Solvent Effects on the Three-Photon Absorption of a Symmetric

J. Phys. Chem. B , 2008, 112 (15), pp 4703–4710 ... Publication Date (Web): March 21, 2008. Copyright ... The Journal of Chemical Physics 2008 128, ...
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J. Phys. Chem. B 2008, 112, 4703-4710

4703

Solvent Effects on the Three-Photon Absorption of a Symmetric Charge-Transfer Molecule Na Lin,†,| Lara Ferrighi,‡ Xian Zhao,| Kenneth Ruud,‡ Antonio Rizzo,§ and Yi Luo*,† Department of Theoretical Chemistry, School of Biotechnology, Royal Institute of Technology, SE-10691 Stockholm, Sweden, Centre for Theoretical and Computational Chemistry, Department of Chemistry, UniVersity of Tromsφ, N-9037 Tromsφ, Norway, Istituto per i Processi Chimico-Fisici del Consiglio Nazionale delle Ricerche (IPCF-CNR), Area della Ricerca di Pisa, Via G. Moruzzi 1, I-56124 Pisa, Italy, and Institute of Crystal Materials, Shandong UniVersity, 250100 Jinan, Shandong, People’s Republic of China ReceiVed: NoVember 9, 2007; In Final Form: January 19, 2008

We present a theoretical study of the solvent-induced three-photon absorption cross section of a highly conjugated fluorene derivative, performed using density functional (DFT) cubic response theory in combination with the polarizable continuum model. The applicability of the often used two-state model is examined by comparison against the full DFT response theory results. It is found that the simplified model performs poorly for the three-photon absorption properties of our symmetric charge-transfer molecule. The dielectric medium enhances the three-photon absorption cross section remarkably. The effects of solvent polarity and geometrical distortions have been carefully examined. A detailed comparison with experiment is presented.

1. Introduction Multiphoton absorption involves the excitation of a species by simultaneous absorption of several photons. Since the mid1990s, two-photon absorption (TPA) has gained considerable attention both in experimental and theoretical studies because of its various promising practical applications, ranging from three-dimensional optical storage, fluorescence excitation microscopy, and optical power limiting to up-converted lasing and photodynamic therapy.1 Higher-order processes, such as threephoton absorption (3PA), are far less popular because of the much more demanding experimental realizations. However, 3PA possesses characteristics that, for many applications, are even better suited than TPA, such as, for example, greater penetration depth and confocality, which can significantly improve the performance of various applications, such as optical limiting,2,3 fluorescence imaging,4-7 and stimulated emission.8 Enhancing the probability of 3PA in molecules may play an essential role for the future applications of 3PA technology. In this context, theoretical modeling may be an effective way of gaining insight into the molecular characteristics that govern high 3PA efficiency of molecular materials. So far, two approaches have been adopted to calculate the 3PA cross section of a molecule. One is to use cubic response (CR) theory,9,10 relating the fifth-order property to the residue of the third-order response function and taking into account the contributions of all excited states. Another approach is a simplified few-state model where the sum-over-states (SOS) expression of the property is truncated, and only a few dominating states and excitation channels are taken into account. Cronstrand et al.11 have studied in detail the convergence behavior, advantages, and shortcomings of the few-state models for 3PA. The authors concluded that, unlike for TPA, few-state models cannot routinely be employed for evaluating 3PA cross * Corresponding author. E-mail: [email protected]. † Royal Institute of Technology. ‡ University of Tromsφ. § Area della Ricerca di Pisa. | Shandong University.

sections of extended systems, though they may still serve as valuable tools for the interpretation of certain observed features. As for other linear and nonlinear optical properties of organic molecules, the understanding of solvent effects is of importance for the design of materials exhibiting strong 3PA rates. Strong solvent dependence has also been observed in the TPA process.12 Recently, Cohanoschi et al. measured 3PA cross sections of a highly conjugated organic dye in four different solvents,13 and found that one-photon absorption (OPA) spectra depend only slightly on the polarity of the solvent, whereas the 3PA cross section of the charge-transfer (CT) state increases as the polarity of the solvent decreases. However, these observations could not be satisfactorily explained by a simplified two-state model. In this paper, we employ a state-of-the-art computational method that combines the polarizable continuum model (PCM)14-19 and CR theory to study the solvent effect on the 3PA cross section of a conjugated symmetric CT molecule very similar to the one on which Cohanoschi et al. performed their analysis.13 The validity of the two-state model for the system under investigation is also carefully examined. In Section 2 we will give the relevant theoretical background. Computational details can be found in Section 3. The results are presented and discussed in Section 4, and the main conclusions are collected in Section 5. 2. Theory 2.1. One-Photon Absorption. The OPA cross section is often expressed through the related oscillator strength for the transition between the initial |0〉 and final |f〉 electronic states (atomic units):

