13512
J. Phys. Chem. B 2006, 110, 13512-13522
Influence of Isomerization on Nonlinear Optical Properties of Molecules T. Kinnibrugh,† S. Bhattacharjee,‡ P. Sullivan,‡ C. Isborn,‡ B. H. Robinson,*,‡ and B. E. Eichinger‡ Department of Chemistry, P.O. Box 351700, UniVersity of Washington, Seattle, Washington 98195-1700, and Chemistry Department, New Mexico Highlands UniVersity, Las Vegas, New Mexico 87701 ReceiVed: February 8, 2006; In Final Form: May 3, 2006
The influence of rotational and geometrical isomerism on the nonlinear optical (NLO) properties, specifically the first-order hyperpolarizability β, of chromophores of current interest has been investigated with density functional theory (DFT). In the first of this two-part study, the rotational isomerism of a linear chromophore was explored. Calculation of the torsion potentials about two of the rotatable and conformation-changing single bonds in a chromophore demonstrated the near equality of the molecular energies at 0° and 180° rotational angles. To explore the consequences of this near conformational energy degeneracy to NLO behavior, the eight low energy rotational isomers of FTC [Robinson, B. H.; et al. Chem. Phys. 1999, 245, 35] were investigated. This study provides the first-reported DFT-based calculation of the statistical mechanical average of β over the conformational space of a molecule having substantial nonlinear optical behavior. The influence of the solvent reaction field on rotameric populations and on the β tensor is reported. In the second part, two molecules having two donors and two acceptors bonded respectively in ortho and meta positions on a central benzene ring are shown to have substantially different β tensors. These two so-called molecular Xs have different highest occupied molecular orbital to lowest unoccupied molecular orbital (HOMO-LUMO) distributions, and consistent with expectations, it is found that the larger βzzz is associated with a large spatial asymmetry between the HOMOs and LUMOs. Large hyperpolarizability correlates with the HOMO concentrated on the donor groups and the LUMO on the acceptor groups.
Introduction The substantial effort that is being directed toward synthesis of chromophores exhibiting desirable nonlinear optical properties is becoming more productive with increased use of both ab initio and semi-empirical quantum chemistry calculations. Using current methods, it is possible to compute the electronic properties of quite large isolated (gas-phase) molecules with modest computer resources in a small fraction of the time that is required to synthesize, purify, and characterize the typical molecules of current interest. Quantum chemical calculations provide insight into the electronic properties, e.g., highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) distributions and the dependence of the electron density on the electric field, that aid in understanding the general features of molecular architecture that are responsible for large nonlinear optical (NLO) behavior. While these calculations do not provide definitive predictions of nonlinear optical properties under experimental conditions (which are dynamic measurements, usually involving dissolution of the chromophore in an appropriate solvent or dispersion in a polymer matrix at a variety of temperatures), they do provide useful guidance to experimental efforts by helping to narrow the choices for synthetic attention. The limitations of the predictive power of computational methods owe much to the size of the molecules and the necessity to treat them approximately, using relatively small basis sets. Furthermore, there are condensed phase effects, such as the reaction field, that * To whom requests for reprints
[email protected]. † New Mexico Highlands University. ‡ University of Washington.
should
be
sent:
significantly perturb the measured properties, causing large departures from gas-phase predictions.1 Control over molecular architecture, including geometrical isomerism, is in the hands of the synthetic chemist, and knowing how orbital structure is related to atomic arrangements is a significant aid to designing NLO materials. Unfortunately, control over rotational isomerization cannot be achieved with the same certainty. Many of the organic NLO materials that are currently envisioned consist of molecules that are expected to have thermal populations of rotational isomers (rotamers). Concomitantly, the different rotamers will most likely have different NLO properties, and this adds a layer of complexity to the theoretical analysis of materials properties and device behavior that is not usually considered. The methods for obtaining an average over the angular distribution of chromophores in poled materials are well-known. Apart from seminal studies on the influence of twist angle on biphenyland stilbenelike systems,1-4 less attention has been paid to the average over rotamers. In the first part of this work, we consider this average for FTC,5 a molecule of sufficient complexity to be interesting for its NLO characteristics yet simple enough so that a thorough exploration of rotational states can be performed. FTC is a candidate NLO material and has been well-studied, making it of great interest for more continuing theoretical work. As chromophores of increasing complexity and sophistication are being developed, molecular designers are confronted with a large number of interactive and potentially conflicting influences on electronic structure. The elegant empirical rules of organic chemistry that have been used so successfully to guide molecular design become increasingly difficult to apply with certainty as the molecular complexity increases. The outcome
10.1021/jp0608271 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/20/2006
Isomerization and NLO Molecules
J. Phys. Chem. B, Vol. 110, No. 27, 2006 13513
Figure 1. Some geometrical isomers of molecular Xs. In type II chromophores, each acceptor is para to one donor and meta to the other, while in type III molecules a second donor is ortho to each acceptor. Type IV has a center of inversion and will not have a nonzero firstorder hyperpolarizability.
