Computational Study of Molecules with High Intrinsic

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Computational Study of Molecules with High Intrinsic Hyperpolarizabilities Cla´udia Cardoso,*,‡ Paulo E. Abreu,† Bruce F. Milne,‡ and Fernando Nogueira‡ Chemistry Department, UniVersity of Coimbra, P-3004-535, Coimbra, Portugal, and Centre for Computational Physics, Physics Department, UniVersity of Coimbra, P-3004-516, Coimbra, Portugal ReceiVed: March 15, 2010; ReVised Manuscript ReceiVed: August 6, 2010

In the current manuscript we present the results of a computational study on a series of chromophores with enhanced intrinsic hyperpolarizability. The high hyperpolarizability values of these molecules were previously reported and were achieved by making use of aromatic moieties in order to modulate the aromatic stabilization energy along the conjugated bridge between the donor and the acceptor. Calculations were performed using semiempirical, DFT, and TDDFT methods, and the results reproduce the trend determined experimentally for the first hyperpolarizability values. Several calculation schemes were used, and the best agreement was achieved when long-range Hartree-Fock exchange corrections and solvent effects are included in the DFT calculations. The long-range corrections proved to be especially important for the azobenzene derivatives, which otherwise have their hyperpolarizability overestimated considerably in the DFT calculations. The results are also analyzed within the framework of a two-level model, which correctly reproduces the trend in the hyperpolarizabilities of the molecules under study. Introduction Photonics is playing an ever-increasing role in today’s technology by efficiently replacing electronics in electro-optic devices with applications in the fields of high speed data transmission, processing, and storage.1-6 In this context, researchers have been focusing on the search for materials with the largest possible optical nonlinearities. Organic molecules are promising candidates since their properties can be readily modified and their dielectric constants are much smaller than those of the most common inorganic molecules. An organic second-order nonlinear optical (NLO) chromophore typically contains a conjugated π-electron system, asymmetrically substituted with electron donor and acceptor groups, through which a charge-transfer occurs. An important part of current research efforts is focused on the optimization of these two components of the system: the π bridge and the donor/acceptor groups. In the early 1990s it was shown that for a given conjugation bridge there is an optimal combination of donor and acceptor strengths or ground state polarization to maximize the dipole moment-first hyperpolarizability product, µβ. Beyond a certain point, increased donor-acceptor strength or further ground state polarization attenuates µβ.7,8 Bond length alternation (BLA), that is, the average difference in length between single and double bonds in the molecule, was also shown to be a relevant parameter in the optimization of the hyperpolarizability of molecules.9-11 The aromaticity of the conjugated rings in the π-bridge is related with BLA. The decrease of the aromatic delocalization energy leads to a decrease of the latter. As the aromatic delocalization energy of the bridge decreases and the mixing of the ground state and charge-transfer state increases, the polarizability and hyperpolarizability increase. The possibility of modulating NLO properties at the molecular level using molecules that respond to electrochemical or chemical inputs such as protons or metal cations has also * To whom correspondence should be addressed: E-mail: cmcardoso@ teor.fis.uc.pt. † Chemistry Department. ‡ Centre for Computational Physics, Physics Department.

been explored.12-15 In chemosensitive systems of the donor/πacceptor type, interaction with a cationic species alters the electron density of the terminal sites, resulting in a modulation of the internal charge-transfer character of the dye molecule, which leads to a change in the optical response.16-18 There have been several reports of modulation of the two-photon absorption and fluorescence properties of linear chromophores upon cation binding or protonation/deprotonation at the terminal donor substituent.19-22 Recently Pe´rez-Moreno et al.23 reported on a series of chromophores with enhanced intrinsic hyperpolarizabilities that breach the apparent limit of all previously studied molecules. Their design focused on modulating the amount of aromatic stabilization energy along the conjugated bridge between the donor and the acceptor. To induce the desired modulation, moieties with differing degrees of aromaticity were employed to construct the asymmetrically substituted π bridge. With the purpose of studying the role of the different contributions involved in modulating the NLO properties of this type of compound, we present in the following sections the results of semiempirical and density functional theory (DFT) calculations performed for the molecules reported in Pe´rez-Moreno et al.23 and shown in Figure 1. Calculation Methods At the semiempirical level, the ground state geometries and hyperpolarizability values were computed using MOPAC2009,24 with the PM325-28 and the PM629 Hamiltonian. Electronic structure calculations were performed using the GAMESS-US code30 at the DFT and time dependent DFT (TDDFT) levels with the 6-311+G(d,p) basis set.31 Test calculations with extra diffusion and polarization functions showed differences of only 2% in the hyperpolarizability. In the DFT calculations we performed geometry optimization of the molecules with the B3LYP exchange correlation potential.32,33 Absorption spectra were calculated using TDDFT, and the static second order hyperpolarizabilities were computed using the finite field method. B3LYP hybrid and BLYP GGA exchange and

