Symmetry Breaking in Octupolar Chromophores: Solvatochromism

Apr 1, 2008 - In this contribution, we adopt an essential-state description for octupolar (AD3 or DA3) chromophores (where A is an electron-acceptor a...
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J. Phys. Chem. B 2008, 112, 5079-5087

5079

Symmetry Breaking in Octupolar Chromophores: Solvatochromism and Electroabsorption Francesca Terenziani,* Cristina Sissa, and Anna Painelli Dipartimento di Chimica GIAF and INSTM-UdR Parma, Parma UniVersity, Parco Area delle Scienze 17/a, 43100 Parma, Italy ReceiVed: October 23, 2007; In Final Form: January 23, 2008

In this contribution, we adopt an essential-state description for octupolar (AD3 or DA3) chromophores (where A is an electron-acceptor and D is an electron-donor) that also accounts for the coupling of electrons to molecular vibrations and for solvation effects. The first excited state of octupolar chromophores is always multistable and can therefore support symmetry breaking. In particular, symmetry is always broken in the relaxed excited state of octupolar dyes in polar solvents, with consequent localization of the excitation on one of the dipolar molecular branches. This rationalizes the common observation of strongly solvatochromic fluorescence spectra for octupolar chromophores. The model is validated through the comparison with experimental data. The essential-state model is also adopted to derive a perturbative expression for the electroabsorption spectrum: if compared with the formalism derived for dipolar molecules, a new term appears for octupolar chromophores, due to the field activation of an otherwise dark transition. The importance and implications of this term are discussed.

Introduction Conjugated octupolar chromophores of AD3 or DA3 structure (where A and D represent electron-acceptor and electron-donor groups, respectively) are currently widely investigated for several reasons. They display high two-photon absorption (TPA) cross sections1-3 and significant second-order nonlinear optical properties, even in the crystalline phase.4 Their highly solvatochromic fluorescence, that can be induced by one- and twophoton excitation, makes them interesting micropolarity probes,3 whereas the possibility to enhance or suppress their two-photon excited fluorescence makes them good candidates for metal sensing.5 On a different perspective, octupolar chromophores are useful model systems to study the energy redistribution and the dynamics of electronic coupling in supramolecular and dendritic structures: several time-resolved spectroscopic measurements (including ultrafast transient absorption and fluorescence anisotropy, photon echo peak shift) on octupolar dyes offered important clues on the size of coherent domains in dendrons and dendrimers.6 Recently, octupolar chromophores have also been studied by electroabsorption measurements to get information on the nature of the excited states7,8 and specifically on the possible appearance of symmetry-breaking phenomena.7 From the theoretical point of view, at least two different approaches have been described in the literature for these chromophores. In the excitonic model, the octupolar AD3 or DA3 systems are described as three dipolar (DA) units only interacting via electrostatic forces,2,9 fully disregarding the charge resonance between the three molecular branches. In spite of this rather crude approximation, qualitatively good results are obtained when comparing the properties of octupolar compounds with those of the corresponding dipoles, but a quantitative agreement is difficult to obtain.2,9 An interesting alternative to the excitonic model is offered by the so-called essential-state or charge-resonance models. In these approaches, * To whom correspondence should be addressed. E-mail: [email protected]. Telephone: +39 0521 905433. Fax: +39 0521 905556.

few states, corresponding to the main resonance formulas for the relevant molecule, are selected to describe the electronic structure. Two electronic states (DA and D+A-) are then relevant to dipolar molecules,10 three states (DAD, D+A-D, and DA-D+) are needed for quadrupolar dyes,11,12 and four states are required for octupolar dyes to account for the charge-transfer processes in the three molecular branches.13 The essential-state model fully describes charge resonance. For octupolar chromophores, an extended version of the model, also accounting for the coupling between electronic degrees of freedom and molecular vibrations, has been adopted to discuss the evolution of static nonlinear optical properties with the bond-length alternation.13 In this contribution, based on an essential-state description of the electronic structure, we define a model for octupolar dyes that accounts for the coupling of electrons to molecular vibrations and for solvation effects. The model describes the spectroscopic behavior of these dyes in solution. The first excited state of octupolar chromophores is always multistable, resulting, in polar solvents, in a localization of the relevant excitation on one of the dipolar branches. This finding rationalizes the observation of a strongly solvatochromic fluorescence for this class of chromophores. The model applies to linear and nonlinear optical spectra (absorption, fluorescence, TPA) of octupolar chromophores and is discussed with reference to a specific representative chromophore. Moreover, the essential-state model offers a good basis to describe electroabsorption spectra: a perturbative treatment along the lines originally suggested by Liptay for dipolar dyes14 leads to an expression for the electroabsorption response of octupolar chromophores where, besides the standard contributions from the linear absorption spectrum and its derivatives, a new contribution appears due to the field-activated absorption toward a dark state. The Model In an essential-state approach, octupolar chromophores can be described as resonating between four limiting structures, as

10.1021/jp710241g CCC: $40.75 © 2008 American Chemical Society Published on Web 04/01/2008

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SCHEME 1: The Four Resonating Structures for AD3 Octupolar Chromophores (Similar Structures Apply to DA3 Dyes, with Interchanged D and A Sites)

where F, the expectation value of the Fˆ operator in the ground state, measures the fractional charge on the central site and is therefore proportional to the molecular octupolar moment. F is fixed by the model parameters as follows:

