Breakdown of the first hyperpolarizability/bond-length alternation parameter relationship

Abstract

We have investigated the dependence of the static first hyperpolarizability on the bond-length alternation (BLA) parameter. Our analysis indicates that the validity of the first hyperpolarizability/BLA parameter relationship is restricted to the no-field, vacuum, limit, while it successively breaks down along with increasing polarity of a surrounding medium, becoming invalid, for instance, in an aqueous solution. This contention is based on a series of TD-DFT, TD-DFT/PCM and hybrid TD-DFT/MM calculations of the first hyperpolarizability for a set of molecular configurations generated from Car–Parrinello hybrid QM/MM simulations of the stilbazolium merocyanine chromophore in chloroform and water solvents, and on a rationalization by means of the two-state model for the first hyperpolarizability. The BLA dependence of the three parameters entering the two-state model is shown to be qualitatively different in vacuum and in solvents. Particularly, in the vacuum case, the difference between ground and excited state dipole moments goes to zero for a vanishing BLA, which is not true in the presence of an aqueous medium. In the aqueous medium, an opposing behavior of the parameters involved in the two-state model results in an almost constant first hyperpolarizability with varying BLA parameter.

Simplifying thumb rules and structure-property relations are of utmost significance in rational design of new materials with prespecified properties. Computational modeling plays an important role in that context because it can provide accurate number prediction of the property in question and can elucidate the merits and limitations of the simplifying thumb rules and so make their utility more precise (1). We find good evidence of this in the research of nonlinear optical (NLO) materials, where several structure-property relations are highlighted (2–12) and where computational modeling has advanced quite far (13). One such important structure-property relation for NLO materials is the relation between the first hyperpolarizability and the bond-length alternation (BLA) parameter for molecules that can exist in resonance between the two, neutral or charge-separated (zwitter-ionic) forms (2–12). The BLA parameter has been defined as the difference between the average single and double bond lengths in the conjugation pathway (14). By convention, it has been defined to be positive for the neutral form of a molecule while it takes negative values for the charge-separated form. Many theoretical studies have been performed with the purpose of investigating the effect of electric fields and solvents in altering BLA (9, 14–16). It has been shown that with increasing field strength and increasing solvent polarity, the BLA parameter decreases and in some cases changes sign with a cyanine-like structure corresponding to a BLA parameter equal to zero (14, 17). The BLA parameter has therefore been proposed to be a structural parameter to be tuned for optimizing nonlinear optical response because this directly quantifies the ground state polarization (2, 10, 11). For a particular molecule, the BLA parameter can be tuned internally by varying the strengths of the donor and acceptor groups or by altering the conjugation length, and externally by using electric fields or solvents of different polarity (6, 9, 14–16, 18).

There are many theoretical and experimental reports that discuss the relationship between the first hyperpolarizability and the BLA parameter (2, 9–11). In theoretical calculations, molecular structures with varying BLA parameter have been generated using negative and positive charges placed close to the donor and acceptor groups and a subsequent optimization of the structure in the finite field due to the charges or “sparkles” (3, 5, 6, 9, 18). It has been indicated by semiempirical calculations that the field strength due to stronger polar solvents is less influential and only brings smaller changes in solute molecular geometry and in its BLA parameter (6, 18). Experimentally, many molecules have been synthesized possessing varying BLA parameters (with negative and positive values) and where the corresponding hyperpolarizability has been used to derive the first hyperpolarizability/BLA parameter relationship (7). The BLA parameter values were determined by X-ray crystallography in these cases. The BLA parameter dependence of the first hyperpolarizability has been reported to have peaks corresponding to intermediate BLA parameter values with a minimum in magnitude corresponding to the charge separated and neutral forms and with a sign change occurring when the BLA parameter is close to zero. For example, in the case of polyenes the extreme bond-alternated, neutral and charge separated forms correspond to a BLA parameter value of ± 0.11, while the extreme in the first hyperpolarizability has been shown to occur for an intermediate BLA parameter, ± 0.05 (3, 16).

