multichromophores for nonlinear optics: designing the material properties by electrostatic...

DOI: 10.1002/cphc.200700368

Multichromophores for Nonlinear Optics:Designing the Material Properties byElectrostatic Interactions**Francesca Terenziani, Gabriele D’Avino, and Anna Painelli*[a]

Introduction

Smart materials are in demand for applications ranging fromthe development of electronic devices[1] to drug delivery[2] orin vivo optical microscopy.[3] By definition, smart materials re-spond in a qualitatively different way to different stimuli ; nonli-nearity is the qualifying property for functional behavior. Delo-calized electrons are an obvious source of nonlinearity and ma-terials based on p-conjugated molecules or polymers combinepromising properties with the flexibility of organic synthesis.Herein we discuss the properties of materials made up of largep-conjugated molecules of interest for nonlinear optics (NLO)applications (NLO-phores). Interchromophore distances largerthan the sum of the Van der Waals radii point to negligibleoverlap of the orbitals on different chromophores, so that elec-trons are localized on the chromophoric units. Nonlinear func-tional behavior is, in these materials, a consequence of thelarge electronic delocalization within each NLO-phore, relatedto the presence of a large p-conjugated backbone.[4,5] The elec-trostatic interactions between different chromophores affectthe material properties greatly, resulting in nonadditive collec-tive and cooperative behavior. The charge distribution oneach, largely (hyper)-polarizable, NLO-phore readjusts in re-sponse to the distribution of charges on surrounding chromo-phores, in a complex nonlinear feedback mechanism responsi-ble for cooperative phenomena. Moreover, electrostatic inter-actions favour energy-transfer processes; whereas electronsare strictly localized on nonoverlapping chromophores, excita-tions are truly delocalized, leading to important collective phe-nomena.[6] On the negative side, the nonadditive behavior ofmolecular materials for NLO has hindered their full exploita-tion; optimizing the properties of NLO-phores in solution doesnot necessarily lead to optimized responses in dense materi-als.[7, 8] However we can invert this situation and exploit inter-

chromophore interactions as a powerful tool to optimize thematerial properties. To achieve this goal, a reliable relationshipbetween material properties and the supramolecular structureis needed. Here we review recent advances along this chal-lenging path.Inserting electron-donor (D) and acceptor (A) groups into

the p-conjugated backbone lowers excitation energies and in-creases relevant transition dipole moments, leading to im-proved NLO responses at the molecular level. Dipolar, D-p-A,[5,9,10] quadrupolar, D-p-A-p-D or A-p-D-p-A,[11,12] and multipo-lar molecules[12–14] (some examples of which are shown inFigure 1) have been devised as optimized structures for variousapplications, and reliable structure–properties relationshipshave been obtained at the molecular level. The interactionwith the surrounding solvent has important consequen-ces.[15–18] In fact, largely (hyper)polarizable NLO-phores readjusttheir charge distribution in response to the local environment,leading to a feedback mechanism that can either amplify orsuppress their NLO responses. The coupling between electronsand molecular vibrations is another source of nonlinearity.[19,20]

The low-energy physics of dipolar and multipolar NLO-phores is governed by electron- or charge-transfer (CT) pro-cesses that involve the motion of the electronic charge along a

[a] Dr. F. Terenziani, G. D’Avino, Prof. A. PainelliDipartimento di Chimica GIAFParma UniversityParco Area delle Scienza 17A, 43100 Parma (Italy)Fax: (+39)0521905556E-mail : [email protected]

[**] Dedicated to Cesare Pecile, Emeritus

Supporting information for this article is available on the WWW underhttp://www.chemphyschem.org or from the author.

To fully exploit the promise of molecular materials for NLO appli-cations, inter- and supramolecular interactions must be account-ed for. We review our recent work on electrostatic interchromo-phore interactions in multichromophores for NLO applications.The discussion is based on a bottom-up modeling strategy: eachchromophore is described in terms of an essential state model,validated and parameterized against spectroscopic data for sol-vated chromophores. The relevant information is then used tobuild a model for clusters of chromophores interacting through

electrostatic forces. Exact NLO responses and spectra calculatedwithin this model fully account for collective and cooperative in-terchromophore interactions, which can either amplify or sup-press NLO responses; supramolecular engineering of multichro-mophores is a powerful tool for the design of NLO materials.Moreover, new features emerge in multichromophores with nocounterpart at the single-chromophore level, offering new excit-ing opportunities for applications.

ChemPhysChem 2007, 8, 2433 – 2444 @ 2007 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim 2433

sizeable distance: CT processes are then strongly affected bythe presence of electric fields. The fields can be externally ap-plied or may be generated inside the sample by molecular vi-brations or by the surrounding solvent, as already discussed inthe literature.[16,18, 19,21,22] Herein we briefly review our recentwork on the linear and nonlinear optical responses of materialssuch as molecular aggregates or crystals, films or supramolec-ular structures where two or more chromophores are kept to-gether by chemical bonds or other forces. Much as with polarsolvation, an intriguing feedback mechanism occurs betweenthe (hyper)polarizable chromophores, each one readjusting itscharge distribution in response to the electric fields generatedby the charge distribution in surrounding chromophores.[6,23, 24]

At variance with polar solvation, not only static, but also fluctu-ating electric fields, as generated by excitations on nearbychromophores, must be considered. The nonlinearity of thechromophores is amplified by the feedback and manifestsitself through amplified or suppressed NLO responses, andmore generally through nonadditive collective or cooperativebehavior.[6, 22–25]

The exciton model was proposed almost 40 years ago[28,29]

to explain why the optical spectra of molecular crystals are notthe sum of the spectra of the molecular units composing thecrystal. The same model was later applied to the optical spec-tra of molecular aggregates.[30] The exciton model offers auseful reference scheme to understand the linear and nonlin-ear spectra of molecular materials. The basic approximation ofthe exciton model is the absence of mixing between stateswith a different number of excitations, which amounts to ne-glecting the molecular polarizability ; the exciton model thenhardly applies to materials made up by the strongly (hyper)po-larizable chromophores of interest for NLO applications.In an effort to provide a simple reference scheme for the de-

scription of optical (linear and nonlinear) properties of molecu-lar materials for NLO applications, we have developed abottom-up modeling strategy for clusters of NLO-phores.[23,24,26, 27,31–33] The first step in the approach is the defini-

tion of an essential-state model for the relevant chromophore.The model is parameterized and validated via a detailed spec-troscopic analysis of the chromophore in solution. Interchro-mophore interactions then enter as classical electrostatic inter-actions. The n essential states needed to describe the electron-ic structure of each chromophore lead to an nN basis to de-scribe N interacting chromophores in a cluster. The total Hamil-tonian can be diagonalized exactly if n and N are not toolarge. The approach is simple; our aim is not a detailed de-scription of the properties of a specific system (as is possiblewith more refined but more demanding quantum chemical ap-proaches), rather we wish to develop some new basic under-standing of interchromophore interactions to offer reliableguidelines for the synthesis of multichromophoric systemswith optimized properties.This Minireview is organized as follows: first we discuss the

role of intermolecular interactions in static NLO responses withreference to the technologically relevant problem of optimiz-ing molecular materials for second-order NLO response. Then,before attacking the more complex problem of responses atoptical frequencies, we discuss the excited states in molecularmaterials, briefly reviewing the exciton model and its limita-tions when applied to materials for NLO applications. In thefollowing two sections our approach is applied to discuss twophoton absorption (TPA) spectra of clusters of dipolar andquadrupolar chromophores. Before concluding with an out-look, we address the extreme consequences of collective andcooperative behavior as recognized in the multielectron trans-fer phenomenon.

