Essential State Models for Solvatochromism in DonorAcceptor

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J. Phys. Chem. B 2009, 113, 4718–4725

Essential State Models for Solvatochromism in Donor-Acceptor Molecules: The Role of the Bridge Luca Grisanti,† Gabriele D’Avino,† Anna Painelli,*,† Judith Guasch,‡ Imma Ratera,‡ and Jaume Veciana‡ Dipartimento Chimica GIAF, Parma UniVersity, and INSTM-UdR Parma, I-43100 Parma, Italy, and Institut de Cie´ncia de Materials de Barcelona (CSIC)/CIBER-BBN, Campus UniV. Bellaterra, 08193-Cerdanyola, Barcelona, Spain ReceiVed: NoVember 5, 2008; ReVised Manuscript ReceiVed: January 19, 2009

Essential state models are presented to discuss absorption spectra of two related donor-acceptor (DA) chromophores that show two solvatochromic bands in the near-infrared spectral region. The two-state model only accounts for the lowest energy band and results in a very small value of µ0, the dipole moment associated with the D+A- state. The model is then extended to account for the active role of the bridge: the resulting three-state model satisfactorily reproduces the double solvatochromism, leading at the same time to a roughly doubled estimate of µ0. This result, supported by a detailed analysis of an N-state model that explicitly accounts for bridge states, rationalizes the well-known discrepancy between the geometrical DA distance and the dipole length extracted from the analysis of optical spectra of DA chromophores as reflecting the active role of bridge states, not explicitly accounted for in essential state models. I. Introduction

τ≈

Donor-acceptor (DA) molecules offer a unique opportunity to investigate the physics of electron transfer, a key process in chemistry with important implications in several fundamental phenomena ranging from photosynthesis to vision. In DA molecules, the lowest energy excitation involves an electron transfer from the D to the A site (or vice versa), and optical spectroscopy is a powerful tool to investigate ET.1-5 Current understanding of ET in DA molecules is rooted in a two-state picture proposed in the 1950s by Mulliken6 to explain the appearance of so-called charge-transfer (CT) absorption bands in DA complexes. The model is simple: a neutral, |DA〉, and an ionic state, |D+A-〉, separated by an energy difference 2z are mixed by a matrix element -τ to give a ground and an excited state, g and e, respectively. The ionic state has a large permanent dipole moment, µ0 ) eR, where R is the D-A distance, and as a result of the mixing between the two states, the g f e transition acquires a sizable intensity, with a transition dipole moment µeg ) µ0(F(1 - F))1/2, where F measures the weight of the |D+A-〉 state into the ground state. Most DA complexes are weakly bound, τ , z, and linear perturbation on τ/z applies. In this approximation, the optical transition energy

pωeg ≈ 2z +

2τ2 ≈ 2z z

(1)

gives direct information on the ET gap, while the squared transition dipole moment, µeg (estimated from the area of the absorption band) is related to the hybridization energy: * Corresponding author. Mailing address: Dip. Chimica GIAF & INSTM UdR-Parma, Parco delle Scienze 17/A, 43100 Parma, Italy. E-mail: [email protected]. † Parma University. ‡ Institut de Cie´ncia de Materials de Barcelona.

2zµeg µ0

(2)

Equations 1 and 2 have been extensively and successfully adopted to describe CT complexes in solution,6-10 and they also proved useful to understand the basic features of optical absorption spectra in charge-transfer crystals,11 where µ0 is safely estimated from the crystallographic DA distance.12 In CT complexes and salts, ET occurs through space, and τ is related to the overlap of the frontier orbitals of the D and A molecules. However the Mulliken model applies quite irrespective of the origin of τ. It was recognized early that the Mulliken model also describes the basic physics of mixed valence systems: the so-called Mulliken-Hush model4 and its extensions to account for the role of solvation or for molecular vibrations proved very successful.1,2,13 More recently, Mulliken and related models have been rediscovered in relation to D-π-A chromophores, an interesting class of molecules where two molecular fragments with electron donating and accepting character are joined by a π-conjugated bridge. These molecules, also called push-pull chromophores, are of interest for nonlinear optical applications,14 and two-state approaches, based on the Mulliken model, are popular in this context.15,16 It should be recognized however that, particularly for push-pull chromophores, where the D and A units are bound by short π-bridges, the weakhybridization limit does not necessarily hold and perturbative expressions 1 and 2 must be considered with care.16-19 Moreover, the estimate of µ0 becomes delicate in bridged systems: charges are delocalized in the molecular fragments and the definition of a crystallographic D-A distance becomes elusive. To overcome this problem, a nonperturbative expression has been introduced to estimate µ0 from spectroscopic data:4,19,20

10.1021/jp809771d CCC: $40.75  2009 American Chemical Society Published on Web 03/18/2009

Solvatochromism in D-A Molecules

µ0 ) √(∆µeg)2 + 4µeg2

J. Phys. Chem. B, Vol. 113, No. 14, 2009 4719

(3)

