′j,R′jRi,Ri = v

Ri,RiR′j,R′j ). In

equation (14) the sums over the sites are generally convergingvery slowly [3] and taking into account only nearest-neighborcontributions is not a good approximation [74]. J is relatedto the scattering process of an electron–hole pair between twoinequivalent molecules and, analogously, I between equivalentmolecules in different unit cells. They are responsible for the

9


dispersion of the FR exciton. The corresponding eigenstatesat q = 0 are symmetric, |�FR+

ex 〉 and antisymmetric, |�FR−ex 〉,

with respect to the exchange of the electron–hole pair betweentwo non-equivalent molecules:

|�FR±ex 〉 = a

†R1b

†R1|0〉 ± a

†R2b

†R2|0〉. (15)

The energy difference (EFR+ex −EFR−

ex ) between symmetric andantisymmetric states is the Davydov splitting and is given by2|J (q = 0)|.

For the CT state, taking into account only the interactionbetween an electron and a hole on nearest neighbors andassuming that the corresponding matrix element is the sameW along all the directions, the secular problem in equation (8)is already in diagonal form with solution:

ECT±ex (q) = E − W. (16)

This solution is at least two-fold degenerate due to thelack of the exchange interaction that, through the Davydovsplitting, removes the degeneracy between symmetric andantisymmetric states (extra degeneracy is related to the numberof nearest neighbors). Moreover, due to the lack of theexchange interaction, the CT exciton does not disperse.

Comparing the two solutions EFRex and ECT

ex , we note thatin this simplified model the condition for the CT state to be thelowest-energy excitation is:

I − W − |J | � −W. (17)

In general, in molecular crystals made by small moleculesthis condition is not satisfied due to the large differencebetween W and W: the lowest excitation is thus a FRexciton. However, in aromatic molecular crystals wherethe size of the molecular units is large enough, the averageelectron–hole distances for an electron–hole pair on the samemolecule or on two adjacent molecules are very close toeach other so that W and W become comparable. Underthese conditions, if the repulsive contribution stemming fromthe exchange electron–hole interaction is strong enough, thecondition in equation (17) is satisfied and the CT solutionbecomes energetically favorable.

In order to verify if this simple model description holdsin practice, we have considered the case of pentacene, solvingthe ab initio BSE in an ideal situation where only transitionsbetween HOMO-LUMO derived bands are taken into account.From the solution of the BSE, we obtain the optical absorptionspectrum, given by the imaginary part of the dielectric functionε2(ω) in the q → 0 limit (in particular, in figure 5 we considerthe a∗ direction). In these calculations we switch on thedifferent contributions one by one in order to analyse in detailstheir effect on the excitation spectra. Since in the calculationswe do not make any assumption about the localisation of thewavefunctions, the fact that the actual numerical results arein agreement with the model description is a validation ofthe hypothesis of negligible overlap between wavefunctionslocalized on different molecules.

In order to recover the simple solution of the model, alsoin the actual calculations in a first step we artificially set thedispersion of HOMO-LUMO bands to zero (we will discussthe effect of the hopping in section 6.2). In this situation weconsider four possible cases (see figure 5):

ε 2(ω)

0 0.5 1 1.5 2 2.5 3 3.5 4Energy (eV)

CT

FR-

FR+

FR

CT

CT

FR-

FR+

(a)

(b)

(c)

(d)

v=W=0

v=0

W=0

_

_

Figure 5. Absorption spectrum of pentacene from the solution ofthe BSE considering only HOMO-LUMO transitions withnon-dispersive (flat) bands. Four situations are considered: (a) withno interactions (v = W = 0); (b) with only the exchangeelectron–hole interaction (W = 0); (c) with only the directelectron–hole interaction (v = 0); (d) with both interactions.Reprinted with permission from [40]. Copyright (2013) by theAmerican Physical Society.

• v = W = 0: without electron–hole interactionsthere is only a single peak in the spectrum at anenergy corresponding to the HOMO-LUMO gap E (seefigure 5(a)). In a molecular picture this means that allthe electron–hole excitations become degenerate and, fora given position of the hole, the electron is delocalizedeverywhere.

