Nonlinear Optical Susceptibilities of Poled GuestHost Systems: A

J. Phys. Chem. B 2004, 108, 8931-8940

8931

Nonlinear Optical Susceptibilities of Poled Guest-Host Systems: A Computational Approach H. Reis,* M. Makowska-Janusika,† and M. G. Papadopoulos* Institute of Organic and Pharmaceutical Chemistry, National Hellenic Research Foundation, Vasileos Constantinou 48, GR-11635 Athens, Greece ReceiVed: January 12, 2004; In Final Form: April 13, 2004

Linear and nonlinear optical susceptibilities of poled guest-host polymer systems are calculated, using structures simulated by molecular dynamics. Three systems are investigated, each consisting of a poly(methyl methacrylate) (PMMA) matrix and doped with either of the one-dimensional nonlinear optical (NLO) chromophores N,N-dimethyl-p-nitroaniline (DPNA) and 4-(dimethylamino)-4′-nitrosilbene (DMANS) or the two-dimensional NLO chromophore N,N′-di-n-propyl-1,5-diamino-2,4-dinitrobenzene (DPDADNB). The electrical properties of the NLO chromophores are computed at the ab initio level, including correlation at the second-order Møller-Plesset (MP2) theory and frequency dispersion at the restricted Hartree-Fock (RHF) level, and with density functional theory (DFT). The permanent local fields in the bulk are small; however, their effects on the first hyperpolarizabilities are comparably large. Refractive indices and second-order susceptibilities corresponding to second harmonic generation (SHG) are calculated using a rigorous localfield (RLF) approach. The system DPNA/PMMA is well-described by the single point-dipole approach, whereas for DPDADNB/PMMA and DMANS/PMMA, only distributed molecular response models yield reasonable results. The Lorentz local field approximation, modified to take partial poling order into consideration, is shown to be approximately consistent with both the experimental results and those of the RLF approach.

1. Introduction Organic molecular materials with large nonlinear optical (NLO) properties, in combination with polymeric materials, have been intensively studied recently, because of their potential use in future technologies as optical switches, in high-speed image processing, and in other photonic applications.1-3 Two quantities essentially determine the magnitude of the macroscopic first nonlinear susceptibility (for example, second harmonic generation (SHG), χ(2)(-2ω;ω,ω)) in these systems: the molecular hyperpolarizability β, together with the density of NLO chromophores, and the degree of noncentrosymmetry. In guesthost polymeric systems with dipolar dopants, the last quantity can be achieved by electric field poling at temperatures above the glass transition temperature (Tg).2 After establishing a steady state, the system is cooled below Tg in the presence of the field, thereby freezing in the noncentrosymmetry and the nonvanishing SHG activity. A lot of work, both experimental and theoretical, has been devoted to investigation of the molecular dynamics that is associated with these processes, especially with the slow return of the guest molecules to an isotropic orientational distribution,4 which is one of the problems limiting the usefulness of these systems for applications. Another problem that has been thoroughly investigated is the dependence of the NLO activity of electric field poled polymeric systems on the (dipolar) chromophore concentration.3,5 Using analytical models of statistical mechanics and simplified Monte Carlo simulations, Dalton et al.3,5 could successfully explain the leveling off at high concentrations and eventual decrease of the NLO activity with increasing chromophore load essentially as a result of the * Author to whom correspondence should be addressed. E-mail address: [email protected]. † Permanent address: Institute of Physics, WSP, Al Armii Krajowej 13/15, PL-42201 Czestochowa, Poland.

competition between the poling field, which tries to align the dipolar chromophores in parallel, intermolecular electrostatic forces, which favors an antiparallel alignment, and thermal randomization. Relatively little work has been done on the accurate determination and prediction of the nonlinear susceptibilities of systems with low chromophore loads. From the experimental point of view, this may be due to the difficulty involved in establishing reproducible conditions, with respect to system composition, field strengths, and other parameters that determine the nonlinear response. Problems on the theoretical side comprise the general difficulty of accurately modeling polymers in electric fields, because of time-scale problems6,7 and the size of typical NLO guest molecules, which prevents the application of accurate ab initio calculations. As a result, many theoretical predictions use the simple Lorentz local field approximation (LFA),1,2,8-10 although the approximation is known to ignore essential intermolecular interaction effects in solute-solvent systems.11 The work of Dalton and others3,5 that was mentioned previously has shown that the LFA fails at high chromophore concentrations, but the approximations used are too severe to draw conclusions about the quantitative performance of LFA at low concentrations. Models that are more accurate than the LFA, taking into account the discrete nature of the molecular environment, but are still amenable to computations, are available; see, e.g., the review of Champagne and Bishop12 for methods used to compute NLO properties for the solid state. The model that we shall use here computes dipolar electrostatic interactions rigorously and has been applied to a large range of properties of molecular crystals,13 including the linear and nonlinear susceptibilities.14-17 Later, it has also been applied to compute the susceptibilities of pure molecular liquids, using molecular

10.1021/jp0498522 CCC: $27.50 © 2004 American Chemical Society Published on Web 06/11/2004

8932 J. Phys. Chem. B, Vol. 108, No. 26, 2004 simulation methods to predict the molecular structure.18,19 In both crystals and liquids, results were compared with the Lorentz LFA and showed partially substantial differences between the two models. Recently, molecular dynamics calculations on doped poled polymers were published.20 Using these results, here, we will apply the discrete local field model for the first time to predict the linear and nonlinear response of several poled guest-host polymer systems and compare the predictions with those of the Lorentz LFA. The model allows the prediction of susceptibilities essentially without any resort to experimental information and may therefore eventually develop into a valuable tool to prescreen potential guest-host systems for applications, after its reliability and range of applicability are established. Following the conventions in computational chemistry, the molecular electrical properties will be given in atomic units (au). However, all other quantities will be given in SI units. The SI unit system will also be followed in the theoretical section. Conversion factors for the electric properties are as follows: for µ, 1 au ) 8.478 × 10-30 C m; for R, 1 au ) 0.16487 × 10-40 C2 m2 J-1; for β, 1 au ) 0.32066 × 10-52 C3 m3 J-2; and for γ, 1 au ) 0.62360 × 10-64 C4 m4 J-3. 2. Methods The guest-host polymer structures for N,N-dimethyl-pnitroaniline (DPNA) and and 4-(dimethylamino)-4′-nitrosilbene (DMANS) in poly(methyl methacrylate) (PMMA) used in the computations in this work were taken from a recent work that reported on molecular dynamics simulations of the electric field poling process of bulk PMMA doped with DPNA (3.6 wt %), DMANS (5.6 wt %), and N,N′-di-n-propyl-1,5-diamino-2,4dinitrobenzene (DPDADNB) (5.9 wt %).20 The molecular structures are shown in Figure 1. A 90-mer of PMMA doped with two molecules of one of the three chromophores (DPNA, DMANS, or DPDADNB) in a unit cell was simulated at 500 K, which is approximately 200 K above the glass transition temperature (Tg), subject to an external poling field with a sufficiently high field strength to achieve a steady-state distribution. The systems then were cooled below Tg in the presence of the poling field, and the sub-Tg structures were simulated for 1.5 ns. Five independent structures were simulated for each system. All simulations were performed using classical Newton dynamics, using nonpolarizable partial charges on all atoms to compute the Coulomb interactions and (12-6)-Lennard-Jones potentials for the repulsion and dispersion interactions. Further details can be found in ref 20. For the present work, we did not use the 5.9 wt % DPDADNB/PMMA simulations of ref 20; instead, we simulated a different system that consisted of 15.9 wt % DPDANB in PMMA, using the same parameters and methods as those used in ref 20, except that only three independent structures were simulated. The motivation for these simulations was the experimental investigation of refractive indices and the SHG signal of a system with a very similar composition by Nalwa et al.9 In the framework of local field theory, linear and nonlinear macroscopic susceptibilities are related to molecular properties by local field factors, which describe the effect of the electric field on a molecular site induced by the externally applied electric field. In discrete local field theory, the local fields are computed by considering the molecular environment rigorously, without resorting to continuum or mean field approximations. Following refs 13 and 16, the components of the first-order susceptibility χ(1) ij (ω) and the second-order susceptibilility corresponding to second harmonic generation, χ(2) ijk (2ω) ≡