A two-state model has been proposed in order to rationalize the first hyperpolarizability/BLA parameter dependence in systems where the dominant contributions to the first hyperpolarizability origin from the ground state and a charge transfer (CT) state: (7, 9):

[1]

where ω_ge is the CT excitation energy, μ_ge is the transition dipole moment for CT excitation, and μ_ee and μ_gg are the excited and ground state dipole moments, respectively. The rationalization of the first hyperpolarizability by the two-state model thus involves three terms: the difference between the ground and excited state dipole moments Δμ = μ_ee - μ_gg, the transition dipole moment, and the energy of the corresponding excitation. The first of these terms, namely, Δμ, gives the important “sinusoidal” variation with respect to the BLA parameter (3) according to the original BLA model (19). The ground state dipole moment is larger than the excited state dipole moment for a molecule in neutral form, while the reverse holds true when the molecule assumes its charge separated form. When the molecule is in a cyanine form (i.e., 50∶50 percentage resonance of neutral and charge separated forms) the ground and excited dipole moments are eventually equal. The sinusoidal behavior of the difference in dipole moments has to be attributed to the change in ground and excited state molecular dipole moments with its change in molecular geometry as defined by the BLA parameter. Overall, for a neutral form Δμ is positive, for the zwitter-ionic form it takes negative value, while for the cyanine form it corresponds to zero. In this model, the other two terms and have been proposed to have an inverted parabolic relationship with a maximum corresponding to a BLA parameter equal to zero (6). The symmetric distribution of electrons in both the HOMO and LUMO leads to the maximization of these two terms for a BLA parameter equal to zero (9). Overall, the first hyperpolarizability dependence on the BLA parameter, which is a product of all three terms, shows maximum corresponding to intermediate BLAs and goes to zero for a zero BLA because then Δμ is zero as well (7). We note here that the two-state model for NLO response is validated for 1D systems with charge-transfer character of the first excited state, while often a complete summation over the excited states is warranted for 2D and 3D systems.

The BLA parameter dependence of the first hyperpolarizability has been studied theoretically for many organic molecules including both negative and positive solvatochromic molecules such as phenol blue, stilbazolium merocyanine, 9-(dimethylamino)-nona-2,4,6,8-tetraenal that all show a significant change in structure, and so in BLA parameter, with varying solvent polarity (12, 18). In these cases, the molecular structures with different BLA values were generated using an electric field or using charges near the acceptor or donor groups. Particularly, in the case of phenol blue, the maximum in the first hyperpolarizability has been shown for an intermediate BLA value corresponding to a charge separated enol like structure (with BLA -0.05) (12). Experimentally, it has been shown that the first hyperpolarizability of phenol blue has a maximum in the case of chloroform, which is a nonpolar solvent (4).

Although the notion of the first hyperpolarizability/BLA parameter has been extensively utilized in design of new NLO materials, there have only been few attempts to verify it in cases where the NLO chromophore is placed in solvent environments (14, 20). The hyperpolarizability of molecules in complex environments poses high computational and methodological demands. An accurate modeling should include contributions such as finite temperature effects and explicit inclusion of site-specific solute–solvent interaction (including the mutual solute–solvent polarization) and so requires a discretization of the solvent description. Moreover, vibrational contributions to the hyperpolarizability is significant, especially in the static limit, and it would be important to include these contributions for accurate predictions of this property (21, 22). To some extent, our use of molecular simulations (as detailed below) will capture these vibrational contributions.

With the availability of ab initio molecular dynamics methods within the hybrid quantum mechanical and molecular mechanical (QM/MM) framework (23, 24) and hybrid QM/MM response property techniques (25–27), it is now a realistic proposition to model finite temperature structures of solutes in complex environments and to compute properties of the solute molecule explicitly including all environmental perturbations. In the present work we have adopted this standpoint and utilized hybrid QM/MM structure and property procedures to analyze the first hyperpolarizability/BLA parameter relationship, using stilbazolium merocyanine (SM) as a test system. We here focus our analysis on 1D (quasi) nonlinear optical materials but note that there certainly also are NLO materials of 2D or 3D character with more complicated structural NLO relationships.