δOP )

2ωf 3

∑a |〈0|µa|f〉|2

(1)

where Ef ) pωf is the transition energy from |0〉 to |f〉. µa denotes the a Cartesian component of the electric dipole operator. We will in the following use the notation µ0f a ) 〈0|µa|f〉 for the

10.1021/jp7107522 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/21/2008

4704 J. Phys. Chem. B, Vol. 112, No. 15, 2008 electric dipole transition moment between the ground state and excited state |f〉. 2.2. Three-Photon Absorption. Though being a fifth-order property, the 3PA cross section can be obtained from the thirdorder transition matrix elements (here given for three photons of identical circular frequency ω ) ωf/3):

Tabc )

〈0|µa|m〉〈m|µb|n〉〈n|µc|f〉

∑Pa,b,c ∑ n,m (ω

m

- ωf /3)(ωn - 2ωf /3)

(2)

where ∑Pa,b,c denotes the permutations with respect to the Cartesian indices a, b, and c. The 3PA cross section, σ3P, is related to the three-photon transition probability, δ3P:

σ3P )

4π4Ra08ω3 2 3

3c0 n

L6δ3P∆(ωf - 3ω,Γf)

(3)

where c0 is the speed of light in vacuo, R is the fine structure constant, a0 is the Bohr radius, and n is the refractive index of the medium. When employing a spherical cavity, the local field factor L can be expressed as a function of the optical dielectric constant opt:

L)

3opt 3n2 ) 2n2 + 1 2opt + 1

(4)

Γf is the broadening of the final state, and the lifetime broadening ∆(ωf - 3ω,Γf) for a 3PA process is usually described by a Lorentzian function

∆(ωf - 3ω,Γf) )

Γf 1 π (ω - 3ω)2 + Γ 2 f f

(5)

[ [

TTS zzz ) 27 ) 27

]

Lin et al.

00 ff 2 0f 3 2µ0f z (µz - µz ) - (µz )

2ωf2

]

2 0f 2 µ0f z [2(∆µz) - (µz ) ]

2ωf2

(10)

Under these assumptions, the total 3PA probability δTS 3P for linearly polarized light becomes TS δ3P

2 (TTS zzz) ) 7

(11)

2.3. PCM Formulation of CR Theory. The PCM formulation14,15,19 yields a representation of the electrostatic interaction of the solute molecule with the surrounding solvent where the solute is treated at the quantum mechanical level, whereas the solvent is modeled as a macroscopic dielectric continuum whose molecular structure is neglected, retaining only a few parameters. For a homogeneous and isotropic solvent, only the static () and optical (opt) dielectric constants are needed to describe the solvent reaction field. In the PCM, a molecular solute is placed in a molecule-shaped cavity, usually described by a set of interlocking spheres centered on the nuclei of the solute, and embedded in a macroscopic dielectric representing the solvent. The PCM has been combined with linear20 and quadratic21 response theory. Very recently, Ferrighi, Frediani, and Ruud22 have derived the theory for CR functions in the PCM formalism, and implemented the theory in the DALTON program package.23 Solvent effects for the degenerate four-wave mixing process on three different classes of heteroaromatic chromophores were analyzed.22 The reader is referred to ref 16 for a detailed discussion of the nonequilibrium formalism and to ref 22 for details on how the CR equations9,24 are modified to incorporate the PCM for the solvent effect. 3. Computational Details

The resulting SI unit for the cross section is cm6 s2/photon2. The unit 10-78 cm6 s2/photon2 is used when reporting our results throughout the paper. The factor (L6/n3) in eq 3 is 1.003, 1.059, 0.991, and 1.038, respectively, for the solvents used in this paper (methyl benzoate, tetrahydrofuran (THF), acetophenone, and cyclohexanone). Its effect is therefore rather negligible. The orientationally averaged values for the 3PA probability δ3P for linearly (L) and circularly (C) polarized light can be written as L ) δ3P

1 (2δG + 3δF) 35

(6)

C ) δ3P

1 (5δG - 3δF) 35

(7)

where

δF)

TiijTkkj ∑ i,j,k

(8)

δG )

TijkTijk ∑ i,j,k

(9)

If only the contributions from two states are taken into account, the dominant component of the third-order transition TS moment Tzzz along the molecular (z) axis, can be written as