Reis et al.

Figure 1. Molecular systems investigated.

χ(2) ijk (-2ω;ω,ω), are, assuming the one-point dipolar approximation, given by -1 χ(1) ij (ω) ) (0V)

χ(2) ijk (2ω) ) (20V)-1

RN,ik(ω)dN,kj(ω) ∑ Nk

∑ dN,li(2ω)βN,lmn(-2ω;ω,ω)dN,mj(ω)dN,nk(ω)

(1)

(2)

Nlmn

where N labels the molecules in the unit cell or simulation box with volume V, 0 is the permittivity of a vacuum, and RN,ij(ω) and βN,lmn(-2ω;ω,ω) are components of the in-bulk molecular polarizability and first hyperpolarizability corresponding to SHG, respectively, of molecule N. The Cartesian axes ijk ... ∈{1,2,3} refer to the laboratory coordinate system. Any reference to the molecular coordinate sytem will be denoted by x, y, z. The factor of 1/2 that appears in eq 2 arises because of the definition of the macroscopic polarization P as a sum of terms without numerical prefactors, whereas the induced molecular dipole moment is defined as a Taylor series.16 The quantities dN,ij(ω) are components of local field tensors, given by

dNi,j(ω) )

[

I∑ N′

]

L·A(ω) (0V)

-1

(3) Ni,N′j

The matrixes L, I, A are of the order 3N, I is a generalized unit tensor with elements INi,N′j ) δijδNN′, whereas A(ω) has elements RN,ij(ω)δNN′ and the components of L are the Lorentzfactor tensors LNN′, which are lattice dipole sums, giving the field at molecule N due to dipole moments on all molecules of type N′ and can be calculated by the Ewald summation technique.21 The point-dipole approximation assumed so far does not take into account the finite size and shape of the molecules, which is known to lead to unrealistic results for molecular crystals that are composed of larger molecules, such as naphthalene or m-nitroaniline.14,17 In a more realistic model that has been developed to Luty,22 each molecule is divided into a set of point submolecules, which are chosen in a way to describe roughly the expected distribution of the polarizabilities. More-accurate schemes to describe the molecular electric response by distributed approaches are available23,24 but are computationally too expensive for our system. More feasible alternatives using additivity schemes for the linear polarizability have usually not been extended to first hyperpolarizabilities,25 although the model developed by Dykstra et al.26,27 has been successfully applied to second hyperpolarizabilities.27,28 The point-dipole formulation can be retained in the submolecule treatment, and the quantities in eqs 1-3 are then appropriate averages over these submolecules. In previous work on one-component systems, the polarizability R was distributed over the submolecules in such a way that each submolecule

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J. Phys. Chem. B, Vol. 108, No. 26, 2004 8933

contributed the same amount to the linear susceptibility χ(1).13 Such a scheme is difficult to extend to mixtures that consist of different molecules, and, in this work, we will therefore assign the same polarizabilty RNk to each submolecule, subject to the n rNk, where n is the number of subcondition RN ) ∑k)1 molecules of molecule N, and, analogously for the first hyperpolarizability tensor, βNk. Because of these auxiliary conditions, polarization effects between submolecules on the same molecule are excluded. The different distribution schemes used will be denoted by n/m, where n gives the number of submolecules used for the chromophores and m is the number of submolecules per monomer of the PMMA chain. Each monomer of the polymer was considered as a single entity. To avoid overcounting the interactions between the monomers, no intrapolymer polarization was allowed, because this interaction is already taken into consideration by the polarizability of each monomer. The simplest scheme used, 1/1, corresponds then to the point-dipole approximation, where the electrical response of each chromophore molecule and each monomer entity of the PMMA chain to the local fields induced by the externally applied optical fields are expressed by induced point dipoles situated in the respective center of mass (COM). In a first approximation, the size and shape of the chromophore molecules are taken into account using n ) 3, n ) 5, and n ) 5 distributions for DPNA, DPDNDAB, and DMANS, respectively, where point dipoles are induced in the COM of the dimethylamino (n-propyl-amino for DPDNDAB), nitro, phenyl, and ethenyl groups. A more extensive distribution was used for DMANS and DPDADNB, in which a submolecule was placed on each second-row atom or on the COM of this atom and any H atoms directly attached to it, leading to n ) 20 for both molecules. Similarly, the response of each monomer of PMMA was varied using an m ) 5 distribution scheme with one submolecule in the COM of each methyl, methylene, and COO group and one at the quartenary C atom, and an m ) 7 scheme, where additionally each atom of the COO group was allowed to carry an induced dipole. Because of the intramolecular flexibilities of each molecule in the simulation, the COM of each group was determined anew at each considered snapshot of the trajectories. The electrical properties of the chromophores were computed only for the optimized geometry, and it was assumed that the properties are the same for each geometric structure adopted during the simulations. The rotation matrixes effecting the transformation of the tensorial properties of the chromophores from the molecular reference system to the laboratory system used in the molecular dynamics simulation were calculated according to the algorithm developed by Kneller,29 which minimizes the sum of the squared deviations of the rotated from the reference site positions, using quaternions to describe the rotations. The electrical response of the monomer of PMMA was approximated by the isotropic, average linear polarizability Rij ) Ravδij. The in-bulk molecular polarizabilities in eqs 1-3 should incorporate environmental effects exerted by the surrounding molecules in the undisturbed medium. Local-field theories take into account the effect due to the permanent local field F(0)N acting on the molecule. In the case where partial charges qNs are assigned to atomic sites s, as in the molecular simulations, whose results will be used in this work, the components of this field can be calculated by