Computational Details

Car–Parrinello molecular dynamics (28) in a hybrid QM/MM framework (23, 24, 29) (CPMD-QM/MM) is used to model the finite temperature structure of SM (see Fig. 1) in water and in chloroform solvents. The hyperpolarizability calculations were performed using either a continuum or a hybrid QM/MM solvent description. All calculations have been performed using the Sadlej–pVTZ basis set at the CAM-B3LYP (30) level of density functional theory. The continuum solvent calculations have been performed using the nonequilibrium polarizable continuum model (PCM) approach (31). For the calculations of the solute properties in discrete solvents, we have used the QM/MM response technique (25–27) where the solvent molecules were described using a classical force-field with charges and polarizabilities. Particularly, we have used the modified Ahlström model (32) for the water solvent which includes both atomic charges and polarizabilities for the water molecules and allows for an explicit treatment of mutual solute-solvent polarization in the property calculations. All calculations have been performed using a development version of the Dalton program (33).

Fig. 1.

The molecular structure of stilbazolium merocyanine.

The CPMD-QM/MM calculations for SM in chloroform and water were initiated from a configuration obtained from a well equilibrated molecular dynamics (MD) run. The details about the MD, CPMD-QM/MM, and property calculations using PCM and QM/MM response techniques have been presented elsewhere (25, 26, 34). In the case of SM in water, the first hyperpolarizability was calculated for 56 configurations extracted at regular intervals from a CPMD-QM/MM trajectory for a total time scale of 23 ps. Three sets of calculations for these configurations were performed: (i) calculations on SM without solvent (referred to as QM/MM-0); (ii) calculations using nonequilibrium PCM solvent response (QM/PCM); (iii) calculations using the QM/MM response technique (QM/MM-1). In the case of SM in chloroform, the calculations were performed for 90 configurations obtained at a regular interval from a CPMD-QM/MM trajectory of a 26 ps total time scale. A single set of property calculations were performed for this system using the nonequilibrium PCM response approach. The solute molecular cavity, used in all the aforementioned PCM response calculations, was constructed using the United Atom for Hartree–Fock (UAHF) radii set (35), where the cavity is constructed from spheres located on the heavy elements only.

Results and Discussion

We have calculated two representatives of the first hyperpolarizability tensor, β, namely β_vec and β_tot, where β_vec refers to the hyperpolarizability along the molecular dipole moment and β_tot refers to total the hyperpolarizability, defined as (13);

[2]

[3]

[4]

where μ is ground state molecular dipole moment. While there are 27 components of β that can be computed and analyzed for each configuration, the quantities of our interest are β_vec and β_tot. The former one is the vector component of β along the dipole moment direction and is sampled experimentally in electric-field-induced second harmonic generation experiments (13).

The results of β_vec computed for SM are plotted versus the BLA parameter in Fig. 2. Panels A, B, and C show the results of quadratic response calculations in gas phase, in water solvent simulated by the PCM model, and by full QMMM calculations (QM/MM-0 QM/PCM and QM/MM-1 models). The BLA parameter has been computed using the formula

[5]