The molecule we choose to study here is (2,2′-(4,4′-(1E,1′E)2,2′-(9,9-diethyl-9H- fluorene-2,7-diyl)bis(ethene-2,1-diyl)bis(4,1-phenylene))dibenzo[d]thiazole, which has the same active part as the one employed in the experimental work in ref 13, but with shorter (ethyl) chains replacing the decyl chains. All geometries were optimized with the 6-31G(d) basis set at both the Hartree-Fock (HF) and Becke’s three-parameter LeeYang-Parr (B3LYP) levels, using the Gaussian 03 program.25 The resulting gas-phase geometries are shown in Figure 1. Solvent effects both on molecular geometries and on molecular properties are modeled by PCM. All property calculations were performed at the B3LYP level with the 6-31G basis set using the parallel implementation of the cubic PCM response module in the DALTON program.23 For the solvent calculations, the cavity has been defined by placing a sphere on all the non-hydrogen elements and on the acidic protons of the amino group when present. The following set of values was employed for the radii: RC ) 2.04 Å (with no H bonded), RC ) 2.28 Å (CH and CH2 carbons), RC ) 2.40 Å (CH3 carbons), RN ) 1.92 Å, RS ) 2.22 Å, and RH ) 1.44 Å, where the usual multiplicative factor R ) 1.2 is already included. A nonequilibrium solvation scheme was adopted for the solvent calculations, and the optical dielectric constant opt was obtained as the square of the refractive index n. In response theory, the transition dipole moment between |0〉 and |f〉 is given by the single residue of a linear response function involving at least one electric dipole operator. The dipole moment fluctuation

Solvent Effects on Three-Photon Absorption

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Figure 1. Geometries of (2,2′-(4,4′-(1E,1′E)-2,2′-(9,9-diethyl-9H-fluorene -2,7-diyl)bis(ethene-2,1-diyl)bis(4,1-phenylene))dibenzo[d]thiazole in gas phase optimized at the (a) B3LYP and (b) HF levels.

TABLE 1: BLA (Å), C1-C2-C5-C6 (θ1, Degree) and C7-C8-C9-C10 (θ2, Degree) Dihedral Angles in the Gas Phase and in Four Different Solvents Computed at Both HF and B3LYP Levels with the 6-31G(d) Basis Set HF geometry

B3LYP geometry

medium

BLA

θ1

θ2

BLA

θ1

θ2

vacuum methyl benzoate THF acetophenone cyclohexanone

0.147 0.147 0.147 0.147 0.147

-141.67 -149.40 -150.65 -149.46 -149.54

-141.68 -149.26 -149.63 -149.48 -149.43

0.110 0.110 0.110 0.110 0.110

-179.75 -177.07 -177.48 -175.76 -176.23

-179.75 -178.19 -177.42 -176.33 -176.82

(µffz - µ00 z ) is evaluated as the double residue of the appropriate quadratic response function. The third-order transition matrix element Tzzz is given by the single residue of a CR function involving three electric dipole operators. We refer to ref 9 for a thorough description of the methodology. 4. Results and Discussion 4.1. Molecular Structures. The molecule synthesized by Cohanoschi et al.13 contains long decyl aliphatic chains, which increase its solubility in the various solvents. We replaced the -C10H21 (decyl) chains with -C2H5 (ethyl) groups in order to reduce the computational cost. Geometry optimizations have been performed for the molecule in the gas phase and in the four solvents used in the experiment.13 Some key structural parameters are collected in Table 1. The molecular geometry is very sensitive to the choice of the computational method, as we have also observed in our previous studies.26,27 For instance, for the gas-phase geometry, at the B3LYP level, the four phenyl rings along the molecular main axis lie almost on the same plane with both dihedral angles C1-C2-C5-C6 (labeled as θ1) and C7-C8-C9-C10 (labeled as θ2) being -179.75°. The HFoptimized geometry exhibits a nonplanar structure with θ1 and θ2 both being -141.7°. The large torsion angle predicted by the HF method agrees well with the experimental findings for many similar organic molecules.28-34 Calculations for the molecule in the gas phase were also carried out to investigate the dependence of the geometrical parameters on the length of the alkyl chains, with n varying from 1 to 9 in the -(CH2)nCH3 group. It was found that the length of the alkyl chain has negligible effect on the remaining structure parameters. In particular, there is no observable change for the bond length alternation (BLA), defined as the average difference between carbon-carbon single and double bonds, neither for the HF nor the B3LYP geometries. The torsion angles θ1 and θ2 vary by -0.04° and -0.13° at the HF level, and by +4.40° and +3.93° at the B3LYP level, as n varies from 1 to 9. For the calculations in solvent we used the values for the dielectric constants  and refractive indices n given in ref 13