where µeg, the transition dipole moment, is estimated from the area below the charge-transfer transition and ∆µeg, the mesomeric dipole moment, that is, the difference between the ground- and excited-state dipole moments, is experimentally accessible from electroabsorption measurements21-25 or can be obtained from the analysis of the solvatochromism of absorption bands.21,26,27 Both approaches suffer from some limitation. Electroabsorption data can only be obtained for neutral molecules, soluble in nonpolar or weakly polar solvents, while the analysis of solvatochromic data requires sophisticated models to discriminate between the effects of solvation and molecular vibrations.1,13,28 In order to avoid limitations of experimental techniques, an approach was proposed that estimates µ0 from µeg and ∆µeg values obtained from quantum chemical calculations.19 Quite irrespective of the adopted approach, µ0 values obtained for bound D-A molecules (either mixed valence or push-pull chromophores) point to D-A distances that are smaller and often considerably smaller than the geometrical DA distance.1,17-19,25,28-30 This delicate point has been discussed with reference to mixed-valence systems and was ascribed to the polarization of the ligands.22 The behavior of a bridged system, either push-pull chromophores or mixed-valence compounds, contrasts strikingly with what is observed for CT complexes and salts, suggesting an active role of the bridge in the definition of the effective dipole length. The mixing between the metal and the bridging ligand orbitals29 was already suggested as a possible explanation for the observed discrepancy between geometrical and dipole lengths in mixed valence systems.23a,29 In a recent paper,31 Veciana et al. reported an extensive spectroscopic study of two DA dyes, Fc-PTM and (Me)9FcPTM. The top panels in Figure 1 show the relevant absorption spectra measured in several solvents. The main charge-transfer (D to A) band of both dyes is located in the near-IR region (around 11 000 and 6000 cm-1 for Fc-PTM and Me9Fc-PTM, respectively) and shows the typical solvatochromic behavior of DA dyes with a largely neutral ground state. However, additional absorption features at higher energy (around 17 000 and 15 000 cm-1) are observed for both compounds, in a spectral region that, particularly for (Me)9Fc-PTM, is well separated from other (localized) absorption bands. These absorptions are assigned to an electron transfer from the π-bridge to the PTM acceptor unit (π to A band).31 Their solvatochromism then offers an interesting opportunity to investigate the role of bridge states in DA chromophores. The paper is organized as follows: the next section summarizes our nonperturbative approach to absorption spectra of DA chromophores and applies it to describe the near-IR absorption band of Fc-PTM and (Me)9Fc-PTM. Relevant model parameters, and particularly the resulting µ0 value, are critically discussed. Section III introduces a three-state model that also describes the higher energy π-bridge to A band. The three-state model leads to a different parametrization, with considerably larger µ0 values, confirming the hypothesis that bridge states may be responsible for the discrepancy between the geometrical dipole length in DA chromophores and the corresponding spectroscopic estimate. To further support this view and to set the physics of essential state models on a firmer basis, section IV discusses a general model for a D-π-A chromophore where the active role of bridge states is explicitly investigated. By tuning the model parameters, the system is driven from a region where bridge states play an important role toward the limit of

Figure 1. Experimental (top panels, from ref 31) and calculated spectra for Fc-PTM (left column) and Me9Fc-PTM (right column). Central panels show spectra calculated in the two-state model with molecular parameters in Table 1 and the εor values in the legend. Bottom panels show spectra calculated in the three-state model with molecular parameters in Table 2 and the ε˜ or values in the legend. The intrinsic bandwidth is set to σ ) 0.07 eV in all calculated spectra.

TABLE 1: Molecular Parameters for Fc-PTM and Me9Fc-PTM Described in the Two-State Model parameter

Fc-PTM

Me9Fc-PTM

z (eV) τ (eV) µ0 (D) εv (eV) ωv (eV)

0.61 0.35 7.5 0.10 0.18

0.36 0.30 8.5 0.12 0.18

TABLE 2: Molecular Parameters for Fc-PTM and Me9Fc-PTM Described in the Three-State Modela parameter

Fc-PTM

Me9Fc-PTM

z˜ x˜ τ˜ µ˜ 0 ε˜ v ω ˜v

0.78 0.87 0.47 15.0 0.06 0.18

0.50 0.69 0.47 15.5 0.07 0.18

a All parameters are in electronvolts, except for µ0, which is in debye.

a pure two-state model: studying the evolution of an essential two-state model that mimics the complete model, we gain information on the role of bridge states in the definition of model parameters as well as on the applicability range or on the limitations of essential state models for D-π-A chromophores. II. Near-IR Absorption of Fc-PTM and Me9Fc-PTM: A Two-State Model Here we focus attention on the main charge-transfer (D to A) band of the two dyes of interest. The minimal model to understand the solvatochromism of the CT band requires two electronic basis states coupled to an effective solvation coordinate that accounts for the orientational motion of polar solvent molecules around a polar solute. However, as originally suggested by Jortner,1,13 in order to properly reproduce spectral band shapes, the model must account at least for a high-

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frequency vibrational mode. The standard analysis of CT absorption bands is then based on a set of parameters that include the two electronic parameters, z and τ, introduced in the previous section, the vibrational relaxation energy, λv, and the corresponding vibrational frequency, ωv, plus the solvent relaxation energy, λs (the corresponding frequency is irrelevant since polar solvation dynamics is described by a classical coordinate).1,13,20 In the weak hybridization limit, τ , z, 2z is set equal to the vertical transition energy (see eq 1), so the 0-0 transition energy, often referred to as ∆G00, is

∆G00 ≈ pωeg - λs - λv

(4)

where λs, λv, and ωv are estimated from a joint analysis of the solvatochromism and of the band shapes. All parameters are treated as solvent dependent, so a set of five (mutually independent) parameters is extracted from the analysis of the absorption spectrum in each solvent. In recent years, we have proposed an alternative approach to optical spectra of organic D-π-A chromophores16-18,28 (push-pull chromophores) that describes exactly the same physics as the standard Marcus-Hush and Jortner models, with two electronic states coupled to a classical solvation coordinate and to a high-frequency vibrational coordinate. At variance with the standard approach, however, we adopt a fully nonperturbative treatment so that we can reliably discuss systems with intermediate and strong hybridization. Nonperturbative treatments of vibronic models have already been discussed in the context of mixed valence systems,32,33 here, based on microscopic considerations and on a rigorous distinction between diabatic and adiabatic states, we are able to identify a set of solVent-independent molecular parameters, considerably reducing the number of free parameters needed to describe the solvatochromism of optical spectra of D-π-A chromophores. The model has already been discussed in the literature;16-18,28 here we shortly summarize it for the sake of clarity. As in the Mulliken-Hush model, the electronic structure of a D-π-A chromophore is defined on the basis of the two basis (diabatic) states, |DA〉 and |D+A-〉, the relevant Hamiltonian being

Hel )

(

0 -τ -τ 2z

)

(5)

The diagonalization of the Hel gives closed expression for the ground and excited states:

|g〉 ) √1 - F|DA〉 + √F|D+A-〉 |e〉 ) - √F|DA〉 + √1 - F|D+A-〉

(6a) (6b)