TABLE 1: Dependence of the Dipole Moment of p-Nitroaniline on Functional and Basis Set (Dipole Moments in Debye) basis set functional
MIN
DN
DND
DNP
BLYP HCTH PBE RPBE
6.734 6.726 6.751 6.660
8.202 8.258 8.180 8.121
7.572 7.603 7.530 7.475
7.550 7.578 7.502 7.448
of a competition between two different effects working in opposition may be difficult to predict using only qualitative, descriptive rules. A case in point is provided by molecules having two donors and two acceptors bonded to a central hub. These molecular Xs of are of considerable current interest. The theme of these molecules is illustrated in Figure 1. In the type II, III, and IV chromophores, two pairs of electron donoracceptor (D-A) groups are attached to a bridging group, most simply, a benzene ring, in different ways to achieve specific NLO properties. In the type II isomer, the D and A groups are meta to one another, and in types III and IV, they are ortho. On looking at the single-double bond alternation alone in the molecules to be considered, one might conclude that the ortho substituted molecule would have a larger βzzz (the component of the β tensor collinear with the dipole vector). Is this correct? Computations can help to answer this question, and in the second part of the paper, we discuss the results bearing on βzzz for type II and III. (If the aim is to design two-photon materials requiring a large second-order hyperpolarizability, but no first-order hyperpolarizability, symmetric molecules such as type IV are of interest. This isomer will not be considered here because the point group of the molecule, C2h, has an inversion operation, which forces the first-order hyperpolarizability to vanish.) Methods Almost all calculations were performed with DMol3, a density functional code developed by Freeman, Delley, and coworkers.6-11 Preliminary calculations using a variety of generalized gradient approximation (GGA) functionals on p-nitroaniline (pNA) showed that one of the most recent functionals, the revised Perdew-Burke-Ernzerhof (RPBE)12 of Hammer et al.13 gave results in close agreement with the MP2 calculations of Sim et al.14 on pNA. In particular, Sim et al. determined the dipole moment of pNA to be in the range 6.87-7.82 D, calculated with MP2 and their best basis set, with variations in the calculated moment depending on whether the geometry of the (constrained) planar molecule was optimized with HF or MP2. We calculated the dipole moment of pNA with all available basis sets, called MIN, DN, DND, and DNP in DMol3, and four different functionals, with the results shown in Table 1. (MIN uses one orbital for each occupied orbital in the free atom; DN is a double-numeric basis set, using approximately two orbitals for each occupied orbital in the free atom; DND includes polarization functions on all atoms except H; and DNP includes polarization on H as well.) Focusing on only the best
available basis set, the RPBE functional gave the smallest dipole moment of the functionals used, giving the best agreement with the experimental value (6.23 D in benzene to 6.84 D in dioxane15). There is perhaps little to recommend one functional over another for the work in this paper in the absence of an extensive comparison with additional experimental dipole data. Furthermore, a reliable set of experimental hyperpolarizabilities for molecules of the types investigated here is unavailable, so we must be content to select one functional and apply it consistently to all the structures of interest. Calculations of the polarizability and hyperpolarizability tensors for pNA with this functional and basis set gave results between the HF/MP2 and MP2/MP2 results of Sim et al.14 The experimental gas-phase value of the product of the dipole moment and hyperpolarizability for pNA has been determined by EFISH.16 Density function theory (DFT) calculations17 give good agreement with the experiment. Unfortunately, good agreement on small molecules does not guarantee similar success with larger ones. The Namur school has seriously questioned the absolute accuracy of DFT for large molecules,18,19 but we believe that the molecules considered here are under the threshold where serious problems arise. Since our interest is in relative comparisons among isomers of various kinds, reliance on absolute accuracy is not required. Furthermore, we want to assess reaction field effects, and this puts severe limitations on computational feasibility. For these reasons, DFT is deemed to be adequate for our purposes. The geometries of the molecules were optimized, sometimes with a cascade of basis sets from the least to the most accurate, using the RPBE functional, with the final structure converged with the DNP basis set to an SCF energy tolerance of 1 × 10-5 hartee and the geometry to an energy tolerance of 2 × 10-5 hartee, gradient to 0.004 hartee/Å, and displacement to 0.005 Å (the medium convergence criteria of DMol3). The convergence criteria are satisfied if the energy is converged together with either the gradient or displacement. Integrals were performed with a cutoff of 5.5 Å. All subsequent calculations of the static electronic properties were performed on molecules with fixed geometries. The accuracy of the calculations was increased to converge the energy to 1× 10-6 hartee with integrations cut off at 7.0 Å. The energies and dipole vectors of a given molecule were calculated by the finite field method, with the procedure of Sim et al.14 being used to determine polarizability and hyperpolarizability tensors from the dipole vectors. Field strengths set at e ) 0.001 au and 2e yielded polarizability tensors, R, symmetric to better than about 1 part in 104. The Kleinman20 symmetry of β was generally not as perfect as that of R, but in no case did the asymmetry amount to more than about 0.01% of the largest component of β. (For the highly polarizable molecules studied here, β is dominated by βzzz.) Further optimization of the field strength was not deemed productive given the intrinsic approximations involved in the comparison of calculated gas-phase electrostatic properties with dynamic condensed-phase experimental results. We use the “phenomenological” convention21
µi(E) ) µi(0) +
βijkEjEk + ... ∑j RijEj + ∑ j,k
(1)
for the data analysis. The linear combinations of dipole vectors were constructed so as to eliminate all terms of orders less than e4 from R and β. In the data analysis, the coordinate frame was rotated to align the dipole vector with the positive z-axis, and values of βzzz to be reported are in this frame. The value of the angular averaged
13514 J. Phys. Chem. B, Vol. 110, No. 27, 2006
Kinnibrugh et al.