• W = 0, v = 0: with respect to the non-interactingcase, the repulsive exchange electron–hole interaction v

shifts the FR states at higher energy and removes thedegeneracy between symmetric and antisymmetric FRstates (see figure 5(b)), indeed the v interaction is atthe origin of a non-zero Davydov splitting DS. Since theexchange electron–hole interaction does not contribute forCT states (see equation (16)), their energy instead remainsunchanged with respect to the previous case (comparefigures 5(a) and (b)).Interestingly, when W = 0 the spectrum for momentumtransfer along the a∗ axis has the largest intensity forthe lowest FR exciton. According to equation (13)the lowest FR exciton must be an antisymmetric state.Thus, from the analysis of the spectra in the present flat-band limit, we can univocally conclude that in pentacenethe antisymmetric and symmetric components of the FRexciton are mainly visible along the a∗ and b∗ axes

10


respectively. This identification instead is not possibleon the basis of geometry arguments alone, as we havediscussed in section 5. This conclusion is in agreementwith [25], while in [39, 44] symmetric and antisymmetriccomponents were inverted.

• v = 0, W = 0: under this condition all the peaksare red-shifted with respect to the non-interacting case(compare figures 5(a) and (c)), as the direct electron–hole interaction W is attractive. Since the on-site W isalways larger than W , the CT exciton is less redshifted,remaining with a energy higher than the FR excitons (seefigure 5(c)). Moreover, with respect to the previous cases,the degeneracy of the CT energy levels is now removed:indeed electron–hole CT pairs characterised by a differentelectron–hole separation feel a different W . Instead,in this case symmetric and antisymmetric FR states aredegenerate (i.e. DS = 0) as v = 0.

• v = 0, W = 0: with respect to the previous case, ingeneral the energy separation between FR and CT statesis reduced since only the FR exciton feels the exchangeelectron–hole interaction v. In pentacene, the order of FRand CT states is even inverted: when v is added to W bothsymmetric and antisymmetric FR excitons result havinga larger energy than the lowest CT exciton (comparefigures 5(c) and (d)). We thus conclude that, in absenceof hopping, the lowest-energy excited state in pentaceneis a pure CT exciton. In contrast, in picene this doesnot happen: the exchange electron–hole interaction is notstrong enough and the lowest excited state remains a pureFrenkel exciton (see section 7.2).

So far we have considered the optical limit correspondingto the vanishing momentum transfer q → 0. Still remaining inthe situation where hopping contributions have been artificiallysuppressed with setting the band dispersion to zero, we havesolved the ab initio BSE as a function of the momentum transferq in pentacene (see figures 6(a)–(b)). We find that while theenergy of pure CT excitons does not change as a function of q,both symmetric and antisymmetric FR excitons have a finitedispersion. We thus see that the investigation of the excitondispersion provides an immediate tool to distinguish FR fromCT excitons. This result can be understood on the basis of oursimple model with neglibible overlap between wavefunctionslocalized on different sites and no hopping contributions.Indeed, since in equations (15)–(16) both W and W do notdepend on the momentum transfer q, the exciton dispersion isgiven only by the exchange electron–hole interaction v throughthe I(q) and J (q) terms.

In particular, from the BSE solution for q along thereciprocal-lattice axis a∗ we find that exciton bandwidth ofthe FR− state is ∼0.1 eV, which is ∼0.4 eV for the FR+ state(see figure 6(a)). This suggests that in the antisymmetricstate the exchange electron–hole terms involving equivalentor inequivalent molecules cancel each other, since for FR−

excitons I(q) and J (q) have opposite sign (see equation (15)).On the contrary, along b∗ the exciton bandwidth is negligible(see figure 6(b)).

To summarize, we have here considered an ideal systemwith bands, derived only from molecular HOMO-LUMO

Γ XMomentum transfer

1

1.5

2

2.5

3

3.5

4

Ene

rgy

(eV

)

Γ Y

(a) (b)

a axis b axis* *

CT

FR-

FR+

CT

FR-

FR+

Figure 6. Exciton dispersion for momentum transfer q (a) along a∗

and (b) along b∗ in pentacene from the solution of the BSE with flatHOMO-LUMO bands. Reprinted with permission from [40].Copyright (2013) by the American Physical Society.

states, that have zero dispersion. From the solution of the BSEfor both the absorption spectra and the dispersion as a functionof the momentum transfer q, we have shown unambigouslythat in pentacene the pure CT state has lower energy than thepure FR state.