(

F ) 0.5 1 -

sketched in Scheme 1: a neutral (|N〉) and three charge-separated (zwitterionic) forms (|Z1〉, |Z2〉, and |Z3〉).13 The energy difference between the |N〉 and the three charge-separated states is 2η, and the three zwitterionic forms are mixed with the neutral one by the charge-transfer integral 〈N|H|Z1〉 ) 〈N|H|Z2〉 ) 〈N|H|Z3〉 ) -x2t. The direct mixing between the three zwitterionic forms is disregarded as a higher-order process. On this basis set, the following operators are conveniently defined:

( ) ( ) ( ) ( )

0 0 Fˆ ) 0 0

0 1 0 0

0 0 1 0

0 0 ; σˆ ) 0 1

0 1 1 1

1 0 0 0

1 0 0 0

0 δˆ 1 ) 0 0 0

0 2 0 0

0 0 -1 0

1 0 ; 0 0

0 0 ; δˆ 2 ) 0 -1

0 0 0 0

0 0 0 0

0 0 1 0

0 0 0 -1

(1)

(2)

The two relevant (in plane) components of the dipole moment operator are (numbering and reference frame as in Scheme 1)

1 µˆ x ) µ0δˆ 1; 2

µˆ y )

x3 µ δˆ 2 0 2

(3)

where µ0 is the magnitude of the dipole moment of each one of the three zwitterionic states or, in other terms, the dipole moment of a D+A- branch. Exploiting the molecular C3 symmetry, the basis set is symmetrized to give two (nondegenerate) totally symmetric (Atype) states |A1〉 ) |N〉 and |A2〉 ) 1/x3(|Z1〉 + |Z2〉 + |Z3〉), and two degenerate states (E-type) |E1〉 ) 1/x6(2|Z1〉 - |Z2〉 |Z3〉) and |E2〉 ) 1/x2(|Z2〉 - |Z3〉). On this basis, the Hamiltonian only mixes the two A-symmetry states, giving the ground (g) and the highest-energy excited state (e), while the two degenerate E-symmetry states stay unmixed. The resulting eigenstates are

|g〉 ) x1 - F|A1〉 + xF|A2〉 |c1〉 ) |E1〉 |c2〉 ) |E2〉 |e〉 ) xF|A1〉 - x1 - F|A2〉

xη

(5)

+ 6t2

For large positive η, the ground state is dominated by A1 and F ∼ 0. For large negative η, instead the ground state is dominated by A2 and F ∼ 1. For η ) 0, A1 and A2 have the same weight in the ground state and F ) 0.5. Transition energies and dipole moments, and hence all quantities of interest for spectroscopy, can be expressed in terms of F:

x1 -F F;

pωgc1 ) pωgc2 ) x6t

x

pωge ) x6t

1 ; F(1 - F)

x1 -F F;

pωc1e ) pωc2e ) x6t

xF2;

(x) ) µ(y) µgc gc2 ) µ0 1

(y) (x) µgc ) µgc )0 1 2

(y) µ(x) ge ) µge ) 0;

x1 -2 F;

µc(x)1e ) µc(y)2e ) -µ0

µc(y)1e ) µc(x)2e ) 0

1 µc(x)1c1 ) µc(x)2c2 ) µc(y)1c2 ) µ0; 2

µc(y)1c1 ) µc(y)2c2 ) µc(x)1c2 ) 0

(6)

The electronic Hamiltonian then reads

Hel ) 2ηFˆ - x2tσˆ

)

η 2

(4)

Due to the specific choice of reference axes system, the diagonal and nondiagonal matrix elements of the dipole moment operator have always just a single nonvanishing component. Specifically, the two degenerate c states with E-symmetry are one-photon allowed, with transition dipole moments pointing along orthogonal (x and y) directions but with the same magnitude: the squared transition dipole moment (and hence the intensity of the transition) linearly increases with F. The e state is dark, that is, it has a vanishing transition dipole moment from the ground state. However, it can be reached from the ground state by two-photon excitation. The degenerate c states are also twophoton active but with a smaller intensity than the e state. Figure 1 summarizes the spectroscopic behavior of octupolar molecules depicted in terms of the essential-state model. Transition frequencies (ωge and ωgc) and the TPA cross section (σ2) are shown as a function of the octupolar character, F. The TPA intensity of the main peak (g f e) steeply increases as F f 0.5; the secondary peak (g f c1/2) has a much smaller intensity, that linearly increases with F. The electronic model for octupolar dyes accounts for the basic spectroscopic features of this class of molecules. However, to fully understand the spectroscopy of octupolar dyes, the model must be extended to account for the coupling between electronic and slow degrees of freedom, including vibrations and polar solvation coordinates. In close analogy with our recent work on dipolar15 and quadrupolar chromophores,12 and in agreement with an early model by Cho,13 to account for the variation of the molecular geometry with the charge distribution, we introduce three independent effective vibrational coordinates, q1, q2, and q3, relevant to the three molecular branches. The three coordinates are equivalent: they have the same harmonic frequency, ωv, and the same relaxation energy, v, that measures the energy gained upon the relaxation of the ith branch when the charge is transferred from D to A in the same branch (|N〉 f |Zi〉 process).