Figure 2. NLO chromophore FTTC. The thiophene and vinyl groups in the backbone are joined by rotatable single bonds, which create the possibility for conformational variety. The rotational potentials for two of these, indicated as A and B in the figure, have been studied.
hyperpolarizability, as is measured by hyper-Rayleigh scattering (HRS), was calculated using eq 12 of Cyvin, Rauch, and Decius,22 i.e.,
βHRS ) x〈βFFF2〉 + 〈βFGG2〉
(2a)
so that
βHRS2 )
6
16
38
βiiiβijj + βiij2 + ∑ βiii2 + 105∑ ∑ 35 i 105 i*j i*j 16
20
βiijβjkk + βijk2 ∑ 105ijkcyclic 35
(2b)
Here, the components βFFF and βFGG are in the lab frame appropriate for analysis of HRS (see ref 20 for details). Calculations of molecular properties with reaction field effects were performed with COSMO in the DMol3 package as well as with the polarized continuum model (PCM) method in Gaussian03. Parameters used in these calculations are reported below in the appropriate section. Rotational Potentials Relatively little attention has been paid to the conformational averaging that typical NLO chromophores are likely to experience at room temperature and above where devices are expected to operate. Conjugated backbones that connect donor and acceptor groups in these chromophores typically consist of a combination of rigid rings and 1,2-vinylic linkages, with rotatable single bonds joining the two. A representative example of such a chromophore, called FTTC, is depicted in Figure 2. While investigating this molecule, it was observed that different conformers yielded different hyperpolarizabilities while their energies were virtually identical. This prompted us to first look at the rotational potentials for a few bonds of FTTC to get an initial understanding of the importance of conformational variability. For this purpose, two bonds were selected for exploration: the bond between the thiophene and vinyl unit on the left (labeled A in Figure 2) and that between the vinyl unit and the substituted furan ring on the right, labeled B. These two bonds are representative of those found in a variety of chromophores under current investigation in these laboratoriess the four thiophene-vinyl single bonds are essentially equivalent, so investigation of the one suffices to characterize the torsion potential of all. For present purposes, the most probable rotational states near 0° and 180° are labeled transoid and cisoid, as these states bring the double bonds adjacent to the rotatable bond into trans and cis orientations relative to one another. (To avoid confusion with the geometrical cis-trans isomerism of the double bonds, which is not considered here, our usage of cisoid and transoid is conventionally descriptive of the rotational states of nominal single bonds.)
The energies of the most probable rotamers were examined by first manually rotating the A bond of the molecule of Figure 2 into states separated by about 20°, with subsequent calculation of the energies of each conformation at a fixed geometry. The two barriers for the A and B bonds, computed for rotamers having fixed internal coordinates save for the rotated bond, are both approximately 12 kcal/mol above the minima. The geometries of the rotamers were subsequently optimized while holding the torsion angle fixed. The low energy geometries near 0° and 180° were then fully relaxed without constraints to give the transoid conformer at -1.6°, being only 0.97 kcal/mol higher in energy than the cisoid at 178.4°. The barrier with the relaxed geometry is near 90° and is about 11 kcal/mol above the minimum. Given this barrier, simple kinetic theory [Rate ) (kT/ h) exp(-E/kT)] suggests that the lifetime of a given rotamer is on the order of 2.5 µs in the gas phase and is probably about the same in a low-viscosity solution. Since there is only a small difference in the geometry of the two stable rotamers, it is likely that crankshaft-type rotations about the vinyl units will occur in condensed phases, and it is therefore necessary to consider the manifold of rotational isomers of chromophores to comprehend their properties in device applications. Changes in the bond length alternation (BLA) that accompany changes in rotational state are also interesting for the insight they provide into the extent of electron delocalization.23,24 The difference of length between the A bond of FTTC in Figure 2 and the adjacent double bond is 0.048 Å in the transoid conformation and 0.053 Å in the cisoid. These values of the BLA are right at the extremum suggested by Marder et al.24 to be optimal for maximizing β in polyene backbones. The correlation between β and BLA suggests that 0.048 Å is closer to the optimum than is 0.053 Å for the heterogeneous backbone studied here. Given that the BLA difference is about 10%, whereas the difference in energy is within kT at room temperature, BLA appears to be a more sensitive signpost to larger β than are the relative energies of the conformers. The rotation about bond B at the furan ring in FTTC is characterized by a slightly higher barrier, about 14 kcal/mol (relaxed geometries), but the minima at 177.9° (cisoid) and 1.3° (transoid) are similar to those of bond A. The cisoid conformer is a mere 0.36 kcal/mol above the transoid. The BLA between the rotating single bond and the adjacent vinyl double bond is essentially constant at 0.034 Å for the two stable conformers, suggesting that there is no significant difference in electronic structure between the two. These observations suggested that a more thorough investigation of the NLO properties of conformers would be informatives we would like to know how much hyperpolarizabilities for molecules of this class change as a function of conformation. To solve this problem, one might be drawn to the rotational isomeric state (RIS) model of polymer science, which enables the calculation of molecular properties over a number of
Isomerization and NLO Molecules
J. Phys. Chem. B, Vol. 110, No. 27, 2006 13515
Figure 3. FTC molecule. The nontrivial rotatable bonds are labeled a, b, and c. Two rotational states, located near 0° and 180° for each of these bonds, are almost equally probable, giving a total of eight lowenergy conformers.
conformations that is exponential in the number of bonds.25 Unfortunately, the RIS method is valid only if the molecular behavior can be described by additive properties localized at the level of bonds. Molecules with delocalized π-systems are not expected to obey this fundamental assumption. (The BLA
is proof enough that one ought not apply RIS to these molecules. The BLA depends on chain length as well as donor and acceptor structures,24 both of which violate the fundamental assumptions of RIS.) We are therefore moved to study each conformer individually, which becomes tedious for more complicated molecules. A method based on conformational searching with a good force field description of the molecules will be useful for this task. However, as pointed out, these molecules are not expected to be well-described by standard force fields that have not been specifically parametrized for delocalized π-systems. We have therefore opted, in this initial study, to to perform a thorough quantum mechanical investigation of a molecule
TABLE 2: Energetic and Electrostatic Properties of FTC Conformers conf
statistical mechanical ave
designat
E + 1775 (hartree)
∆E (kcal/mol)
µ (D)
/A3
Rzz/A3
βHRS/10-30 (esu)
βzzz/10-30 (esu)
cct(1)
-0.565611
1.30
25.97
119.9
267.7
176.0
386.8
ttt(2)
-0.567121
0.36
21.29
108.2
192.5
136.9
168.2
ttc(3)
-0.567690
0.00
25.78
111.5
242.1
130.6
314.5
tct(4)
-0.566249
0.90
26.15
119.3
271.1
175.5
414.4
ccc(5)
-0.564259
2.15
25.12
117.2
249.0
160.5
309.5
tcc(6)
-0.565850
1.15
23.36
111.4
222.9
162.8
255.1
ctt(7)
-0.567075
0.39
23.64
114.8
230.9
158.2
257.1
ctc(8)
-0.567451
0.15
26.24
115.7
255.2
142.5
330.4
0.32
24.74
113.3
237.3
144.8
291.8
13516 J. Phys. Chem. B, Vol. 110, No. 27, 2006
Kinnibrugh et al.