Results and Discussion

Static NLO Responses of Clusters of Polar and PolarizableChromophores

The static NLO responses of a molecule are defined via the ex-pansion of the molecular dipole moment, m, on the appliedfield F, as follows:

mðFÞ ¼ mð0Þ þ aF þ 12

bFF þ 13!

gFFF þ ::: ð1Þ

where a is the linear polarizability, and b and g are the firstand second hyperpolarizabilities, respectively. This equation iswritten in a simplified form: both m and F are vectors (withthree components along the three reference directions). Thepolarizabilities are tensors: a has two indices (referring to thedirections of m and of the applied field F), b has three indices,and so on. The static polarizabilities can be expressed, usingthe sum-over-state (SOS) expressions,[34] in terms of the excita-tion energies and transition dipole moments of the system inthe absence of applied fields. On the other hand, Equation (1)defines the static polarizabilities as the successive derivativesof the ground-state dipole moment versus the applied fields.Even if SOS expressions relate the polarizabilities to excited-state properties, static polarizabilities are ground-state proper-ties and can be calculated from the derivatives of the ground-

Figure 1. Structures of dipolar and quadrupolar chromophores. Top: two di-polar D-p-A chromophores: a dye[26] with a neutral ground state (DANS,commercial, for R=R’=CH3) and a zwitterionic dye (Q3CNQ[27]). Middle andbottom: two quadrupolar D-p-A-p-D chromophores classified as class I andclass II according to ref. [18] (see also Figure 7).

2434 www.chemphyschem.org @ 2007 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim ChemPhysChem 2007, 8, 2433 – 2444

A. Painelli et al.

www.chemphyschem.org

state dipole moment versus an applied field (the finite-fieldmethod).[35] Equation (1) holds true quite irrespective of thecomplexity of the system at hand; it applies to molecules con-taining a single chromophoric unit as well as to multichromo-phoric assemblies. In extended systems, such as crystals orfilms, the dipole moment in Equation (1) should be replacedby the polarization, the dipole moment per unit volume.A common belief maintains that the properties of a cluster

of nonoverlapping chromophores are the sum of the proper-ties of the components. The oriented gas model formalizesthis belief by describing the material (typically a molecularcrystal) as a gas of noninteracting chromophores in a frozenorientation. However, real molecular materials hardly show ad-ditive behavior. In fact the chromophores readjust their chargedistributions in response to the instantaneous charge distribu-tion in surrounding molecules, leading to a feedback interac-tion that can profoundly alter the properties of the material,with collective and cooperative effects whose importancegrows with the (hyper)polarizability of the chromophoric unitsand with the order of nonlinearity.[23,36]

To appreciate this important point we make reference to thetechnologically relevant example of molecular materials forsecond-order NLO response. By symmetry, b vanishes in cen-trosymmetric systems; dipolar D-p-A chromophores were sug-gested early on as molecules of choice for b-applications.[10,37]

The basic properties of these molecules are well understood,and reliable structure–properties relationships have been de-vised relating b to the charge distribution on the chromophoreand hence to the geometry of the chromophore.[38] Several D-p-A molecules with optimized structures and large b havebeen synthesized.[10,37, 39] However, defining reliable relation-ships between b and the supramolecular structure is more dif-ficult, and the lack of a clear understanding of the role of inter-molecular interactions has so far hindered the search for mate-rials with optimized b-response.[8,40]

The electronic structure of isolated D-p-A chromophores canbe described in terms of the two-state model sketched inFigure 2.[19,41, 42] Each chromophore resonates between a neu-tral and a zwitterionic structure, D-p-A$D+-p-A� , so that inthe ground state a fraction 1 of electrons is transferred from Dto A, leading to a state that can be described as D+1-p-A�1.The properties of the chromophores are governed by 1; in thebottom right panel of Figure 2 we show the 1 dependence ofthe first hyperpolarizability. b is positive for neutral dyes (1<0.5), negative for zwitterionic dyes (1>0.5), and vanishes inthe cyanine limit (1=0.5).[19,42, 43] The maximum jb j=0.107 (inunits with m0=1 and

ffiffiffi2p

t=1) is obtained for isolated chromo-phores having 10.28 and 0.72. 1The two-state model for

D-p-A chromophores has been quite extensively and success-fully validated by the comparison with spectroscopic proper-ties of solvated dyes.[17,26,27, 31,33] In our bottom-up strategy itrepresents a simple and reliable starting point to discuss elec-trostatic intermolecular interactions in clusters of D-p-A chro-mophores.We start with the instructive example of a pair of D-p-A

chromophores. We consider two different geometries, a and b,sketched in Figure 3. For a pair of chromophores we have fourbasis states: a zero-energy state with both chromophores neu-tral (D-p-A, D-p-A), two degenerate states at energy 2z with aneutral and a zwitterionic chromophore (D+-p-A� , D-p-A andD-p-A, D+-p-A�), and a state with both chromophores in thezwitterionic form (D+-p-A� , D+-p-A�) with energy 4z+V. As inFigure 2, 2z measures the energy required to ionize a D-p-Aunit, whereas V is the electrostatic interaction between thetwo zwitterionic chromophores. V depends on the actual ge-ometry of the dimer, it is positive (repulsive interaction) for adimers and negative (attractive interactions) for b dimers. Theprecise dependence of V on the dimer geometry requires amodel for electrostatic interactions. The dipole approximation

Figure 2. Two-state model for D-p-A chromophores. The neutral and zwitter-ionic resonating structures correspond to the two basis states, f0 and f1.The neutral state has a negligible dipole moment, whereas the zwitterionicform has a large dipole moment, m0. The two states are separated by anenergy gap 2z. A matrix element �

ffiffiffi2p

t� �

mixes the two states in theground (g) and the excited state (e). A fraction 1 of electron is transferredfrom D to A in the ground state, leading to D+1-p-A�1. The amount of trans-ferred charge, 1, measures the weight of the zwitterionic state in theground states, and its dependence on z (measured in units with

ffiffiffi2p

t=1) isshown in the bottom left panel. Specifically, 1 ranges from 0 to 1 as z de-creases from large positive to large negative values (exchanging the relativeenergy of the two basis states). In the cyanine limit the two basis states aredegenerate (z=0) and have the same weight, 1 =0.5. The dotted and fullarrows signal that the transition from the ground to the excited state is al-lowed both in one-photon and in two-photon absorption processes. Thefirst hyperpolarizability b has a nonvanishing component bzzz with the z-axisaligned along m0. When measured in units with

ffiffiffi2p

t=1 and m0=1, b onlydepends on 1, as shown in the bottom right panel.

1 The fairly bulky notation for this matrix element has its roots in quantum cellmodels for charge-transfer complexes and salts. The neutral DA states is asinglet state with two paired electrons on the D site: j›fl,0> , where the twoarrows represent the spins of the two electrons on the D site and the zerorepresents a void A site. The charge separated state, D+A� , has an electrontransferred from the D to the A site, with the two electrons in a singlet config-uration: (j›,fl>+jfl,›> )/

p2. In standard quantum cell models, t measures

the hopping probability for an electron on adjacent sites :t= <›fl,0 jH j›,fl> , then leading to

ffiffiffi2p

t=<DA jH j D+A�> .

ChemPhysChem 2007, 8, 2433 – 2444 @ 2007 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim www.chemphyschem.org 2435

Multichromophores for Nonlinear Optics


is not applicable because the molecular length l of NLO-phores(cf. Figure 1) is of the same order of magnitude and possiblylarger than typical intermolecular distances, r, in aggregates orcrystals. In the same spirit of the extended dipole approxima-tion often adopted in excitonic models,[44] we adopt a dumb-bell model for the D+-p-A� chromophore, with positive andnegative point charges located at the ends of a segment oflength l.[23] With this approximation, V is given by Equation (2)