where R_C11–C12 is the bond length between C11 and C12 and using similar definition for the other bond lengths (see Fig. 1). As expected, SM remains mostly in the charge-separated form (based on the negative BLA parameter values for most of the configurations) with only 2 out of 56 configurations with positive or zero value for the BLA parameter (corresponding to cyanine-like and neutral configurations). Overall, the BLA parameter calculated for SM in water varies in the range -0.15 to 0.05. In Fig. 2, open circles represent a scattering diagram of β_vec versus the BLA parameter, while filled circles show the average β_vec for each value of the BLA distance parameter where a value of 0.02 has been used as a bin value for BLA. We note that the results show a typical slope of β_vec in terms of the BLA parameter in the vacuum calculations as predicted by the β_vec-BLA two-state model (2–13), while the slope is significantly reduced in the polarizable continuum model, just to become depleted in the QM/MM calculations, indicating that the structure-property relation is invalid at the high polarity of an aqueous solution. The scattering of data around the straight line reflects incomplete statistics, thus values at the extreme of the BLA parameter are more uncertain than those at the more frequent BLA values covered by the CPMD dynamics. Despite the fluctuation, the general picture of the depletion of the BLA effect is clear. It is also seen that the absolute value of β_vec is significantly increased by the solvent and that PCM overestimates this increase with respect to the more accurate QM/MM model. As seen in Fig. 2, the vacuum case (QM/MM-0) is qualitatively different from the QM/PCM and QM/MM-1 cases; β_vec decreases as the molecule becomes more zwitter-ionic, thus at extreme negative BLA values, while in other cases β_vec is roughly constant as function of BLA. We point out that β_vec approaches zero for cyanine-like structures in the case of the QM/MM-0 model, while it remains nonzero for the other two models. Moreover, the gradient of the QM/MM-1 model is approximately zero, suggesting that there is no correlation between BLA and the molecular response. The larger spread in the QM/MM results as compared to the PCM results reflects the use of explicit solvent molecules, which leads to larger solvent anisotropies. In the present calculations, the QM/MM-0 model reproduces the already established relationship between β_vec and the BLA parameter where the latter goes to zero for the geometry corresponding to a BLA parameter equal to zero. However, the other two models show that there is no longer any BLA parameter dependence of β_vec as it remains roughly constant for all values of the BLA parameter.

Fig. 2.

BLA dependence of first hyperpolarizability calculated for SM in water from (A) QM/MM-0, (B) QM/PCM, and (C) QM/MM-1 models.

The almost constant behavior of β_vec in solution can possibly be rationalized considering internal and external electric fields perturbing the SM molecule. In the vacuum case a zwitter-ion is built up as the BLA parameter decreases, meaning that an internal electric field is created over the molecule. This perturbs the system and changes β_vec with a decrease following the internal field. In solution this zwitter-ion will induce a large charge and dipole distribution in the solvent with the consequence of a build-up of an external (solvent) field in the opposite direction as the internal field. Thus, in solution the effective field will be reduced in magnitude as compared to the vacuum case. The net field effect will be small, leaving only the residual solvent field created in the case where the BLA parameter is approximately zero. Thereby, the combined effect of the internal and external fields will cancel. For a highly polar solvent such as water this is more effective thereby rationalizing the qualitative difference between the results obtained in vacuum and in a highly polar medium.

The spread of the individual β_vec values in Fig. 2 can be interpreted as follows: The spread of the β_vec values Fig. 2—small in A, large in B, and even larger in C—refer to that the zwitter-ion charges are stabilized in the PCM medium, making it preferable for the system to locate the residual molecular charges as close as possible to the dielectric medium. Thus these internal charges do not fluctuate much in magnitude/location, giving a small spread of the β_vec values (a rather constant electric field). In vacuum (Fig. 2A) nothing can stabilize the charges, and these are correspondingly fluctuating in magnitude and location, which produces a fluctuating electric-field that eventually leads to the fluctuating of β_vec. In the case of QM/MM, the internal charges are also stabilized, but due to the explicit sampling of the solvent (which is implicit in PCM) the location and magnitude of the internal charges will fluctuate even more than in the case of vacuum (the location of the internal charges is coupled to the position of the explicit solvent molecules). Thus in the case of QM/MM the explicit solvent anisotropy leads to even larger fluctuations than in vacuum.