6.2. Effect of hopping

The hopping contribution appearing in the single-particlehamiltonians he and hh in equation (6) are responsible for themixing of CT and FR states. Thus, in any real system that hasa non-zero band dispersion, excitons are always a mixture ofthe two states.

We can estimate the correction E±ex to the energy levels

(at q = 0) of the pure FR and CT states in a perturbativescheme where we consider only the hopping between nearestneighbors. For the lowest energy state we thus find:

E±ex ∝ − (tc ± tv)2

|EFR±ex − ECT±

ex | (18)

where t c and tv are the hopping integrals for conduction andvalence bands, respectively. We note that the energy correctionare larger and the CT and FR states are more strongly mixedwhen the hoppings are stronger and when the energies of pureCT and FR are closer each other. Morever only states with thesame ± symmetry are coupled at q = 0.

The hopping has different effects for the CT and FR states.For simplicity, we assume here that the absolute value ofthe hopping terms are the same for valence and conduction:|tv| = |t c|. We consider the two possible cases: tv = t c ortv = −t c. In figure 7 we represent the energy-level diagramscorresponding to the two situations. FR± and CT± are the pureFrenkel and charge transfer states (i.e. without hopping); while(FR+CT)± and (CT+FR)± are the mixture due to the hoppingof Frenkel and charge-transfer states, which originate from thepure FR± and CT± states, respectively. Finally, the Davydovsplitting DS calculated without hopping becomes DS′ whenthe hopping is considered.

11


+FR

(b)

DS

(a)

(FR+CT)+

FR-

DS´

FR-

DS´

CT-

(CT+FR)+

+FR

FR- FR

-

(FR+CT)+

(CT+FR)+

CT-

=CT-+CT

=CT-+CT

(c) (d)

=CT-+CT

+CT

DS´

(FR+CT)-

+FR+FR

FR-

(FR+CT)-

(FR+CT)-

+FR+FR

FR-

DS´DS

(FR+CT)-

=CT-+CT+CT

Figure 7. Energy-level diagram for the singlet channel showing the coupling between Frenkel (FR) and charge-transfer (CT) state when thelowest excitation is (a) and (c) a Frenkel or (b) and (d) a charge-transfer exciton. In (a) and (b) we have assumed that the hopping integralsfor valence and conduction bands are the same: tv = t c while in (c) and (d) they have opposite sign tv = −t c.

When tv = t c (see figures 7(a) and (b)) the antisymmetricstates are not altered by the hopping as E−

ex ∝ (tc − tv)2 = 0.Moreover, if the lowest state is a pure FR exciton, as infigure 7(a), the hopping reduces the Davydov splitting: DS′ <

DS. When the hopping is strong enough, the symmetric state(FR+CT)+ can even be shifted below the antisymmetric stateFR−, inverting the order of the Davydov components of theFR levels, i.e. DS ′ < 0. If instead the lowest state is a pureCT exciton and DS = 0, as in figure 7(b), the consequenceof mixing with FR states is also to induce a finite Davydovsplitting DS ′ > 0.

On the contrary, for tv = −tc (figures 7(c) and (d)) thesymmetric states become those that are not influenced by thehopping. In particular, if the lowest state is a pure FR exciton,as in figure 7(c), the hopping increases the Davydov splitting:DS ′ > DS. Finally, if instead the lowest state is a pure CTexciton, as in figure 7(d), the situation is similar to the case inwhich tv = t c. The only difference is that the order betweensymmetric and antisymmetric states is inverted.

From this simple analysis we can conclude that theDavydov splitting has a different origin for CT and FR excitons.For CT states it is induced by the hopping while for Frenkelstates it is related to the interplay between the exchangeelectron–hole interaction J and the hopping term. In general,when tv = tc we expect that it is smaller in FR states, sincein this case the exchange electron–hole interaction and thehopping term have opposite effect. When tv = −t c bothterms contribute to increase the Davydov splitting that as aconsequence will be larger for FR excitons.