V ¼ w 1~rþ 1

~r þ 2 sinf� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið~r þ sinfÞ2 þ cos2 f

p" #

ð2Þ

where the plus and minus sign apply to the a and b dimers, re-spectively, ~r ¼ r=l is the intermolecular distance in units of themolecular length, f is half the interchromophore angle (cf.Figure 3), and w=e2/l is the unit of electrostatic energy. Theexplicit expression for the 4I4 Hamiltonian matrix can befound in the Supporting Information. The numerical diagonali-

zation of the matrix yields the eigenstates of the interactingsystem. The calculation of b is most easily done via a finite-field approach, that is, entering a static electric field in the de-sired orientation (see the Supporting Information) and thencalculating the F dependence of the ground-state dipolemoment. Its second derivative yields the required b. In a and bdimers the dipole moment has nonvanishing components onlyin the z and x direction (cf. Figure 3), respectively ; as a resultbzzz and bxxx are the only nonvanishing components of the b

tensor for a and b dimers, respectively.Figure 3 summarizes the results obtained for systems with

w=1.8 (which, for typicalffiffiffi2p

t1 eV, corresponds to a molecu-lar length l7–10 J) and ~r=0.8 (r5–9 J). The bottom panelsof Figure 3 show the exact bzzz and bxxx for the two dimers, as afunction of f and 1chr, where 1chr is the 1 relevant to the isolat-ed chromophore. The central panels show the same results inthe oriented-gas approximation. The oriented-gas results areeasily understood; for aligned chromophores (either f=0 forthe a dimer or f=90o for the b dimer) b is simply twice aslarge as that expected for the isolated chromophore. It rangesfrom �0.21 to 0.21 (using units with m0=1 and

ffiffiffi2p

t=1), andsmoothly decreases toward 0 for antiparallel orientation (eitherf=90o for a dimer or f=0 for b dimer). Deviations from theadditive result are apparent for both geometries and clearlypoint to cooperative behavior.[23] For the a dimer, the exact b

never attains the values expected in the oriented-gas model.Irrespective of 1chr and of the angle between the chromo-phores, the maximum b is always lower than expected for apair of noninteracting chromophores. Specifically, the maxi-mum jb j achievable in the a geometry is about one half thatexpected in the oriented-gas model. On the contrary, for the bgeometry the b response may be amplified by a factor of twowith respect to noninteracting chromophores.Results obtained for dimers exemplify the situation for larger

clusters of D-p-A chromophores. Systems where D-p-A chro-mophores have parallel relative orientation, such as a dimers,have suppressed b responses. This result was first obtainedworking on one-dimensional clusters of 16 chromophores,modeling linear aggregates.[23] The same result was more re-cently confirmed by studying a model system for a calix[4]ar-ene functionalized with 4 D-p-A chromophores.[45] Quite inter-estingly, in this case the same qualitative behavior is observedas for the a dimer, but with an even larger suppression of b—the cooperative suppression of b nonlinearly increases withthe size of the system. Indeed, several attempts to obtain largeb by preparing multichromophoric systems with severalaligned chromophores[45,46] proved to be not very successful.Similarly, attempts to orientate dipolar chromophores embed-ded in different matrices via poling techniques were less suc-cessful than expected.[8,47] In this case problems related to theorientation process[47,48] add to the effects of electrostatic inter-actions that in the poled sample may suppress the b response.So a careful study of intermolecular interactions helps to un-derstand why progress in the preparation of good molecularmaterials for b responses was so frustrating. On the positiveside, however, the same study offers new and promisingguidelines; aligning two or more chromophores as in b dimers

Figure 3. Two different geometries of D-p-A chromophore dimers. In the ageometry, a rigid and nonconjugated bridge keeps the two D-sides of thetwo molecules at fixed distance r (exactly equivalent results are obtained fortwo chromophores kept at constant A–A distance). In b, the nonconjugatedbridge connects the D end of a molecule with the A end of the other mole-cule. The dye length is l, and 2f is the interchromophore angle. For the aand b geometries bzzz and bxxx are, respectively, the only nonvanishing com-ponents of the b tensor. The central and bottom panels show the nonvan-ishing components of b for the two dimers, calculated as a function of fand of 1chr. We fix

ffiffiffi2p

t=1 as the energy unit, w=1.8, r/l=0.8 and vary z asto tune 1chr from 0 to 1. The central and bottom panels refer to oriented-gasand exact results, respectively. In the oriented-gas model, b is calculated asthe sum of the responses of the non-interacting dyes, and hence corre-sponds to the addictive result. Suppressed/enhanced exact responses calcu-lated for a and b geometries point to suppression/amplification of the re-sponse by collective and/or cooperative effects. b is measured in units withm0=1 and

ffiffiffi2p

t=1.


A. Painelli et al.


can lead to a response that is larger than the additive resultand, quite interestingly, the amplification increases with thenumber of chromophores.[23]

Deviations from the additive oriented-gas model can be as-cribed to two different sources. First of all, due to interchromo-phore interactions, the charge distribution on each chromo-phore readjusts; in the dimer each chromophore has a differ-ent 1 value than the isolated chromophore. This phenomenoncan easily be accounted for by calculating the oriented-gas b

for two chromophores with the same 1 as they have in thecluster. This simply deforms the b ACHTUNGTRENNUNG(1,f) surfaces calculated inthe oriented-gas approximation, but cannot affect the maxi-mum attainable jb j values. So the overall suppression or am-plification of b has a different and more subtle origin. The b re-sponse is the second derivative of the total dipole momentversus the applied field—for interacting molecules this dipolemoment self-consistently depends on the fields generated bynearby molecules, so the suppression/amplification of NLO re-sponse is due to polarization fields.[23] From a different (butequally correct) perspective the SOS expressions relate the po-larizabilities of a cluster of chromophores to the cluster transi-tion frequencies and dipole moments. Excitation energies anddipole moments in the cluster differ from those of an isolatedchromophore due to excitonic and ultraexcitonic effects. Thesuppression/amplification of the b response is then related tothe delocalization of excited states.

Beyond the Exciton Model

Clusters of Large and Largely Polarizable Chromophores

A detailed description of excited states is required for the cal-culation of linear and/or nonlinear responses at optical fre-quencies. Well-known SOS formulas[34] express the optical re-sponses in terms of transition frequencies and dipole mo-ments. The exciton model offers a simple description of excit-ed states in molecular materials and hence represents a power-ful tool to rationalize the linear and nonlinear spectroscopicproperties of molecular crystals or aggregates or, more gener-ally, of clusters of molecules interacting via classical electrostat-ic forces.[30]

The exciton model is briefly introduced in Figure 4, whichdescribes the excitations in clusters of two equivalent chromo-phores, and in Figure 5, which generalizes the discussion toseveral chromophores. The exciton is a molecular excitationthat, due to electrostatic intermolecular interactions, delocaliz-es over the cluster with important effects on optical spectra.The concept of exciton requires weak intermolecular interac-tions so that only degenerate states, that is, states with thesame number of excitations, are mixed by the interactions.Only in this approximation is the exciton number a goodquantum number. Other approximations concern the modelfor electrostatic interactions which are commonly described inthe dipolar approximation, truncated to nearest-neighbor mol-ecules. With these additional approximations, J, the interactionterm responsible for the exciton delocalization (or hopping), isproportional to the squared transition dipole moment from

Figure 4. Exciton model for two equivalent chromophores. Each chromo-phore can be in its ground state (*) or in the excited state (star). The fourstates relevant to the noninteracting chromophores are shown in the centralcolumn: in the lowest-energy state both chromophores are in the groundstate. An energy gap �hw is required to promote one of the chromophoresto the excited state, leading to two degenerate states. Another energy gap�hw separates the singly excited states from the state with both excited chro-mophores. Interchromophore electrostatic interactions mix up the non-inter-acting states. At the lowest order (exciton approximation) only degeneratestates (i.e. states with the same number of excitations) are mixed up by amatrix element called J. As a result, the two states with a single excitationare split by an energy gap 2 j J j , with the symmetrical combination beingthe lowest/highest energy for negative/positive J. Depending on the relativeorientation of the two chromophores, different selection rules apply, so thateither one of the two linear combinations is active in one-photon (or two-photon) processes, or in some cases both of them.