10.1021/jp105707q  2010 American Chemical Society Published on Web 09/10/2010

Molecules with High Intrinsic Hyperpolarizabilities

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Figure 1. Molecules under study. Compound numbering taken from Pe´rez-Moreno et al.23

correlation functionals were used. Because the symmetry group of all the molecules was C1, no use of symmetry was made during the calculations. Calculation of vibrational frequencies for these molecules in C1 symmetry with the basis sets and theoretical methods employed was not feasible, so verification of the nature of the stationary points obtained was not performed. Due to the fact that these functionals are expected to reproduce charge-transfer excitations poorly and the hyperpolarizabilities of π-conjugated systems are overestimated by conventional pure DFT (and neither hybrid nor asymptotic farnucleus correction methods improve the overestimations),34 we recalculated the linear and nonlinear optical properties with the long-range corrected BLYP functional (LC-BLYP). The LC scheme explicitly includes long-range orbital-orbital interactions in the exchange functional by combining DFT exchange with the HF exchange integral.35-37 In contrast to density partitioning schemes such as B3LYP, the proportion of nonlocal HF contribution varies according to the range of the interaction. This is done by dividing the electron repulsion operator into short-range and long-range parts by using the standard error function. In this way the ratio of the nonlocal HF part to the local DFT part becomes larger for greater distances. To better reproduce the experimental conditions, solvent effects were also considered, the geometries were reoptimized, and the linear and nonlinear responses were recalculated. With the semiempirical methods the COSMO model was used,38 whereas for the (TD)DFT calculations the polarizable continuum method (PCM) was employed.39,40 The solvent considered was dimethyl sulfoxide (DMSO), the same used in the hyperRayleigh scattering measurements reported in Pe´rez-Moreno et al.23 Results and Discussion Molecular Geometries and Orbitals. The molecular geometries were first optimized with semiempirical methods, namely

with the PM3 and PM6 Hamiltonians. The optimizations were then repeated with DFT methods at the B3LYP/6-311+G(d,p) level of theory both in vacuo and with solvent effects (COSMO for the semiempirical methods and PCM for the DFT calculations). The geometries calculated with the semiempirical methods are mainly planar, but all of them show some distortion around the NdN bond. Furthermore, in MOL5 steric hindrance between the terminal doubly bonded O and the Cl atom attached to the thiazole ring leads to slight deviation from planarity at this end of the molecule. A similar effect was expected to happen in MOL6,23 but the Cl and N atoms were sufficiently far apart to cause no observable hindrance. Both in vacuo and when solvent effects were included the azobenzene dyes MOL1, MOL2, and MOL4 were found to have DFT/B3LYP geometries that are more planar than the geometries obtained with the semiempirical methods, as expected for the B3LYP functional. Orbitals. Visual inspection of the frontier molecular orbitals (FMO) leads to the conclusion that the molecules can be divided into two subsets with different FMO properties. To illustrate the differences between the two groups, the HOMO and LUMO isosurfaces for MOL4 and MOL7 are shown in Figure 2. This pair of molecules are very similar in their structure and geometry, differing only in one of the rings of the conjugation path. The HOMO and LUMO of MOL4, and likewise MOL1 and MOL2, are localized in opposite ends of the molecule, reducing the overlap between these orbitals. These molecules compose the first subset, and all of them have an azobenzene group in their conjugation path. Molecule MOL7 and also MOL3, MOL5, and MOL6, have orbitals displaying greater delocalization. The resulting increased HOMO-LUMO overlap places these molecules in a distinct subset of their own. The small HOMO-LUMO overlap of the first subset indicates a greater charge-transfer character for the first excited

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Figure 2. HOMO (left) and LUMO (right) isosurfaces of the molecules MOL4 (top) and MOL7 (bottom).