TABLE 3: Comparison of Energies and Dipole Moments for FTC Conformers Calculated in Vacuum and in Chloroform and Water with COSMO vacuum
COSMO chloroform
COSMO water
conf
E + 1775 (hartree)
∆E (kcal/mol)
µ (D)
E + 1775 (hartree)
∆E (kcal/mol)
µ (D)
E + 1775 (hartree)
∆E (kcal/mol)
µ (D)
1 2 3 4 5 6 7 8
-0.565611 -0.567121 -0.567690 -0.566249 -0.564259 -0.565850 -0.567075 -0.567451
1.305 0.357 0.000 0.904 2.153 1.155 0.386 0.150
25.97 21.29 25.78 26.15 25.12 23.36 23.64 26.24
-0.592635 -0.593995 -0.594210 -0.593191 -0.591246 -0.592599 -0.594370 -0.594090
1.089 0.235 0.100 0.740 1.960 1.111 0.000 0.176
33.82 27.76 33.49 34.08 32.72 30.38 30.91 34.04
-0.608589 -0.609547 -0.609605 -0.608990 -0.607350 -0.608069 -0.610245 -0.609698
1.039 0.438 0.402 0.788 1.817 1.365 0.000 0.343
38.53 31.64 38.09 38.88 37.31 34.55 35.31 38.71
analogous to FTTC but containing only three rotatable bonds. The molecule with one fewer vinyl-thiophene unit, FTC, has eight low-energy conformers. Furthermore, FTC has been important in the development of a large class of chromophores and is interesting to study in its own right.5 Rotational Averages for FTC The eight conformers of FTC, obtained by rotating about the three nominally single bonds labeled a, b, and c shown in Figure 3, were constructed in successionseach conformer was subsequently optimized without constraints to the nearest local miminum. The hyperpolarizability of each was then determined with the use of the procedure described above. Table 2 contains the results. Inspection of the table reveals that the dipole moments range from 21.3 to 26.2 D, while the hyper-Rayleigh β varies from 130.6 × 10-30 to 176.0 × 10-30 esu, which is a band of values amounting to 30% of the statistical mechanical average (calculated from simple application of Boltzmann probabilities calculated from the relative energies in Table 2). The variation of βzzz is larger, ranging from a low of 168.2 × 10-30 esu to a high of 414.4 × 10-30 esu. Inspection of the structures of the isomers quickly reveals the reason for the trends: as the molecule winds up on itself, the zzz component of β is partially transferred to the transverse components of the tensor, and these contribute substantially to the hyper-Rayleigh average. Thus, the hyper-Rayleigh β is less sensitive to conformational variation than is βzzz. At 298.16 K, the statistical mechanical average of the hyperRayleigh βHRS, 〈βHRS〉, is 144.8 × 10-30 esu, as shown at the bottom of Table 2. At the same temperature, the statistical mechanical average of βzzz is 〈βzzz〉 ) 291.8 × 10-30 esu, more than 25% less than the maximum value in Table 2. This difference is significant when comparing two different molecules with the aim of deciding which is apt to be the better experimental target, especially when the two are not homologous so that equivalent rotational conformers are not obvious. It is standard practice to report values for the hyper-Rayleigh scattering intensities, βHRS, for NLO chromophores relative to the CHCl3 solvent. In a separate study on small molecules,17 higher accuracy DMol3 calculations on chloroform at a variety of field strengths gave the best estimate βHRS ) 0.039 × 10-30 esu, with the field strength e giving optimal numerical accuracy at 0.008 au. The dipole moment was found to be 1.01 D, in close agreement with the tabulated value of 1.04 D.26 The same set of calculations gave the polarizability r ) diag(9.3 9.3 6.3) Å3. Using the computed βHRS for CHCl3, the calculated ratio of the averaged hyper-Rayleigh hyperpolarizability of FTC to that of chloroform is 5300, while the measured ratio using an excitation wavelength of 780 nm is 9229.27 This discrepancy of a factor of about 74% between the measured and predicted βHRS ratios is not uncommon and is most likely a reflection of
TABLE 4: COSMO Results for Polarizabilities and Hyperpolarizabilities of FTC Conformers in Chloroform conf
µ (D)
〈R〉/A3
βHRS/10-30
βzzz/10-30
1 2 3 4 5 6 7 8 SM ave
33.82 27.76 33.49 34.08 32.72 30.38 30.91 34.04 31.87
173.1 154.8 160.1 173.0 168.4 165.4 164.9 165.6 163.2
582.3 427.8 441.1 573.5 538.3 527.9 509.4 489.8 485.5
1401.2 533.5 1056.0 1348.5 1045.0 847.1 833.8 1132.9 950.8
the acknowledged shortcomings arising from the application of static gas-phase quantum calculations to the prediction of the time-dependent behavior of solvent and solute. To address the reaction field contribution to the static hyperpolarizabilities of the conformers, we performed additional calculations using established methods. Reaction Field Calculations for FTC Conformers The polarization of the medium in which a polar molecule is dispersed acts to enhance the dipole moment of the molecule, as was shown many years ago by Onsager.28 Newer methods, particularly the COSMO29 functionality in DMol3 and PCM30,31 in Gaussian03,32 provide estimates of electrostatic solvent effects based on distributed molecular charges. These theories, furthermore, provide for inhomogeneous reaction fields that result from the atomistic nature of the solvent and are doubtless superior to the older Onsager theory. The input parameters required for COSMO calculations are the solvent dielectric constant and radius; default values for all other parameters were used. The dielectric constant was set to 4.8 for CHCl3, and the radius was 2.0 Å. External fields were applied just as for the vacuum calculations to give the results shown in Tables 3 and 4. (We are not aware of previously reported calculations of hyperpolarizability with the COSMO option in DMol3. Given the absence of a precedent, we proceeded cautiously by attempting a determination of the terms in the perturbation expansion of the dipole vector from finite differences around a homogeneous external field applied in vacuo, with a strength that was selected to reproduce the dipole moment induced by the COSMO reaction field on the zero field molecule. This proved to be a bad idea, as the polarizability calculated from the slope of the dipole vector at the offset field was severely contaminated by higher order terms in the expansionsdiagonal elements were not positive. Happily or fortuitously, the COSMO option worked well with the finite field method and required no further adjustments.) The most remarkable aspect of these results is that the relative energies of the conformers do not change very much as a result of the reaction field. The relative ordering of the conformers from lowest to highest energy changes a bit in going from