for the four solvents. The BLA was found to be 0.147 Å for HF and 0.110 Å for B3LYP, in gas phase as well as in the four solvents. The dielectric medium does not affect the BLA of the molecule, in contrast to what has been observed for push-pull molecules,35 and in agreement with the observations in ref 27. Acetophenone shows the largest solvent effect on the dihedral angles for the B3LYP geometries, with a change of 3.99° for θ1 and 3.42° for θ2. When the geometry optimization is carried out at the HF level in the solvents, both torsion angles become larger than those resulting in the gas phase. The geometry in THF shows the largest deviation from the gas-phase structure, with an increase of 8.98° for θ1 and 7.95° for θ2. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), both in gas phase and in acetophenone, are shown in Figure 2. The HOMOs obtained with the HF and B3LYP geometries are very similar, whereas the LUMO obtained at the HF geometry displays a higher electron density on the two phenyl rings at the edges of the molecule (see Figure 2). In general, the solvent has a negligible effect on the electron density of the molecular orbitals. 4.2. One-Photon Absorption. Employing equilibrium geometries, OPA has been computed at the B3LYP/6-31G level using linear response theory. The excitation to the first excited state dominates the OPA spectrum for both HF- and B3LYPoptimized geometries. The dominant effect comes from the HOMO-LUMO transition, which corresponds to the CT from the π-center to both acceptors at the ends, as clearly indicated in Figure 2. The calculated excitation energy Ef, the oscillator strength δOP, as well as the corresponding wavelength λOP are collected in Table 2. The static dielectric constant  and the optical dielectric constant opt are also given. The excitation energies obtained for the B3LYP geometries are generally lower than those for the corresponding HF geometries. Recently we have found that excitation energies calculated with the HF geometry on a one-dimensional CT system similar to that studied in this work are in better agreement with the experimental value than those obtained with B3LYP geometries.27 The present results appear to confirm this evidence. For example, for methyl

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Figure 2. Charge densities of the HOMO and LUMO in gas phase (left) and dissolved in acetophenone (right), for both HF- and B3LYPoptimized geometries.

TABLE 2: Excitation Energy Ef (eV), OPA Oscillator Strength, δOP, as well as the Corresponding Wavelength λOP (nm) Computed at the B3LYP/6-31G Level for Geometries Optimized with a 6-31G(d) Basis Set at Both HF and B3LYP Levels in the Gas Phase and Four Different Solventsa HF geometry

B3LYP geometry

medium



opt

Ef

λOP

δOP

Ef

λOP

δOP

vacuum methyl benzoate THF acetophenone cyclohexanone

1 6.6 7.6 17.3 18.2

1 2.298 1.974 2.363 2.105

3.064 2.924 2.929 2.922 2.924

405.2 424.5 423.9 424.9 424.6

6.201 6.576 6.546 6.563 6.542

2.703 2.592 2.602 2.586 2.593

459.4 478.9 477.2 480.2 478.8

6.995 7.326 7.278 7.327 7.295

a The static, , and optical,  , dielectric constants are given. Note that the optical dielectric constant is given by the square of the refractive opt index opt ) n2.

benzoate, the experimental value of 412 nm is rather well reproduced using the HF geometry (424.5 nm), whereas the B3LYP counterpart is found to be too large (478.9 nm). In solution, the excitation energy of the CT state is red-shifted with respect to the gas phase, the value of the shift being almost the same for all four solvents taken into account here: around 18-20 nm for HF geometries and 17-21 nm for B3LYP geometries. Experimentally, the Stoke’s shifts for the four solvents are around 79-86 nm. However, the absolute absorption shifts in the experimental spectra are smaller than the Stoke’s shifts.13 In agreement with the experimental values, we observe that the excitation energy of the CT state is not very sensitive to the choice of the solvent. The absorption intensity is slightly enhanced upon solvation. The solvent effect on the oscillator strength is similar for all solvents, increasing by about 0.3 for both geometries employed. 4.3. Three-Photon Absorption. 4.3.1. Cubic Response Theory Results. The maximum 3PA

cross sections and the corresponding fundamental wavelengths for the first CT state, obtained from the single residue of appropriate CR functions for both HF- and B3LYP-optimized geometries, are collected in Table 3. A lifetime of 0.1 eV is employed for the Lorentzian band shape of the 3PA cross sections, in accordance with the assumptions usually made when analyzing the experimental data. We note, however, that our results will depend strongly on the choice of this lifetime, as recently pointed out in an analysis of TPA.12 The 3PA cross section is clearly enhanced in solution. For example, using the HF geometry in THF, the 3PA cross section increases by a factor of 1.6 compared to the gas-phase value, from 115.36 to 182.32. Similar conclusions can be drawn from Table 3 for the B3LYP-optimized geometries. The strongest solvent effect is found for cyclohexanone for both geometries: the cross section increases from 115.36 to 185.34 when employing the HF geometry, and from 161.56 to 261.42 for the B3LYP geometry. It is also clear that the 3PA cross section