TABLE 1: Excitation Energies in eV of the Azobenzene Chromophores Calculated with the B3LYP and LC-BLYP Levels of Theory, in the Gas Phase and in DMSO (Using the PCM Model)a gas phase 1 2 3 4 5 6 7

DMSO

TABLE 2: Average Hyperpolarizability Values in Atomic Units, Computed with Semi-Empirical and DFT Methods gas phase

exp.

B3LYP

B3LYP

LC-BLYP

E01

E02

2.56b 2.57b-3.43 2.62 2.16 2.34 2.06 2.08

2.18 2.09 2.36 1.88-2.72 2.05 1.90 1.73

2.87b 2.64-2.67 2.54 2.59-2.74 2.38 2.30 2.06

2.25 2.29 2.06 2.19 1.78 1.80 1.83

2.40 2.40 2.25 2.41 1.88 1.95 2.00

a E01 and E02 are the experimental values.23 E01 is determined experimentally, and E02 is deduced from the experimental β and µ10 values. b Corresponds to the second excitation since the oscillator strength of the first is much smaller.

state of the azobenzene dyes when compared with the molecules belonging to the second subset. This is in agreement with the fact that MOL1, MOL2, and MOL4, that belong to the first subset, have larger bond length alternation (BLA) than the other molecules (the second subset). As the aromatic delocalization energy decreases (benzene, thiophene, and thiazole are 36, 29, and 25 kcal/mol, respectively)41 bond-length alternation across the bridge decreases. As the aromatic delocalization energy of the bridge decreases, the mixing of the ground state and chargetransfer state increases, indicating that the first subset has a stronger charge transfer character. The same separation into two groups is seen in the linear and nonlinear optical properties of the molecules, as discussed below. Differences in the quality of the results of the (TD)DFT hyperpolarizability calculations for the two subsets highlights the problems of the traditional exchange-correlation functionals in dealing with charge-transfer excitations. Linear Response. We computed the linear response for the series of molecules under study in DMSO solution using the PCM method with the B3LYP and LC-BLYP functionals. In order to investigate the influence of solvent on the calculated absorption spectra the calculations were repeated in vacuo. In Table 1 are presented the calculated and experimental23 absorption energies of the two subsets of molecules mentioned above. The molecules show essentially one large absorption peak at 2-2.5 eV and in the case of the azobenzene molecules a smaller peak at higher energy. The position of the main peak computed with B3LYP is different in the gas phase and when PCM solvent effects are included. The solvent causes a bathochromic shift for the azobenzene dyes, slightly smaller for the second set of molecules. In the gas phase the main peaks lie at around 2.06-3.4 eV, and shifting to 1.73-2.72 eV when solvent effects are considered.

1 2 3 4 5 6 7

DMSO

PM6

B3LYP

PM6

B3LYP

LC-BLYP

exp.

10286 11542 13201 15949 20972 22144 20505

11247 15151 7480 33194 10866 19905 29926

13099 14685 18202 30441 21139 35249 37420

52056 84183 26100 153542 48646 96392 147680

25422 40333 33868 44200 42774 72620 128860

12729 12729 27773 39345 32402 85054 92576

In the LC-BLYP scheme with PCM solvent effects, the positions of the peaks change again, this time toward higher energies when compared with the PCM solution-phase B3LYP spectra. The LC-BLYP main peaks lie closer to the B3LYP peaks in the gas phase. For the first subset there is a decrease in the spacing between the main peak and the peaks at higher energy, and also an increase of the oscillator strength in the latter. In the case of MOL2, the two peaks show the same height and are almost superimposed. In the last column of Table 1 are summarized the energy of the first two absorption peaks reported experimentally.23 The first value, E01, corresponds to the measured energy of the first absorption peak; E01 and the second value, E02, are obtained using experimental data and will be discussed later in this paper. The LC-BLYP peaks lie about 0.5 eV above the experimental peak for all the molecules. The main peak calculated within B3LYP lies below the experimental value for the azobenzene molecules and above for the molecules belonging to the second set, anticipating the NLO results presented in the next section. Overall, B3LYP appears to provide better agreement with the experimental peak energies. The experimental results also show for MOL4, MOL5, MOL6, and MOL7, a minor peak around 3 eV. This peak is not present in the calculated spectra, except for MOL4. Nonlinear Optical Properties In this section we present the values of the hyperpolarizabilities obtained with several computational methods for the seven molecules under study. In order to compare the results to the experimental values obtained in solution, we have employed the isotropic average of the first nonlinear response tensor throughout. Semiempirical Results. Although semiempirical methods are not expected to perform well in the calculation of properties such as the hyperpolarizabilities, for systems larger than the systems under study these are some times the only affordable method. Similarly, semiempirical methods are fast enough to permit the calculation of hyperpolarizabilities for large databases


Figure 3. Experimental and calculated hyperpolarizability values. The latter were computed for the gas phase and with COSMO solvent effects using PM3 and PM6 Hamiltonians.