∑s′ Lsi,s′qs′

FNsi ) (0V)-1

(4)

where Lsi,s′ are components of dipole-charge lattice sums giving the Coulomb field at site s due to all charges at site s′. They may be calculated as outlined, for example, in refs 23 and 30. No induced fields are considered in this expression, which may generally lead to a serious underestimation of the field effect. However, in our case, the polarizability of the polymer is rather small and the more strongly polarizable chromophore molecules are too far apart to influence each other considerably. In addition, the partial charges used in the force field are generally derived in such a way to incorporate polarization effects in an average fashion.31 Therefore, we may expect eq 4 to be an acceptable approximation for the permanent local field. The molecular polarizabilitities R and β of the chromophores needed to evaluate eqs 1-3 were calculated by ab initio and DFT methods, using geometries optimized at the restricted Hartree-Fock (RHF) level with the standard 6-31G** basis set in C1 symmetry for all chromophores. Static electrical properties of DPNA, DMANS, and DPBDANB at the RHF and at the correlated second-order Møller-Plesset (MP2) level were calculated using the finite-field perturbation method of Cohen and Roothan,32 using the GAUSSIAN98 program package.33 The base electric field value used was 0.003 au. Frequencydependent polarizabilities R(ω) and first hyperpolarizabilities β(-2ω;ω,ω) were computed with the random phase approximation (RPA), using the Dalton program package.34 The ratio of frequency-dependent RPA values to the static ones were used as scaling factors for the static MP2 values to yield approximate frequency-dependent MP2 values. Different basis sets were used for the three molecules: for DPNA, we used the polarizationconsistent basis set of Sadlej;35 for DMANS and DPDADNB, we used the standard 6-31G** basis set; and for DPDADNB, the more-extensive 6-31++G** basis set was additionally used. DFT calculations of electric properties were performed with the Amsterdam density functional (ADF)36 program package. Values for the polarizability R, first hyperpolarizability β, transition dipole, and excitation energy can be obtained in ADF in the framework of time-dependent density functional theory (TDDFT). The exchange-correlation (xc) functional used was the gradient-regulated asymptotic correction (GRAC) potential,37 which is a gradient-regulated connection between the generalized gradient approximation (GGA) due to Becke and Perdew (BP86) and the asymptotically correct Leeuwen-Barends potential (LB94).38 Additional calculations using the basic local density approximation (LDA), which corresponds to the local Slater exchange functional,39 together with the uniform local gas correlation functional due to Vosko, Wilk and Nusair,40 yielded similar values to those obtained with GRAC, whereas the results obtained using only LB94 were considered to be too large. The ionization potentials necessary for GRAC were computed from the difference of the energy of the neutral and the singly charged cation, computed at the BLYP level, and were found to be 10.26, 7.76, and 6.65 eV for DPNA, DPDADNB, and DMANS, respectively. The adiabatic local density approximation (ALDA) was used for the xc kernel, which describes the first-order response of the xc-potential to the external field. ADF uses Slater-type orbitals (STOs), and we chose the all-electron doubly polarized valence triple-ζ basis set TZ2P. 3. Results and Discussion 3.1. Electric Properties of the Chromophores. 3.1.1. The Molecular Electrical Properties. In Table 1, the computed electronic properties µz and Rav(ω) and selected components βiiz(-2ω;ω,ω) of the free molecules DPNA, DPDADNB, and DMANS are shown. For β, the dominant terms βzzz for DPNA

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TABLE 1: Molecular Electronic Dipole Moment (µz), Average Polarizability (rav(ω)), Dominant Components of the First Hyperpolarizability (βijk(ω) ≡ βijk(-2ω;ω,ω)), and the Vector Component βz of DPNA, DMANS, and DPDADNB Computed at Different Levels and Basis Sets (BS1 ) Pol, BS2 ) 6-31G**, BS3 ) 6-31++G**)a µz

Rav(0)

Rav(ω)

Rav(2ω)

βzzz(0)

βyyz(0)

βz(0)

SCF/BS1 MP2/BS1b GRAC

3.08 2.93 3.28

117.83 131.78 130.70

119.47 133.65 134.65

124.60 139.57 154.06

DPNA, λ ) 1064 nm 1219 2392 2473

SCF/BS2 MP2/BS2b GRAC

3.49 3.02 4.06

206.16 215.14 309.35

207.77 216.85 318.27

212.94 222.37 357.21

DMANS, λ ) 1907 nm 5103 8999 35855

SCF/BS2 MP2/BS2b SCF/BS3 GRAC

3.47 3.22 3.58 3.58

169.44 184.90 187.46 224.25

171.58 187.27 189.97 230.30

178.90 195.38 198.58 257.69

DPDADNB, λ ) 1064 nm 934 1326 1109 1266

a

βzzz(ω)

βz(ω)

581 1300 1287

1629 3197 5634

798 1756 3108

2882 5344 21075

5798 10223 66089

3289 6076 39200

362 731 437 426

βyyz(ω)

1175 1669 1410 2538

βzyy(ω)

1142 1621 1356 2077

486 925 585 1148

All values given in units of au. b Frequency-dependent property obtained by muliplicative scaling with the dispersion at the SCF level.

and DMANS and βyyz and βzyy for DPDADNB are given, together with the vector component of βz(-2ω;ω,ω) along the z-axis:

βz(-2ω;ω,ω) ) 1

∑i (βzii(-2ω;ω,ω) + 2βiiz(-2ω;ω,ω))

5

(5)

Listed are static and frequency-dependent values for λ ) 2πc/ω ) 1064 nm for DPNA and DPDADNB and λ ) 1907 nm for DMANS. All molecules were oriented in such a way that the dipole moment at the 6-31G**/SCF level was directed along the z-axis. The y-axis lies in the plane of the ring in DPNA and DPDADNB, whereas in DMANS, the y-axis bisects the angle of ∼40° between the two planes formed by the phenyl rings in DMANS. The ab initio values of DPNA and DMANS are generally consistent with comparable previous ab initio calculations of the same or similar molecules as p-nitroaniline (pNA), e.g.,41,42 p-amino-p′-nitrostilbene,41 DMANS,43 and 1,5-diamino-2,4dinitrobenzene,44 and also approximately agree with semiempirical results of Nalwa et al. for di-N,N′-ethyl-diaminodinitrobenzene,9,10 assuming that the same convention for the hyperpolarizabilities was used in their computations. Frequency dispersion at the RPA level is moderate at the frequencies considered. Correlation effects for DPDADNB are smaller than those for DPNA and DMANS but are not negligible, especially for β. Comparison of the values obtained with the basis sets 6-31G** and 6-31++G** shows that the use of diffused functions changes the dominant component of β by ∼15%, which is a larger effect than that found previously for DMANS.43 The static polarizabilities and first hyperpolarizabilities of DPNA at the DFT level are in good agreement with the corresponding static MP2 values. The same holds to a lesser extent for the static (hyper)polarizabilities of DPDADNB. The larger discrepancies are partially due to the less-complete basis set used for the ab initio calculations in this case (6-31G** instead of Pol for DPNA), as shown by comparison of the results obtained with 6-31G** and the more-complete 6-31++G** basis sets. However, the dipole moments of all three molecules as well as the polarizabilities and hyperpolarizabilities of DMANS, at the DFT level are much too large, compared to the ab initio results. This behavior is in agreement with previous work of Champagne et al. on push-pull polyenes41 and polyacetylenes45 with increasing chain length and has been traced to an incorrect electric field dependence of the response