Figure 5. Exciton model for several equivalent molecules (for graphical rea-sons only four molecules are shown). To ensure the equivalence of all sites,periodic boundary conditions are imposed so that the N chromophores lieat the vertices of a regular polygon. As in Figure 4 each molecule can be inits ground state (*) or in an excited state (star). On the left, some of the 2N

states for the N noninteracting molecules are shown: the lowest-energystate has all chromophores in the ground state; the N states with a singleexcitation have energy �hw ; the N ACHTUNGTRENNUNG(N�1)/2 states with two excited chromo-phores have energy 2�hw ; states with a higher excitation number at energies3�hw, 4�hw are not shown. As for the case of two chromophores (cf. Figure 4),in the excitonic approximation J mixes only degenerate states. For singly ex-cited states exciton bands are formed (following the same scheme as HLckelorbitals in cyclic polyenes) with an overall bandwidth of 2 j J j . Due to thesymmetry of the system only the symmetrical state is active in one-photonabsorption processes. This state lies at the bottom/top of the exciton bandfor negative/positive J. Then absorption spectra are red-shifted by j J j for J-excitons (J<0) or blue-shifted by the same amount for H-excitons (J>0).The states with two excitations are also mixed up by the J-interaction,giving rise to two-exciton bands (not shown). Depending on the model pa-rameters, states with two nearby excitations (i.e. two adjacent stars) cansplit from the two-exciton band, forming bound biexciton states.




the ground to the excited state of the isolated molecule. Theformation of delocalized excitons is then expected for opticallyallowed states, whereas dark states are not affected by inter-molecular interactions.Only one-exciton states are relevant to linear spectroscopy,

and were the only states discussed in the early days of the ex-citon model.[28,29] Two-exciton (and n-exciton) states becomeimportant for NLO processes and have attracted attentionmore recently.[23,32, 49–51] What makes the two-exciton states spe-cial is the possibility for interchromophore interactions to splitbiexciton states out from the two-exciton band.[50] Specifically,states with two nearby excitations (shown as two nearby starsin Figure 5) can be stabilized or destabilized by electrostatic in-teractions.[52] In the dipolar approximation, the interactionenergy between two nearby excitons is proportional to thesquared difference between the permanent dipole moments inthe ground and excited state. Bound biexcitons are thereforeonly possible in clusters of polar chromophores. Of course thebiexciton is stabilized (destabilized) for attractive (repulsive) in-terchromophore interactions. Similarly, exciton strings (i.e.states with several nearby excited molecules) may split downin clusters of dipolar molecules from bands with many excita-tions.[24, 51]

The noninteracting states in Figures 4 and 5 can be definedin terms of the ground and excited states of the isolated chro-mophore, in the crudest version of the exciton model. In abetter approximation, the ground and excited states are calcu-lated for an isolated chromophore experiencing the averageelectric field generated by the surrounding chromophores intheir ground-state configuration.[23] With this mean-field choiceof the basis states[23] one partly accounts for the (hyper)polariz-ability of the chromophores that readjust their charge distribu-tion in response to the average distribution of surroundingcharges. However, not allowing for the mixing of states with adifferent number of excitations, the exciton model disregardsany response of the chromophore to the instantaneous fluctu-ations of the charge distribution on surrounding chromo-phores; as far as excitations are concerned the (hyper)polariza-bility of the chromophores is neglected. In other words, thebasic approximation of the exciton model requires j J j!�hw,an easily met condition for chromophores with high-energyexcitations and/or small transition dipole moments. NLO-phores are typically extended p-conjugated molecules withlow-energy excitations and large transition dipole moments.For this special class of chromophores the exciton model is ex-pected to fail.[23]

The dipolar approximation for electrostatic interactions,commonly adopted in current implementations of the excitonmodel, is also questionable; it applies to the description ofelectrostatic interactions between charge distributions on twoglobally neutral objects provided that the dimension of theobjects is negligible with respect to their distance. NLO-phoresare large molecules with typical dimensions of the order of (atleast) 10 J, usually larger than intermolecular distances.Our bottom-up modeling strategy for clusters of NLO-

phores overcomes the basic problems of the exciton model inits standard implementation. As discussed above, we rely on

essential-state models for relevant chromophores. Thesemodels, validated against experiment, are used to build upmodels for molecular clusters of predefined geometry as tosimulate the supramolecular structure of interest. Electrostaticinterchromophore interactions are introduced in a point-charge model that relaxes the dipolar approximation. The re-sulting Hamiltonian can be diagonalized exactly for not toolarge clusters. All approximations of the exciton model are re-laxed, the molecular (hyper)polarizability is fully accounted forand both cooperative and collective effects are described onthe same footing.

Supramolecular Engineering: Optimizing the TPACross-section of Bichromophores

Materials for TPA are attracting considerable attention in viewof many interesting applications. In recent years, after the in-vestigation of simple dipolar chromophores,[53] attention hasprogressively moved toward more complex structures,[54] in-cluding quadrupolar (D-p-A-p-D or A-p-D-p-A)[11] and octupo-lar chromophores,[14,55] as well as dendrimeric structures. Herewe concentrate on TPA responses of multichromophoric struc-tures, that is, of clusters of chromophores that only interact viaelectrostatic interactions. In this section we focus on dipolardyes, whereas the next section discusses quadrupolar dyes.We discuss the TPA spectra calculated for the same dimers

of D-p-A chromophores shown in Figure 3. Figure 6 summariz-es the main results, showing TPA spectra (cross section perchromophore) calculated for dimers of neutral dyes (1chr

0.10). In dimer a, the electrostatic interactions are repulsive

Figure 6. TPA cross sections per chromophore for dimers of D-p-A chromo-phores. The calculations are performed for the same parameters as inFigure 3, fixing z=1.3, corresponding to 10.10 for the isolated chromo-phore. The wavelength l in nm, and the TPA intensity s per chromophore,in Goppert–Mayer units, are obtained by setting m0=20 D,

ffiffiffi2p

t=1 eV. Thecontinuous lines in the main panels show the f dependence of the exactTPA calculated for an isotropic distribution of dimers (solution phase). Forcomparative purposes, the insets show the same spectra calculated in theexcitonic approximation. In all panels the dashed lines shows the TPA spec-trum of the isolated chromophore.


A. Painelli et al.


at any angle and the polar and polarizable chromophoresreduce their polarity to partially release the electrostatic strain.This mean-field effect explains part of the blue-shift of the TPAband observed for the dimer with respect to the isolated chro-mophore. Repulsive interactions imply J>0, so that, accordingto Figure 4, the out-of-phase combination of the two one-exci-ton states is lower in energy than the in-phase combination.At f=0 only the in-phase combination is active in TPA so thatthe J interaction is responsible for an additional blue-shift ofthe TPA band. At f=908 only the out-of-phase state is activein TPA and the blue-shift with respect to the isolated chromo-phore is reduced. At intermediate angles both states contrib-ute to the TPA signal giving rise to structured TPA bands. Inthe b geometry, intermolecular interactions are attractive. As aresult, the polarity of the polarizable chromophores slightly in-creases in the dimer, explaining part of the red-shift of the TPAband. Moreover, in this case J<0 and the in-phase combina-tion of the one-exciton states is lower in energy that the out-of-phase combination. At f=0 only the out-of-phase combina-tion is active, while at f=908 the in-phase combination ap-pears. Again at intermediate angles both components showup, leading to a splitting of the TPA band.What is more important for practical purposes are the TPA

intensities. With reference to the maximum cross-section,Figure 6 demonstrates the importance of the interchromo-phore angle and indicates that the linear arrangement (f=908in both cases) is the most promising for amplified responses inboth dimers. On the other hand, the parallel arrangement (f=

0) leads in both cases to a suppressed response with respectto the isolated chromophores. The amplification/suppressionof the response amounts to factors of the order of two, butmuch larger effects (up to factors of 10 or higher) can be ob-tained for more polar chromophores (1chr0.3). Zwitterionicchromophores (1chr0.9) show a similar behavior as neutralones as far as TPA cross-sections are concerned, even if theTPA frequencies behave differently.For comparative purposes the insets in Figure 6 show the

TPA spectra calculated in the excitonic approximation. In thiscase we use exactly the same model for electrostatic interac-tions as that discussed above (i.e. we relax the dipolar approxi-mation) and construct the exciton model starting with theground and excited states that the chromophores have in thedimers, so as to account for mean-field effects.[23] The transitionfrequencies are well captured in this very refined version ofthe exciton model, demonstrating that the energies of the ex-cited states are approximately correct. However the depend-ence of the cross-section on the interchromophore angle ismarginal even in this version of the exciton model. To appreci-ate fully the potential of intermolecular interactions in the am-plification/suppression of nonlinear responses the very basicapproximation of the exciton model must be relaxed to allowthe ultraexcitonic mixing of states with a different number ofexcited molecules. In other words the molecular (hyper)polariz-ability must be fully taken into account to reliably estimate thenonlinear spectra of multichromophoric assemblies.The results shown in Figure 6 show the effects that electro-

static intermolecular interactions can have on the TPA spectra

of clusters of dipolar molecules. Ref. [33] reports a more de-tailed study on specific dimers of D-p-A chromophores, wherethe strategy of bottom-up modeling is successfully applied.The work starts with the spectroscopic analysis of the isolatedchromophores in solution to extract reliable molecular parame-ters. Electron-vibration coupling is introduced as required toreproduce absorption and fluorescence band shapes. The mi-croscopic information relevant to the D-p-A chromophores isthen transferred to the models for the TPA spectra of the inter-acting chromophores. The calculated spectra, which also in-clude the vibronic structure, are in good agreement with ex-perimental data, giving confidence in the validity of the pro-posed model.