Symmetry Breaking in Octupolar Chromophores

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Figure 1. Four-state model for AD3 (or DA3) octupolar chromophores. The neutral (A1) and the three zwitterionic (A2, E1, and E2) states are separated by an energy gap 2η. A matrix element -x2t mixes the two totally symmetric states to give the ground state, g, and the highest excited state, e. The E-symmetry states stay unmixed and correspond to the excited states c1 and c2, active in linear absorption processes (dotted arrow) and weakly active in TPA (full red arrows). The totally symmetric e state is active in TPA (full blue arrows). A fraction F/3 of electron is transferred from each of the three D sites toward the A site in the ground state. The total amount of charge on the central site, F, is proportional to the octupolar moment in the ground state. The top graph shows the F-dependence of the transition frequencies to states c and e (units with x2t ) 1); the bottom panel shows the TPA cross section (arbitrary units) evaluated on the maximum for the two TPA transitions to the c (red line) and e (blue line) states.

It is convenient to define symmetry-adapted vibrational coordinates:

qA )

1 (q1 + q2 + q3) x3

qE1 )

1 (2q1 - q2 - q3) x6

qE2 )

1 (q2 - q3) x2

(7)

so that the total Hamiltonian reads

x

x

2v v ωqAFˆ ωq δˆ -  ωq δˆ + 3 3 E1 1 x v E2 2 1 2 2 1 1 (ω qA + pA2) + (ω2qE12 + pE12) + (ω2qE22 + pE22) (8) 2 2 2

H ) Hel -

where the pA, pE1, and pE2 operators are the conjugate momenta of the q-coordinates and p is set to 1. The totally symmetric coordinate, qA, couples to the Fˆ operator, so that vibrations along qA only modulate the mixing between |A1〉 and |A2〉, that is, the symmetrical charge transfer from the periphery groups to the center (or backward). The two degenerate coordinates, qE1 and qE2, are coupled to δˆ 1 and δˆ 2 operators: vibrations along these coordinates mix states of different symmetry and drive an asymmetric charge distribution on the periphery groups. In the adiabatic approximation, the vibrational kinetic energy (p2 terms in the Hamiltonian above) is neglected to define a

Figure 2. Phase diagram for octupolar chromophores as a function of F (the octupolar character) and v (the electron-vibration coupling, in units of x2t). Region I corresponds to stable g and e states, and multistable c1 state, as shown in the inset; in region III, both the ground state and the c1 PESs are multistable (PESs are not drawn for graphical reasons).

Figure 3. Magnification of the conical intersection between the two PESs relevant to c1 and c2 shown in Figure 2, along the qE1 and qE2 coordinates.

q-dependent Hamiltonian. The eigenvalues of the corresponding matrix, plotted against the vibrational coordinates, define the potential energy surfaces (PESs). The inset in Figure 2 shows typical PESs obtained for η > 0. Energy is plotted against the two E-symmetry coordinates (qE1 and qE2) that are of interest for symmetry-breaking phenomena. The totally symmetric coordinate qA (not shown) is kept at its local equilibrium value, (qA)eq ) 2v〈F〉/x3. The presence of two degenerate excited states makes the PESs of octupolar dyes particularly interesting: any whatever tiny perturbation splits the two states and reduces the symmetry of the system. As a result, when plotted against qE1 and qE2, the PESs relevant to c1 and c2 show the typical structure of a conical intersection (cf. Figure 3) with the two states exactly degenerate only at the origin (qE1 ) qE2 ) 0) and the lowest surface developing three equivalent minima along the directions of the three molecular branches.16 The first excited state of octupolar dyes always has a multistable PES and is prone to symmetry-breaking phenomena. This result, related to the special symmetry of octupolar dyes, contrasts sharply with the behavior of quadrupolar dyes. The symmetry of quadrupoles in fact does not allow for degenerate representations: as a result, symmetry breaking in either the ground or the excited states is conditional, and a large portion of the phase diagram for quadrupolar dyes is occupied by class II chromophores, that do not undergo symmetry breaking in any electronic state.12

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Whereas the c1 state is always multistable for octupolar chromophores, the ground state is stable as long as

v
0.5. As already discussed in ref 12, this is probably due to the intrinsic instability of such systems. In fact, for octupolar systems, as F f 1, the lowest-energy state becomes almost degenerate with the two c states, giving an almost threefolddegenerate multistable ground state. In the following, we will only discuss the spectroscopic behavior of chromophores belonging to class I, as relevant to experimental systems. Figure 3 shows a magnification of the c1 and c2 PESs relevant to an octupolar dye belonging to class I. As discussed above, the two states are degenerate for the undistorted molecule (qE1 ) 0, qE2 ) 0), where the two PESs create a conical intersection. Away from this special point, degeneracy is lost, and the lower surface develops three equivalent minima along the directions of the three molecular branches. The height of the barrier between the three minima, or, equivalently, the depth of the three valleys, depends on the strength of the electron-vibration coupling, v. Upon photoexcitation, a population is created on the vertical (unrelaxed) excited state: this state is unstable, and it relaxes toward one of the stable minima. Depending on the ratio between the barrier height and the phonon frequency, the relaxed state can be truly localized in one of the three minima or it can be delocalized between the three minima as a result of a fast tunneling. Discriminating between the two situations, known as true and false symmetry breaking, respectively, is a fairly delicate problem.18 Here, to calculate optical spectra of octupolar chromophores, we make resort to numerically exact nonadiabatic diagonalizations of the coupled electron-phonon problem, avoiding the problematic calculation of the dynamics on adiabatic PESs.12,19 Solvatochromism. The analysis of the solvatochromic behavior of polar and multipolar chromophores is a powerful way to gain basic information on the nature of the molecule at hand. Octupolar molecules have a nonpolar ground state, so that no major solvatochromism is expected in their absorption spectra. This observation is in line with available experimental data.3,9,20-23 More surprisingly, the fluorescence band of octupolar chromophores shows a large bathochromic shift with increasing solvent polarity, with effects most often as large as those found for dipolar, DA chromophores.3,9,20-22 This behavior is strongly reminiscent of what is observed for quadrupolar dyes of class I, where the solvatochromic fluorescence is related to symmetry breaking in the excited state and therefore to the formation of a polar (and hence solvatochromic) relaxed excited state.12 In the spirit of the continuum model for the solvent, and adopting exactly the same approach originally developed for