Concerning the exciton energies as a function of themomentum transfer q, we can immediately realise that thehopping is responsible for the dispersion of CT excitons, whichotherwise would be zero for pure CT excitons (see figures 6(a)–(b)). At the same time the hopping modifies the intrinsicdispersion of FR excitons that is given by I(q) and J (q) (seeequation (15)). Moreover, also the exchange electron–holeinteraction has an (indirect) effect on CT excitons: the termsI(q) and J (q) in equation (15) change the energy separationbetween FR and CT excitons as a function of q, which in turnmodulates the mixing between FR and CT (which is larger forexcitons closer in energy). Therefore, we can understand thatthe exciton dispersion is the result of the competition betweenhopping and exchange electron–hole interaction. While forFR+CT states the dispersion is mainly related to the exchangeelectron–hole interaction, in CT+FR states it is mainly relatedto band-structure effects.

The ab initio BSE results for pentacene, taking intoaccount the full dispersion of the same HOMO-LUMO bandsalready considered in the previous section, confirm this picture.We find (see figures 8(a)–(b)) that the lowest exciton inpentacene is a (CT+FR)− state, originating from the lowest CTexciton. It is located at 1.55 eV at q = 0. The correspondingsymmetric (CT+FR)+ state is located at 1.76 eV at q = 0,Therefore the calculated Davydov splitting is 0.2 eV for thelowest exciton of pentacene. The bandwidths of these CTstates ranges between 0.05 and 0.20 eV, matching the order ofmagnitude of the HOMO-LUMO bandwidths. In particularthe bandwidth of the first exciton is 0.14 eV along the a∗ axis

12


Γ XMomentum transfer

1.5

1.6

1.7

1.8

1.9

Ene

rgy

(eV

)

Γ Y

(a) (b)

* b axisa axis

(CT+FR)+

(CT+FR)-

*

Figure 8. Exciton dispersion for momentum transfer q (a) along a∗

and (b) along b∗ in pentacene from the solution of the BSE withHOMO-LUMO bands in which their original dispersion is fullytaken into account. Reprinted with permission from [40]. Copyright(2013) by the American Physical Society.

(see figure 8(a)), which is about half of the dispersion along theb∗ axis (see figure 8(b)). This difference is an indirect effectof the exchange electron–hole interaction. Indeed, because ofit, pure FR excitons have an intrinsic positive dispersion alonga∗, which is instead negligible along b∗ (see figures 6(a)–(b)).Therefore, as q along a∗ increases, FR and CT get further apart,reducing the mixing induced by the hopping, while this doesnot happen along b∗.

7. Excitons and their dispersion

7.1. Optical absorption

Figure 9 shows the absorption spectra of picene, pentacene,tetracene and coronene calculated from the solution of theab initio Bethe–Salpeter equation equation (2) (including thecontribution from all the bands). The energy range chosen infigure 9 corresponds to excitations within the band gap of eachmaterial, i.e. the peaks are by definition associated to boundexcitons. The band structures, which are the input of the BSEcalculations (see section 3), have been calculated in the GWapproximation, which corrects the LDA underestimation ofband gaps14 (see table 2).

In figure 9 we have selected the absorption spectra forpolarizations along the reciprocal lattice axes a∗ and b∗, lyingin the herringbone plane, which characterise the low-energyabsorption properties of the various materials. The spectraare highly anisotropic as a consequence of the molecularcharacter of electron–hole transitions that give rise to thedifferent excitations. In all the considered systems an intensepeak is present at higher energies (not shown) beyond theband gap and is related to an unbound exciton. This peakis instead quite isotropic dominating the spectra especially forpolarization along the c∗ axis [39, 41, 81]. The bound excitonsrepresented in figure 9 are mainly related to HOMO-LUMOtransitions, with the exception of picene. In fact in picenealso higher energy transitions give a remarkable contribution:

14 For the details of the GW calculations we refer to [39, 41, 128].

the exciton binding energy evaluated including only HOMO-LUMO bands would be 0.3 eV smaller than the value obtainedfrom the full calculation. This is linked to the peculiar bandstructure of picene (see figure 3).

Comparing pentacene and tetracene we observe that forthe lowest-energy exciton the binding energy, defined as thedifference between the absorption onset in the independent-particle picture (i.e. the GW band gap) and the peak positionof the exciton, significantly decreases with the molecular size(see table 2), in agreement with the findings of [80]. At thesame time the exciton binding energy is larger in picene than inpentacene, where for the larger screening the direct electron–hole interaction W is weaker.