portion of the xc-potential in local density approximations (LDAs) and generalized gradient approximations (GGAs).46-48 A missing counteracting term in this potential is responsible for increasingly larger errors of the linear and nonlinear polarizabilities using LDA/GGA approximations in molecules of increasing chain length,48 whereas predictions for more “standard” molecules (such as pNA) are in reasonable agreement with experiment and/or ab initio computations.49 The frequency dispersion of the polarizabilities at the DFT level is much stronger than that at the RPA level at the frequencies considered for all molecules. This may be due to an inadequate description of the dispersion at the RPA level or inadequacies at the DFT level, which, in turn, may be associated with the aforementioned problem, which is known to affect lowlying excitation energies also.46 An indication of the reliability of the computed dispersion can be gained from the lowest dipole-allowed vertical electronic transition energies calculated by the different methods. The computed values at the RPA and TDDFT/GRAC levels for the first dipole-allowed excited state of DPNA and DMANS and the first three states of DPDADNB are shown in Table 2, together with comparable experimental values obtained in nonpolar solvents. The RPA method clearly overestimates the position of the lowest excited state considerably, even considering that the gas-phase first excitation will be located at slightly higher energies than that in nonpolar solvents.50 The RPA results for DPNA and DMANS seem to be basis-set independent, because the Pol basis set yields essentially the same result as the 6-31++G** basis set. On the other hand, TDDFT/GRAC strongly underestimates the first excitation energy for DMANS. Therefore, we conclude that the DFT results for DMANS, both for static properties as well as for the frequency dispersion, are unreliable and they will not be considered further in this work. For DPNA, however, DFT only slightly overestimates the first excitation energy. Static properties of DPNA are equally well-described by MP2 and DFT/GRAC, whereas the “correct” frequency dispersion is bracketed by the RPA and the DFT/ GRAC estimates, probably being closer to the DFT results, at least for βzzz, which is dominated by the contribution stemming from the first excited dipole-allowed state (“two-level approximation”). Comparison of the DFT computed excitation energies for the first three excited states of DPDADNB with experimental values of the related molecule N,N′-di-n-hexyl-1,3-diamino-4,6-dinitrobenzene44 shows that the energies of the two low-lying perpendicular excitations are predicted quite reasonably; how-

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TABLE 2: Computed Vertical Excitation Energy (Ee0)a for the First Dipole-Allowed Electronic Excitations of DPNA, DMANS (e ) 1), and DPDADNB (e ) 1, 2, 3) (Approximate Polarization, with Respect to Ground-State Dipole Moment: | ) Parallel, ⊥ ) Perpendicular) at Different Levels of Theory and Experimental Values in Nonpolar Solvents Ee0 (eV)a DPNA

DMANS

e ) 1(|)

e ) 1(|)

DPDADNB e ) 1(⊥)

e ) 2(|)

e ) 3(⊥)

RPA/Pol 5.011 (λe0 ) 238.7 nm) 4.110 (λe0 ) 291.0 nm) RPA/6-31++G** 5.015 (λe0 ) 238.5 nm) 4.122 (λe0 ) 290.2 nm) 4.890 (λe0 ) 244.6 nm) 5.416 (λe0 ) 220.9 nm) 5.767 (λe0 ) 207.4 nm) DFT/GRAC 3.291 (λe0 ) 363.5 nm) 2.052 (λe0 ) 583 nm) 3.111 (λe0 ) 384.5 nm) 3.663 (λe0 ) 326.6 nm) 3.624 (λe0 ) 330.1 nm) experiment 3.388 (λe0 ) 353.1 nm)b 2.910 (λe0 ) 411 nm)c 3.00 (λe0 ) 399 nm)d 3.55 (λe0 ) 337 nm)d 3.86 (λe0 ) 310 nm)d a Note: 105 J/mol ) 1.0364 eV. The value given in parentheses is the corresponding wavelength. b DPNA in n-hexane (from ref 56). c DMANS in n-pentane (from ref 57). d N,N′-di-n-hexyl-1,3-diamino-4,6-dinitrobenzene in methylcyclohexane (from ref 44).

ever, the parallel excitation is predicted at an energy that is too low. This leads to an incorrect prediction of the energetic sequence of the excited states, assuming that this sequence is not affected by the substitution of the n-propyl groups on the amino nitrogens in DPDADNB by n-hexyl groups in DHDADNB. At the RPA level, on the other hand, the correct sequence is predicted, but again at energies that are much too high. At least three excited levels contribute considerably to the first hyperpolarizability of DHDADNB,44 and without knowledge of further quantities that enter into the sum-overstates expression, it is difficult to estimate the effect of the deviations of the two computational methods on β(-2ω;ω,ω), in comparison with the experimental values. To estimate the linear electric field response of the PMMA chain used in the molecular simulations, we calculated the polarizabilities of chains composed of increasing numbers of monomers, using the semiempirical method AM1.51 The geometries of the chains were taken from the simulated data files. The polarizabilities were averaged over several different chains with the same number of monomers. The computed mean polarizability per monomer Rav(ω)/N at λ ) 2πc/ω ) 1064 nm, as a function of the number of monomers N, was approximately constant at N ) 33. The value obtained was then scaled with the ratio of the single monomer polarizabilities at the multiplicatively scaled MP2/6-31++G** level and at the AM1 level to yield Rav/N ) 62.55 au. This value was assigned to each monomer of the polymer chain in the computation of the susceptibilities. This approach of using the isotropic component of the polarizability to describe the linear response of a monomer works well because Rav is, by construction, independent of the orientation and, furthermore, dependent relatively little on different conformations of the monomers adopted in the polymer. Odd-order dipole response properties, such as the dipole moment and the first hyperpolarizability, do not have isotropic components and therefore cannot be treated in this way. Although the orientational dependence of these properties could easily be treated by rotational matrices, with respect to a reference monomer, the conformational dependence could be more problematic, being possibly much stronger than for the isotropic component of the polarizability. To investigate this, we first optimized the geometry of a monomer at the AM1 level, after adding two H atoms at the free valence sites. Using this optimized geometry with the corresponding electric properties as a reference, the first hyperpolarizability component in the direction of the poling field β333 of each monomer in a pentamer, randomly chosen from a simulation snapshot, was computed using the quaternion approach mentioned previously to compute the rotational matrices, yielding -63, -78, -17, 14, and -15 au. These values were compared with the corresponding values computed for each monomer cut out of the pentamer, with two H atoms added at the free valence sites, thus preserving the