Figure 4. Potential energy surface and isopotential lines relevant to the relaxed c1 state calculated along the qE1 and qE2 coordinates in the case of an apolar solvent (top panel, or ) 0) and a polar solvent (bottom (y) panel, or > 0). The two components of the reaction field, F(x) R and FR , are fixed to their equilibrium value for the c1 state.

dipolar molecules15,24 and later extended to quadrupolar chromophores,12 we account for the solute-solvent interaction in the framework of the reaction-field approach.25 Treating the solvent as an elastic medium, the relevant Hamiltonian reads

Hsolv )

-µ0F(x) ˆ1 R δ

-

µ0F(y) ˆ2 R δ

µ02 (x) 2 2 + [(F ) + (F(y) R ) ] 4or R

(10)

(y) where F(x) R and FR are the x- and y-components of the reaction field, FR. The solvent relaxation energy, or, measures the energy gained upon solvent relaxation following the vertical N f Zi (i ) 1, 2, 3) process. This parameter increases with the solvent polarity and relates the equilibrium amplitude of the reaction ˆ x/y〉/ field to the dipole moment of the solute: F(x/y) R,eq ) 2or〈µ µ02.26 Figure 4 illustrates the importance of polar solvation on fluorescence spectra, in connection with the multistable nature of the first excited-state PES. Specifically, in the upper panel of Figure 4, the PES relevant to the first excited state, c1, is drawn as a function of qE1 and qE2 for an octupolar dye in a nonpolar solvent (or ) 0): the multistable nature of c1 is clearly seen in the presence of three equivalent minima. In the bottom panel, we show the same PES as calculated in a polar solvent (or ) 0.1) while keeping qA and the reaction field FR fixed at their local equilibrium value for the c1 state (F(x/y) ) 2or〈c1|δˆ 1/2|c1〉/µ0). As a result of the R interaction with a polar solvent, and depending on the orientation of the reaction field, one of the three minima is stabilized, as shown in Figure 4. Of course, the equivalence of the three minima is globally regained, considering the different orientations of FR. Tunneling between the three equivalent solventstabilized minima involves the reorientation of the solvent molecules, that is, a motion along the very slow coordinate associated with polar solvation. Adopting a classical description

Symmetry Breaking in Octupolar Chromophores

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Figure 5. Experimental (panel a) and calculated (panel b) absorption and fluorescence spectra of octupolar chromophore 3b. Experimental (panel c) and calculated (panel d) absorption and fluorescence spectra of dipolar chromophore 1b. Parameters for calculated spectra (eV): η ) 1.30/1.44 for 3b/1b, x2t ) 0.6, ωv ) 0.21, v ) 0.19, half-width at half-maximum (HWHM) ) 0.11, and or values reported in the legend. µ0 ) 33 D has been fixed to reproduce molar extinction coefficients of both chromophores.

SCHEME 2: Molecular Structure of Chromophores 3b and 1b27

for this slow coordinate, tunneling processes are strictly forbidden: in polar solvents, the excited state always undergoes a true symmetry breaking. In other terms, in polar solvents, the excitation localizes on one of the three molecular branches upon relaxation along the solvation coordinate. This symmetry breaking in the excited state has important consequences in fluorescence spectra: the polar relaxed excited state is stabilized in polar solvents, and the fluorescence band red-shifts with increasing the solvent polarity, in agreement with experimental data.3,9,20-23 The case of absorption spectra is different. The vertical absorption process occurs at the equilibrium geometry relevant to the ground state, where, for octupolar dyes of class I, symmetry is never broken. All states involved in absorption are therefore nonpolar, and the absorption frequency is barely affected by the solvent polarity, in agreement with experimental observation. Figure 5a shows absorption and fluorescence spectra of the octupolar chromophore 3b, whose complete spectroscopic characterization was reported in ref 27 (see Scheme 2). The dye has a largely neutral ground state (F = 0.11) and shows a solvent-independent absorption frequency. On the contrary, a large red-shift of the fluorescence band is observed (about 150 nm) when the solvent polarity increases from toluene to acetonitrile.27 Panel b of Figure 5 reports absorption and fluorescence spectra calculated for compound 3b by using the

Figure 6. Experimental (dots) and calculated (full lines) TPA spectra (normalized per branch) of compound 3b (black) and 1b (red) in toluene. Calculated spectra have been obtained with the same parameters given in the caption of Figure 5 (or ) 0.2/0.27 eV for 3b/1b). 1 GM ) 10-50 cm4 s photon-1.