The spectra of tetracene and pentacene have a similarshape, with a strong peak at the excitation onset for thea∗ polarization followed by two less intense peaks. Alongthe b∗ axis there are only two main structures, with theabsorption onset located at higher energy than along a∗. Piceneand coronene instead are more isotropic, with similar onsetenergies for the two polarizations. In particular, coroneneis characterised by a weak structure at 2.9 eV, having thelargest binding energies among the four studied compunds (seetable 2), followed by more intense peaks within the band gap.

In the various systems the lowest-energy exciton in thea∗ polarization has the same character as the correspondingone along the b∗ direction. Therefore we can understand theenergy shift observed moving from the a∗ to the b∗ axis asthe Davydov splitting [21] related to the first bound exciton.The two excitonic states that are symmetric and antisymmetricwith respect to the exchange of an electron–hole pair betweenthe two inequivalent molecules in the unit cell have differentexcitation energies. Each of them is (mainly) visible along adifferent polarization direction (see section 5). We note thatthe Davydov splitting is larger in pentacene and tetracene thanin picene where it is almost negligible, suggesting that it has adifferent nature (see section 7.2).

In these materials the comparison with experiment andalso between different calculations, even at the same level ofapproximation, is not always straightforward. In fact in thevarious calculations different crystal structures are adopted(see table 1), which can be also theoretically relaxed (or not,as in our case). Moreover, the calculated spectra are oftenreported along different polarization directions (i.e. along thecartesian axes, as in [80, 81], or along the axes of the real-spaceunit cell [78] or of the reciprocal lattice, as in the present case).Also in the experiments the details of the crystal structure of themeasured samples are not always fully accessible, in particularconcerning the herringbone angle between the molecules inthe unit cell (see e.g. [41]). As the electronic and opticalproperties are highly anisotropic and sensitive to the mutualorientation of the molecules, those differences and uncertantiesdemand care for a quantitative comparison of the spectra[41]. In any case an overall agreement between the differentBSE calculations and ellipsometry and EELS experiments[41, 74, 75, 128, 134–138] can be verified.

As a matter of example, in figure 10 we report the detailedcomparison, taken from [41], between the experimentaland calculated loss function −Imε−1

M (q, ω) of tetracene for

13


3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4Energy (eV)

0

0.2

0.4

0.6Im

[ε]

a axisb axis

*

*Picene

DS

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2Energy (eV)

0

2

4

Im[ε

]

a axisb axis

*

*Pentacene

DS

2.6 2.8 3 3.2 3.4 3.6 3.8 4

Energy (eV)

0

1

2

3

4

5

6

7

Im[ε

]

a axisb axis

*

*Coronene

1.6 1.8 2 2.2 2.4 2.6 2.8

Energy (eV)

0

1

2

3

4

5

Im[ε

]

a axisb axis

*

*Tetracene

DS

Figure 9. Absorption spectra, expressed in terms of ImεM(ω), of the four prototypical compounds along the reciprocal lattice axes a∗ andb∗. The energy range is restricted to values below the band gap of each system. This implies that all the peaks correspond to bound excitons.Panels (a)–(b) are adapted from [39]. Copyright (2012) by the American Physical Society.

vanishing momentum transfer q with direction spanning thewhole a∗b∗ plane in the first Brillouin zone. The noticeablevariation of the spectra as a function of the direction of qclearly shows the high anisotropy of the loss function oftetracene (which is linked to the polarization properties ofthe Davydov components [41]). The calculated spectra(right panel) share the main features of the experimentalspectra (left panel), with a prominent peak at 2.3–2.4 eValong a∗ (defined as 0◦, see also figure 9), whose positionin the calculations is underestimated by ∼0.1 eV. Being thecalculations carried out without any adjustable parameters, thequantitative agreement is remarkable. Both in the experimentand theory the first prominent peak along a∗ is then followedby two smaller structures at higher energy up to 2.8 eV. Inthe opposite side of the map, i.e. along b∗, however, whilein the experimental spectra three main structures are visible,only two peaks are present in the same energy range in thesimulated spectra (see also figure 9). The missing peak islocated in the experiment at an energy very close to that of thefirst prominent peak along a∗ and begins to become visible inthe calculations at an angle ∼80◦, i.e. 10◦ from the (theoretical)b∗ direction. Since already the angle between a∗ and b∗ isdifferent in the experimental sample by 5◦ [41] with respect tothe crystal structure used in the calculations, this suggests thatthe herringbone angle could also be slightly different, affectingthe comparison of the results for the loss function as well. Thisalso indicates that more combined theoretical-experimental