Isomerization and NLO Molecules
J. Phys. Chem. B, Vol. 110, No. 27, 2006 13517
Figure 4. (A) Correlation between the energies calculated with DMol3/COSMO29 for FTC conformers in chloroform and energies of the same conformers in vacuo. (B) Same as part A, except that the solvent is water. (C) DMol3/COSMO dipole moments of FTC rotamers in chloroform correlated with the vacuum dipole moments. (D) Correlation between vacuum zzz-components of hyperpolarizabilities calculated with Gaussian03 (ordinate) and DMol3 (abscissa).
vacuum to chloroform to water, but nothing very dramatic happens. While the absolute energy differences between vacuum and COSMO energies are many times RT, the magnitudes of the energy differences between the conformers in the solvents are about the same as in vacuo. Experience with molecular dynamics simulations suggests that solvent packing effects can easily be as large a perturbation of the relative conformer energies as reaction field effects of this magnitude, meaning that one should not place too much faith in the direct applicability of the COSMO calculations to the statistical population of conformers in any real system. The correlation of COSMO energies with vacuum energies can be seen in Figure 4A for chloroform and Figure 4B for water. The correlation of dipole moments in chloroform and vacuum is displayed in Figure 4C. It is these fairly strong correlations that preserve, to a large extent, the relative energy ordering of the conformers. A plot (not shown) of the vacuum dipole moment against vacuum energy from Table 3 reveals that the two are not perceptibly correlated, which reinforces the observation that the population of conformers in the presence of a reaction field is not biased toward those conformers having the larger dipole moments. This is a reflection of the fact that, in this theory, the reaction field responds to the entire charge distribution of the molecule, not to just the overall dipole moment.
The next observation to be gleaned from the results shown in Table 4 is that the average βHRS of the conformers increases approximately 2.4-fold in chloroform compared with the vacuum value in Table 2, while the average βzzz is larger by a factor of about 3.25! A difference between the experimental and calculated hyperpolarizability βHRS cited above is far less than thiss apparently, COSMO overestimates the reaction field enhancement of the electrostatic moments of the chromophore. To test whether this is endemic to the method, we also performed calculations with Gaussian03, using the B3LYP functional and 6-31G* basis set with keywords SCRF ) PCM and solvent ) chloroform. The results are shown in Table 5. The polarized continuum model (PCM) reaction field method in Gaussian03 gives an even stronger dependence on the reaction field, with βHRS larger than the vacuum value by a factor of about 3.5. We find excellent agreement between the B3LYP (Gaussian03) and PBE (DMol3) values for vacuum hyperpolarizability tensors of the FTC conformers, as a comparison of Tables 2 and 5 reveals. A graphical representation of the correlation between the zzz-components of the tensors for the two different computational methods is shown in Figure 4D. Clearly, the difference between the reaction field results with the PCM functionality in Gaussian03 and COSMO in DMol3 is not
13518 J. Phys. Chem. B, Vol. 110, No. 27, 2006
Kinnibrugh et al.
TABLE 5: Comparison of Gaussian Vacuum and Chloroform Dipole Moments and Hyperpolarizabilities gaussian vacuum
gaussian chloroform
conf
∆E (kcal/mol)
µ (D)
βHRS/10-30 (esu)
βzzz/10-30 (esu)
∆E (kcal/mol)
µ (D)
βHRS/10-30 (esu)
βzzz/10-30 (esu)
1 2 3 4 5 6 7 8 SM ave
2.24 1.02 0.00 1.30 2.75 1.52 1.21 0.56
24.27 20.11 24.10 24.34 23.57 21.96 22.28 24.60 23.63
172.4 136.8 134.0 173.8 159.4 162.5 156.7 145.5 142.2
378.1 168.4 321.5 409.0 304.6 254.6 252.8 336.8 308.7
1.92
31.65
581.0
1282.6
0.00 1.13 2.26 1.33 0.46 0.35
31.16 31.68 30.91 28.54 29.34 31.87 30.89
453.6 585.4 536.2 534.2 512.4 496.9 490.5
1086.8 1379.3 1041.2 1279.0 825.9 1156.6 1082.4
derived from differences in the vacuum values but results instead from the different treatments of the reaction field. Whether one relies on vacuum calculations or reaction field results, the results suggest that statistical averaging over rotamers will be required to understand the properties of these types of semiflexible NLO chromophores. Is this also to be expected in device applications, where poling of the molecules will be required to achieve macroscopic NLO behavior? To answer this question, we note that, for a given orientation of the dipolar axis, the conformers with the largest static dipole moments will have the lowest energy in a static poling field. Conversely, application of a field will perturb the energies of the conformers, favoring those having the largest dipole moments. Will this preferentially enhance χzzz, the first-order macroscopic hyperpolarizability? The answer seems to be yes, as inspection of Figure 5 reveals a strong correlation between βzzz and the dipole
Figure 5. Rotamers with the large dipole moments apt to have large hyperpolarizability tensor components in the direction of the dipole vector, shown by this correlation for FTC.
