Polarity/polarization hyperpolarizability

Ah initio methods can yield reliable, quantitatively correct results. It is important to use basis sets with diffrise functions and high-angular-momentum polarization functions. Hyperpolarizabilities seem to be relatively insensitive to the core electron description. Good agreement has been obtained between ECP basis sets and all electron basis sets. DFT methods have not yet been used widely enough to make generalizations about their accuracy. 

Expressions for the sixth- through tenth-order coefficients are given in the Appendix. In ESHG experiments with the optical field polarized perpendicular to the static electric field, the measured second hyperpolarizability is [13]... 

In all the variational methods, the choice of trial function is the basic problem. Here we are concerned with the choice of the trial function for the polarization orbitals in the calculation of polarizabilities or hyperpolarizabilities. Basis sets are usually energy optimized but recently we can find in literature a growing interest in the research of adequate polarization functions (27). 

Table 6 Effect of the polarization functions on the polarizabilities and hyperpolarizabilities of He...

Table 7 Polarizabilities and hyperpolarizabilities of H2 at = 1.4 au, with optimization of C in polarization functions (Cipl = l-l) from Ref. 6...

This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field—induced polarization functions have been constructed from the first- and second-order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion. We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities. 

In any event, we are confident that the computational approach developed in this study, owing to its efficient use of molecular symmetry, can help develop large basis sets for first and second hyperpolarizabilities. An important aim would be that of estimating, at least at empirical level, Hartree-Fock limits for these quantities. To this end the use of basis sets polarized two times, according to the recipe developed by Sadlej [37], would seem very promising. 

In a previous work [1,2], we were interested in the calculation of second order hyperpolarizabilities of eonjugated systems including substituted benzenes, pyridine N-oxydes and vinyl oligomers, in relation with non linear optical activity [3]. We showed that MNDO ealeulations were in good agreement with SCF ab initio results obtained using a double zeta basis set plus polarization and diffuse orbitals. 

The SH signal directly scales as the square of the surface concentration of the optically active compounds, as deduced from Eqs. (3), (4), and (9). Hence, the SHG technique can be used as a determination of the surface coverage. Unfortunately, it is very difficult to obtain an absolute calibration of the SH intensity and therefore to determine the absolute number for the surface density of molecules at the interface. This determination also entails the separate measurement of the hyperpolarizability tensor jS,-, another difficult task because of local fields effects as the coverage increases [53]. However, with a proper normalization of the SH intensity with the one obtained at full monolayer coverage, the adsorption isotherm can still be extracted through the square root of the SH intensity. Such a procedure has been followed at the polarized water-DCE interface, for example, see Fig. 3 in the case of 2-( -octadecylamino)-naphthalene-6-sulfonate (ONS) [54]. The surface coverage 6 takes the form ... 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

Based on the fundamental dipole moment concepts of mesomeric moment and interaction moment, models to explain the enhanced optical nonlinearities of polarized conjugated molecules have been devised. The equivalent internal field (EIF) model of Oudar and Chemla relates the j8 of a molecule to an equivalent electric field ER due to substituent R which biases the hyperpolarizabilities (28). In the case of donor-acceptor systems anomalously large nonlinearities result as a consequence of contributions from intramolecular charge-transfer interaction (related to /xjnt) and expressions to quantify this contribution have been obtained (29). Related treatments dealing with this problem have appeared one due to Levine and Bethea bearing directly on the EIF model (30), another due to Levine using spectroscopically derived substituent perturbations rather than dipole moment based data (31.) and yet another more empirical treatment by Dulcic and Sauteret involving reinforcement of substituent effects (32). 

The large molecular hyperpolarizability of the merocyanine chromophore (4,5) and the highly polar environment of the quasicrystals has prompted studies of the second order nonlinear optical properties of these materials (6). 

Figure 9. Variation of the real (m) and imaginary (o) parts of the two photon hyperpolarizability (yr) with two-photon energy (2aJt) for the yellow solution. The error bars represent 90% confidence limits. All data are taken in a 1111 geometry (all beams polarized ). The solid lines are theoretical fts to the data according to Equation 3. (Reproduced with permission from Ref. 24. Copyright 1979, American Institute of...

We have shown in this paper the relationships between the fundamental electrical parameters, such as the dipole moment, polarizability and hyperpolarizability, and the conformations of flexible polymers which are manifested in a number of their electrooptic and dielectric properties. These include the Kerr effect, dielectric polarization and saturation, electric field induced light scattering and second harmonic generation. Our experimental and theoretical studies of the Kerr effect show that it is very useful for the characterization of polymer microstructure. Our theoretical studies of the NLDE, EFLS and EFSHG also show that these effects are potentially useful, but there are very few experimental results reported in the literature with which to test the calculations. More experimental studies are needed to further our understanding of the nonlinear electrooptic and dielectric properties of flexible polymers. 

The induced polarization is important in the calculation of molecular properties, such as the hyperpolarizability discussed earlier in this chapter, and for the prediction of molecular packing and macromolecular folding. The diffraction... 

Besides the linear polarization contribution, the hyperpolarizabilities may be accounted for, as well as the field gradients (and higher derivatives of the multipolar fields) taken at the molecular center if the fields are very non-uniform. The total dipole induced by electric fields in a molecule may be written [89]... 

As has been discussed in part I of this review,1 the microscopic hyperpolarizabilities of the ith order have their corresponding quantities at the macroscopic level in the form of nonlinear susceptibilities The macroscopic polarization is then given by... 

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