ωeg ) τ(F(1 - F))-1/2

(8)

|µeg | 2 ) µ02(F(1 - F))

(9)

∆µeg ) µ0(1 - 2F)

(10)

µg ) Fµ0

(11)

where µg is the ground-state dipole moment. To account for the relaxation of the molecular geometry that accompanies the CT transition, we introduce an effective adiabatic vibrational coordinate Q. As sketched in the left panel of Figure 2, harmonic potential energy surfaces (PESs) with same frequency, ωv, but with different equilibrium geometries are assigned to the two basis states. The resulting vibrational relaxation energy, εv, sometimes called the small polaron binding energy, is defined in left panel of Figure 2. In this approximation, the adiabatic Hamiltonian reads

H(Q) )

(

)

0 -τ 1 + ωvQ2 -τ 2z + ωv√2εvQ 2

(12)

The diagonalization of the above Q-dependent Hamiltonian gives Q-dependent g and e states, that is, Q-dependent F in eq 6. The corresponding Q-dependent energies define the groundand excited-state PESs, shown in the right panel of Figure 2. The adiabatic PESs in the right panel of Figure 2 (and not the diabatic PES in the left panel) in principle contain all the information relevant to the calculation of optical spectra in a nonpolar solvent. Just as an example, the relative displacement along Q of the g and e PES (and not of the DA and D+Adiabatic PES) is responsible for the appearance of a vibronic structure in the absorption band. The actual calculation of the spectra is however nontrivial. The exact PESs (see Figure 2b) are anharmonic, and the calculation of vibrational eigenstates on an anharmonic PES is a tricky job. Even worse, in cases when several minima occur in the same PES (as is often the case for multibranchend chromophores34), the adiabatic approximation itself becomes dangerous and it is necessary to resort to numerical nonadiabatic approximation.34,35 However, for not too strongly anharmonic PESs, the linear absorption spectrum can be calculated in the local harmonic approximation, as follows:28,36

ε(ω) )

Lωegµeg2

e-S 60ln10pc2ε0 σ√2π

[

exp -

n

∑ √Sn!

×

n

(ωeg - Sωv + nωv - ω)2 2σ2

]

(13)

where

F)

(

1 z 1- 2 2 (z + τ2)1/2

)

(7)

measures the fractional charge transferred from D to A in the ground state. Nonperturbative expressions for spectral properties can be given as a function of F:

where a Gaussian shape is assigned to each vibronic line with intrinsic linewidth σ. The Huang-Rhys factor S ) λv/(pωv) is related to the vibrational relaxation energy relevant to the g and e state, λv (cf Figure 2b) that represents just a fraction of the bare relaxation energy as defined for the two basis states:

λv ) εv(1 - 2F)2

(14)

Solvatochromism in D-A Molecules

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Figure 2. The diabatic PESs (left) relevant to the two basis states and the adiabatic PESs (right) relevant to the ground and excited states as obtained from the diagonalization of H(Q) in eq 12 with z ) 1 eV, τ ) 0.5 eV, ωv ) 0.2 eV, and εv ) 0.4 eV. In the left panel, εv marks the vibrational relaxation energy relevant to the basis states, while λv, in the right panel, shows the same quantity for the adiabatic states.

where F is the equilibrium value of the ionicity (i.e., that corresponding to the minimum of the ground-state PES in the right panel of Figure 2). The large redistribution of the electronic charge that accompanies the optical transition implies a coupling to solvation degrees of freedom. In the framework of the reaction field approximation, the solvent is described as an elastic polarizable medium that reacts to the presence of solute molecules exerting at the solute location an electric field proportional to the solute dipole moment. The electronic polarization of solvent molecules occurs on a very fast time scale: the electronic degrees of freedom of solvent molecules react instantaneously to the CT transition and can be accounted for by a renormalization of model parameters. In the simplest approximation, only the solvent refractive index enters the renormalization. In view of the marginal variation of the refractive index in common organic solvents, solvent-independent molecular parameters are safely assumed.16 The reorientation of the polar solvent molecules around a polar solute is a slow motion that can be accounted for by introducing an effective solvation coordinate, proportional to the orientational component of the reaction field, F, coupled to the electronic system in a similar way as Q. The total Hamiltonian, also accounting for polar solvation, reads:16,17

H(Q, F) )

(

)

0 -τ + -τ 2z + ωv√2εvQ - µ0F µ02 2 1 F (15) ωvQ2 + 2 4εor

The solvation relaxation energy, εor, can be related to the solvent dielectric constant and refractive index if microscopic models are adopted for the cavity occupied by the solute,16,26 but we treat εor as an adjustable parameter.16-18 The diagonalization of the Q- and F-dependent Hamiltonian gives Q- and F-dependent PESs for the g and e states, as needed for the calculation of optical spectra. The vibrational coordinate and (the orientational component of) the reaction field enter the Hamiltonian in a similar way, but their different dynamic suggests a different treatment. In particular, the orientational component of the reaction field, F, is treated as a classical coordinate: the Hamiltonian in eq 15 is diagonalized for several F values, obtaining for each F different ground- and excited-state PESs, similar to those shown in the right panel of Figure 2. Relevant optical spectra are then calculated according to eq 13, and the total spectrum is obtained