proposed model and fixing appropriate parameters (listed in the caption). Calculated spectra well reproduce the observed solvatochromic behavior, as well as the evolution of the band shapes, by tuning a single parameter, or, that increases from 0.2 to 0.6 eV, to account for the increase of the solvent polarity from toluene to acetonitrile. The same parameters can be used to calculate the TPA spectrum as shown in Figure 6: the agreement with respect to the experimental spectrum is good for position, band shape, and intensity. Data in Figure 5 (panels a and b) are representative of the behavior of octupolar dyes with a largely neutral ground state (F < ∼0.1). Molecules with higher values of F are difficult to synthesize, but they would be very interesting for amplified TPA responses (see Figure 1). Figure 7 (panel a) displays absorption and fluorescence spectra obtained by fixing model parameters to get F ) 0.44. The relevant parameters (listed in the caption of Figure 7) are the same as estimated for the squaraine-based quadrupolar chromophore 2 discussed in ref 12. The quadrupolar squaraine belongs to class II according to the phase diagram relevant to quadrupolar dyes; that is, it does not show symmetry breaking, as demonstrated by the negligible solvatochromism observed in either absorption or fluorescence spectra.12 For

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Terenziani et al. corresponds to 8/15 times the response obtained by applying a perturbation field in the molecular x-direction and evaluating the x-polarized tensorial component of the variation of absorbance or, equivalently, 8/15γxxxx(-ω;ω,0,0).2 Perturbative expansion in the static field is taken up to the second order, and the EA spectrum is calculated by taking the difference between the perturbed and unperturbed spectrum, divided by the squared amplitude of the field. The perturbative expression contains all contributions quadratically dependent on the field amplitude, that is, all terms contributing to the EA spectrum. Indeed, the exact EA spectra calculated as the difference between the spectra in the presence and in the absence of a (weak) static field numerically coincide with the spectra obtained via the perturbative expansion. The perturbative expression obtained for the orientationally averaged EA response relevant to octupolar chromophores in solution is

[ ] [ ]

d S (ω) EA(ω) ) A S (ω) + Bω + dω ω d2 S (ω) + Dω u (ω) (11) Cω 2 dω ω Figure 7. (panel a) Absorption and fluorescence spectra calculated for an octupolar compound having F ) 0.44 in solvents of different polarity (or values in the legend, eV units). (panel b) TPA spectrum of the same chromophore for or ) 0. The dashed part of the TPA spectrum has been multiplied by 50 for graphical reasons. Parameters for calculations (eV): η ) 0.28, x2t ) 1.2, ωv ) 0.16, v ) 0.16, HWHM ) 0.05, and µ0 ) 21 D.

octupolar chromophores, class II is missing, and in fact for the corresponding (hypothetical) octupolar dye the calculated fluorescence spectra show an important solvatochromism (Figure 7), even if less remarkable than that observed for more neutral chromophores (cf. Figure 5). As expected, the chromophore has a stable (nonpolar) ground state, and the main effect of solvent polarity on absorption spectra is recognized in the inhomogeneous broadening of the band. The calculated TPA spectrum for this octupolar dye having intermediate F (panel b of Figure 7) is 2 orders of magnitude more intense than that calculated (and observed) for the more neutral dye (Figure 6), as expected based on results in Figure 1. Electroabsorption. In the framework of the proposed model, exact electroabsorption (EA) spectra of octupolar dyes can be calculated in two different ways: (i) from the difference between linear absorption spectra calculated in the presence and in the absence of a static electric field and (ii) as the relevant thirdorder hyperpolarizability γ(-ω;ω,0,0) calculated via sum-overstate expressions.28,29 Both approaches allow for the calculation of EA spectra accounting for vibrations and polar solvation.29 Including slow degrees of freedom is important to reproduce in detail experimental spectra, but this leads to complex equations that obscure the basic physics of EA. Here, we discuss the basic features of EA spectra of octupolar dyes extending to this class of molecules the well-known and widely adopted formalism originally developed by Liptay for polar dyes.14,30 To this aim, we limit attention to the purely electronic problem and derive, via a perturbative calculation, an analytical expression for the EA response. Specifically, we use perturbation theory to calculate the X-polarized EA spectrum (oscillating field polarized along the laboratory X-axis) in the case of a weak static electric field applied along the same X-direction. By symmetry, the orientationally averaged response for the solution

As proposed by Liptay for dipolar molecules,14,31 the analytic expression for the EA response is conveniently written in terms of the linear absorption spectrum, S (ω), and its first and second derivatives; however, at variance with the Liptay result for dipolar dyes, an additional term (the last one in eq 11) appears, unrelated to the linear absorption spectrum and due to the fieldinduced absorption toward the e state. The relevant normalized band shape function, u (ω), is taken as a Lorentzian, and a Lorentzian line shape is similarly assumed for the absorption band. The coefficients A, B, C, and D are given by the following:

A)

[

]

2 µc1e2 µc1e2 µc1e2 µgc12 4 µc1c1 3 + 2 + 2 4 5 ω 2 ωgc1ωc1e ωgc1ωge ω 2 ωgc12 gc1 c1e

[

] [

]

2 µc1c12 µgc12 µc1c12 4 µc1e 4 1 (xx) ∆R + 2 B) +2 -2 ) 5 ωc1e ωgc1 ωgc1 5 2 c1,g ωgc1

2 2 C ) µc1c12 ) µc1c22 5 5 D)

(

)