moment for the FTC conformers. That is, a poling field will enhance the population of conformers having the largest dipole moments and not only will these be most strongly aligned, but they are the conformers that are most apt to have the largest βzzz. This argument is mitigated by the overall magnitudes of the fields and temperatures that are feasible. Typical laboratory poling fields are on the order of 100 V/µm (ca. 2 × 10-4 au, some 10 times smaller than the smallest fields used in these calculations), so that µE/kT = 2 at room temperature for dipoles of the magnitudes discussed here. Fields of this magnitude are insufficient to induce more than a small field-enhanced population of favorable conformers having dipole moments that differ by only a few debye. (The reaction field does not contribute to orientation, as shown by Onsager.28)
To circumvent these complications arising from thermal statistics, it may be beneficial to design molecules with the greatest possible rigidity. One way that this can be done is to construct backbones that are rigidified by rings.33 Another, potentially more accessible, method to enhance rigidity is to engineer steric constraints into molecules. This is being done currently with multiple donor/acceptor molecules, the molecular Xs referred to above. Geometrical Isomerism of Molecular Xs Three geometrical isomers of molecular Xs, each consisting of a pair of identical donors and identical acceptors arranged symmetrically, are shown in Figure 1. (We exclude the symmetrical substitution 1,2,3,4 and the asymmetrical 1,2,3,5 in the expectation that these will be too crowded for the donor and acceptor considered below.) In type II structures, the adjacent D and A groups are meta to one another, and in type III, they are ortho. Structures of this type have attracted attention for several years34-36 in the expectation that the components of β transverse to the dipolar axis will be large. Large transverse components in β are presumed to enhance the macroscopic hyperpolarizability, χzzz, because the molecules will contribute to the z-axis hyperpolarizability in a device even if imperfectly aligned. There may be an added advantage to these structures, as pointed out above, in that the molecules tend to be sterically crowded and may, therefore, have restricted conformational freedom. On the other hand, too much steric crowding can lead to twisting of the conjugated backbone, which might decrease conjugation and diminish β. However, recent theoretical work37 shows that twisting enhances the magnitude of β. Given all these factors, we would like to separate out just the substitution pattern, exemplified in the type II and type III chromophores, and ask which is most conducive to large β. The several competing influences mentioned above are very difficult to analyze in the absence of molecular models and quantum calculations. Previous work on structures of this class has been limited to semi-empirical methods.34-36 Dipole moments varying from 1.9 to 11.4 D35 for a wide range of structures were calculated with AM1. One might be skeptical of these results, as several of the molecules in this set might be expected to be significantly more polar than pNA. The small dipole moments that were likewise calculated by Qin, Bai, and Ye36 were ascribed to nonplanarity and steric hindrance as well as to the properties of the donors and acceptors. Given these somewhat surprising results, it is important that ab initio methods be applied to molecules such as these to determine whether the observations are repeatable or are artifacts of the chosen method and whether the explanation offered is supported. DFT calculations on these molecules help to complete the picture of their electronic structure and NLO properties. Molecular Xs have been of considerable experimental interest, so we anticipate that any additional information that we provide
Isomerization and NLO Molecules
J. Phys. Chem. B, Vol. 110, No. 27, 2006 13519 TABLE 6: Electrostatic Moments of Type II and Type III Molecular Xs
Figure 6. Structure of the type I chromophore. This is just one-half of the type II chromophore, obtained from the latter by deletion of one set of donor and acceptor groups.