as the sum of spectra calculated at different F, weighted according to the Boltzman law.17 As expected on a physical basis, polar solvation is responsible for the appearance of inhomogeneous broadening that smears out the vibronic structure.17 The local harmonic approximation for optical spectra leads to an expression for the absorption spectrum (eq 13) formally equivalent with expressions currently in use. In particular, λv is the vibrational relaxation energy that defines the Huang-Rhys factors in the linear absorption spectrum and corresponds exactly to the same parameter that enters the Marcus-Hush or Jortner models. However in these models, λv represents a freely adjustable parameter in each solvent, whereas in our model its solvent dependence is embedded in the F-dependence of F, the ground-state ionicity calculated at the bottom of the groundstate PES for each F value. Our model then predicts solventdependent Franck-Condon structures. At the same time, in polar solvents the spectrum is calculated by summing up several spectra calculated according to eq 13, with different frequencies, ωeg, transition dipole moments, µeg, and relaxation energies, λv: the resulting inhomogeneous broadening of the spectrum leads to more subtle spectroscopic effects than those modeled in terms of a renormalized σ in eq 13. Spectra calculated for Fc-PTM and Me9Fc-PTM are reported in the middle panels of Figure 1. The spectra have been obtained for the molecular parameters listed in Table 1, while adjusting εor for each solvent as shown in Figure 1. Both compounds present a largely neutral ground state, as confirmed by the positive solvatochromism typical of neutral dyes: we estimate that F increases from 0.068 in C6H12 to 0.077 in DMSO for Fc-PTM and from 0.103 in C6H12 to 0.132 in ArNO2 for Me9FcPTM. The evolution with the solvent polarity of the main CT band is well reproduced, a particularly impressive result because for each compound just a single parameter, εor, is allowed to change with the solvent. Adopting solvent-independent molecular parameters largely reduces the number of free parameters with respect to the standard treatment based on the Marcus-Hush approach. Specifically, in the standard treatment five solvent-dependent parameters (the optical frequency, the transition dipole moment, the vibrational frequency, and relaxation energy, plus the width of the Gaussian lineshapes) are required to calculate the absorption spectrum in each solvent: 20 parameters are then needed to reproduce the four spectra for each compound in the top panels of Figure 1. In our approach, instead the same spectra are calculated fixing six solvent-independent parameters (the five molecular parameters in Table 1 and the Gaussian line width σ in eq 13) plus a single parameter, εor, which is adjusted in each solvent, for a grand-total of 10 adjustable parameters. Despite the dramatic reduction of free parameters, the calculated spectra in the middle panel of Figure 2 satisfactorily compare with experimental data, confirming the validity of the model. Moreover, the global fit of optical spectra in solvents of different polarity allows for a reliable partitioning of the relaxation energy into a vibrational and a solvation contribution, a delicate issue in the standard treatment where the spectra measured in each solvent are analyzed separately. In particular, we notice that for both compounds εor vanishes in the nonpolar solvent and smoothly increases with the solvent polarity. The smaller εor values estimated for the methylated compound are in line with a larger cavity required to accommodate the bulkier solute.16 The lower ionization potential of the methylated-Fc units implies a lower z value for Me9Fc-PTM with respect to Fc-PTM (and hence a lower transition frequency) while other parameters are

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similar in both compounds. The spectra do not show a resolved vibronic structure, hindering a precise estimate of the vibrational frequencies ωv and of the intrinsic linewidths σ, which are set to 0.18 and 0.07 eV, respectively, for both compounds. The parameter µ0 only enters the calculation in the definition of the oscillator strength of the CT transition (or equivalently, it fixes the absolute scale of calculated absorption spectra), while it is irrelevant for band shapes and frequencies. The µ0 values in Table 1 are set to reproduce the experimental extinction coefficients and correspond to dipole lengths of 1.56 and 1.77 Å for Fc-PTM and Me9Fc-PTM, respectively. These values are unreasonably small and point to the need to relax the two-state approximation.

Figure 3. Diabatic (left panels) and adiabatic (right panel) PES for the three-state model in eq 17 with z˜ ) 1 eV, x˜ ) 2 eV, τ˜ ) 0.5 eV, ω ˜ v ) 0.2 eV, ε˜ v ) 0.4 eV, and ε˜ or ) 0.

III. Three-State Model The shoulder around 15 000 cm-1 in the absorption spectrum of Me9Fc-PTM is safely assigned to a secondary CT absorption implying a π-bridge to A transition.31 This assignment is supported by the absence of any absorption feature in this spectral range in either the methylated-Fc unit or in the PTM unit and is further corroborated by the weak solvatochromism of this band.31 Similar features are observed for Fc-PTM around 15 000-17 000 cm-1, but in this case, the overlap with the localized absorption of the Fc unit makes the analysis delicate. To account for secondary CT bands in Me9Fc-PTM and FcPTM at least three resonating structures must be accounted for D-π-A T D+-π-A- T D-π+-A-, where the first two structures (corresponding to the neutral and charge-separated structures of the previous section) largely dominate over the third, higher energy state. The CT occurs through the bridge, so on the basis of the three states, |D-π-A〉, |D-π+-A-〉, and |D+-π-A-〉 the electronic Hamiltonian reads

H(3)

(

0 -τ˜ 0 ) -τ˜ 2x˜ -τ˜ ′ 0 -τ˜ ′ 2z˜

)

(16)

where a tilde identifies the three-state model parameters with respect to the two-state model parameters in the previous section. Specifically, 2x˜ and 2z˜ measure the energy of the states |D-π+-A-〉 and |D+-π-A-〉, respectively, having set to zero the energy of the neutral state. As discussed above, the CT state involving the bridge is higher in energy that the main CT state: x˜ > z˜. The two hopping integrals, τ˜ and τ˜ ′ describe the electron hopping from the bridge to the acceptor and from the donor to the bridge, respectively. Adding an electronic state to the model greatly increases the number of model parameters: four of them are required just for the electronic Hamiltonian in eq 16. A detailed parametrization of the three-state model is difficult, particularly in view of the large overlap of the second absorption band with higher energy absorptions in both compounds. In the absence of additional data (for example, photoinduced absorption spectra from the first excited state) we reduce the number of molecular parameters by arbitrarily imposing equal hopping integrals in eq 16 (τ˜ ) τ˜ ′). The dipole moment of the |D-π+-A-〉 state is a fraction of µ˜ 0, the dipole moment of |D+-π-A-〉: to avoid the proliferation of adjustable parameters, we set it to µ˜ 0/2; that is, we locate the centroid of positive charge on the bridge just halfway between the D and A centers. As in the two-state model, the coupling to an effective molecular vibration is introduced assigning harmonic PESs with same frequency but different equilibrium geometries to the basis (diabatic) states, as shown in the left panels of Figure 3. In

principle, different geometries are expected for each of the three diabatic states leading to two independent vibrational relaxation energies for |D-π+-A-〉and |D+-π-A-〉 states. However, getting reliable information on the vibrational coupling of the third state is difficult, and we impose the same geometry on the two charge-separated states (see Figure 3, left panel), so that the same relaxation energy, ε˜ v, applies to both states. Different choices are of course possible but do not alter the main results. Polar solvation is treated again in the framework of the reaction field model, with the reaction field proportional to the molecular dipole moment. Since the dipole moment of the third state is set to a fixed fraction of µ0, polar solvation is described by the single parameter ε˜ or that measures the solvation relaxation energy of the |D+-π-A-〉 state. The total Hamiltonian, accounting for both vibrational coupling and polar solvation, then reads