Biexcitons and TPA Spectra of Clusters of QuadrupolarChromophores

Quadrupolar D-p-A-p-D or A-p-D-p-A chromophores are ac-tively investigated because of their large TPA cross-sections.The essential-state model for the electronic structure of quad-rupolar dyes is based on three resonating structures,[53,56] as il-lustrated in Figure 7. The combined effect of polar solvationand molecular vibrations leads to interesting behavior.[18]

Largely neutral (class I) chromophores have a small c–e gap(Figure 7), resulting in a bistable one-photon absorption (OPA)state. In polar solvents the c-state is a broken-symmetry polarstate, a phenomenon that shows up in largely solvatochromicfluorescence spectra. Dyes with a large charge separation(class III) have a small g-c gap and hence a bistable groundstate; for these dyes solvatochromic behavior is expected inabsorption but not in fluorescence. Finally, class II dyes, with 1

0.5, are not prone to symmetry breaking and their spectrado not show any major solvatochromism. At present severalquadrupolar dyes can be safely assigned to class I and II,[18]

whereas quadrupolar dyes of class III are not known. Class IIdyes are expected to have much larger TPA cross-sections thanclass I dyes (cf. Figure 7 bottom right panel), but the frequencyof the TPA photon almost coincides with the OPA frequency(cf. Figure 7 bottom left panel) so that the strong TPA signal ismasked by the concomitant OPA.The three-state model for D-p-A-p-D or A-p-D-p-A chromo-

phores has been validated via an extensive spectroscopicstudy of several dyes in solution.[18] In a bottom-up modelingapproach we adopt the same model to discuss one-dimension-al clusters of quadrupolar chromophores. Specifically we are in-terested in the possibility of improving the TPA response ofthese molecules through supramolecular engineering. Wefocus on class I dyes with 10.1–0.2,

ffiffiffi2p

t1 eV and m030–40 D, corresponding to TPA cross-sections for the isolatedchromophore as high as ~1000 GM,[18] and consider an array ofchromophores arranged in attractive geometry as sketched inthe left panel of Figure 8. We impose periodic boundary condi-tions, that is, we arrange the N (typically six) chromophores atthe N vertices of a regular polygon (cf. Figure 8, right panel) toensure that all chromophores are equivalent and that the leftand right arms of each chromophore are equivalent (i.e. thesymmetry of the chromophore is preserved). Results are only




marginally affected by the number of chromophores (weobtain similar results for four, six, and eight chromophores) sothat our discussion applies to linear aggregates that are large

enough to make finite-size effects negligible. The number ofbasis states is 3N, and the corresponding Hamiltonian matrixcan be numerically diagonalized for not too large N. The Ham-iltonian is defined in terms of the three-state Hamiltonian foreach chromophore plus electrostatic interactions. We only ac-count for electrostatic interactions between nearest-neighbordyes and, much as was done for dipolar dyes, we relax the di-polar approximation and describe electrostatic interactions be-tween point charges at the D and A locations.[32] Explicit ex-pressions for the Hamiltonian and electrostatic interactions aregiven in the Supporting Information. The Hamiltonian for thecluster is defined in terms of the two parameters, z and t, char-acteristic of the dye, and by the parameters that define theelectrostatic interaction, that is, w=e2/l, the basic unit of elec-trostatic energy (l is the D-A distance, see Figure 8) and r, theintermolecular distance. We fix

ffiffiffi2p

t as the energy unit (a par-ticularly convenient choice because it typically amounts to~1 eV, so that dimensionless energies roughly correspond tothe same number of eV), z=1.5 to have 10.17 for the isolat-ed chromophore, and w=1.8 that for typical t values amountsto l7–11 J.The colour map in Figure 9 summarizes the main results.

The intensity of the TPA absorption is reported on a logarith-mic scale versus the photon frequency (horizontal axis) withvarying intermolecular distance r (vertical axis). In the samefigure, the dashed white line shows the evolution of the fre-quency of the OPA photon with r. For comparative purposes,we also discuss the results obtained in the standard excitonic

Figure 7. Three-state model for D-p-A-p-D chromophores (the same schemeapplies with obvious modifications to the A-p-D-p-A case). The neutral andthe two charge-separated structures on top correspond to the three basisstates f0, f1, and f2. The neutral state has a negligible dipole moment,whereas the two charge-separated structures have large dipole moments ofequal magnitude, m0, and opposite direction. The two charge-separatedstates, f1 and f2, are conveniently combined into a symmetric and an anti-symmetric wavefunction, f . The neutral and charge-separated states areseparated by an energy gap 2z. A matrix element �2t mixes the two sym-metrical states f0 and f+ to give a ground state g and an excited state e.The antisymmetric states f� stays unmixed and corresponds to the excitedstate c, active in OPA processes (dotted arrows). The symmetrical e state isactive in TPA (full arrows). A fraction 1/2 of electron is transferred from eachof the two D sites towards the A site in the ground state. The amount oftransferred charge, 1, is proportional to the quadrupolar moment in theground state and only depends on the ratio z

� ffiffiffi2p

t. All spectroscopic prop-erties can be expressed as a function of 1. The left bottom panel shows the1-dependence of the frequency of the photon absorbed in OPA and TPAprocesses (units with

ffiffiffi2p

t=1); the right bottom panel shows the TPA cross-section (arbitrary units) evaluated at the maximum of the band.[53] Quadru-polar dyes are conveniently classified in three groups: class I dyes have largepositive z, small 1 (<~0.2) and wOPA much larger than wTPA ; class II dyeshave small j z j values, intermediate 1 (~0.3–0.7) and wTPAwOPA. Class IIIdyes have large negative z, large 1 (>~0.7) and wTPA@wOPA.

[18]

Figure 8. Sketch of the geometry of the cluster. a) Relative arrangement ofnearby molecules. b) The six-molecule cluster with periodic boundary condi-tions.

Figure 9. Plot of the TPA cross-section (in a logarithmic scale) for a cluster ofsix molecules with z=1.5 (1chr0.17) and w=1.8 (all energies in

ffiffiffi2p

t units).The horizontal axis shows the energy of the absorbed photon (dimension-less ; for typical

ffiffiffi2p

t=1 eV values, the energy on the horizontal axis may beread in eV). The vertical axis shows, on a reciprocal scale, the intermoleculardistance, r, over half the effective molecular length (l, see Figure 8). Forgraphical purposes, a Gaussian line shape, with s =0.03, is assigned to eachtransition, and weak transitions (with cross-sections more than one order ofmagnitude smaller than the TPA of the isolated molecule) are disregarded.The dashed line marks the OPA energy. The continuous white lines show re-sults obtained in the exciton model with dipolar interactions: the lines atlower and higher energy refer to TPA and OPA processes, respectively. TPAprocesses related to 2c excitations in the exciton model are not shown sincethey deviate by less than 0.1 (in the energy scale) from the OPA line.


A. Painelli et al.


model, adopting the dipolar approximation for electrostatic in-teractions. The relevant results are shown as continuous whitelines in Figure 9.We start the discussion with singly excited states c and e.