To test the β_vec-BLA parameter relationship in the case of a nonpolar solvent, we have computed β_vec for SM in chloroform, a solution with small dielectric constant (ϵ = 4.15). The nonequilibrium PCM model used is a good approximation for nonpolar solvents that are not involved in site-specific interaction with solute molecules (36–38). The scatter diagram of β_vec-BLA parameter and a plot of 〈β_vec(BLA)〉 against the BLA parameter are shown in Fig. 3. A linear dependence between β_vec and the BLA parameter can be discerned. As shown in other semiempirical finite-field calculations, β_vec reaches zero for a nonzero value of BLA, which corresponds to 0.02 in the case of SM in chloroform. Another interesting observation is that the range of BLA values picked up by the SM molecule in chloroform varies between -0.05 to 0.04, meaning that the average molecular structure is rather close to a cyanine-like structure instead of a zwitter-ionic form as it has been observed in the case of the water solvent. This is in complete agreement with the observed behavior of solvatochromic molecules, which structurally transform to a zwitter-ionic state in polar solvents when compared to the neutral or cyanine like structure in nonpolar solvents and in gas phase (12, 34, 39). It is instructive to analyze the gradient of this plot, which actually is larger than in the plots corresponding to the QM/MM-0, QM/PCM, and QM/MM-1 models for SM in water. In fact, the gradient related to the results predicted by the QM/PCM model of SM/CHCl₃ is even larger than that for the QM/MM-0 model of SM/water. This can be rationalized based on the charge distribution of SM in chloroform and water solvents. SM in water is in charge-separated form while there is no significant charge separation of SM in chloroform; see Fig. S1 showing donor and acceptor group charge distributions for SM in these solvents. This means that for SM in water the external electric field generated by the PCM model largely cancels the internal electric field, resulting in smaller dependence of the structure to hyperpolarizability. In contrast, for SM in chloroform the local field due to SM survives chloroform, being a nonpolar solvent, does not generate an equally opposing or canceling external electric field. So in that case the internal electric field is highly sensitive to the molecular geometry itself and this leads to the observed dependence of hyperpolarizability to the BLA parameter.

Fig. 3.

BLA dependence of first hyperpolarizability calculated for SM in chloroform from (A) QM/PCM model.

In order to rationalize the results in terms of the two-state model for the first hyperpolarizability, Eq. 1, we evaluated the ingoing parameters, i.e., the ground and excited state dipole moments, the transition dipole moment, and the transition energy directly using the QM/MM-0, QM/PCM and QM/MM-1 models. Fig. 4 A–C shows the , and Δμ as a function of BLA, respectively. The difference of the ground and excited state dipole moments largely dictates the β_vec-BLA parameter model and the sign in β_vec. In vacuum, Δμ has a negative minimum for negative BLA parameters, then increases, reaching first an inflexion point at zero BLA, and further increases to positive values for positive BLA. This exactly reproduces the earlier reported behavior of Δμ versus the BLA parameter (5, 7, 9). SM, being a negative solvatochromic molecule, has a larger ground state dipole moment compared to the excited state dipole moment due to its zwitter-ionic nature (39, 40). In a more polar solvent like water the zwitter-ionic form can be naturally stabilized (37, 40). So one would expect Δμ to be negative for the molecular structures with negative BLA something that is verified in Fig. 4C. As predicted, for the cyanine-like structure generated by equal mixing of charge-separated zwitter-ionic and the neutral canonical forms corresponding to a zero BLA parameter, the ground and excited state dipole moments are equal and Δμ eventually becomes zero (9), which also is seen in Fig. 4C. For the neutral form, the excited state dipole moment is larger than the ground state dipole moment, which leads to a sign change of Δμ, being now positive. The QM/MM-0 results for the vacuum case also give insight into the dependence on μ_ge and of the BLA parameter (5, 7, 9). The squares of these parameters have been assumed to have inverted parabolic behavior for different BLA parameters; however, we show that is decreasing with decreasing value of the BLA parameter, while remains more or less constant. For the vacuum case, out of the three contributions to the total β_vec, remains almost constant and Δμ decreases with decreasing value of the BLA parameter. For zero BLA, Δμ is zero and so β_vec becomes zero independent of the values of ω_ge and μ_ge. However, with the QM/PCM and QM/MM-1 models, the three variables show different behavior with increasing BLA parameter; ω_ge and μ_ge increase with increasing BLA parameter while the value of Δμ increases with increasing BLA parameter. Overall, these variables show opposing behavior with increasing BLA parameter resulting in the product, β_vec, that remains constant with increasing BLA. The results show that the β_vec-BLA parameter relationship works for the gas phase and in the case of nonpolar solvents and that it eventually breaks down for more polar solvents like water. The assumptions made for the BLA dependence of ω_ge and μ_ge are then no longer correct.