(

˜ (Q, F) ) H 0

-τ˜

)

0 µ˜ 0 -τ˜ 2x˜ + ω -τ˜ ′ ˜ v√2ε˜ vQ - F + 2 0 -τ˜ ′ 2z˜ + ω ˜ V√2ε˜ vQ - µ˜ 0F µ˜ 02 2 1 F (17) ω ˜ vQ2 + 2 4ε˜ or

where again the tilde marks symbols relevant to the three-state model. For each F, the diagonalization of the Q-dependent Hamiltonian leads to three Q-dependent eigenstates, that describe the PES relevant to the ground state, g, and to the first and second excited states, e1 and e2, respectively. Right panel of Figure 3 shows the results obtained for a specific set of parameters. These adiabatic curves define the relaxation energies, λv1 and λv2, relevant to the two excited states (cf Figure 3, right panel). Absorption spectra are calculated extending the summation in eq 13 to the n and n′ vibrational states of the first and second excited states, accounting for the relevant Huang-Rhys factors, Si ) λvi/(pωv), with i ) 1, 2. The calculation is repeated for different F, and the calculated spectra, Boltzmann-weighted on the total ground-state energy, are summed up to give the total absorption spectrum. The spectra calculated within the three-state model are reported in bottom panels of Figure 1. Spectra are obtained for the molecular parameters in Table 2 and the ε˜ or values in the captions. The vibrational frequencies and the intrinsic line width are set to the same values as in the two-state model. As expected, the three-state model results in two absorption bands, which

Solvatochromism in D-A Molecules reproduce the experimental observation of two solvatochromic CT absorption bands. A detailed comparison of calculated and experimental spectra is hindered, particularly for Fc-PTM, by the overlap of the secondary CT band with nearby localized absorption bands. For Me9Fc-PTM an improved fit can be obtained relaxing some of the arbitrary constraints imposed above. As an interesting example in the Supporting Information, we show the spectra calculated for Me9Fc-PTM treating both τ˜ and τ˜ ′ as freely adjustable parameters. These data demonstrate that it is certainly possible to fine-tune the model, increasing the number of adjustable parameters, but that the basic role of the bridge state is captured already in the simplest model. From Table 2, it turns out that x˜ > z˜ for both compounds, as required on physical basis. Moreover τ˜ , which measures the direct charge hopping from either the D or the A site to the bridge, is larger than the bridge-mediated hopping, τ, of the two-state model. The effective strength of the vibrational coupling is roughly halved in the three-state model, suggesting that the effective εv estimated in the two-state model is roughly the sum of the contributions from the two excited states. The values of the solvent relaxation energy as obtained in the three-state model are instead larger than the corresponding two-state model results (see Figure 1). This increase compensates for the reduction of the mesomeric dipole moment in the three-state dipole moment. The sizable weight of the |D-π+-A-〉 in the first excited state in fact leads to a decrease of the relevant dipole moment with respect to the two-state dipole moment so that larger solvent relaxation energies are required in the three-state model to reproduce the same solvatochromism. This is an interesting results: microscopic models in fact relate the εor value to the solvent dielectric constant and refractive index and to the size and shape of the cavity occupied by the solute.16,26 Our analysis demonstrates instead that εor is best treated as an adjustable parameter, whose specific value also depends on the model adopted to describe the electronic structure of the solute. More important for the aim of this paper is the observation that the effective dipole length extracted in the three-state model, µ˜ 0 in Table 2, is about twice the corresponding estimate in the two-state model, µ0 in Table 1, corresponding to a DA distance of 3.1 and 3.3 Å for Fc-PTM and Me9Fc-PTM, respectively. These values are still small compared with the geometrical D-A distance (the distance between the Fe atom and the central C atom in the PTM radical is 9.5 and 9.7 Å for Fc-PTM and Me9Fc-PTM, from crystallographic data), but they considerably improve over the corresponding estimates obtained in the twostate approach, leading to a ratio between the geometrical and spectroscopic estimate of the DA distance in line with similar results for other DA molecules.1,17-19,25,28-30 This result quite unambiguously demonstrates that the unphysically small DA distance extracted from the two-state model analysis of optical spectra for both compounds results from the strong effects of the low-lying secondary CT absorption in these systems. IV. Dipole Length: The Role of Bridge States The three-state model described above does not add much to our understanding of the main (low-energy) CT absorption band whose basic features are well captured by the two-state model.39 However, the three-state model leads to a more reasonable estimate of the dipole length compared with the unphysically small dipole length obtained for both Fc-PTM and Me9Fc-PTM within the two-state approximation. This result confirms, in a system where the role of a bridge-state is experimentally demonstrated by the observation of a secondary CT absorption, that the well-documented discrepancy between the effective dipole length and the geometrical D-A distance in bridged DA

J. Phys. Chem. B, Vol. 113, No. 14, 2009 4723

Figure 4. Schematic representation of the basis state for an N ) 5 molecule with corresponding energies and dipole moments.