The OPA state, c, has a finite transition dipole moment fromthe ground state and the standard exciton model predicts, inthe chosen attractive geometry, a red-shift of the relevant tran-sition frequency with increasing intermolecular interactions,that is, with decreasing r. Indeed the effect is very small ; onthe scale of the figure the continuous white line relevant tothe c-exciton appears to be almost vertical. In the completemodel instead a sizeable decrease of the OPA frequency withdecreasing r is observed (dashed line in Figure 9) that is mainlydue to the slight increase of the average molecular 1 resultingfrom the attractive intermolecular interactions. In fact 1 in-creases from ~0.17 for the isolated molecule (r!1) to ~0.2for r l/2. Notice that the variation of 1 cannot be accountedfor in the dipolar approximation, because in this approxima-tion the average electric field generated by non-dipolar mole-cules exactly vanishes.The TPA state, e, is a dark state; the transition dipole

moment from g to e vanishes. In the dipolar approximationthen the J interaction vanishes for the e-excitation. As a resultthe energy of the TPA absorption calculated in the exciton ap-proximation is not affected by intermolecular interactions, asshown in the figure by the white vertical line at lower energy.In contrast, our model, by relaxing the dipolar approximation,predicts a red-shift of the TPA frequency with decreasing r, asshown by the blue trace marked as e in the colour plot. In facttwo main contributions to the red-shift of the e-band can berecognized, and both are missed in the dipolar approxima-tion.[32] The increase of 1 with decreasing r, which is responsi-ble for the red-shift of the OPA frequency, accounts for aboutone half of the total red-shift of the TPA band. The other con-tribution is due to the dispersion of the e-exciton band, drivenby an interaction proportional to the squared transition quad-rupole moment, [1ACHTUNGTRENNUNG(1�1)]2. It should be noticed that sizeabletransition quadrupole moments do not lead to any appreciableOPA intensity of the e-exciton state—the dipole moment is byfar the dominant term in the perturbation induced by the elec-tromagnetic field. However, as far as the strength of intermo-lecular interactions is concerned, quadrupolar terms are large,at least when the molecular length is sizeable with respect tointerchromophore distances.[32]

Beside e, several very intense TPA transitions appear inFigure 9. The red–yellow traces occurring near the whitedashed line correspond to very intense TPA absorptions due tostates with two c-excitations. These doubly excited states, atabout twice the energy of a single c excitation, can be reachedin TPA processes by absorption of a photon of approximatelythe same frequency as absorbed in the OPA process. The re-sulting TPA signal is largely amplified by resonance, but it ismasked by the concomitant OPA process. More interesting forpractical purposes are the intense TPA processes described bythe blue/yellow traces marked b1 and b2. These processes in-volve states with two c-excitations, and in fact, at large r, bothtraces converge toward the OPA frequency (dashed white line).

With decreasing r, the b1 and b2 states split down from the 2cband moving in a frequency region not obscured by OPA. Asthese states move away from the 2c band, their resonant en-hancement decreases. Their TPA cross-section however remainslarge in absolute terms, being more than one order of magni-tude larger than for the e-state (notice the logarithmic scaleused for the TPA cross-section in Figure 9).[32] This interestingphenomenon is again missed in the exciton model with dipo-lar interactions; in this approximation, in fact, exciton–excitoninteractions vanish for non-dipolar molecules and biexcitonscannot split from the 2c states.The b1 and b2 states are bound biexciton states that split

down from the 2c band due to electrostatic interactions. Spe-cifically, the energy required to bind the excitons results fromthe increase of the quadrupolar moment in the molecule uponexcitation. Due to attractive intermolecular interactions, a statewith two nearby excitations is expected to split down from the2c band. Indeed we find two bound biexcitons and, as ex-plained in ref. [32] , this results from ultraexcitonic mixing, thatis, from the mixing of states with a different number of exci-tons. Once again, the standard excitonic picture fails ; on oneside it does not account for ultraexcitonic mixing; on the otherside, exciton–exciton interaction necessarily vanishes for non-dipolar dyes in the dipolar approximation for intermolecular in-teractions.

Exciton Strings and Multielectron Transfer

Supramolecular engineering is a powerful tool to amplify mo-lecular responses, but to exploit the promise of molecular ma-terials the new physics emerging from intermolecular interac-tions has to be fully appreciated. Biexcitons in clusters of quad-rupolar chromophores represent an interesting example ofphenomena appearing in clusters of molecules due to intermo-lecular interactions, with no counterpart at the molecular level.As another intriguing example, here we briefly review the phe-nomenon of multielectron transfer, as recently discussed inclusters of D-p-A chromophores.[24]

Consider a one-dimensional cluster of N D-p-A chromo-phores, aligned as illustrated in Figure 10. Intermolecular inter-actions are attractive and favor an increase of polarity of thepolarizable chromophores as the interchromophore distance rdecreases. Starting with a lattice of largely neutral chromo-phores at long interchromophore distance (l/r!0 in Figure 10)and decreasing the distance (i.e. compressing the lattice), twolarge interactions compete in the definition of the groundstate of the system. The large energy gap (2z) required toionize a chromophore (cf. Figure 2) favors the neutral state,while intermolecular interactions favor the zwitterionic state.For proper values of the model parameters a discontinuouscharge crossover is found by varying the interchromophoredistance, as shown in Figure 10.[23,24] At large distances in fact2z dominates and the ground state is largely neutral, while atshort distances the electrostatic energy dominates, leading toa largely zwitterionic ground state. The abrupt change of po-larity of the chromophores from a largely neutral to a largelyzwitterionic value implies a big and abrupt variation of the ma-




terial properties. The discontinuous charge crossover in clus-ters of D-p-A chromophores represents the extreme conse-quence of cooperativity, and has no counterpart for isolatedmolecules. Moreover, in close proximity to the crossover, a bist-ability region appears; any tiny variation of external conditionscan in fact switches the material between two phases withmacroscopically different properties, an appealing phenomen-on for applications in molecular-based switches.[23,24]

The properties of the material in the proximity of the discon-tinuous crossover are unusual. Focusing on the optical excita-tion with the largest oscillator strength (which, for the chosensystem, always coincides with the lowest-energy excitation) wecalculate the variation of the population of zwitterionic statesupon photoexcitation. In the familiar excitonic approximationthe absorption of a photon creates a single excitation, switch-ing a single molecule from the ground to the excited state; inthis approximation, the maximum variation in the populationof zwitterionic species clearly amounts to one. In fact, asshown in the bottom panel of Figure 10, the number of zwit-terions created upon photoexcitation is roughly one for largeintermolecular distances. At large distances intermolecular in-teractions are weak, and the cluster is made up of largely neu-tral molecules (1~0); upon photoexcitation, one molecule isturned zwitterionic or, in other terms, one electron is trans-ferred from D to A within a molecule (D1). In the oppositelimit of short distances, electrostatic interactions are strong,and the cluster is made up by zwitterions (1~1); upon photo-excitation one chromophore is turned neutral, that is, one elec-tron is back-transferred from A� to D+ within a moleculeACHTUNGTRENNUNG(D�1). However, near the discontinuous charge crossover,the number of zwitterions created or destroyed upon photoex-

citation becomes much larger than one, suggesting that theabsorption of a single photon may induce the motion of sever-al (up to six for the parameters in Figure 10) electrons at atime.[6,22,25] Detailed analysis of the states involved in the ab-sorption process indicates that, close to the crossover, the pho-toexcitation produces several zwitterions on a background ofneutral molecules or vice versa. These molecules with reversedionicity cluster together forming a droplet of zwitterions on abackground of neutral molecules (or vice versa). Multielectrontransfer has already been discussed in different contexts, butusually describes a cascading effect related to the relaxation ofsome slow degree of freedom following a more traditional op-tical excitation.[57] Here instead multielectron transfer repre-sents the primary photoexcitation event—the absorption of asingle photon directly drives the concerted motion of severalelectrons residing in several nearby molecules.[24]

Multielectron transfer is another manifestation of extremecollective and cooperative behavior due to large competing in-teractions. It survives in models with delocalized electrons[22]

and is currently a hot topic in the field of nanocrystals.[58] Thepossibility of transferring several electrons upon absorption ofa single photon is extremely appealing to understand photoin-duced phase transitions,[22,24] and, if properly exploited, canlead to important improvements in the yield of photoconver-sion devices.