individual geometric conformation adopted by each monmer. This gave the values (in the same corresponding order of monomers) -31, -36, -6, 1, and -6 au. It can be seen that the conformational dependence is indeed quite large. We therefore decided to ignore the first hyperpolarizability of the host entirely. In further tests, we computed the first hyperpolarizability of several randomly chosen 30-mers, at the AM1 level, and found that the value of |β333| rarely exceeded 100 au, probably because of entanglement effects, so that our approximation probably does not introduce a large error. The conformation dependence of the dipole moments is also quite pronounced; the dipole moment |µ| of the five monomers taken from the pentamer ranges from 1.44 D to 2.08 D, whereas the corresponding value of the reference molecule is 4.4 D. Therefore, it is probably more accurate to use the partial charges (as in eq 4) instead of the dipole moments for the computation of the permanent local field. 3.1.2. The Permanent Local Field. Using eq 4, we estimated the permanent electric local field on the chromophores in the poled, glassy polymers due to the surrounding partial charges. The field was computed on each atom of the chromophore molecule and then averaged, over the trajectories of each different MD run, over all atoms, over all molecules in one structure simulation, and finally over all simulations with different starting structures. The externally applied field was ignored, because the purpose was the computation of the local field in a case similar to an experimental determination of the nonlinear response, which is generally done after the poling process. The results for the fields, expressed in the molecular coordinate system, are shown in Table 3. There is a clear tendency for the field to be directed parallel to the molecular dipole moment, although perpendicular components are comparably large. This shows that even in the highly entangled polymeric environment in the liquid phase, which becomes “frozen” by the poling process,20 the molecular components have a tendency to align themselves in such a way as to enhance electrical dipolar interactions. Overall, the absolute field values are rather small, compared with values computed for crystals such as urea in dipolar approximation (∼7 GV/m16) and are comparable to values computed by molecular dynamics simulations for weakly (multi)polar liquids such as nitrobenzene52 and benzene,18 where peak values in the distributions of |F| of 1.7 GV/m and 1.3 GV/m, respectively, were found. In those cases, the induced effects, both linear and nonlinear, were taken into account, and these would have probably increased the fields in the PMMA solutions slightly. The effect of the local fields on the polarizabilities of the chromophores is quite different for the different chromophores, as shown in Table 3 by the average values Rav(F) and βz(F) computed for the chromophores in the respective local field. The effect on the linear polarizabilities is negligible for all molecules; the βz values of DPNA and DPDADNB, at the scaled

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TABLE 3: Average Local Fields (F) on the Chromophores in the Systems DPNA/PMMA, DMANS/PMMA, and DPDADNB/ PMMA, Estimated Using eq 4, and Electrical Properties rii and βijk of the Chromophores in This Field, Computed at the MP2 and TDDFT/GRAC Levelsa DPNA/PMMA

DMANS/PMMA

DPDADNB/PMMA

Fx,Fy,Fz ) 0.13,0.12,0.73a

Fx,Fy,Fz ) -0.07,-0.39,0.65a

Fx,Fy,Fz ) 0.14,0.61,1.30a

MP2/Pol

GRAC

MP2/6-31G**

90.61 250.63 216.69 185.96 (0.6%) -51 1415 -35 798 (9.1%)

λ ) 1064 nm 90.93 122.85 255.06 307.87 219.18 263.06 188.39 (0.6%) 231.26 (0.4%) 196.71 (0.7%) 260.45 (1.1%) -54 -195 -53 -214 1791 2789 1728 2078 -39 -33 1007 (8.8%) 1391 (21%)

λ ) ∞ nm

Rxx Ryy Rzz Rav βxxz βyyz βzzz βz

74.86 126.00 196.28 132.38 (0.5%) -56 -157 2674 1477 (14%)

67.34 120.53 207.23 131.70 (0.8%) -87 -228 2639 1395 (8%)

λ ) ∞ nm 83.80 172.01 397.22 217.68 (1.2%) -31 -64 11228 6679 (25%)

Rxx Ryy Rzz Rav(ω) Rav(2ω) βxxz βzxx βyyz βzyy βzzz βz

λ ) 1064 nm 75.34 67.74 127.21 121.92 200.38 218.24 134.31 (0.5%) 135.97 (1.0%) 140.14 (0.4%) 157.85 (2.5%) -71 -109 -72 -145 -179 -279 -201 -418 3601 6244 2011 (15%) 3478 (12%)

λ ) 1907 nm 83.92 172.52 402.05 219.50 (1.2%) 225.43 (1.4%) -32 -34 -66 -71 12873 7663 (26%)

MP2/6-31G**

GRAC

λ ) ∞ nm 122.05 296.23 256.68 224.98 (0.3%) -167 1359 -215 586 (38%)

a Units of measure for average local fields are GV/m. Rii and βijk values are given in units of au. Changes to the field-free properties are given in parentheses.

(2) (2) TABLE 4: Refractive Indices (n⊥(ω) and n3(ω)) and SHG Susceptibilities (χ(2) 333(-2ω;ω,ω), χ⊥⊥3(-2ω;ω,ω), and χ3⊥⊥(-2ω;ω,ω)) of Guest-Host Systems Poled in the 3-Direction, Computed Using Input Electrical Properties at Different Computational Levels and Applying Different Partitioning Schemes (n/m)

SHG Susceptibility (pm/V) level

n/m

n⊥

1/1 1/1 3/1 3/7

1.479 ( 0.004 1.478 ( 0.004 1.477 ( 0.005 1.478 ( 0.005

MP2/6-31G** MP2/6-31G** MP2/6-31G** MP2/6-31G**

1/1 5/1 20/1 20/5

1.6 ( 0.1 1.49 ( 0.01 1.479 ( 0.006 1.473 ( 0.005

MP2/6-31G** GRAC GRAC GRAC GRAC

5/1 1/1 5/1 20/1 5/5a

1.474 ( 0.007 1.540 ( 0.015 1.499 ( 0.007 1.499 ( 0.007 1.499 ( 0.011

MP2/Pol GRAC GRAC GRAC

a

(2) χ⊥⊥3

(2) χ3⊥⊥

χ(2) 333

0.21 ( 0.06 0.43 ( 0.12 0.38 ( 0.11 0.48 ( 0.11

0.20 ( 0.07 0.36 ( 0.13 0.32 ( 0.11 0.41 ( 0.12

0.94 ( 0.16 1.78 ( 0.31 2.02 ( 0.26 2.15 ( 0.26

DMANS, λ ) 1907 nm 1.48 ( 0.03 300 ( 300 1.53 ( 0.02 -270 ( 300 1.512 ( 0.008 12 ( 8 1.505 ( 0.005 0.72 ( 0.20

900 ( 600 -130 ( 260 12 ( 8 0.72 ( 0.20

2000 ( 2000 -120 ( 260 27 ( 20 4.90 ( 0.93

1.20 ( 0.08 4.3 ( 0.7 1.42 ( 0.14 1.41 ( 0.13 1.50 ( 0.15

0.51 ( 0.08 2.2 ( 4.4 0.62 ( 0.15 0.63 ( 0.14 0.58 ( 0.30

n3 DPNA, λ ) 1064 nm 1.505 ( 0.010 1.506 ( 0.010 1.508 ( 0.010 1.509 ( 0.009

DPDADNB, λ ) 1064 nm 1.502 ( 0.008 1.570 ( 0.039 1.524 ( 0.008 1.526 ( 0.008 1.523 ( 0.009

1.25 ( 0.08 9.9 ( 2.4 2.09 ( 0.14 2.07 ( 0.14 2.23 ( 0.15

Averages taken over two independently simulated structures.