systems can be solved if bridge states are explicitly accounted for. To generalize this result and to set it on a firmer basis, we discuss here an electronic model for a D-π-A dye where several bridge states are accounted for. The model has been recently introduced in the literature2,37 to describes the electron transfer from D to A as occurring via seVeral successive hops through bridge sites. As described in ref 2, we consider the bridge states as virtual states, which being higher in energy than the relevant CT state are not actually populated. We then limit attention to the purely electronic model.2 Specifically, an N-site molecule has N - 2 sites in the bridge, and the electron hops from D to A via N - 1 hops involving only adjacent sites. The resulting N states are schematically shown in Figure 4 for N ) 5. The same energy, 2x˜, is assigned to all bridge states, that is, to all the states with a charged bridge, while D+-π-A- has energy 2z˜ (see Figure 4). The same hopping integral, τ˜ , describes the charge transfer between all adjacent sites along the chain. The relevant Hamiltonian is a trivial extension of the three-state electronic model Hamiltonian in eq 16:

(

0 -τ˜ 0 0 -τ˜ 2x˜ -τ˜ 0 0 -τ˜ 2x˜ -τ˜ 0 -τ˜ 2x˜ HN ) 0 · ·. l l l 0 0 0 ··· 0 0 0 ···

··· ··· ··· · ·. ·

0 0 0 l

0 0 0 l

· . -τ˜ 0 -τ˜ 2x˜ -τ˜ 0 -τ˜ 2z˜

)

(18)

Consistent with the three-state model described in the previous section, the dipole moment of the D+-π-A- state is set to µ˜ 0, while the dipole moments of bridge states are fractions of µ˜ 0, as relevant to a system with equally spaced sites (see Figure 4), leading for the general case in eq 18 to the following values: 0, µ˜ 0/(N - 1), 2µ˜ 0/(N - 1), 3µ˜ 0/(N - 1), ..., (N - 2)µ˜ 0/(N - 1), µ˜ 0. We consider systems with an almost neutral ground state, in which bridge sites are weaker donors than D, so that 0 < z˜ < x˜. In the limit x˜ - z˜ . τ˜ bridge states become very high in energy and a perturbative treatment on τ˜ /(2(x˜ - z˜)) reduces the N-state model to an effective two-state model with z f z˜, µ0 f µ˜ 0, and τ f τ˜ N-1/(2x˜ - 2z˜)N-2 ≈ τ˜ N-1/(2x˜)N-2 (as before the tilde applies to symbols relevant to the N-state model, while bare symbols refer to the two-state model). The N-site (or equivalently N-state) molecules have N - 1 optical excitations whose energies and transition dipole moments are shown in Figure 5 as a function of the energy gap x˜ - z˜ for systems with N ) 3, 4, and 5. In all cases, model parameters

4724 J. Phys. Chem. B, Vol. 113, No. 14, 2009

Figure 5. Transition energies (top panels) and squared transition dipole moments (bottom panels) calculated for N ) 3, 4, and 5 state models (left, center, and right panels, respectively) with z˜ ) 1, µ˜ 0 ) 1, variable x˜, and τ˜ ) (τ(2x˜)N-2)1/(N-1), to converge, for large x˜, to a two-state model with z ) τ ) 1 and µ0 ) 1. Red lines refer to the lowest energy (main CT) transition. Dotted lines show the (x˜-independent) results relevant to the limiting two-state model.

have been chosen as to converge, in the large x˜ limit, to the two-state model with z ) 1, τ ) 1, and µ0 ) 1 (in this section, we work with dimensionless quantities, fixing τ as the energy unit and µ0 as the unit dipole moment). With this choice, we expect convergence, in the large x˜ limit, to a two-state model with F ≈ 0.15 (see eq 7). The corresponding limiting values of the transition frequency and squared transition dipole moments (ωeg ≈ 2.8 and µeg2 ≈ 0.13; see eqs 8 and 9) are shown as dotted lines in Figure 5. The lowest energy or main CT transition (marked by red lines in Figure 5) is well separated from higher energy transitions involving bridge states and has by far the largest intensity: the main CT transition dominates the lowenergy portion of the spectrum. The corresponding energies and transition dipole moments (red lines in Figure 5) do properly converge toward the two-state limit (dotted lines) for x˜ - z˜ f ∞ or equivalently for x˜ f ∞; the convergence is rather slow and particularly so for large N. To investigate the effect of bridge states in the definition of an effective two-state model and in particular on the estimate of the relevant dipole length, we focus attention on this lowest energy main CT transition, disregarding higher energy excitations involving the bridge. In other terms, we analyze the data relevant to the lowest energy transition as obtained from the diagonalization of the N-state model to extract an effective two-state model, in a similar way as usually done by analyzing experimental absorption spectra. In particular, the parameters of the effective two-state model, τ, z (or equivalently F, see eq 7), and µ0, can be estimated from three spectral properties. Following the standard procedure,4,19,20 eqs 8-11 can be used to extract F, τ, and µ0 out of the ωeg, µeg, and ∆µeg values calculated in the N-state model. The right panel of Figure 6 shows the (x˜ - z˜) dependence of the effective µ0 estimated along these lines for the N ) 3, 4, and 5 models in Figure 4. In all cases, the effective µ0 converges toward the exact limit, µ0 ) 1, for x˜ f ∞, but it is always underestimated for any finite x˜, the deviation increasing with N. Of course, the N-state model exactly converges to the two-state limiting model only for τ˜ /(2(x˜ - z˜)) f 0, and away from this limit, different estimates of the effective model parameters can be obtained if a different choice is made about the reference spectral properties. Just as an example, with eqs 8, 9, and 11, the effective two-state model parameters, τ, F, and µ0, can be extracted out of the transition frequency and dipole moment (ωeg, µeg) and the permanent dipole moment (µg), calculated in the N-state model. While the effective µ0 obtained according to this alternative procedure (right panel of Figure 5) shows a qualitatively similar behavior

Grisanti et al.