Conclusions and Outlook

In recent years research on molecular materials for NLO appli-cations has been mainly devoted to the development of newNLO-phores with optimized properties. However the interac-tions of each NLO-phore with its surroundings can profoundlyalter its properties. A detailed understanding of interchromo-phore and/or supramolecular interactions is required to reliablyguide the synthesis of optimized materials. Textbooks on solid-state physics traditionally devote little attention to molecularcrystals: chemical bonds within each molecular unit are muchstronger than weak intermolecular forces and the properties ofthe crystal are therefore expected to be the sum of the proper-ties of the composing molecules, as in the oriented-gas model.Davidov shifts and splittings are small deviations from the ad-ditive behavior due to excitonic interactions.[28] The discoveryof excitonic effects in molecular aggregates in solution,[59] withtechnologically relevant applications, boosted renewed interestin the exciton model, which provides a simple referencescheme to understand linear spectra and properties of molecu-lar materials.[30] Nowadays, following the increasing interest to-wards molecular materials for NLO applications, a new para-digm for intermolecular interactions is required. In the firstplace the excitonic model must be extended to discuss stateswith a high number of excitons to exploit the new physics ofbiexcitons and exciton strings. More fundamentally, the basicapproximation of the exciton model must be relaxed to ac-count for the large (hyper)polarizability of chromophores forNLO applications. The exciton model describes the non-addi-tive behavior of molecular materials accounting for collectivebehavior, that is, the delocalization of local excitations due to

Figure 10. Top: sketch of a linear array of D-p-A chromophores in the attrac-tive geometry. Middle: ground-state polarity of the chromophores in a clus-ter of 16 molecules with z=2 and w=2 (in units of

ffiffiffi2p

t) versus the inverseinterchromophore distance (l/r). Bottom: for the same cluster, the averagenumber of zwitterions created (D>0) or destroyed (D<0) upon photoexci-tation to the state with the largest dipole moment.


A. Painelli et al.


energy-transfer processes mediated by electrostatic interac-tions. Cooperative phenomena add to collective effects ampli-fying the nonlinearity of molecular materials made up by large-ly (hyper)polarizable chromophores. Not only do excitationsdelocalize, but each chromophore readjusts its charge distribu-tion in response to the electric fields generated by the instan-taneous charge distribution on the surrounding chromo-phores.[6] Understanding the complex interplay between collec-tive and cooperative interactions in molecular materials forNLO applications is a challenging task, but is a prerequisite forthe development of reliable supramolecular structure–proper-ties relationships for molecular materials for NLO applications.To guide the synthesis of new materials with optimized NLO

responses, general indications, as provided in terms of simplemodels, are required. Here we adopt a bottom-up modellingstrategy: essential-state models are defined for the isolatedchromophores, to be validated against the spectroscopic datafor the chromophore in solution. These models represent thestarting point to build up a description of interacting chromo-phores in molecular crystals, aggregates or supramolecular ar-chitectures, where electrostatic interchromophore interactionsare explicitly accounted for. The model is simple and can be di-agonalized exactly. The validity of common approximationschemes can be tested, and indeed we demonstrate in severalcases the failure of the excitonic approximation and/or of thedipolar model for electrostatic interactions. On a more positivenote, several clusters of different sizes and geometries can beinvestigated to relate the material properties to the nature ofthe chromophores composing the cluster and to their supra-molecular arrangement. Best geometries can be suggested foroptimized responses, making supramolecular engineering apowerful tool for material scientists.The work reviewed in this paper represents the first step of

a more ambitious project. The explicit inclusion of molecularvibrations in models for interacting chromophores is demand-ing, but is a prerequisite for a complete modeling of optical re-sponses at finite frequencies in terms of band shapes and in-tensities.[31,33] The adopted model for electrostatic interactionshas the main merit of relaxing the dipolar approximation, butis clearly oversimplified. Inputs on interaction energies fromquantum-chemical calculations can improve the reliability ofthe results with reference to specific materials. More generally,the validation of the model against experimental data is in itsinitial stages;[17, 18,26,27, 31,33] more work in this direction isneeded, a challenging task requiring a tight collaboration be-tween synthetic chemists, spectroscopists, and theoreticalchemists.The message we want to convey in concluding this work is

that interchromophore interactions are extremely important indefining the properties of molecular materials for NLO applica-tions. Disregarding their role while designing new materialsmay lead to quite frustrating results. On the other hand, onceproperly understood, interchromophore interactions offer anew and extremely powerful handle for optimizing the materi-al properties. Supramolecular engineering can amplify (or sup-press) by orders of magnitude the nonlinear responses of thematerial at hand, and new physics with no counterpart at the

molecular level can emerge, offering new exciting opportuni-ties for applications.

Acknowledgements

Work supported by Italian MIUR through PRIN 2006031511,G.D’A. acknowledges support from NE MAGMANET NMP3-CT2005-515767. A.P. thanks Z. G. Soos for discussions and corre-spondence.

Keywords: chromophores · cooperative phenomena · donor–acceptor systems · electrostatic interactions · nonlinear optics

[1] E. M. Vogel, Nat. Nanotechnol. 2007, 2, 25.[2] S. K. Sahoo, S. Parveen, J. J. Panda, Nanomedicine 2007, 3, 20.[3] A. Diaspro, G. Chirico, M. Collini, Q. Rev. Biophys. 2005, 38, 97.[4] Z. G. Soos, D. Mukhopadhyay, A. Painelli, A. Girlando in Handbook of

Conducting Polymers, Vol. 1 (Eds. : T. A. Skotheim, R. L. Elsenbaumer, J. R.Reynolds), Marcel Dekker, New York, 1998, pp. 165.

[5] J.-L. BrQdas, K. Cornil, F. Meyers, D. Beljonne in Handbook of ConductingPolymers, Vol. 1 (Eds. : T. A. Skotheim, R. L. Elsenbaumer, J. R. Reynolds),Marcel Dekker, New York, 1998, pp. 1.

[6] A. Painelli, F. Terenziani in Nonlinear Optical Properties of Matter : FromMolecules to Condensed Phases, Vol. 1 (Eds. : M. G. Papadopoulos, J.Leszczynski, A. J. Sadlej), Kluwer, Dordrecht, 2006, pp. 251.

[7] A. Painelli, L. Del Freo, F. Terenziani in Molecular Low Dimensional andNanostructured Materials for Advanced Applications, Vol. 1 (Eds. : A. Graja,B. R. Bulka, F. Kajsar), Kluwer, Dordrecht, 2002, pp. 113.

[8] L. R. Dalton, A. K.-Y. Jen, P. Sullivan, Y. Liao, B. Eichinger, B. H. Robinson,A. Chen, Nonlin. Opt. Quantum Opt. 2006, 35, 1.

[9] R. A. Marcus, Rev. Mod. Phys. 1993, 65, 599.[10] D. R. Kanis, M. A. Ratner, T. J. Marks, Chem. Rev. 1994, 94, 195, and refer-

ences therein.[11] M. Albota, D. Beljonne, J.-L. BrQdas, J. E. Ehrlich, J.-Y. Fu, A. A. Heikal, S. E.

Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W.Perry, H. Rçckel, M. Rumi, G. Subramaniam, W. W. Webb, X.-L. Wu, C. Xu,Science 1998, 281, 1653; O. Mongin, L. PorrTs, M. Charlot, C. Katan, M.Blanchard-Desce, Chem. Eur. J. 2007, 13, 1481.

[12] S. R. Marder, Chem. Commun. 2006, 131.[13] J. Zyss, I. Ledoux, Chem. Rev. 1994, 94, 77; J. D. Weibel, D. Yaron, J. Zyss,

J. Chem. Phys. 2003, 119, 11847; H. Kang, A. Facchetti, H. Jiang, E. Caria-ti, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S.-T.Ho, E. C. Brown, M. A. Ratner, T. J. Marks, J. Am. Chem. Soc. 2007, 129,3267.

[14] D. Beljonne, W. Wenseleers, E. Zojer, Z. Shuai, H. Vogel, S. J. K. Pond,J. W. Perry, S. R. Marder, J.-L. BrQdas, Adv. Funct. Mater. 2002, 12, 631.

[15] S. Di Bella, T. J. Marks, M. A. Ratner, J. Am. Chem. Soc. 1994, 116, 4440; C.Reichardt, Chem. Rev. 1994, 94, 2319; A. Abbotto, L. Beverina, S. Brada-mante, A. Facchetti, C. Klein, G. A. Pagani, M. Redi-Abshiro, R. Wort-mann, Chem. Eur. J. 2003, 9, 1991; F. Terenziani, A. Painelli, Chem. Phys.2003, 295, 35.