on these aspects of the molecules will be useful. Furthermore, relatively few comprehensive studies of NLO materials have been done to compare semi-empirical and DFT results, with Prezhdo’s38 work on the Marder et al.24 polyenes being an exception. While we will not compare DFT calculations directly withtheprevioussemi-empiricalresultsonequivalentmolecules,34-36 the following results should contribute to the understanding of structure-NLO property relations for this class of molecules. The Xs considered here are based on a sequence variant of FTC (a type I chromophore, shown in Figure 6), with type II and type III molecules shown in Figure 7. Geometries were optimized as described above, and the structures shown in this figure are the optimized geometries. Both isomers are somewhat twisted from planarity, with the twist of the type III molecule being a little greater than that of the type II, owing perhaps to repulsion between the proximal cyano groups on the two acceptors. These structures are the most stable of a variety of conformers that were investigated. Alternative rotation angles about the single bonds pendant to the phenyl ring precipitate an unfavorable steric interaction between the phenyl hydrogen and the R-H atom of the vinyl group and result in a greater twist out of planarity. The rotation angles in the optimized structures range from about 11° to 19° for type II and from 17° to 24° for type III. Other rotational isomers obtained by rotating about the vinylthiophene bonds may have acceptable energiesswe have not explored all conformers for these molecules, in the sense of the FTC study above. Alternative rotations about the vinylfuranyl single bonds appear not to be favorable, as the cisoid conformer (see the FTC conformer section for nomenclature) has a steric interaction between the R-H atom of the vinyl group and the 3-cyano group of the dihydrofuranyl ring as well as between the β-H of the vinyl group and the nearest phenyl H that together are only relieved by twisting out of plane. (The steric hindrance around the thiophene groups of FTC and FTTC
isomer
µ (D)
βHRS/10-30 (esu)
βzzz/10-30 (esu)
FTC (cct conf) type I type II type III
26.0 25.3 28.8 39.0
176 196 309 250
387 411 492 137
is substantially less severe than that for their vinyl-furanyl bonds, which accounts for the different behavior of the vinylfuranyl rotations in the linear and X molecules.) The electronic properties of the two X isomers, as well as the type I molecule and FTC for comparison, are shown in Table 6. The differences between the dipole moments and hyperpolarizabilities of the two isomers are substantial. The fact that the dipole moment of the type II molecule is about 35% larger than that of the type III variant is surprising, given that the angle between the two acceptors is smaller for the type III molecule than it is for the type II. This may be due, at least in part, to the head-to-head orientation of the two tricyanofuranyl acceptor groups in the type III chromophore (see Figure 7). To help understand the electronic structure of the two chromophores, the HOMO and LUMO orbitals of each are shown in Figure 8A and B. It is tempting to conclude that the larger value of βzzz for the type II chromophore (see Table 3) is a consequence of the strong asymmetry between the HOMO and LUMO orbitals of this isomer. This is consistent with a two-state model in which the difference between the LUMO and HOMO dipole moments is dominant.39 The orbital asymmetry of the type III isomer is less pronounced, as is consistent with a reduced tendency of electrons to migrate to the acceptor end of the molecule when the electronic structure is perturbed by an external field. If one looks just at the sequence of conjugated bonds through the phenyl ring, just the opposite conclusion might be reached. When donor and acceptor groups are ortho to one another, strict alternation of single and double bonds through the ring can be maintained. This is not the case with the type II chromophore. Here is an example of theory giving a different result from elementary π-bond conjugation arguments. One might question whether the twist out of plane of the type III chromophore is responsible for the smaller βzzz. To help to answer this question, a conformer of the type I chromophore was constructed with torsion angles about the vinyl-phenyl single bonds of 27.7° (donor side) and 21.1° (acceptor side). These angles are comparable to the twist angles in the type II and III chromophores, as noted above. The hyperpolarizability
Figure 7. Structures of the type II and type III molecular Xs that are investigated here. The para-related donor-acceptor pairs, together with the central phenyl group, comprise the type I chromophore. This is a sequence isomer of FTC, as comparison with Figure 3 will show.
13520 J. Phys. Chem. B, Vol. 110, No. 27, 2006
Kinnibrugh et al.
Figure 8. (A) HOMO (left) and LUMO (right) orbitals for the type II chromophore. Note the concentration of the HOMO on the donor side and that of the LUMO on the acceptor side. In this figure, the modulus of all orbitals is 0.03. (B) HOMO (left) and LUMO (right) orbitals for the type III chromophore. The concentration of the HOMO and LUMO on the donor and acceptor, respectively, is not as pronounced as in the type II structure. This is believed to be the primary reason for the type III chromophore having a smaller βzzz.
calculations yielded βHRS ) 249 × 10-30 esu and βzzz ) 518 × 10-30 esu, which are both larger than those reported in Table 6 for the minimum energy conformations. The same phenomenon was observed for FTC, the sequence isomer of the type I chromophore. A conformer with a twist angle about the vinylthiophene single bonds of 15.9° (donor side) and -15.5° (acceptor side) gave βHRS ) 198 × 10-30 esu and βzzz ) 436 × 10-30 esu, which are enhancements over the values in Table 6 of about 13%. Given these comparative results, it appears that the type III chromophore is not deficient on account of the outof-plane twist. If anything, the linear molecule results suggest that a planar version of the type III chromophore would have an even smaller hyperpolarizability. For the twisted versions of both FTC and the type I isomer, bond lengths were left at the values in the energy minimized structures; i.e., the structures were not re-optimized for the changed torsional conformers. This was done so as to isolate the dependence on twist angle. The HOMOs and LUMOs of the two isomers are shown in Figure 9. There is little to be read from visual inspection of the orbitalssthey appear not to be very different for the planar and twisted conformers of the two molecules. (There may be a slight enhancement of the LUMO density on the acceptor side for the twisted conformers.) It appears that the dominant influence of the twist, at least for
this range of angles, is to raise the energy of the HOMO for the type I chromophore, which leads to an apparent narrowing of the gap. According to the two-state model,39 this should increase β. Changes in the energies of the orbitals of FTC are less strongly perturbed by the twist, which may be simply a reflection of the fact that smaller twist angles were imposed on the FTC for this calculation. The observation that a twist enhances hyperpolarizability was not anticipated in the early work on NLO chromophores,40 but our findings of enhancement for both the type I and FTC twisted chromophores are consistent with the results of Albert et al.2 and Keinan et al.37 on the influence of twist on the hyperpolarizability of merocyanine dyes, as well as with the somewhat less conclusive results of van Walree et al.3 on twisted conformers. The average BLA between the single bonds pendant from the aromatic ring and the adjacent vinylic double bonds for the type II chromophore is 0.0715, while the similar average for the type III chromophore is 0.0768. Once again, it appears that the most sensitive structural correlator with hyperpolarizability within a homologous series of molecules is the BLA.40 Thus, it appears that the difference between the hyperpolarizabilities of the type II and type III chromophores is solely the result of orbital symmetry through the central benzene ring. Inspection of the orbitals in Figure 8A and B suggests that the
Isomerization and NLO Molecules
J. Phys. Chem. B, Vol. 110, No. 27, 2006 13521
Figure 9. LUMOs (above) and HOMOs (below) for the type I chromophore and FTC. There is little discernible difference between the orbitals for the “planar” and “twisted” conformers. The energy, in electronvolts (eV), of each orbital is given.