Solvent Effects on Three-Photon Absorption

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TABLE 3: Three-Photon Transition Tensor Element Tzzz (au), 3PA Cross Section σ3P (10-78 cm6 s2/photon2) and Corresponding Wavelength λ3P (nm) Computed with a CR Theory Approach at the B3LYP/6-31G Level Employing Geometries Optimized with a 6-31G(d) Basis Set at Both HF and B3LYP Levels, in Gas Phase and in Four Different Solventsa HF geometry

B3LYP geometry

medium



opt

Tzzz × 10-6

λ3P

σ3P

Tzzz × 10-6

λ3P

σ3P

vacuum methyl benzoate THF acetophenone cyclohexanone

1 6.6 7.6 17.3 18.2

1 2.298 1.974 2.363 2.105

-0.99733 -1.3313 -1.3025 -1.3488 -1.3304

1213 1271 1269 1272 1271

115.36 179.43 182.32 181.55 185.34

-1.4243 -1.9029 -1.8474 -1.9345 -1.8918

1375 1434 1428 1437 1433

161.56 244.43 257.15 258.96 261.42

a

The static, , and optical, opt, dielectric constants are given.

is strongly dependent on the geometrical structure of the molecule, particularly the torsion angles θ1 and θ2. To fully understand the effect of the torsion angles on the 3PA cross sections, we have performed a set of constrained geometry optimizations in gas phase for different values of θ1 ) θ2. For each constrained geometry, we have collected in Figure 3 the total energy increase with respect to the minimum, the change in the excitation wavelength to the first CT state, and the corresponding 3PA cross section. Similar approach has also been employed in a recent work on solvent-induced effects on TPAs.27 It is found that the change of torsion angle is thermodynamically possible. In the case of a B3LYP-optimized geometry, a (-50°) change from the equilibrium geometry yields an increase in energy by 2 kcal/mol, which implies that, at room temperature in solution, one might expect to observe a rather wide distribution of structures with different torsion angles rather than a

Figure 3. Changes with respect to optimized geometry in the total energy ∆E (upper panel), in the wavelength of the excitation to the first excited state, ∆λ (middle panel), and corresponding cross section σ3P (bottom panel) are presented as functions of torsion angles θ1 ) θ2. All results are obtained from B3LYP/6-31G calculations in gas phase starting from both HF- (diamonds) and B3LYP- (triangles) optimized geometries.

unique equilibrium structure. The excitation wavelength of the first CT absorption state increases as the torsion angle decreases. As we can see in Figure 3 (middle panel), a change of the dihedral angle by (-50°) with respect to the equilibrium geometries, results in the CT state being blue-shifted by as much as 78.7 and 52.8 nm for the HF- and B3LYP-optimized geometries, respectively. The 3PA cross section is dramatically increased when the molecule becomes planar. For example, with B3LYP geometries, the largest cross section is found to be 161.56 for the geometry θ1 ) θ2 ) 180°, which is 1.2 times larger than the value of 138.83 obtained for θ1 ) θ2 ) 130°. For the HF-optimized geometries, the same trend can be found with the 3PA cross section of the θ1 ) θ2 ) 142° geometry, which is 1.8 times larger than that evaluated for θ1 ) θ2 ) 90°. From Table 3 we see that, for the B3LYP-optimized geometries, the 3PA cross sections follow the trend of the static dielectric constant rather than that of the optical dielectric constant of the solvent. This is quite different from what was found by Frediani et al.36 for TPA cross sections in solution, modeled using B3LYP-optimized geometries as in the present study. For HF-optimized geometries, on the other hand, the 3PA cross section shows a non-monotonic behavior with respect to the dielectric constant. This may be a result of the relatively large geometrical changes observed for HF structures in the solvent. In the experiment, the σ3P values are inversely proportional to the static dielectric constant, and large differences appear for the four different solvents.13 Our calculations cannot reproduce the values nor the ordering of the 3PA cross sections observed experimentally. This may be partly due to the simplification introduced in our calculation by only considering the intrinsic properties of a single solute molecule and its interaction with the solvent. By using this simplified model, even the geometry of the ground state obtained in different solvents might not be completely reliable, since the interaction actually involves a large number of solvent and solute molecules, which can induce distortions of the target solute molecule. Furthermore, vibronic contributions, which also could be an important factor, were not taken into account in our calculations. We have recently studied the vibronic effects on the TPA of a CT stilbene derivative, and we have shown how HerzbergTeller (HT) borrowing mechanisms among different states can work efficiently.26 The calculation of a three-photon transition moment involves far more intermediate states than those entering the two-photon transition moment (cf. eq 2 above and eq 1 in ref 27). One can therefore expect that the borrowing mechanism originating from the vibrational HT term might become far more efficient in 3PA processes, where the coupling between many more states has to be taken into account. 4.3.2. Two-State Model and Its Limitations. For large systems, the calculation of the 3PA cross section via a direct analytical CR approach may easily become computationally prohibitive.