Figure 6. Dipole length extracted from the analysis of the main CT band for the same systems described in Figure 5. Left panel shows results obtained by extracting the two-state model parameters (τ, z, and µ0) from the frequency (ωeg), the squared transition dipole moment (µeg2), and the mesomeric dipole moment (∆µeg) calculated for the lowest energy transition in the N-state models; right panel shows similar results obtained using the permanent dipole moment instead of the mesomeric dipole moment.

to the previous one, the deviations from the limiting value are even larger than before. V. Discussion and Conclusions Essential state models offer a simplified description of the low-energy states of D-A molecules that helps to understand and predict the behavior of these interesting molecules. We have recently shown that a nonperturbatiVe treatment of a two-state model that describes the same physics as the well-known Marcus-Hush1-4 and Jortner models13 proves particularly useful in this context.16-18,27 The joint analysis of optical spectra collected in different solvents leads in fact to the definition of a reliable set of enVironment-independent molecular parameters, while the effect of polar solvation on the transition frequencies and bandshapes is reproduced tuning just a single parameter, the solvent relaxation energy. In comparison with the standard analysis,4,19,20 our procedure leads to a large reduction of the number of free parameters and hence to a more reliable parametrization. What is even more important, however is that the resulting set of molecular parameters can be adopted to describe the same molecular unit in different systems, setting the basis for the bottom-up modeling of molecular materials.18b,38 Very recently, this strategy has been applied to Fc-PTM, leading to the rationalization of the bistability observed in Fc-PTM crystals in terms of cooperative effects induced by intermolecular electrostatic interactions.39 The lowest energy, main CT absorption is observed in FcPTM and (Me)9Fc-PTM at very low energies, in the nearinfrared spectral region. Other absorption features are found at higher energy, still below the excitations localized in the ferrocene or PTM units, and are reliably assigned to CT processes involving bridge states.31 Here we have extended the two-state model for DA chromophores to include a third essential state involving a CT from the π-bridge to the PTM unit. This model reproduces well the solvatochromism of both the main CT band and the higher energy features. The threestate model results in a larger set of molecular parameters and, apart from the possibility to extend the analysis of optical spectra up to higher energy, does not add much to the description of the main CT process. It however sheds light on the empirical nature of the two-state model parameters: the direct electron hopping between the D and A sites described by τ in the twostate model represents a simplified picture of a bridge-mediated hopping as described in the three-state model. Indeed, the threestate model itself represents a highly simplified picture of the molecular system. Exactly as it occurs for the most famous empirical parameters β in the Huckel model or ∆q in the crystal field model, the

Solvatochromism in D-A Molecules parameters of essential state models are difficult to estimate a priori. The molecular dipole length, µ0, however has an immediate experimental counterpart, being related to the D-A distance, and it is quite natural to compare the estimated dipole lengths with crystallographic distances. As is well established in the literature1,17-19,25,28-30 this comparison is quite disappointing, with too small µ0 values. The two-state analysis of optical spectra of Fc-PTM and (Me)9Fc-PTM leads to unphysically small µ0 values, pointing to DA distances less than 2 Å. This uncomfortable result is partly cured in the three-state model, which leads to DA distances twice as large. While µ0 estimates obtained in the three-state model are still small compared with the geometrical DA distance, they are more in line with typical estimates in other DA molecules.1,17-19,25,28-30 A detailed study of N-state models for DA chromophores that explicitly account for bridge states demonstrates that the well-documented discrepancy between the geometrical DA distance and the dipole length estimated from the analysis of optical spectra of DA molecules in solution can be rationalized as reflecting the active role of (virtual) bridge states not accounted for in the essential state model. Acknowledgment. This work was supported by NE MAGMANET Grant NMP3-CT2005-515767, and by MIUR through Grant PRIN2006031511. Supporting Information Available: Calculated spectra and molecular parameters for Me9Fc-PTM in the three-state model with adjustable τ and τ′. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Bixon, M.; Jortner, J. AdV. Chem. Phys. 1999, 106, 35–202. (2) Nitzan, A. Chemical Dynamics in Condensed Phases; Oxford University Press: New York, 2006. (3) Marcus, R. A. ReV. Mod. Phys. 1993, 65, 599–610. (4) Hush, N. S. Prog. Inorg. Chem. 1967, 8, 391. Hush, N. S. Coord. Chem. ReV. 1985, 64, 135. Hush, N. S.; Reimers, J. R. Chem. ReV. 2000, 100, 775. (5) Myers Kelley, A. B. J. Phys. Chem. A 1999, 103, 6891. (6) Mulliken, R. S. J. Am. Chem. Soc. 1952, 74, 811–824. (7) Mulliken, R. S.; Person, W. B. Molecular Complexes; Wiley: New York, 1969. (8) Friedrich, H. B.; Person, W. B. J. Chem. Phys. 1966, 44, 2161– 2170. (9) Foster, R. Nature 1958, 181, 337. (10) Markel, F.; Ferris, N. S.; Gould, I. R.; Myers, A. B. J. Am. Chem. Soc. 1992, 114, 6208–6219. (11) Herbstein, F. H. In PerspectiVe in Structural Chemistry; Dunitz, J. P., Ibers, J. A., Eds.; Wiley: New York, 1971, Vol. IV, p 166. Soos, Z. G.; Klein, D. J. In Molecular Associations; Foster, R., Ed.; Academic Press: New York, 1971, Vol 1, p 1. (12) Tanaka, M. Bull. Chem. Soc. Jpn. 1977, 50, 2881. Jacobsen, C. S.; Torrance, J. B. J. Chem. Phys. 1983, 78, 112. Akhtar, S.; Tanaka, J.; Nakasuji, K.; Murata, I. Bull. Chem. Soc. Jpn. 1985, 58, 2279. Yakushi, K.; Kuroda, H. Chem. Phys. Lett. 1984, 111, 165. Painelli, A.; Girlando, A. J. Chem. Phys. 1986, 84, 5655. Painelli, A.; Girlando, A. J. Chem. Phys. 1987, 87, 1705. Pecile, C.; Painelli, A.; Girlando, A. Mol. Cryst. Liq. Cryst. 1989, 171, 69–87. Brillante, A.; Philpott, M. R. J. Chem. Phys. 1980, 72, 4019.