Table 2. Quasiparticle gaps obtained from band structurescalculated in LDA or with the GW approximation are compared tothe optical gaps, defined as the absorption spectrum onset. Hencethe difference between the GW gap and the optical gap correspondsto the binding energy of the lowest-energy exciton for which thevalue of the Davydov splitting is also reported.

Gap Binding DavydovSystem LDA GW Optical energy splitting

Picene 2.4 4.1 3.3 0.8 0.03Pentacene 0.7 2.0 1.5 0.5 0.24Tetracene 1.2 2.9 2.3 0.6 0.27Coronene 2.4 4.0 2.9 1.1

investigations would be needed to definitely settle thisissue.

In [81] the coupling between resonant and antiresonanttransitions, beyond the Tamm–Dancoff approximation (TDA)that is adopted here and in the other BSE calculations, wasalso considered. While the prominent peak at energy abovethe band gap is redshifted by roughly 0.3 eV with respect tothe TDA calculation, hardly any change in the position of thelowest-energy peaks was observed, supporting the results thatare reviewed here.

We remark that the various structures in the spectra infigure 9 have all a pure electronic origin and do not derivefrom a coupling with vibrational excitations that is neglectedhere. Therefore the nature of those peaks qualitatively differs

14


Figure 10. Measured (left panel) and calculated (right panel) EELS spectra of tetracene for momentum transfer q → 0 in the a∗b∗ plane.For both experimental and calculated spectra, the angle is defined with respect to the a∗ direction. In the experiment the b∗ direction is at 95◦

while it is at 90◦ in the calculation. In both cases the spectra are normalised to their maximum intensity. Adapted from [41].

from their interpretation as ‘vibronic satellites’ of the firstmain peak that is based on molecular state models (see e.g.[44]). Nonetheless, one still expects that the electron-phononinteraction plays a non-negligible role in organic materialslike these. This effect can be indeed directly monitored byinvestigating the variation of the spectra as a function of thetemperature. It has been done for example in the case oftetracene in [41], where we found that, while the temperatureis not affecting much the position of the peaks (less than0.1 eV from 20 K to 350 K), their shape changes considerably.By increasing the temperature, the various spectral featuresbecome broader and the differences between the intensities ofthe peaks become smaller.

7.2. Exciton localisation

In figure 11 we have drawn the exciton wavefunction (seeequation (4)), representing the electronic charge distribution|�λ(rh, re)|2 for a fixed position of the hole rh, for the lowest-energy exciton in the four systems, which is visible only forlight polarized along the a∗ axis (see figure 9).

In agreement with other BSE calculations[54, 55, 78, 80, 81], in pentacene and tetracene we find acharge-transfer exciton: with respect to the position of thehole, the electronic charge is mainly localized on the nearest-neighbor molecules. The charge-transfer character of the ex-citon in pentacene and tetracene arises from the interplay be-tween the exchange electron–hole interaction and the band dis-persion that makes the hopping term large enough to causea strong mixing between FR and CT states. Moreover, thecharge-transfer character increases in pentacene with respectto tetracene (see also [80]), where the localisation of electroniccharge around the hole position is larger. This is a consequenceof the smaller size of the tetracene molecule with respect topentacene (four benzene rings instead of five). In fact, as thenumber of benzene rings decreases, the screening is weaker.Moreover the hopping processes related to the electronic banddispersion are less efficient in mixing FR and CT states and, inturn, the exciton wavefunction is more localised15. In picene,

15 It is also important to note that we are not claiming here that the excitoncharacter in pentacene and tetracene is purely charge-transfer [57], seesection 6.2.

instead, we find a Frenkel exciton with both the hole and elec-tron charges mainly localized on the same molecule. The mix-ing of HOMO-LUMO transitions with higher energy excita-tions makes the contributions from the direct electron–holeinteraction W larger, while it does not affect the kinetic termin equation (2). As a result, the exciton binding energy is alsoincreased, giving rise to a strongly localized FR exciton, wellseparated in energy from the CT ones. As a consequence, themixing with higher energy CT excitons is negligible in picene.