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TABLE 4: Dipole Moment Fluctuation ∆µz (au), z Component of Transition Dipole Moment µz01 (au) Between the Ground TS -78 cm6 s2/ State and the First Excited State, Three-Photon Transition Tensor Element TTS zzz (au), 3PA Cross Section σ3P (10 TS 2 photon ) and Corresponding Fundamental Wavelength λ3P (nm) Computed Within a Two-State Model at the B3LYP/6-31G Level Employing Geometries Optimized With the 6-31G(d) Basis Set at Both HF and B3LYP Levels, in Gas Phase and Four Different Solvents HF geometry

B3LYP geometry

medium

∆µz

µ01 z

TS λ3P

-5 TTS zzz × 10

σTS 3P

∆µz

µ01 z

TS λ3P

-5 TTS zzz × 10

σTS 3P

vacuum methyl benzoate THF acetophenone cyclohexanone

0.00 -0.01 0.00 -0.03 0.01

6.20 6.58 6.55 6.56 6.54

1213 1271 1269 1272 1271

-2.5380 -3.3235 -3.2687 -3.3108 -3.2733

7.45 11.16 11.45 10.91 11.19

-0.00 0.01 0.01 -0.01 0.01

7.00 7.33 7.28 7.33 7.29

1375 1433 1428 1437 1433

-4.6842 -5.8494 -5.6919 -5.8824 -5.7722

17.44 24.09 24.35 23.88 24.28

On the other hand, the number of intermediate states involved in the summations of eq 2 soon becomes huge, and, in an SOS approach, truncations often become mandatory. Because the transition to the first excited-state is dominant in the 3PA process, it could be reasonable to assume that a two-state model including the ground- and the first excited-state should be sufficient for a good description of the cross section. For the molecule under analysis in this work, the dipole 00 moment fluctuation ∆µz ) µ11 z - µz , the transition dipole 01 moment µz , the 3PA cross sections obtained from a two-state model, and the corresponding fundamental wavelengths are presented in Table 4. Both HF- and B3LYP-optimized geometries were considered. A comparison of the two-state model results with CR theory results for the three-photon transition tensor element Tzzz can be made by comparing results in Table 3 with those given in Table 4. The poor performance of the two-state model for Tzzz is very evident. The mean value of the deviation with respect to the CR results exceeds 300% for the HF-optimized geometries. For the B3LYP geometries, the values obtained from the two-state model are one-third of those yielded by the CR approach, with a mean deviation as large as 220%. These results are in disagreement with what was found by Cronstrand and co-workers.11 These authors found that the twostate model for symmetric molecules overestimates the response values. In our case, the two-state model for Tzzz underestimates the results obtained via full CR response theory, for both HFand B3LYP-optimized geometries. The quality of the two-state model is therefore strongly system dependent. The cross section values, which in the two-state model are proportional to the square of Tzzz (see eq 11), are about 16 and 11 times smaller than those yielded by the CR approach, for the HF- and B3LYP-optimized geometries, respectively. For example, in THF with an HF-optimized geometry, the CR value of σ3P is 182.32, which is 16 times that yielded by the twostate model (11.45). The value obtained in the same solvent by optimizing the geometry with a B3LYP wave function is 257.15, which is 11 times larger than the value of 24.35 obtained with the two-state model. The molecule under investigation has symmetrical substitutions at its ends. Its permanent dipole moment should vanish for the component lying along the molecular z-axis. The same applies to the dipole fluctuation ∆µz. Therefore, the 3PA matrix TS within a two-state model should be determined element Tzzz solely by µ0f . z As can be seen from Table 4, the permanent dipole moment fluctuation along the z-axis is quite small (less than 0.03 au for all geometries). The z-component of the permanent dipole moment µ00 z (not shown in the table) is also very small, as expected. The transition dipole moment along the z-axis is dominant since the CT mainly takes place along the conjugation, and it is enhanced upon solvation. This confirms that, as also argued by Cronstrand and co-workers,11 for this