MP2 level, increase by a modest amount of ∼10%, whereas for DMANS, which experiences the smallest permanent local field, the βz value increases by ∼25%. The apparently very large field effect on βz of DPDADNB at the DFT/GRAC level, especially in the static case, is due to the occurrence of negative components in the sum defining βz (eq 5). The effect on the largest component, βyyz, is much smaller; in the static case, it increases by 7% to 1359 au for TDDFT/GRAC. The same relative increase as found at the scaled MP2/6-31G** level (7% to 1415 au). 3.2. The Linear and Nonlinear Susceptibilities. The refractive indices ni(ω), which are related to the linear susceptibilities 1/2 by ni(ω) ) [χ(1) ii (-ω;ω) + 1] , and the nonlinear susceptibili(2) ties χijk (-2ω;ω,ω) ≡ χijk of the different simulated PMMA/ dopant combinations, calculated according to eqs 1-3, are given in Table 4. For these calculations, the frequency-dependent

electrical properties of the chromophores computed with the permanent local field were used, whereas the electrical response of the monomers was approximated by the isotropic polarizability Rav ) 62.55 au, and their first hyperpolarizability was neglected. Only susceptibility components that are not expected to vanish because of the uniaxial symmetry of the system are shown, and averages were taken over components expected to (2) be equal by symmetry, i.e., n⊥ ) (n1 + n2)/2, χ⊥⊥3 ) (χ(2) 113 + (2) (2) (2) (2) χ223)/2, and χ3⊥⊥ ) (χ311 + χ322)/2. The errors of the mean value reported in the table were calculated over the values obtained for each simulated structure and over all components equal by symmetry; the errors resulting from the average over each trajectory were not taken into consideration. The influence of different partitioning schemes in the ad hoc submolecule treatment mentioned in Section 2 was investigated through the use of different n/m partitionings, where n denotes

NLO Susceptibilities of Poled Guest-Host Systems the number of submolecules used for the chromophores and m is the number of submolecules into which each monomer of the PMMA chain was divided. The table shows that, for DPNA, the point-dipole approximationsthat is, the 1/1 partitioning schemesis already adequate, because more-extensive response function distributions do not change the susceptibilities appreciably. For the considerably larger DPDADNB molecule, it is necessary to distribute the response over at least five subgroups; however, further refinement of the monomer and/ or chromophore distribution has no effect. Note, however, that one of the three independently simulated structures was not taken into account in the average for the 5/5 partition scheme, because the susceptibilities obtained for this structure differed very much from those of the other two structures. In the case of DMANS, finally, a distribution of the chromophore response alone is not sufficient. It is necessary to partition additionally the monomer response to get reasonable results: only a five-submolecule approximation for each monomer, together with a 20-submolecule treatment for each DMANS molecule, leads to plausible χ(2) values. This quite different behavior of the different chromophores in the same host polymer may be explained by the different shape and size of the three molecules. It is known that the pointdipole approximation increasingly fails with increasing size of the molecule,14,17 which explains the different behavior of DPNA, as compared to DMANS and DPDADNB. In addition, there seems to be a shape effect, because the less bulky, but prolate, ellipsoidal DMANS molecules need a more distributed description of the response functions, including those of the monomers, than the larger, but oblate, DPDADNB molecules. We note that, in previous calculations on m-nitroaniline in the crystal, only the most distributed scheme with 10 submolecules gave results in qualitative agreement with experiment, even for the refractive indices.17 Here, we found that the larger DPNA molecule is already adequately described by the point-dipole scheme. This shows that the effect of distributing response is strongly dependent on the molecular environment. We also note that the effect of insufficiently distributed response functions is much more pronounced for the SHG susceptibilities than for the linear refractive indices. The differences in the computed SHG susceptibilities in those cases where different sets of molecular data were used (DPNA and DPDADNB) are approximately equal or slightly greater than the differences of the corresponding predominant β (2) components: for DPNA, for example, χ(2) 333(GRAC)/χ333(MP2) ) 1.89 for n/m ) 1/1, whereas βzzz(GRAC)/βzzz(MP2) ) 1.73. A numerical comparison of the computed second susceptibility values with experimental data is not useful because of several factors, such as the use of very different field strengths (leading to different order parameters),7,20 the deposition of charges on the films in some of the experimental poling techniques (leading to strongly inhomogeneous applied fields), neglect of the contribution due to the second hyperpolarizability to χ(2) (estimated to be ∼18% for DMANS),53 neglect of the susceptibility due to PMMA,8 and different values used for the experimental calibration factor.54 Specifically, the first two problems are due to limited knowledge of the experimental conditions and/or inherent limitations of the molecular dynamics approach, as the application of very high field strengths is necessary to arrive at a steady state, with respect to the poling field in the limited time scale accessible to molecular simulations.7 However, we do note that the predicted refractive indices, which are mainly determined by the linear polarizability density and are not very dependent on the degreee of noncentrosym-

J. Phys. Chem. B, Vol. 108, No. 26, 2004 8937 metry, are in reasonable agreement with the experimental values for pure PMMA (nD ) 1.49 )55 as well as the experimental refractive index of DPDADNB in PMMA (16.6 wt %) of 1.488 at λ ) 1064 nm, as determined by Nalwa et al.9 The refractive indices at 2ω, which are not shown in the table, are only marginally larger (0.0-1.0%) than those at ω. This, of course, is due to the use of the same polarizability input data of PMMA for both frequencies, because only the polarizabilities of the chromophores are different. This observation is also corroborated by experiment: Nalwa et al. reported a value of n(2ω) ) 1.498 for DPDADNB in PMMA, which is a difference of only 0.6%, compared to n(ω). Generally, in the cases of converged distribution schemes, the refractive index in the direction parallel to the applied field (n3) is ∼2% larger than the refractive index in the direction perpendicular to the field (n⊥), which is due to the larger polarizability in the direction of the dipole moment, compared to the average polarizability value perpendicular to this direction. The relative magnitude of the components of the predicted macroscopic nonlinear responses corresponds to what may be expected from the dominant components of the molecular first hyperpolarizability. For the one-dimensional NLO molecules (2) DMANS and DPNA, we have χ(2) 333 . χ⊥⊥3; however, for the (2) two-dimensional chromophore DPDADNB, we find χ⊥⊥3 . (2) 9 χ333. Nalwa et al. reported experimental SHG values of χ333 ) 2.4 pm/V (in our convention) for a poled film of 16.6 wt % DPDADNB in PMMA, which is much larger than the value obtained for the nondiagonal component, χ(2) 311 ) 0.31 pm/V. This apparent contradiction to our calculations can be explained by the different order parameters obtained due to different poling field strengths, as we will show in the Lorentz LFA. In the LFA, the local fields on the molecules are given by local field factors f(ω):

f(ω) )

n2(ω) + 2 3

(6)