Figure 4. Experimental and calculated hyperpolarizability values. The latter were computed for the gas phase and with PCM solvent effects using DFT/B3LYP functional.

of compounds, which is a task that is beyond any of the other electronic structure methods available. For this reason we show here the results obtained with the PM3 and PM6 semiempirical Hamiltonians, the latter being a fairly recent development and less extensively studied. We first optimized the geometries and used them in the calculation of the nonlinear response. The calculations in vacuum with both Hamiltonians (Figure 3) underestimate the hyperpolarizability values, particularly in the case of molecules MOL6 and MOL7, which have the largest experimental values. The PM3 values are smaller than the values obtained with PM6. When COSMO solvent effects are included the calculated values increase. Although both methods still underestimate hyperpolarizability values and fail to reproduce the large increase in hyperpolarizability from MOL5 to MOL6, with PM6 the trend of the series is well reproduced and the values are closer to the experiment. DFT Results. In Figure 4 are plotted the hyperpolarizability values obtained in the gas phase using the DFT/B3LYP scheme. The experimental values are also plotted for comparison. The calculations in vacuum underestimate the hyperpolarizability values, especially in the case of molecules MOL6 and MOL7, which have the largest experimental values. The values for the azobenzene moleculessMOL1, MOL2, and MOL4sare closer to the experimental values. Also shown in this plot are the calculated values for DFT/B3LYP with PCM solvent effects included. When solvent effects are considered the calculated


Figure 5. Experimental and calculated hyperpolarizability values. The latter were computed for the gas phase and with PCM solvent effects using DFT/B3LYP functional.

values increase. DFT/B3LYP considerably overestimates hyperpolarizability, with the worst performance being for the azobenzene chromophores MOL1, MOL2, and MOL4, but also MOL7. We now replot the gas phase and solution results, separating the two subsets of molecules. This depiction of the results can be seen in Figure 5. The different behavior of the two sets of molecules becomes more evident. For the azobenzene dyes, the gas phase values are very close to the experimental values, whereas the values obtained with PCM solvent effects overestimate the hyperpolarizability considerably. In the case of the second set of molecules, the gas phase values are smaller than the experimental ones. With the exception of MOL7, the solvent effects improve significantly the agreement with experiment. MOL7 is still largely overestimated by B3LYP and PCM solvent effects; however, studies on similar molecules42 led us to consider the possibility that different rotamers contribute to the measured first hyperpolarizability of MOL7, producing an average value that is lower than the value computed above. The influence of the geometry on the nonlinear optical properties is well recognized, deviations from planarity being particularly important. It is also known that the determination of the ground state geometry of similar molecules like the transazobenzene by means of theoretical methods is difficult, particularly when solvent effects are considered. Geometries are basis and functional dependent, and the total energies of geometries with different torsion angles are close. In solvated trans-azobenzene the energetic difference between the planar and twisted forms is either zero or is totally negligible at room temperature, once sufficiently large basis sets are selected.43 To rule out the possibility that the discrepancy between the calculated and the measured hyperpolarizability values is due to an error in the determination of the ground state geometry, we scanned for MOL2 the torsion angle between the two benzene rings for values between 0 and 90° and reoptimized the other atomic positions. Figure 6 shows a plot of the total energy of the various rotamers, versus the dihedral angle between the two benzene rings. It is important to notice that the lack of symmetry for small angles around the zero degree torsion angle is due to the fact that the phenyl group attached to the thiazole ring is slightly out of plane, and in this way the reflection symmetry expected for a purely planar conformation is broken. The energy differences between geometries with small torsion angles are larger than the values computed for azobenzene,43 suggesting that the ends of MOL2 have a role in the

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Figure 6. Energy differences and average hyperpolarizability computed for MOL2 with B3LYP, for different torsion angles.