kind of one-dimensional symmetric molecule, only the transition dipole moment contributes to the total 3PA cross section. This statement conflicts with the conclusions of Cohanoschi and coworkers,13 who pointed out that, for this molecule, both terms of ∆µz and µ0f z should be taken into account. We believe, however, that this analysis is based on the total dipole moment of the system, which, for the nonplanar geometries, may have significant dipole moment components perpendicular to the molecular conjugation axis (y-axis). Within the conjugation axis, our calculations clearly show only a very small ground- and excited-state dipole moment component along the z-axis. In the experimental study13 it was reported that the value of 00 2 3 (µ11 z - µz ) /hca estimated from Stoke’s shifts is 243.8, 190.5, 189.7 and 174.1 nm in the four solvents, respectively. h is here Planck’s constant, c ) c0/n is the speed of light in the medium, and a is the cavity radius around the dye molecule. The experiment shows, therefore, a large change in polarity of the molecule upon absorption in the solvents, with values of µ11 z of 2.083, 1.842, 1.838, and 1.760 au in methyl benzoate, µ00 z THF, acetophenone, and cyclohexanone, respectively, when we set a to be the coordinate of the outmost atom along the z-axis increased by its van der Waals radius, 18.6 Å, for all solvents in our calculation. It should be stressed that the dipole fluctuations derived from the Stoke’s shifts correspond to the difference between dipoles taken at the minima of the groundand excited-state energy surfaces. Calculations show that total dipole fluctuations ∆µ ) x∆µx2+∆µy2+∆µz2 for the vertical excitation with HF geometries are all around 0.42 au for the four solvents used in experiment. This is consistent with the fact that the molecule exhibits a high degree of symmetry. By comparing the calculated and experimental dipole fluctuations, it is apparent that the excited-state geometry is quite different from the ground-state geometry. Ideally, the absorption processes are mostly related to vertical excitations, and the use of Stoke shifts to estimate the dipole fluctuation is not well justified for the 3PA. It is noted that the experimental spectra were recorded with the excitation of a 25 ps laser, hence the possible involvement of excited-state absorptions cannot be ruled out. The importance of excited-state absorption for TPA has been widely discussed in the literature.37 It has been shown that a full description of pulse propagation and molecular properties is required to model such processes.38,39 The extension of this method to the 3PA of the system under investigation is beyond the scope of the present work. It is nevertheless still possible to obtain an understanding of the experimental evidence when the excited-state of our molecule displays a large dipole fluctuation value. Let us investigate a possible excited geometry of the molecule under analysis yielding in the solvent a dipole fluctuation of the same size as that measured by Cohanoschi and co-workers.13

Solvent Effects on Three-Photon Absorption For this analysis, we concentrate on the methyl benzoate solution. We showed in Section 4.3.1 that torsions around the ethyl double bonds are thermodynamically affordable. These torsions, corresponding to rotation of the two dihedral angles θ1 and θ2 in opposite direction, can be shown to yield negligible dipole fluctuations. By rotating θ1 and θ2 in the same direction, instead, the polarity of the molecule can be enhanced. We have therefore calculated the dipole fluctuation as a function of the torsion angle θ1 ) -θ2. At the B3LYP level, when θ1 ) -θ2 ) -95°, the dipole moment of the excited state, µ11 (constrained), has values of -0.651, -1.290, and -1.562 au, for three components, (x, y, z), respectively. The ground-state permanent dipole moment, µ00 (ground), is, in contrast, 0.012, -0.468, and 0.000 au, respectively. The total dipole fluctuation, ∆µ, is around 2.437 au, which is quite close to the experimental value of 2.083 au. A constrained B3LYP geometry with θ1 ) -θ2 ) -95°, would therefore also yield a remarkable dipole fluctuation, in agreement with the experimental observations. Unfortunately, the excitation energy of the S1 state at the θ1 ) -θ2 ) -95° constrained geometry, corresponding to the emission energy from the equilibrium of S1 to the ground state, is found to be 0.5 eV higher than the absorption energy from the equilibrium of S0 to the S1 state. This anti-Stoke behavior can thus not explain the Stoke’s shift observed in the experiment. Similar description can also be extended to the other solvents. The important message coming out from this simplified analysis is, however, that the geometry relaxation of the excited-state could be a very important and challenging (from the point of view of computational model) factor controlling the 3PA processes of our CT molecule in solution. 5. Conclusions We have applied a CR theory approach in combination with the PCM to study solvent effects on the 3PA properties of a highly conjugated, symmetric molecule with a A-π-π-π-A structure. The experimental values and ordering of the 3PA cross sections as well as the dipole fluctuations in solution cannot be reproduced by calculations made on a single molecule, even with the inclusion of solvent effects. Additional mechanisms, such as vibronic coupling and excited-state absorption, might need to be included to improve agreement with experiment. The effects of geometry distortions, especially the increase of the torsion angle between the phenyl rings, on the 3PA cross sections have been analyzed. It is shown that a simplified twostate model, as already anticipated by Cronstrand and coworkers,11 has difficulty reproducing the results of the full analytical response theory. We finally comment on some of the possible limitations of our study. An exchange-correlation (XC) functional such as B3LYP, despite being very popular and showing, in some instances, good performance in studies of mixed electricmagnetic frequency-dependent high-order properties as well,40,41 might be somewhat inadequate when delocalization effects or CT excitations, as is the case studied here, play a major role. Nevertheless, these drawbacks are, in general, more evident for really extended, or highly conjugated systems. Recently, the Coulomb attenuating model B3LYP42 has been shown to yield good performance in two-photon studies,43 and this may be a promising note for future studies of nonlinear spectroscopic properties. A second cause of possible concern in our study may be the use, both for geometry optimization [6-31G(d)] and spectroscopic properties (6-31G), of rather limited basis sets, missing diffuse and polarization functions. This concern might be particularly serious considering that we are analyzing high-