Assuming uniaxial symmetry of the macroscopic system, the nonlinear susceptibilities are computed by orientational averaging of the molecular hyperpolarizabilities over the orientational distribution function.2,8 Following the procedure in ref 8, the nonvanishing components χ(2)(-2ω;ω,ω) for SHG can be shown to be

χ(2) 333

f(2ω)f(ω)2N ) [2βzzz〈cos3 θ〉 + 40

∑ (βzii + 2βiiz)〈cos θ - cos3 θ〉]

(7)

i)x,y

(2) ) χ⊥⊥3

f(2ω)f(ω)2N [2βzzz〈cos θ - cos3 θ〉 + 80

∑ βiiz〈cos θ - cos3 θ〉 - i)x,y ∑ (βzii + βiiz)〈cos θ - cos3 θ〉]

i)x,y

(8) 2

(2) ) χ3⊥⊥

f(2ω)f(ω) N [2βzzz〈cos θ - cos3 θ〉 + 80

βzii〈cos θ + cos3 θ〉 - ∑ 2βiiz〈cos θ - cos3 θ〉] ∑ i)x,y i)x,y

(9)

where we have neglected the small differences between n⊥ and n3. N is the number density of the chromophores and the

8938 J. Phys. Chem. B, Vol. 108, No. 26, 2004

Reis et al.

Figure 2. SHG susceptibilities of DMANS/PMMA in the Lorentz local field approximation (LFA) as a function of K ) µ‚E/kT.

Figure 3. SHG susceptibilities of DPDADNB/PMMA in the Lorentz LFA as a function of K ) µ‚E/kT.

functions 〈cosn θ〉 are ensemble averages of cosn θ, with θ being the angle between the molecular dipole moment µ and the local field F. If the system is in a steady state, with respect to the poling field, the averages 〈cosn θ〉 can be analytically calculated, neglecting polarizabilities, to yield1

〈cos θ〉 ) coth K -

(

〈cos3 θ〉 ) 1 +

)

1 K

(10)

(

)

6 3 2 coth K - 1 + 2 2 K K K

(11)

where K ) µ‚F/kT, with k being the Boltzmann constant and T being the temperature. Using these expressions in eqs 7-9, the susceptibilities can be computed. Figures 2 and 3 show χ(2) 333 (2) and χ3⊥⊥ as a function of K for the one- and two-dimensional chromophores DMANS and DPDADNB, respectively, using the computed, field-dependent values of β at the 6-31G**/MP2 level for DMANS and β at the GRAC level for DPDADNB. The (2) figures show that, for DPDADNB, χ(2) 333 is larger than χ3⊥⊥ only for small to moderate large K, whereas at large K, it decreases (2) is reaching a plateau value. For the oneto zero, and χ3⊥⊥ dimensional chromophore DMANS, the two components show (2) opposite behavior, with, additionally, χ3⊥⊥ , χ(2) 333 throughout. This explains the discrepancies mentioned previously, because

Figure 4. Comparison of the normalized χ(2) 333 signal of DPDADNB and DMANS in PMMA as a function of K ) µ‚E/kT.

field strengths achievable in experiments as conducted by Nalwa et al. are rather small, compared with those applied in our simulations, yielding a small K value and the two sets of values therefore end up on different sides of the crossing point in Figure (2) 3. We mention that the experimental ratio χ(2) 333/χ3⊥⊥ ≈ 8 is approximately consistent with the computed ratio of βyyz and βzyy at the TDDFT/GRAC level, as numerical tests with eqs 7 and 9 at low K show. Although the magnitude of the experimental values are much larger than those predicted by the Lorentz model at small K, computed with our molecular property values, this may be due to the factors already mentioned previously. For example, the use of the newer calibration value for quartz would reduce the experimental values by ∼40% and inclusion of the second hyperpolarizability in the theoretical values may lead to an enhancement of the simulated values of 10%-30%, bringing simulated and experimental values much closer to each other. The plot of the susceptibility components as a function of K for DPDADNB also helps to explain why the χ(2) 333 signal of polymers doped with two-dimensional chromophores decays more slowly with time, after switching off the poling field, than in the case where the dopants are one-dimensional chromophores, as reported by Nalwa et al.10 The dependence of the χ(2) 333 signal near the maximum of the curve in Figure 3 is weak; therefore, the value of χ(2) 333 will change more slowly in time, when K decreases, than the corresponding quantity of a onedimensional chromophore. This is shown in Figure 4, where the signals of DMANS and DPDADNB as a function of K are compared for the region up to K ) 5, normalized to the value of χ(2) 333 at K ) 5. For this specific case, the signal for DPDADNB/PMMA even slightly increases initially with decreasing K, before finally decreasing toward zero. However, we do note that this is dependent at least on two system-specific factors. The first factor is the relative magnitude of the β components of the molecule, especially of βzzz as compared to the dominant nondiagonal terms, as they shift the position of the maximum in Figure 3 to lower or higher K. Numerical tests using eq 7 show that, with a increasingly negative βzzz value and positive βyyz, βzyy values, the maximum shifts to lower K, whereas a positive βzzz value shifts it to higher K. The second variable is the maximum value of K achieved in the poling process. If an experimentally more accessible starting value of K ) 1 is assumed, the decay curves for the two systems considered here are virtually indistinguishable.

NLO Susceptibilities of Poled Guest-Host Systems

J. Phys. Chem. B, Vol. 108, No. 26, 2004 8939

TABLE 5: Order Parameters 〈cos θ〉 and 〈cos3 θ〉 and SHG Susceptibilities χ(2), Computed in the Lorentz Local Field Approximation SHG Susceptibility (pm/V)

a

system

〈cos θ〉

DPNA DMANS DPDADNB

0.79 ( 0.86 ( 0.14a 0.85 ( 0.11 0.16a

〈cos3

θ〉

0.50 ( 0.66 ( 0.27a 0.59 ( 0.18 0.23a

(2) χ⊥⊥3

(2) χ3⊥⊥

χ(2) 333

0.7 ( 0.9 0.9 ( 1.8 1.3 ( 1.1

0.6 ( 0.9 0.9 ( 1.8 0.7 ( 1.1

2.2 ( 1.3 6.2 ( 2.2 1.9 ( 2.2

Taken from ref 20.