Interestingly, comparing the excitons of coronene andpicene we observe that, although the binding energy is larger incoronene, the exciton in coronene is more delocalised than inpicene. As in tetracene and pentacene the lowest excited statein coronene presents a large CT component. On the other handthe exciton at higher energy related to the feature at 3.24 eVhas a large FR component, as can be seen from the shape ofthe excitonic wave function in figure 12. This shows that theknowledge of the exciton binding energy is not sufficient tounderstand the exciton character. The localisation nature ofthe exciton in fact is set by the energy difference between FRand CT states, which is due to the competition between directand exchange electron–hole interaction (see equation (17)) andby the efficiency of the hopping terms in mixing FR and CTstates.

The BSE calculations carried out by different groups ontetracene and pentacene [39, 54, 55, 78, 80, 81] all agree on thequalitative observation of a charge-transfer nature of the firstexciton. This conclusion is in agreement with the interpretationof EELS experiments [74, 75], but in contrast with traditionalexpectations based on molecular state models that assume thelowest-energy excitons to be pure intramolecular excitations.The comparison with electroabsorption experiments wouldrequire the calculation of the Stark shifts which has not beencarried out in the BSE framework yet.

For picene and pentacene we have also considered thelowest spin-triplet excitons that are dipole forbidden andthus not accessible in absorption experiments. To thisend, we diagonalized the excitonic hamiltonian (2) withoutthe exchange electron–hole interaction 2v. As for thesinglet, also the lowest-energy triplet excitons involve mainlyHOMO-LUMO transitions. Since the repulsive exchangeelectron–hole interaction 2v is missing, the triplet exciton is

15


)b()a(

enecatnePeneciP

Tetracene Coronene

(d)(c)

Figure 11. Electronic charge distribution |�λ(rh, re)|2 for a fixed position rh of the hole (blue ball) for the lowest-energy excitons in thevarious compounds. Adapted from [39] and [41], (a)–(b) copyright (2012) by the American Physical Society ([39]).

characterized by a higher binding energy: 1.3 eV for picene and1.1 eV for pentacene (see table 2). Moreover, in both systemsit is a strongly localized Frenkel exciton, as it can be inferredfrom figure 13. Therefore, the comparison between singlet andtriplet excitons in pentacene shows how the exchange electron–hole interaction plays a key role in setting the charge-transfercharacter of the singlet exciton that is experimentally visiblein the absorption spectrum.

7.3. Exciton dispersion

In figure 14 we report the spectra of ε2 as a function of q.For pentacene we find the same results as those obtained byconsidering only the HOMO-LUMO bands (see figure 8),confirming that the HOMO-LUMO bands give by far themost important contribution to these excitons. In tetraceneand pentacene the lowest antisymmetric (CT+FR)− state isvisible for a polarization along the a∗ axis, while its symmetric(CT+FR)+ counterpart is visible only along the b∗ axis. Bothexcitons remain visible up to � point of the second Brillouinzone, where their oscillator strengths drop to zero and thespectral weight is transferred to higher energy (CT+FR) states.

Interestingly, at large momentum transfer along the b∗ axis anew peak appears at energy smaller than that of the symmetricstate (CT+FR)+ that determines the onset at q = 0. Thisnew peak corresponds to the lowest antisymmetric (CT+FR)−

exciton, which is dipole forbidden at small momentum transferalong the b∗ axis, but becomes visible at larger q.

As the mixing with higher energy CT excitons is negligiblein picene, the lowest excited state has an intrinsic FR characterand its dispersion is set by the exchange electron–holeinteraction only. In figure 14 we also see that for small qbelonging to the first Brillouin zone, the FR− exciton hasa positive dispersion, while the FR+ state has a negativedispersion. For both excitons the bandwidth is about 0.02 eV,which is one order of magnitude smaller than in pentacene,suggesting a completely different mechanism of the excitondispersion. We found a similar behaviour in coronene (seefigure 14) where the FR exciton represented in figure 12 has anegligible dispersion along both the a∗ and b∗ axes16.