J. Phys. Chem. B, Vol. 113, No. 14, 2009 4725 (13) Jortner, J.; Ulstrup, J. J. Chem. Phys. 1975, 63, 4358. Jortner, J.; Bixon, M. J. Chem. Phys. 1988, 88, 4167. Cortes, J.; Heitele, H.; Jortner, J. J. Phys. Chem. 1994, 98, 2527. (14) Kanis, D. R.; Ratner, M. A.; Marks, T. J. Chem. ReV. 1994, 94, 195. Marder, S. R.; Kippelen, B.; Jen, A. K.-Y.; Peyghambarian, N. Nature 1997, 388, 845. Bre´das. J. L.; Cornil, K.; Meyers, F.; Beljonne, D. In Handbook of Conducting Polymers, 2nd ed.; Skotheim, T. A., Elsenbaumer, R. L., Reynolds, J. R., Eds.; Marcel Dekker: New York, 1998; p 1. (15) 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. (c) Lu, D.; Chen, G.; Perry, J. W.; Goddard, W. A., III J. Am. Chem. Soc. 1994, 116, 10679. Thompson, W. H.; Blanchard-Desce, M.; Alain, V.; Muller, J.; Fort, A.; Barzoukas, M.; Hynes, J. T. J. Phys. Chem. A 1999, 103, 3766–3771. (16) Painelli, A. Chem. Phys. Lett. 1998, 285, 352. Painelli, A. Chem. Phys. 1999, 245, 185. (17) Painelli, A.; Terenziani, F. J. Phys. Chem. A 2000, 104, 11041– 11048. Terenziani, F.; Painelli, A.; Comoretto, D. J. Phys. Chem. A 2000, 104, 11049–11054. (18) (a) Boldrini, B.; Cavalli, E.; Painelli, A.; Terenziani, F. J. Phys. Chem. A 2002, 106, 6286–6294. (b) Terenziani, F.; Painelli, A.; Girlando, A.; Metzger, R. M. J. Phys. Chem. B 2004, 108, 10743–10750. (19) Cave, R. J.; Newton, M. D. Chem. Phys. Lett. 1996, 249, 15–19. Nelsen, S. F.; Newton, M. D. J. Phys. Chem. A 2000, 104, 10023–10031. (20) Heckmann, A.; Lambert, C.; Goebel, M.; Wortmann, R. Angew. Chem., Int. Ed. 2004, 43, 5851–5856. Heckmann, A.; Lambert, C. J. Am. Chem. Soc. 2007, 129, 5515. (21) Liptay, W. Angew. Chem., Int. Ed. Engl. 1969, 8, 177–188. LiptayW. In Excited States Lim, E. C., Ed.; Academic: New York, 1974; Vol. 1, pp 129-229. (22) Reimers, J. R.; Hush, N. S. J. Phys. Chem. 1991, 95, 9773. (23) (a) Oh, D. H.; Sano, M.; Boxer, S. G. J. Am. Chem. Soc. 1991, 113, 6880. (b) Bublitz, G. U.; Boxer, S. G. Annu. ReV. Phys. Chem. 1997, 48, 213. (24) Brunschwig, B. S.; Creutz, C.; Sutin, N. Coord. Chem. ReV. 1998, 177, 61–79. (25) Vance, F. W.; Karki, L.; Reigle, J. K.; Hupp, J. T.; Ratner, M. A. J. Phys. Chem. A 1998, 102, 8320–8324. (26) Reichardt, C. Chem. ReV. 1994, 94 (8), 2319–2358. (27) Paley, M. S.; Harris, J. M.; Looser, H.; Baumert, J. C.; Bjorklund, G. C.; Jundt, D.; Twieg, R. J. J. Org. Chem. 1989, 54, 3774. Lombardi, J. R. J. Phys. Chem. A 1998, 102, 2817. (28) Painelli, A.; Terenziani, F. Chem. Phys. Lett. 1999, 312, 211–220. (29) Zhang, L.-T.; Ondrechen, M. J. Inorg. Chem Acta 1994, 226, 43. (30) Coropceanu, V.; Andre´, J. M.; Malagoli, M.; Bre´das, J. L. Theor. Chem. Acc. 2003, 110, 59–69. Coropceanu, V.; Gruhn, N. E.; Barlow, S.; Lambert, C.; Durivage, J. C.; Bill, T. G.; Noll, G.; Marder, S. R.; Bredas, J.-L. J. Am. Chem. Soc. 2004, 126, 2727–2731. (31) Ratera, I.; Sporer, C.; Ruiz-Molina, D.; Ventosa, N.; Baggerman, J.; Brouwer, A. M.; Rovira, C.; Veciana, J. J. Am. Chem. Soc. 2007, 129, 6117–6129. (32) Piepho, S. B.; Krausz, E. R.; Schatz, P. N. J. Am. Chem. Soc. 1978, 100, 2996. (33) Reimers, J. R.; Hush, N. S. Chem. Phys. 2004, 299, 79. (34) Terenziani, F.; Painelli, A.; Katan, C.; Charlot, M.; BlanchardDesce, M. J. Am. Chem. Soc. 2006, 128, 15742–15755. (35) Del Freo, L.; Terenziani, F.; Painelli, A. J. Chem. Phys. 2002, 116, 755. (36) Del Freo, L.; Painelli, A. Chem. Phys. Lett. 2001, 338, 208–216. (37) Berlin, Y. A.; Grozema, F. C.; Siebbeles, L. D. A.; Ratner, M. A. J. Phys. Chem. C. 2008, 112, 10988–11000. (38) Painelli, A.; Terenziani, F. In Non-linear optical properties of matter: From molecule to condensed phases; Papadopoulos, M. G., Leszczynski, J., Sadlej, J. Eds.; Springer: Dordrecht, The Netherlands, 2006; pp 251-282. Painelli, A.; Terenziani, F.; Angiolini, L.; Benelli, T.; Giorgini, L. Chem.sEur. J. 2005, 11, 6053. Terenziani, F.; D’Avino, G.; Painelli, A. ChemPhysChem 2007, 8, 2433. (39) D’Avino, G.; Grisanti, L.; Guasch, J.; Ratera, I.; Veciana, J.; Painelli, A. J. Am. Chem. Soc. 2008, 130 (36), 12064–12072.

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