stabilization of the planar geometry. At room temperature the geometries with angles of 30° have Boltzmann probabilities of about 15%, indicating that the configurations with larger torsion angles have a small contribution to the average hyperpolarizability at room temperature. Moreover, the hyperpolarizabilities calculated for all the geometries, even the ones with largest deviations from planarity, are still much larger than the experimental values. This suggests that, although the conformation plays a role in the determination of the hyperpolarizability values, such a large overestimation of hyperpolarizability must be explained by something else than a configurational problem. The poor results for the azodyes can be attributed to the stronger charge-transfer character of the excitations of these molecules, as indicated by the frontier orbitals and the absorption spectra data presented above. In an attempt to improve the DFT results, we recalculated the hyperpolarizabilities within the LCBLYP scheme including PCM solvent effects. Long-Range Corrections Previous calculations of the charge-transfer excitation energies of ethylene-tetrafluoroethylene dimer show that the excitation energies are significantly underestimated with increasing intermolecular distance by TDDFT employing conventional functionals including LB94 and B3LYP.44 It was reported that the LC scheme improves the accuracy of the excitation energies and oscillator strengths of Rydberg and charge-transfer excitations calculated by TDDFT.37 Because charge-transfer in the present context involves HOMO and LUMO orbitals with poor spacial overlap, the problems of TDDFT may be due to the insufficient exchange interaction between distant orbitals in exchange functionals rather than incorrect far-nucleus asymptotic behavior.36 Therefore, although both hybrid and long-range corrected functionals correct the asymptotic behavior of the LDA and GGA functionals, the latter are expected to further improve the estimation of NLO properties.35,45 As mentioned above, the vacuum DFT/B3LYP results for the azobenzene molecules of the series are close to the experimental values, but when PCM solvent effects are added, the hyperpolarizabilities are hugely overestimated. MOL1, MOL2, and MOL4 have a small overlap between the HOMO and LUMO orbitals, suggesting charge-transfer excitations, poorly described by the exchange-correlation functionals used. Figure 7 compares the values obtained using the LC-BLYP method, with the B3LYP and experimental results. Solvent effects within the PCM method were included. For the azoben-

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Figure 7. Hyperpolarizability values computed with DFT using the BLYP, B3LYP, and LC-BLYP functionals and including PCM solvent effects.

Figure 8. Experimental and calculated hyperpolarizability values. The latter were computed with the LC-BLYP functional and including PCM solvent effects. The results for different rotamers of MOL1, MOL2, and MOL7 are plotted.

zene dyes long-range correction leads to smaller values when compared with DFT/B3LYP, and therefore closer to experiment. For MOL1 and MOL2 values are still twice the experimental values, but this could be due to the presence of different rotamers, the contributions of which are averaged in experiment. For the rest of the molecules, the LC-BLYP values do not differ much from the values calculated with B3LYP, suggesting a weaker charge-transfer character in their excitations. Without performing a systematic search for the lowest energy conformer, we tested some rotamers of MOL1, MOL2, and MOL7. In Figure 8 are plotted the hyperpolarizabilities of the tested geometries computed, including PCM solvent effects with the LC-BLYP functional. Although all the values are still larger than the experimental values, this indicates that the rotamers have considerably different hyperpolarizabilities. Thus, the difference between the computed and measured values is likely explained, at least in part, by the contribution of alternative rotameric forms to the experimentally observed hyperpolarizabilities. In fact, the calculations for MOL2 with B3LYP shown in Figure 6 show large differences in the hyperpolarizability values for different torsion angles. To assess the importance of the solvent effects for this particular functional, we plotted in the same graphic the LC-BLYP values obtained for the gas phase. As for B3LYP, the inclusion of the PCM solvent effects


Figure 9. Difference between the calculated and experimental hyperpolarizability values for MOL3 and MOL4. The calculated values were obtained using the BLYP, B3LYP, PBE hybrid, and the LC-BLYP functionals, and also the HF method. The calculation include PCM solvent effects.