To compare the predictions of the Lorentz approximation with those of the RLF model, it is necessary to consider that the simulated systems are not in a steady state, with respect to the external poling field,20 which is a condition for the applicability of the Lorentz approximation, as used previously. However, the order parameters 〈cos θ〉 and 〈cos 3θ〉 entering eqs 7-9 can easily be calculated from the trajectory files and need not be evaluated assuming the steady-state condition. Using these values, the predictions using the Lorentz approximation may then be compared with the values obtained from the RLF model. In Table 5 we show the values computed using the Lorentz LFA obtained with the computed dominant β values at the DFT/ GRAC level for DPNA and DPDADNB and at the 6-31G**/ MP2 level for DMANS, and using the corresponding computed refractive indices in Table 4 for the calculation of the local field factors. Comparison with the values obtained in RLFT in Table 4 shows that, mainly due to the large errors of 〈cos θ〉 and 〈cos3 θ〉, the values in the two approximation schemes agree with each other within the error margins. However, for DPDADNB, the ordering of diagonal and nondiagonal components is reversed, although all values still agree within the error limits. 4. Conclusions Second harmonic generation (SHG) susceptibilities for poled polymers doped with two one-dimensional nonlinear optical (NLO) chromophores (N,N-dimethyl-p-nitroaniline (DPNA) and 4-(dimethylamino)-4′-nitrosilbene (DMANS)) and one twodimensional NLO chromophore (N,N′-di-n-propyl-1,5-diamino2,4-dinitrobenzene, DPDADNB) were calculated in the rigorous local-field theory (RLFT) as well as in the Lorentz local field approximation (LFA), without any recourse to experimental data. Although the comparison between susceptibilities predicted by the two approximations is hampered by several factorsschiefly among them, the large error margins for the order parameterss our calculations suggest that the Lorentz LFA yields results that are surprisingly similar to those predicted by the RLFT, at least for the low chromophore concentrations of the systems investigated here. Considering the ease of application, however, it is clear that the Lorentz approximation has a large advantage over RLFT. A possible exception to the comparability of the two approximations may be the case of the two-dimensional chromophore DPDADNB in poly(methyl methacrylate) (PMMA), where the two approximations yield a different ordering of the nonvanishing components of the SHG susceptibility. It may also be different if more-polarizable and/or more-dipolar polymer systems are used than PMMA, because, in these cases, the permanent local field on the chromophores may be much larger, leading to strong changes of the in-phase electrical properties, as compared to the gas-phase properties, as has been shown previously for small molecules.16,17,19 Such permanent local field effects are not taken into consideration by the Lorentz approximation at all.11 In this case, the Onsager approximation may lead to better predictions, while still being easier to apply than RLFT. We should stress that these conclusions are only

valid at low chromophore concentrations. Previous work by other groups has already shown that approximations predicting a linear relationship between the first nonlinear susceptibility and the product of dipole and first hyperpolarizability (for onedimensional chromophores) at low field strengths break down at high chromophore concentrations3,5 From this work, we conclude that, as a routine screening tool of NLO chromophores for suitability in guest-host polymeric systems with low chromophore load, the Lorentz approximation may be adequate, as long as the polymer is not expected to create large permanent local fields. Acknowledgment. M.G.P. and the Fellow M. M.-J. acknowledge a Marie Curie Host Development Fellowship (HPMD-CT-2001-00091). H.R. thanks R. W. Munn (UMIST) for his hospitality during the period when parts of the programs were written and the Royal Society of Great Britain for his grant. References and Notes (1) Burland, D. M.; Miller, R. D.; Walsh, C. A. Chem. ReV. 1994, 94, 31. (2) Williams, D. J. In Nonlinear Optical Properties of Organic Molecules and Crystals; Chemla, D. S., Zyss, J., Eds.; Academic Press: Orlando, FL, 1986; Vol. 1. (3) Dalton, L. R.; Steier, W. H.; Robinson, B. H.; Zhang, C.; Ren, A.; Garner, S.; Chen, A.; Londergan, T.; Irwin, L.; Carlson, B.; Fifield, L.; Phelan, G.; Kincaid, C.; Amend, J.; Jen, A. J. Mater. Chem. 1999, 1905. (4) See, e.g., Goodson, T.; Wang, H. Macromolecules 1993, 26, 1837. Teraoka, I.; Jungbauer, D.; Reck, B.; Yoon, D. Y.; Twieg, R.; Willson, C. G. Appl. Phys. 1991, 69, 2568. Walsh, C. A.; Burland, D. M.; Lee, V. Y.; Miller, R. D.; Smith, B. A.; Twieg, R. J.; Volksen, W. Macromolecules 1993, 26, 3720. Schussler, S.; Richert, R.; Bassler, H. Macromolecules 1994, 24, 4318. Kohler, W.; Robello, D. R.; Dao, P. T.; Willand, C. S.; Williams, D. J. J. Chem. Phys. 1990, 93, 9157. Kim, W.-K.; Hayden, L. M. Macromolecules 2000, 33, 5747. Dhinojwala, A.; Wong, G. K.; Torkelson, J. M. Macromolecules 1993, 26, 5943. Hayden, L. M.; Kim, W.-K.; Chafin, A. P.; Lindsay, G. A. J. Polym. Sci. Part B: Polym. Phys. 2001, 39, 895. Brower, S. C.; Hayden, L. M. J. Polym. Sci. Part B: Polym. Phys. 1998, 36, 1013. Dhinojwala, A.; Wong, G. K.; Torkelson, J. M. Macromolecules 1992, 25, 7395. Strutz, S. J.; Brower, S. C.; Hayden, L. M. J. Polym. Sci. Part B: Polym. Phys. 1998, 36, 901. Hooker, J. C.; Torkelson, J. M. Macromolecules 1995, 28, 7683. (5) Dalton, L. R. J. Phys.: Condens. Matter 2003, 15, R897. Pereverzev, Y. V.; Prezdho, O. V.; Dalton, L. R. Chem. Phys. Lett. 2003, 373, 207. Pereverzev, Y. V.; Prezdho, O. V.; Dalton, L. R. J. Chem. Phys. 2002, 117, 3354. Robinson, B. H.; Dalton, L. R. J. Phys. Chem. A 2000, 104, 4785. Pereverzev, Y. V.; Prezdho, O. V. Phys. ReV. E 2000, 62, 8324. Robinson, B. H.; Dalton, L. R.; Harper, A. W.; Ren, A.; Wang, F.; Zhang, C.; Todorova, G.; Lee, M.; Ansizfeld, R.; Garner, S.; Chen, A.; Steier, W. H.; Houbracht, S.; Persoons, A.; Ledoux, I.; Zyss, J.; Jen, A. K. Y. Chem. Phys. 1999, 245, 35. Dalton, L. R.; Harper, A. W.; Robinson, B. H. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 4842. (6) Theodorou, D. N. In Diffusion in Polymers; Neogi, P., Ed.; Marcel Dekker: New York, 1996; p 67. (7) Kim, W.-K.; Hayden, L. M. J. Chem. Phys. 1999, 111, 5212. (8) Singer, K. D.; Kuzyk, M. G.; Sohn, J. E. J. Opt. Soc. Am. B 1987, B4, 968. (9) Nalwa, H. S.; Watanabe, T.; Ogino, K.; Sato, H.; Miyata, S. J. Mater. Sci. 1998, 33, 3699. (10) Nalwa, H. S.; Watanabe, T.; Miyata, S. AdV. Mater. 1995, 7, 754. (11) Wortmann, R.; Bishop, D. M. J. Chem. Phys. 1998, 108, 1001. (12) Champagne, B.; Bishop, D. M. AdV. Chem. Phys. 2003, 126, 41. (13) Munn, R. W. Mol. Phys. 1988, 64, 1. (14) Bounds, P. J.; Munn, R. W. Chem. Phys. 1981, 59, 47. (15) Hurst, M.; Munn, R. W. J. Mol. Electron. 1986, 2, 35/43/101.

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