1 8 1 Kµ 2µ 2 15 gc1 c1e ωgc1 ωc1e

2

(12)

where, for the sake of simplicity, we have dropped the x or y apex on the dipole matrix elements because for each matrix element only one component is nonvanishing (see eq 6); K is the dimensional factor that transforms the dimensionless band shape u (ω) into an absorbance. The coefficients A, B, and C of the linear spectrum and of its first and second derivatives, calculated for octupolar chromophores, are similar to those obtained by Liptay for dipolar molecules.14,31 A is a fairly complex combination of permanent and transition dipole moments and transition frequencies, so that it is hardly possible to extract specific information out if it. B and C are more straightforward: C is proportional to the squared permanent dipole moment of the c1 state (or, equivalently, to the squared transition dipole moment between c1 and c2). Because the permanent dipole moment of the ground state vanishes for octupolar chromophores, C is also proportional to

Symmetry Breaking in Octupolar Chromophores

J. Phys. Chem. B, Vol. 112, No. 16, 2008 5085 If interpreted based on the classical expressions obtained by Liptay for dipolar molecules, the EA spectrum for octupolar chromophores with a small F value could lead to an overestimated C coefficient and hence to an overestimation of the variation of the dipole moment upon excitation. For larger F values (middle panel in Figure 8), the g f e transition moves away from the g f c1/2 transition, so that two EA features can be distinguished (at least for sharp spectral shapes). Moreover, the intensity of the u (ω) contribution lowers, as well as the contribution from the linear spectrum, S (ω). Finally, when F exceeds ∼0.2 (bottom panel), the allowed and forbidden transitions are well separated in energy, the contribution from the u (ω) term becomes negligible, and the main EA feature can be described as a linear combination of the first- and secondderivative of the linear spectrum. Recently, EA spectra of an octupolar chromophore have been measured, and molecular parameters have been derived by the use of the Liptay expression.7,8 As a result of the fit, the estimated difference between the dipole moment in the excited state and in the ground state is of the same magnitude for the octupolar dye and for the dipolar molecule that mimics a branch of the octupole: on this basis, a localization of the Vertical excited state on a single branch of the octupole has been inferred. With the lack of information on the position of the g f e transition (e.g., through TPA measurements), we cannot univocally fix the model parameters required for the calculation of the spectra; however, based on information available for similar chromophores,22 the relevant F value is estimated to be very low, on the order of 0.03, shedding doubt on the reliability of molecular parameters obtained from a fit based on the Liptay expression.

Figure 8. Electroabsorption spectra (black lines) calculated for octupolar chromophore with different F: F ) 0.03 (top panel), F ) 0.09 (middle panel), and F ) 0.21 (bottom panel). Color lines: decomposition of the EA response in contributions from the absorption spectrum (orange lines), from its first derivative (green lines), its second derivative (blue lines), and the u (ω) term (magenta lines). Parameters for calculations (eV): x2t ) 0.4, HWHM ) 0.05, η ) 2.0, 1.0, and 0.5 from top to bottom, the coupling to slow degrees of freedom is disregarded (v ) or ) 0).

the variation of the permanent dipole moment upon photoexcitation, just as that found for dipolar chromophores.31 The B coefficient is composed of two terms: one is proportional to the squared permanent dipole moment of c1, which is known from C, so that the other term can be evaluated to get information on the variation of the molecular polarizability upon excitation from the ground state to the c1 state. The u (ω) contribution to the EA signal, due to the field-induced activation of the forbidden g f e transition (whose location can be deduced from TPA spectra), has no counterpart for dipolar chromophores. The intensity of the relevant signal is governed by D. The relative importance of the four contributions to the EA signal can be inferred from Figure 8, where calculated EA spectra are shown for molecules with different octupolar character. For chromophores with a low octupolar character (small F, top panel), all of the four terms in eq 11 give appreciable contributions to the EA spectrum. In particular, the electric field induced transition to the e state has similar energy as the allowed transition to the c states (the energy difference between state e and states c1/2, pωc1e, vanishes as F f 0; see eq 6). As a consequence, the contribution due to u (ω) partly overlaps to S (ω) and its derivatives, apparently resulting in a second-derivative-like EA spectrum.

Discussion and Conclusions A model for octupolar (AD3 or DA3) chromophores is presented based on an essential-state description for the electronic structure and accounting for the coupling between electrons and molecular vibrations and an effective polar solvation coordinate. The adiabatic solution of the coupled electron-vibration problem shows that the potential energy surfaces corresponding to the two E-symmetry states always form a conical intersection. The lower-energy excited state of octupolar dyes is therefore always multistable, with three minima in the directions of the three molecular branches. The absorption of visible light occurs along the vertical excitation, that is, with all slow coordinates fixed at their equilibrium position for the ground state. As a result, absorption spectra are not affected by symmetry-breaking phenomena occurring in the excited state. On the opposite, steady-state fluorescence occurs from the relaxed excited state. In polar solvents, the emitting (relaxed) excited state is localized on one of the molecular branches, corresponding to a broken-symmetry, dipolar state, whose energy lowers with increasing solvent polarity: symmetry breaking in the excited state manifests itself most clearly with a strongly solvatochromic fluorescence. In nonpolar solvents, the behavior of the chromophore depends on the ratio between the vibrational frequency and the height of the barrier separating the three equivalent minima. In the case of a high barrier, the system is truly localized on a minimum of the PES, with negligible tunneling probability, and symmetry is effectively broken. In the opposite case of a low barrier with respect to the vibrational frequencies, tunneling is fast and effectively restores the original symmetry (false symmetry breaking). The spectroscopic behavior expected in the two cases is different. In the case of true symmetry breaking induced by vibrations, the