largely increases hyperpolarizability, significantly improving the agreement with experiment. To illustrate the role of the HF exchange we show in Figure 7 the hyperpolarizability computed with BLYP, B3LYP, and LC-BLYP with PCM solvent effects. The hyperpolarizabilities consistently decrease and get closer to the experimental values when HF exchange is added. The range-dependent variable HF content of the LC-BLYP functional is provides results that are superior to those of the hybrid B3LYP functional with its fixed, range-independent HF component. Comparison with Other Methods In order to compare the accuracy of the results obtained with the LC-BLYP method with other methods, eventually with less computational cost, we chose one molecule of each subset, and performed two extra calculations: a Hartree-Fock calculation and a DFT calculation using the PBE0 hybrid.46 In Figure 9 is plotted the difference between the calculated and experimental hyperpolarizability values for MOL3 and MOL4, summarizing the results obtained with all the first-principles methods with PCM solvent effects used during this study. For MOL4, which has a stronger charge transfer character, it was found that the larger the amount of the HF contribution included in the DFT functionals, the smaller the difference with respect to the experimental value. However, when only HF is considered, the hyperpolarizability values are underestimated. Although the HF values are much closer to experiment than the BLYP, B3LYP, and PBE hybrids, the long-range corrected functional LC-BLYP further improves the agreement with differences from experiment that are three times smaller than the differences obtained with HF, indicating that both exchange and correlation play a role. For MOL3, all the DFT functionals show a similar degree of accuracy, being closer to experiment than the HF results. The LC-BLYP method shows the best hyperpolarizability results when we consider the two types of molecules. Fundamental Limits In Pe´rez-Moreno et al.23 the β, µ10, and E10 experimental values were used to determine the second excitation peak, E20. The values were introduced in the simplified sum over states (SOS) expression for the off-resonant hyperpolarizability as obtained within the three-level-model ansatz.47


Figure 10. Experimental and LC-BLYP hyperpolarizability in solution, compared with the two-level model prediction.

In Table 1, the two energy values in the “exp.” column correspond to the experimental peak, E10, and to the calculated E02.23 In general, the computed absorption spectra do not show two peaks with a spacing close to the SOS predictions.23 The exception are the spectra computed in solution within the LCBLYP scheme for the azobenzene molecules. In these cases, MOL1 shows two peaks, the first being weaker than the one at higher energy. The spacing is slightly larger than predicted. MOL2 has two peaks with similar magnitudes that are superimposed. MOL4 shows two peaks with a slightly smaller spacing. As mentioned before, the peaks computed in solution within LC-BLYP lie at about 0.5 eV above the experimental energies. The three-level model used to obtain the SOS expression in the calculation of E02 assumes that there is a second transition that plays a role in the onset of the hyperpolarizability. However, the absence of the second peak from the calculated spectra suggests that the two-level model could be a more adequate approximation for this system. In a two-level model approximation, it can be shown that hyperpolarizability depends on the transition dipole moment, the oscillator strength and the energy of the first excitation 3 according to β ∝ ∆µ10f10/E10 . Figure 10 plots the right-hand side of the expression, and this is compared with computed and experimental hyperpolarizabilities. Since the hyperpolarizability values closer to experiment where obtained with TDDFT/LCBLYP in solution, we again used the same functional and include PCM solvent effects. Although the two-level model values were computed using TDDFT, whereas the hyperpolarizabilities were computed using the finite field method within DFT, the two-level model approximation predicts the right trend for the hyperpolarizability. According to this model the increase of the hyperpolarizability for MOL6 and MOL7 coincides with the increase of both the transition dipole and the oscillator strength, when compared with MOL4 (see Figure 11). From MOL6 to MOL7 there is no increase of the transition dipole or oscillator strength, leading to the conclusion that the decrease of the excitation energy is reflected in the hyperpolarizability. Comparing MOL2 and MOL5, it is again the transition dipole and oscillator strength that play a role in the increase of the hyperpolarizability value. More generally, comparing the two subsets of molecules, the replacement of the benzene ring by a thiophene or thiazole ring increases the transition dipole and oscillator strength, and through them the hyperpolarizability.

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Figure 11. Excitation energy in eV, transition dipole in Debye, and oscillator strength calculated with B3LYP (top) and LC-BLYP (bottom).