[16] A. Painelli, Chem. Phys. 1999, 245, 185.[17] F. Terenziani, A. Painelli, D. Comoretto, J. Phys. Chem. A 2000, 104,

11049; B. Boldrini, E. Cavalli, A. Painelli, F. Terenziani, J. Phys. Chem. A2002, 106, 6286.

[18] F. Terenziani, A. Painelli, C. Katan, M. Charlot, M. Blanchard-Desce, J. Am.Chem. Soc. 2006, 128, 15742.

[19] A. Painelli, Chem. Phys. Lett. 1998, 285, 352.[20] V. Chernyak, S. Tretiak, S. Mukamel, Chem. Phys. Lett. 2000, 319, 261; A.

Painelli, L. Del Freo, F. Terenziani, Chem. Phys. Lett. 2001, 346, 470; L. DelFreo, F. Terenziani, A. Painelli, J. Chem. Phys. 2002, 116, 755; A. Painelli,L. Del Freo, F. Terenziani in Nonlinear Optical Responses of Molecules,Solids and Liquids: Methods and Applications, Vol. 1 (Eds. : M. Papado-poulos), Research Signpost, Trivandrum, 2003, pp. 39.

[21] K. V. Mikkelsen, Y. Luo, H. Jgren, P. Jørgensen, J. Chem. Phys. 1994, 100,8240; D. M. Bishop, Adv. Chem. Phys. 1998, 104, 1; A. Painelli, F. Terenzia-




ni, Chem. Phys. Lett. 1999, 312, 211; J. Tomasi, R. Cammi, B. Mennucci, C.Cappelli, S. Corni, Phys. Chem. Chem. Phys. 2002, 4, 5697.

[22] A. Painelli, F. Terenziani, Z. G. Soos, Theor. Chem. Acc. 2007, 117, 915.[23] F. Terenziani, A. Painelli, Phys. Rev. B 2003, 68, 165405.[24] A. Painelli, F. Terenziani, J. Am. Chem. Soc. 2003, 125, 5624.[25] A. Painelli, Z. G. Soos, F. Terenziani in Lecture Series on Computer and

Computational Sciences, Vol. 3 (Eds. : G. Maroulis), Brill, Leiden, 2005,pp. 129.

[26] F. Terenziani, O. Mongin, C. Katan, B. K. G. Bhatthula, M. Blanchard-Desce, Chem. Eur. J. 2006, 12, 3089.

[27] F. Terenziani, A. Painelli, A. Girlando, R. M. Metzger, J. Phys. Chem. B2004, 108, 10743.

[28] A. S. Davidov, Theory of Molecular Excitons, Plenum, New York, 1971.[29] V. M. Agranovich, M. D. Galanin, Electronic Excitation Energy Transfer in

Condensed Matter, North-Holland, Amsterdam, 1982.[30] J. Knoester in Proceedings of the International School of Physics “Enrico

Fermi” 2001, Course CXLIX, Organic Nanostructures: Science and Appli-cations, Vol. 1 (Eds. : V. M. Agranovich, G. C. La Rocca), IOS Press, TheNetherlands, 2002, pp. 149 (and references therein).

[31] A. Painelli, F. Terenziani, L. Angiolini, T. Benelli, L. Giorgini, Chem. Eur. J.2005, 11, 6053.

[32] G. D’Avino, F. Terenziani, A. Painelli, J. Phys. Chem. B 2006, 110, 25590.[33] F. Terenziani, M. Morone, S. Gmouh, M. Blanchard-Desce, ChemPhys-

Chem 2006, 7, 685.[34] B. J. Orr, J. F. Ward, Mol. Phys. 1971, 20, 513.[35] B. Kirtman, J. M. Luis, D. M. Bishop, J. Chem. Phys. 1998, 108, 10008.[36] K. Clays, E. Hendrickx, T. Verbiest, A. Persoons, Adv. Mater. 1998, 10, 643.[37] S. R. Marder, D. N. Beratan, L. T. Cheng, Science 1991, 252, 103.[38] S. R. Marder, B. Kippelen, A. K.-Y. Jen, N. Peyghambarian, Nature 1997,

388, 845.[39] V. Alain, L. Thouin, M. Blanchard-Desce, U. Gubler, C. Bosshard, P.

Gunter, J. Muller, A. Fort, M. Barzoukas, Adv. Mater. 1999, 11, 1210; V.Alain, S. RQdoglia, M. Blanchard-Desce, S. Lebus, K. Lukaszuk, R. Wort-mann, U. Gubler, C. Bosshard, P. GLnter, Chem. Phys. 1999, 245, 51.

[40] J. R. Heath, M. A. Ratner, Phys. Today 2003, 56, 43; C. E. Dykstra, Adv.Chem. Phys. 2003, 126, 1.

[41] R. S. Mulliken, J. Am. Chem. Soc. 1952, 74, 811; D. Lu, G. Chen, J. W.Perry, W. A. Goddard III, J. Am. Chem. Soc. 1994, 116, 10679.

[42] M. Barzoukas, C. Runser, A. Fort, M. Blanchard-Desce, Chem. Phys. Lett.1996, 257, 531.

[43] M. Blanchard-Desce, M. Barzoukas, J. Opt. Soc. Am. B 1998, 15, 302.[44] S. Engelhard, F. H. M. Faisal, J. Chem. Phys. 1999, 110, 3596, and referen-

ces therein.[45] A. Datta, F. Terenziani, A. Painelli, ChemPhysChem 2006, 7, 2168.[46] E. D. RekaX, J.-B. Baudin, L. Jullien, I. Ledoux, J. Zyss, M. Blanchard-Desce,

Chem. Eur. J. 2001, 7, 4395.[47] Y. V. Pereverzev, O. V. Prezhdo, L. R. Dalton, ChemPhysChem 2004, 5,

1821.[48] Y. V. Pereverzev, O. V. Prezhdo, L. R. Dalton, Chem. Phys. Lett. 2001, 340,

328.[49] F. C. Spano, S. Mukamel, Phys. Rev. Lett. 1991, 66, 1197; J. Knoester, F. C.

Spano, Phys. Rev. Lett. 1995, 74, 2780; A. Chakrabarti, A. Schmidt, V. Va-lencia, B. Fluegel, S. Mazumdar, N. Armstrong, N. Peyghambarian, Phys.Rev. B 1998, 57, R4206.

[50] F. C. Spano, V. M. Agranovich, S. Mukamel, J. Chem. Phys. 1991, 95, 1400.[51] M. Kuwata-Gonokami, N. Peyghambarian, K. Meissner, B. Fluegel, Y.

Sato, K. Ema, R. Shimano, S. Mazumdar, F. Guo, T. Tokihiro, H. Ezaki, E.Hanamura, Nature 1994, 367, 47.

[52] F. Terenziani, A. Painelli, J. Lumin. 2005, 112, 474.[53] M. Barzoukas, M. Blanchard-Desce, J. Chem. Phys. 2000, 113, 3951.[54] B. Strehmel, V. Strehmel, Adv. Photochem. 2006, 29, 111.[55] W.-H. Lee, H. Lee, J.-A. Kim, J.-H. Choi, M. Cho, S.-J. Jeon, B. R. Cho, J.

Am. Chem. Soc. 2001, 123, 10658; L. PorrTs, C. Katan, O. Mongin, T.Pons, J. Mertz, M. Blanchard-Desce, J. Mol. Struct. 2004, 704, 17.

[56] S. Hahn, D. Kim, M. Cho, J. Phys. Chem. B 1999, 103, 8221.[57] H. Tributsch, L. Pohlmann, Science 1998, 279, 1891; S. Iwai, S. Tanaka, K.

Fujinuma, H. Kishida, H. Okamoto, Y. Tokura, Phys. Rev. Lett. 2002, 88,057402.

[58] R. D. Schaller, V. M. Agranovich, V. I. Klimov, Nat. Phys. 2005, 1, 189.[59] E. E. Jelly, Nature 1936, 138, 1009; G. Scheibe, Z. Angew. Chem. 1936, 48,

563.

Received: May 24, 2007

Published online on November 12, 2007


A. Painelli et al.


multichromophores for nonlinear optics: designing the material properties by electrostatic...

Documents