16 In coronene the lowest energy exciton is too weak and thus not consideredin figure 14.

16


Figure 12. Electronic charge distribution |�λ(rh, re)|2 for a fixed position rh of the hole (blue ball) for the exciton at 3.24 eV energy incoronene (see figure 9).

)b()a(

enecatnePeneciPFigure 13. Lowest-energy triplet exciton wavefunction for picene and pentacene (see figure 11). Reprinted with permission from [39].Copyright (2012) by the American Physical Society.

These findings are confirmed by recent electron energy-loss spectroscopy (EELS) experiments [75, 138]. While anexciton band structure with bandwidths of about 100 meVhas been observed for pentacene, picene has not shown ameasurable dispersion for q belonging to the a∗b∗ plane. Thepresent work thus provides the tools for the interpretation alsoof these recent experimental results.

8. Summary and perspectives

By solving the ab initio many-body Bethe–Salpeter equation,we have analysed the low-energy exciton properties of fourprototypical molecular crystals: picene, pentacene, tetraceneand coronene. In these materials, localized Frenkel excitonscompete with delocalized charge-transfer excitons. Theinterplay between the attractive direct and repusive exchangeelectron–hole interactions and the effect of the hoppingintegrals (which give rise to non-zero bandwidths) is the

key to understand the result of the competition betweenlocalisation and delocalisation for the two-particle electroniccorrelation, which determines the optical properties of thematerials.

Moreover, we have discussed the anisotropy of theabsorption spectra, together with the different origin ofthe Davydov splitting, establishing a link with traditionalmodels based on molecular states. In the future, newcalculations in the BSE framework will be needed in orderto include in the theoretical description also the couplingwith vibrational excitations and to describe the Stark shiftsmeasured by electroabsorption spectroscopy. Experiments onsamples with well-characterised crystal structures (includingthe intermolecular angles) will allow a more direct comparisonwith those calculations.

Finally, we have explained also the exciton dispersionon the basis of the interplay between electron and holehoppings and the electron–hole exchange interaction. Ouranalyis, unraveling a simple microscopic description to

17


Figure 14. Map of the imaginary part of the macroscopic dielectric function ε2 evaluated as a function of the energy (vertical axis, theenergy range shown is smaller than the band gap) and momentum transfer (horizontal axis) along the a∗ axis and the b∗ axis of the variouscompounds. The black circles are guides to the eye indicating the behaviour of the lowest bound excitons. Panels (a)–(d) are reprinted withpermission from [40]. Copyright (2013) by the American Physical Society.

distinguish Frenkel and charge-transfer excitons, is generaland can be applied to other systems in which the electronwavefunctions are strongly localized, as in strongly correlatedinsulators.

In fact, the calculation of exciton properties at finitemomentum transfer within the Bethe–Salpeter approach can beused to interpret (and predict) spectra that can be measured withgreat accuracy by different spectroscopies, namely electron-energy loss spectroscopy (EELS) and inelastic x ray scattering(IXS) [139, 140]. The combination of these experiments withBSE calculations is a very promising tool to investigate theelectronic properties in a wide range of materials, from wide-gap insulators [76, 141, 142] to layered materials [143], alsoincluding dipole-forbidden d–d excitations in transition-metal

oxides (see e.g. [144–150]) and their dispersion (i.e. for thedescription of ‘orbitons’ [151]).

Acknowledgments

This research was supported by a Marie Curie FP7 IntegrationGrant within the 7th European Union Framework Programme.Computational time was granted by GENCI (Project No.544). We also acknowledge financial support from theEuropean Research Council Advanced Grant DYNamo (ERC-2010- AdG-267374), Spanish Grant (FIS2013-46159-C3-1-P), Grupos Consolidados UPV/EHU del Gobierno Vasco(IT578-13) and European Community FP7 project CRONOS(Grant number 280879-2) and COST Actions CM1204 (XLIC)

18


and MP1306 (EUSpec). For DFT and GW calculations wehave used Abinit [152], for BSE calculations Exc [153] and forthe plots of the crystal structures and the exciton wavefunctionsXcrysden [154] and Vesta [155].

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