Conclusions We have studied a series of molecules with high hyperpolarizabilities. Two sets of molecules were considered due to different frontier orbital properties. The first subset is composed of azobenzene derivatives and the second set by thiophene- and thiazole-containing molecules. The first subset of molecules are expected to possess a charge-transfer character stronger than the second subset, based on the FMO spacial separation. Also, the replacement of the benzene ring by thiophene or thiazole rings with lower aromaticity, suggests the presence of stronger charge transfer in the first subset. Moreover, the larger BLA expected for benzene in comparison with thiophene or thiazole suggest that the second set of molecules will have higher hyperpolarizability values.9-11 The two sets also showed different behaviors when different exchange-correlation functionals were used. For the azobenzene molecules, the HF exchange contribution improves the hyperpolarizability results that are otherwise overestimated by a large amount. The HF contribution is less important for the molecules of the second subset, which also display the largest hyperpolarizabilities. For the first subset the values obtained with only the HF exchange interaction are much closer to experiment than the BLYP, B3LYP, and PBE hybrids. However, the long-range corrected functional LC-BLYP further improves the agreement, with differences from experiment that are three times smaller than the differences obtained with HF, indicating that both exchange and correlation are important.

Cardoso et al. Solvent effects are also important in the calculation of the hyperpolarizability, as was shown in the difference between the values obtained with the PCM method when compared with the gas phase calculations. For the azobenzene dyes, the gas phase values obtained with the B3LYP functional, are closer to the experiment, whereas the values obtained with PCM solvent effects overestimate the hyperpolarizability considerably. This indicates that the solvent effects enhance the charge transfer effect, which the B3LYP functional is known to fail to reproduce. The LC-BLYP scheme is able to overcome this weakness, largely improving the agreement with experiment when PCM solvent effects are considered. On the basis of the present results it can be said that LC-BLYP is the most reliable functional for the calculation of hyperpolarizabilities in sets of molecules displaying varying charge-transfer character. The influence of the geometry on the nonlinear optical properties is well recognized, with deviations from planarity being particularly important. For this reason we scanned the torsion angle between the two benzene rings in MOL2 between 0 and 90°, using the B3LYP functional. Although hyperpolarizability decreases with increasing torsion angle, the hyperpolarizabilities calculated for all the geometries are much larger than the experimental values, suggesting that the overestimation of hyperpolarizability is due to the calculation method. The total energy increases rapidly with the torsion angle, suggesting a small contribution of the less planar conformations to the hyperpolarizability at experimental temperatures. The results are well described by a two-level model approximation. Comparing the two subsets of molecules, the replacement of the benzene ring by a thiophene or thiazole ring increases the transition dipole and oscillator strength, and through them the hyperpolarizability. The agreement with experiment could possibly be improved with a systematic search and respective averaging of hyperpolarizability values for those rotamers with significant populations at room temperature; however, such a study is beyond the aim of the present work. Although semiempirical methods are not expected to perform well for hyperpolarizabilities, for systems larger than the systems under study these are sometimes the only computationally affordable approach. For this reason we show here the results obtained with the PM3 and PM6 semiempirical Hamiltonians. Although both methods underestimate hyperpolarizability values and fail to reproduce the large increase in hyperpolarizability from MOL5 to MOL6, when COSMO solvent effects are combined with the PM6 Hamiltonian the trend of the series is well reproduced and the values are closer to those from experiment. References and Notes (1) Zyss, J.; Ledoux, I. Chem. ReV. 1994, 94, 77. (2) Nalwa, H. S.; Miyata, S., Eds.; Nonlinear Optics of Organic Molecules and Polymers; CRC Press: FL, 1997. (3) Kuzyk, M. G. SPIE Proc. 1997, 3147. (4) Marder, S. R.; Perry, J. W.; Schaefer, P. W. Science 1989, 245, 626. (5) Clays, K.; Wostyn, K.; Olbrechtes, G.; Persoons, A.; Watanabe, A.; Nogi, K.; Duan, X.-M.; Okada, S.; Oikawa, H.; Nakanishi, H.; Bredas, J. J. Opt. Soc. Am. B. 2000, 17, 256. (6) Marder, S. R. Chem. Commun. 2006, 2, 131. (7) Marder, S. R.; Beratan, D. N.; Cheng, L.-T. Science 1991, 252, 103. (8) Marder, S. R.; Gorman, C. B.; Tiemann, B. G.; Cheng, L.-T. J. Am. Chem. Soc. 1993, 115, 3006. (9) Gorman, C. B.; Marder, S. R. Proc. Natl. Acad. Sci. U.S.A. 1993, 90, 11297. (10) Marder, S. R.; Perry, J. W.; Tiemann, B. G.; Gorman, C. B.; S. Gilmour, S. L. B.; Bourhill, G. J. Am. Chem. Soc. 1993, 115, 2524.

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Computational Study of Molecules with High Intrinsic

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