5086 J. Phys. Chem. B, Vol. 112, No. 16, 2008 symmetry is already broken in nonpolar solvents, and the fluorescence solvatochromism is similar to that observed for dipolar chromophores, with an almost linear dependence of the Stokes shift on the solvent polarity descriptor, according to the Lippert-Mataga relationship.32 In the opposite case, when vibrations are responsible for a false symmetry breaking, the excited state is non-dipolar in nonpolar solvents, and symmetry effectively breaks only in polar solvents. In this case, the slope of the Lippert-Mataga curve is not linear, pointing to smaller (or vanishing) excited-state dipole moments in nonpolar solvents. Examples are known in the literature of octupolar dyes showing a linear solvatochromic fluorescence behavior3,9,22 as well as of dyes with distinctively nonlinear Lippert-Mataga plots.20,33 The observation of emission from species with two different dipole moments in nonpolar and polar solvents might suggest the presence of two different emissive species whose relative concentration is governed by the solvent polarity. This anomalous fluorescence behavior, often ascribed to twisted intramolecular charge-transfer (TICT) phenomena, has been excluded for a family of amino-substituted triphenyl-benzene dyes, based on detailed measurements of fluorescence lifetimes and on a careful analysis of side-group effects.33 Our model provides a natural rationalization of nonlinear Lippert-Mataga plots: the dipolar character of the emitting excited state depends on the solvent polarity due to symmetry-breaking phenomena driven by polar solvation. In this framework, there is no need to introduce any special twist coordinate to break the conjugation in the dye. The proposed model describes not only linear but also nonlinear spectra. Octupolar chromophores are intensively investigated for their high TPA cross sections, and increasing the number of branches is a well-established strategy for amplified TPA.2,9 Comparing the behavior of a dipolar DA chromophore with the corresponding octupolar AD3 or DA3 dye is very attractive theoretically but is in practice a somewhat delicate task. First of all, the way to cut a DA branch out of an octupolar dye is far from unique. Moreover, the set of microscopic parameters defining the essential-state model for the dipolar chromophore is not strictly transferable to octupolar molecules. Just as an example, when going from DA to DA3, one expects that 2η, the energy difference between the neutral and charge-separated states, increases as a result of the decreased ionization potential of the donor group in the octupolar dye, where it is chemically connected to three acceptor groups. Based on this simple idea, we undertook the analysis of the DA branch (1b) relevant to the octupolar dye 3b discussed above.27 We kept all molecular parameters fixed to the values obtained from the analysis of the octupolar counterpart, slightly increasing η from 1.30 to 1.44 eV to best reproduce absorption and fluorescence spectra of the polar DA compound. Calculated linear spectra for 1b compare very favorably to experimental spectra (cf. panels c and d of Figure 5), giving confidence to the essential-state models for dipolar and octupolar chromophores and to the possibility to export many molecular parameters from one class of molecules to the other. The molecular parameters relevant to 1b are used to calculate its TPA cross section, reported in Figure 6, where it is compared to the response of 3b. The comparison with 3b is very good for frequencies and band shapes. A qualitative agreement is also observed for cross sections. However, the ratio between the intensity of the octupolar and dipolar species is underestimated in the calculated spectra. In the proposed model, and for the specific dye, the amplification of the TPA intensity (per branch)

Terenziani et al. amounts to a factor not larger than 2, whereas a factor on the order of 4 is experimentally observed. We notice in this context that state-of-the-art quantum chemical calculations based on the time-dependent density functional technique similarly underestimate the amplification of the TPA intensity for the same compounds.9,27 A popular model for the interpretation of linear and nonlinear spectra of branched chromophores is based on an excitonic description of the low-lying excitations.2 The excitonic approach relies on the joint analysis of absorption spectra of dipolar and octupolar dyes. The excitation on a single branch has energy ω0, as obtained from the spectrum of the dipolar chromophore. In the octupolar dye, excitonic basis states are obtained by a single excitation of each branch. The three states are mixed up by an interaction term (the excitonic coupling, V) so that they split into a couple of degenerate excitations and into a third nondegenerate excited state at higher energy. V can be estimated as the energy difference between the absorption frequency measured for the dipole and for the octupole, and it should correspond to 1/3 of the energy difference between states e and c1/2 in the octupole. The excitonic model leads to a qualitatively correct order of the excited states for octupolar chromophores, and it may even lead to quantitative or semiquantitative estimates for other properties.2 However, the excitonic approximation applies, strictly speaking, to excited states that only interact via electrostatic forces, a hypothesis that hardly holds for excitations on DA branches that share a D or an A group in octupolar DA3 or AD3 dyes. The splitting between excited states stems from electrostatic interactions in the exciton model, whereas it is due to the delocalization of electronic charges in the essential-state charge-resonance model. Of course, the applicability of the exciton model becomes poorer as F f 0.5, that is, as the charge is more delocalized across the three branches. On the other hand, the exciton model in its application to octupolar dyes (or more generally to multibranched dyes) does not lend itself to be extended to account for electron-vibration interaction and/or for solvation effects. These interactions are instead easily managed in the essential-state model, and, as discussed above, they are very important to obtain a complete spectroscopic characterization of multipolar dyes. EA spectra of chromophores in solution offer important information on the molecular properties. Limiting attention to the electronic problem, we have obtained an analytic expression for the EA spectrum of octupolar chromophores that results in a combination of the linear absorption spectrum, its first and second derivatives, plus a contribution from the field-induced transition toward the dark e state. The expression for the EA spectrum is similar to that originally derived by Liptay14 and usually adopted to estimate molecular properties from EA spectra.31 The main qualitative difference with respect to the classical Liptay expression is the appearance of a term due to the field-activated g f e transition. This transition, forbidden in absence of a field, becomes allowed when a static field is applied, because the field itself breaks the global symmetry of the system. The importance of this term depends on the molecular parameters. For chromophores with low F values, the field-induced band gives a contribution to the EA signal comparable to that related to the linear spectrum and its derivatives. Moreover, the relevant spectral feature is partly superimposed to the main EA feature, hindering the standard analysis of the EA signal. At increasing F values, the fieldinduced band moves away from the main EA feature, and its intensity lowers, making the standard analysis of the EA signal more reliable.