Fig. 4.

BLA dependence of (A) , (B) , and (C) Δμ calculated for SM in water where ω_ge and μ_ge are excitation energy and transition dipole moment, respectively, while Δμ is difference in excited and ground state dipole moments.

Finally, we compare the β_vec computed from our theoretical calculations with the result obtained from electric-field-induced simple harmonic (EFISH) measurements (41). β_vec is the quantity that is sampled in these experiments. In order to match with the hyperpolarizability corrected for the frequency dispersion, as given in ref. 41, we also computed the average frequency dependent hyperpolarizability for SM in chloroform and in water solvents; the results are presented in Table S1 and show the expected increase in magnitude of the first hypoerpolarizability with increasing frequency.

As mentioned earlier, the QM/MM-1 results include mutual solute–solvent polarization, which is an important effect to be incorporated in the case of a more polar solvent like water (26, 42). While in the case of SM in chloroform, we can use QM/PCM response, which includes an adequate solvent description due to the nonpolar nature of chloroform (36, 37). The comparison between experimental and calculated β_vec values are presented in Table 1. The theoretical β_vec values for SM in chloroform and in water solvents were calculated from the average of the property computed for many configurations. Because the simulations are carried out in an isothermal-isobaric condition, we have not used any weights for different configurations. We have verified the convergence of the property by plotting the N-point average of the hyperpolarizability as a function of the number of configurations (see Fig. S2). Indeed it shows that in all cases, the average hyperpolarizability is converged with respect to the number of configurations.

Table 1.

Average β_vec and β for SM in different solvents

Solvent	Method	βvec, 10-30esu	β, 10-30esu
Water	QM/MM-1	−110.20	183.80
CHCl3	QM/PCM	−38.10	65.56
Methanol	Expt.*	−51	—

^*Ref. 41.

Comparing the theoretical and experimental β_vec values in different solvents, we find, as expected, that the experimental β_vec for SM in methanol is larger in magnitude when compared to SM in chloroform and smaller than for SM in water. SM, being a negative solvatochromic molecule, has a ground state that is stabilized by polar solvents when compared to its excited state. So ω_ge increases in magnitude with increasing solvent polarity. However, the solvent dependence of β_vec can not be rationalized based on ω_ge, one of the three parameters used in the two-state model. From the results presented in Table 1, we see that β_vec increases with increasing solvent polarity, which is in contradiction to the reports on solvent dependence of β_vec in the case of phenol blue (5), which showed a maximum for the intermediately polar solvent chloroform. An important aspect to consider here is that the former molecule is a negative solvatochromic molecule, while the latter is a positive solvatochromic molecule (39). Probably this can be addressed once we have many such hybrid QM/MM studies carried out for a sample of negative and positive solvatochromic molecules.

Conclusions

We have applied multiscale QM/MM techniques to explore the validity of the cherished BLA model to design materials with high NLO coefficients. We find that the role of a solvent is crucial, and that in fact this model only strictly holds in the vacuum limit, while it gradually deteriorates with the polarity of a surrounding medium and becomes invalid in the high polarity limit. We have rationalized the results by means of vacuum and polarizable continuum model calculations, and by QM/MM evaluation of the quantities (i.e., ω_ge, μ_ge and Δμ) that make up for the two-state NLO model. In the vacuum case, the BLA dependence of the Δμ parameter is reproduced according to earlier understanding (5, 7, 9), while in the presence of a polar solvent like water the BLA dependence of all these parameters appears differently from that predicted earlier showing that the BLA dependence of β_vec is no longer valid. In particular, the difference in dipole moments between ground and excited states (Δμ) does not go to zero for vanishing BLA. The opposing behavior of these parameters with BLA results in a cancellation of the individual contributions leading to a nondependence of β_vec with BLA. From the results of our study the clear picture emerges that the first hyperpolarizability/BLA parameter relation becomes restricted to the vacuum limit, and that it progressively deteriorates with increasing polarity of the solvent surrounding.