J. Phys. Chem. B, Vol. 112, No. 15, 2008 4709 order properties, such as 3PA. It is a fact that larger, more adequate basis sets are still prohibitive for systems of the size and properties on the order of those studied here. Also, it is known that, with extended systems, the need for large, polarized, and diffuse functions included in the basis set becomes less stringent. Their importance diminishes as the size of the molecule grows larger, since basis functions placed on neighboring atoms act as polarization functions for close-lying atoms. As long as the chromophores under study are primarily located along the molecular axis, and not directly at the edges of the molecule, missing extensive sets of polarizing and diffuse functions should not be considered a serious deficiency. Finally, we note that we are primarily interested in the relative change of the 3PA cross section in different solvents. The choice of basis set can certainly affect the absolute value of the 3PA cross section, but it is difficult to see how it could drastically change the relative values as observed experimentally. Nevertheless, in spite of these arguments, it is true that the whole community of scientists working on the computational analysis of highorder spectroscopic properties of extended systems is looking with great anticipation at future much awaited theoretical and technological developments, linear scaling high-order response theory in particular, which might constitute the likely solution to this problem. Acknowledgment. This work is supported by the Swedish Research Council (VR) and the National Nature Science Foundation of China (20473046). Support from the Norwegian Research Council through a Ph.D. grant to L.F. within the Nanomat program (Grant No. 158538/431), a Centre of Excellence Grant (Grant No. 179568/V30), and a YFF grant to K.R. (Grant No. 162746/V00), as well as a grant of computer time from the Norwegian Supercomputing Program are also acknowledged. References and Notes (1) Reinhardt, B. Photonics Sci. News 1998, 4, 21. (2) He, G. S.; Bhawalkar, J. D.; Prasad, P. N. Opt. Lett. 1995, 20, 1524. (3) Zhou, G.; Wang, X.; Wang, D.; Shao, Z.; Jiang, M. Appl. Opt. 2000, 41, 1120. (4) Gu, M. Opt. Lett. 1996, 21, 988. (5) Lakowicz, J. R.; Gryczynski, I.; Malak, H.; Schrader, M.; Engelhardt, P.; Kano, H.; Hell, S. W. Biophys. J. 1997, 72, 567. (6) Maiti, S.; Shear, J. B.; Williams, R. M.; Zipfel, W. R.; Webb, W. W. Science 1997, 275, 530. (7) Naskrecki, R.; Menard, M.; Van der Meulen, P.; Vigneron, G.; Pommeret, S. Opt. Commun. 1998, 153, 32. (8) He, G. S.; Markowicz, P. P.; Lin, T.; Prasad, P. N. Nature (London) 2002, 415, 767. (9) Olsen, J.; and Jørgensen, P. J. Chem. Phys. 1985, 82, 3235. (10) Norman, P.; Jonsson, D.; Vahtras, O.; Ågren, H. Chem. Phys. Lett. 1995, 242, 7. (11) Cronstrand, P.; Norman, P.; Luo, Y.; Ågren, H. J. Chem. Phys. 2004, 121, 2020. (12) Ferrighi, L.; Frediani, L.; Fossgaard, E.; Ruud, K. J. Chem. Phys. 2007, 127, 244103. (13) Cohanoschi, I.; Belfield, K. D.; Toro, C.; Hernandez, F. E. J. Chem. Phys. 2006, 125, 161102. (14) Miertusˇ, S.; Scrocco, E.; Tomasi, J. Chem. Phys. 1981, 55, 117. (15) Tomasi, J.; Persico, M.; Chem. ReV. 1994, 94, 2027. (16) Cammi, R.; Cossi, M.; Tomasi, J. J. Chem. Phys. 2006, 125, 161102. (17) Cammi, R.; Cossi, M.; Mennucci, B.; Tomasi, J. J. Chem. Phys. 1996, 105, 10556. (18) Cammi, R.; Frediani, L.; Mennucci, B.; Tomasi, J.; Ruud, K.; Mikkelsen, K. V. J. Chem. Phys. 2002, 117, 13. (19) Tomasi, J.; Mennucci, B.; Cammi, R. Chem. ReV. 2005, 105, 2999. (20) Cammi, R.; Frediani, L.; Mennucci, B.; Ruud, K. J. Chem. Phys. 2003, 119, 5818. (21) Frediani, L.; Ågren, H.; Ferrighi, L.; Ruud, K. J. Chem. Phys. 2005, 123, 144117.

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