conjugation through the ring in the type II isomer, with the donors being ortho to one another, localizes the HOMO on the donor side of the molecule and the LUMO on the acceptor side. In the type III molecule, the donor-acceptor pairs are ortho to one another, which delocalizes both the HOMO and LUMO to a greater extent than in the type II structure. This reinforces the theme found for twisted conformations of quinopyranes2 and the linear chromophores studied here, that breaking the π-conjugation between donor and acceptor is beneficial to development of a large charge-transfer contribution to the hyperpolarizability. Sequence Isomers The different sequences of backbone groups in FTC and the type I chromophores provide yet an additional example of isomerism. In this particular case, interchanging the locations of the thiophene and p-phenylene groups in the conjugated backbone joining the donor and acceptor groups gives a relatively small change in the dipole moment and hyperRayleigh hyperpolarizability, as can be seen in Table 6. The differences between these sequence isomers are probably not outside the range of the uncertainty in the present method, and it would be imprudent at this early stage of development of long wavelength DFT calculations to predict which of these molecules would likely show the stronger hyper-Rayleigh scattering or other NLO property measured in solution.
Conclusions In this work, we have considered three types of isomerisms rotational, geometrical, and sequencesfor molecules that exhibit large nonlinear optical behavior. If the conjugated backbone joining the donor and acceptor groups possesses rotatable single bonds, analogous to those in stilbene, it is probable that a manifold of rotational isomers will exist, with different rotamers having different hyperpolarizablities β. By enumerating the lowenergy conformers of FTC and calculating the hyperpolarizability of each, we have shown that conformational averaging of flexible molecules should not be ignored. The results indicate the magnitude of the range of values of hyperpolarizability one might expect for a typical chromophore. FTC is quite linear in all conformations, which is perhaps one of the reasons that it performs well in applications. Calculation of FTC conformer energies and electrostatic moments with both COSMO in DMol3 and with the PCM method in Gaussian03 reveals that the relative energies of the conformers are perturbed by relatively small amounts; solvent packing effects are expected to be at least as large as the reaction field effects in perturbing the relative energies of conformers away from the values in vacuo. That is, one should not be too hasty in dismissing rotameric populations deduced from vacuum calculations alone; both reaction field and packing effects can be expected to operate in real systems, and the latter can only be assessed with molecular dynamics methods at the present time.
13522 J. Phys. Chem. B, Vol. 110, No. 27, 2006 Both COSMO and PCM predict very large enhancements to the hyperpolarizability from the reaction field, with enhancements ranging from factors of about 2.5-4.5 depending on the property and method of interest. This has a profound influence on conclusions about the relation between macroscopic properties, order parameters, and the molecular properties. The finding that the reaction field does not drastically alter the statistical population of rotamers is somewhat surprising, given that both COSMO and PCM predict that solvent electrostatics significantly perturb total energies and molecular dipole moments (µ for FTC changes by ca. 30% from the vacuum to CHCl3). To understand this result, we note that the molecular dipole moment is produced by additive constituent bond dipoles. This view suggests that the inhomogeneous solvent reaction field that is predicted in these theories acts fairly uniformly at the level of bond dipoles rather than at the level of the overall molecular dipole, creating local perturbations on the potentials of mean force for rotation that are much the same for all bonds. This leaves the population of rotamers largely unaffected. The geometrical isomerism of molecular Xs has been shown to have a profound influence on the predicted hyperpolarizability. It appears that the most important aspect of the conjugation pattern in Xs is that there be maximal conjugation between donor groups in the HOMO and acceptor groups in the LUMO and to have reduced conjugation between donors and acceptors. While the static field polarizability is a ground-state property in DFT, this result can be understood within the two-state model.40 In this picture, the separation of HOMOs and LUMOs in these molecules effectively promotes charge transfer in the excited state. A similar effect is seen in twisted linear molecules. This study of varied types of isomerization has shown that the influence of electronic structure on the development of a large hyperpolarizability consists of several competing factors, making electronic structure calculations very valuable when designing the next generation of chromophores. Acknowledgment. This material is based upon work partially supported by the STC Program of the National Science Foundation No. DMR 0120967. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Support from the Air Force Office of Scientific Research is also gratefully acknowledged. References and Notes (1) Lipinski, J.; Bartkowiak, W. Chem. Phys. 1999, 245, 263. (2) Albert, I. D. L.; Marks, T. J.; Ratner, M. A. J. Am. Chem. Soc. 1997, 119, 3155. (3) van Walree, C. A.; Maarsman, A. W.; Marsman, A. W.; Flipse, M. C.; Jenneskens, L. W.; Smeets, W. J. J.; Spek, A. L. J. Chem. Soc., Perkin Trans. 2 1997, 809. (4) Bartkowiak, W.; Lipinski, J. J. Phys. Chem. A 1998, 102, 5236. (5) Robinson, B. H.; Dalton, L. R.; Harper, A. W.; Ren, A.; Wang, F.; Zhang, C.; Todorova, G.; Lee, M.; Aniszfeld, R.; Garner, S.; Chen, A.; Steier, W. H.; Houbrecht, S.; Persoons, A.; Ledoux, I.; Zyss, J.; Jen, A. K. Y. Chem. Phys. 1999, 245, 35. (6) Delley, B.; Ellis, D. E.; Freeman, A. J.; Baerends, E. J.; Post, D. Phys. ReV. B 1983, 27, 2132. (7) Delley, B. J. Chem. Phys. 1990, 92, 508. (8) Delley, B. DMol, a Standard Tool for Density Functional Calculations: Review and Advances. In Modern Density Functional Theory: A
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