Symmetry Breaking in Octupolar Chromophores The perturbative expression in eqs 11 and 12 has been derived for the purely electronic problem. Its extension to include vibrational and solvation degrees of freedom is not straightforward. This limitation is not specific to octupoles in particular, and it can hinder the analysis of EA spectra of dipolar and quadrupolar dyes as well. In this respect, the essential-state model presented here for octupolar dyes, and already discussed in the literature for dipolar and quadrupolar dyes, allows one to treat on the same footing electronic, vibrational, and solvation degrees of freedom, offering a very powerful tool to discuss linear absorption and emission spectra and nonlinear spectra (with TPA and EA spectra being interesting examples) in the same theoretical framework to build a reliable model for the molecular unit itself. Complex phenomena, including false and true symmetry breaking in the excited state, are quite naturally accommodated in the model, leading to a true understanding of the basic physics of the system at hand. Even more importantly, the molecular model, including its basic parameters, is independent of the local environment and can be transferred to build models for molecular materials based on the same molecular units, opening the way to the bottom-up modeling of molecular materials.22,34 Acknowledgment. The authors thank Mireille BlanchardDesce (Rennes 1 University, France) for making data on chromophores 1b and 3b available. The work was supported by MIUR (Italian Ministry for University and Research) through Grant PRIN2006-031511. References and Notes (1) Joshi, M. P.; Swiatkiewicz, J.; Xu, F.; Prasad, P. N.; Reinhardt, B. A.; Kannan, R. Opt. Lett. 1998, 23, 1742. Chung, S.-J.; Kim, K.-S.; Lin, T.-C.; He, G. S.; Swiatkiewicz, J.; Prasad, P. N. J. Phys. Chem. B 1999, 103, 10741. Cho, B. R.; Lee, S. J.; Lee, S. H.; Son, K. H.; Kim, Y. H.; Doo, J.-Y.; Lee, G. J.; Kang, T. I.; Lee, Y. K.; Cho, M.; Jeon, S.-J. Chem. Mater. 2001, 13, 1438. Mongin, O.; Brunel, J.; Porre`s, L.; Blanchard-Desce, M. Tetrahedron Lett. 2003, 44, 2813. Porre`s, L.; Mongin, O.; Katan, C.; Charlot, M.; Pons, T.; Mertz, J.; Blanchard-Desce, M. Org. Lett. 2004, 6, 47. (2) Beljonne, D.; Wenseleers, W.; Zojer, E.; Shuai, Z.; Vogel, H.; Pond, S. J. K.; Perry, J. W.; Marder, S. R.; Bre´das, J.-L. AdV. Funct. Mater. 2002, 12, 631. (3) Le Droumaguet, C.; Mongin, O.; Werts, M. H. V.; Blanchard-Desce, M. Chem. Commun. 2005, 2802. (4) Zyss, J.; Ledoux, I. Chem. ReV. 1994, 94, 77. Le Floc’h, V.; Brasselet, S.; Zyss, J.; Cho, B. R.; Lee, S. H.; Jeon, S.-J.; Cho, M.; Min, K. S.; Suh, M. P. AdV. Mater. 2005, 17, 196. (5) Bhaskar, A.; Ramakrishna, G.; Twieg, R. J.; Goodson, T., III. J. Phys. Chem. C 2007, 111, 14607. (6) Varnavski, O. P.; Ostrowski, J. C.; Sukhomlinova, L.; Twieg, R. J.; Bazan, G., C.; Goodson, T., III. J. Am. Chem. Soc. 2002, 124, 1736. Varnavski, O.; Yan, X.; Mongin, O.; Blanchard-Desce, M.; Goodson, T., III. J. Phys. Chem. C 2007, 111, 149. (7) Bangal, P. R.; Lam, D. M. K.; Peteanu, L. A.; Van der Auweraer, M. J. Phys. Chem. B 2004, 108, 16834. (8) Stampor, W.; Mro´z, W. Chem. Phys. 2007, 331, 261. (9) Katan, C.; Terenziani, F.; Mongin, O.; Werts, M. H. V.; Porre`s, L.; Pons, T.; Mertz, J.; Tretiak, S.; Blanchard-Desce, M. J. Phys. Chem. A 2005, 109, 3024. (10) Oudar, J. L.; Chemla, D. S. J. Chem. Phys. 1977, 66, 2664. Marder, S. R.; Beratan, D. N.; Cheng, L. T. Science 1991, 252, 103. Lu, D.; Che, G.; Perry, J. W.; Goddard, W. A., III. J. Am. Chem. Soc. 1994, 116, 10679. Barzoukas, M.; Runser, C.; Fort, A.; Blanchard-Desce, M. Chem. Phys.

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