Polarity/polarizability hyperpolarizability

Here p, Po, and E are vectors and a, and y tensors, normally referred to as polarizability, hyperpolarizability and second order hyperpolarizability, respectively. Similarly, the polarization in bulk or macroscopic media is given by ... 

Here, P is the bulk polarization, E is the electromagnetic field vector, and the fill in are, respectively, the 1st, 2nd, and 3rd order bulk susceptibilities of the material. These bulk polarizabilities are correlated to the molecular-level polarizability, hyperpolarizability and second hyperpolarizability, a, p and y respectively. However, exact relationships between the bulk and corresponding molecular parameters are still not firmly established. 

The classic absorption, scattering, reflection, or refraction, the intensity of the light reaching the detector is proportional to the intensity of the incident radiation. When one or more laser beams propagating in materials are large enough, the induced polarization fields are proportional to the product of the incident fields. The polarization p. induced in an atom or a molecule by an external field E can be written as Eq. (1). Where the vectors of p- and E are related by the tensors a, 3, and 7, which are often referred to as the polarizability, hyperpolarizability, and second hyperpolarizability, respectively. Similarly, the polarization induced in a medium by an optical field, can be represented by a power series in the optical fields [Eiq. (2)] where X " is the nth-order susceptibility. 

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. 

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). 

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 molecular hyperpolarizabilities are / , 7, and a is the molecular polarizability. Typical values of / are 10 30 esu (esu units mean that the dimensions are in CGS units and the charge is in electrostatic units, thus / in esu means / in units of cmzesuz /erg2) [1-4]. For an electric field typical of Q-switched laser light, 104 statvolts/cm, the contribution to - //(0) from /3S2 is 10 4 debye. These polarizations are infinitesimal on the scale of our usual chemical thinking. Yet, these small polarizations are responsible for the exotic effects described throughout this volume. The perturbation theory approach used to describe these properties is justified by the fact that so little charge actually moves. 

Two of the most important nonlinear optical (NLO) processess, electro-optic switching and second harmonic generation, are second order effects. As such, they occur in materials consisting of noncentrosymmetrically arranged molecular subunits whose polarizability contains a second order dependence on electric fields. Excluding the special cases of noncentrosymmetric but nonpolar crystals, which would be nearly impossible to design from first principles, the rational fabrication of an optimal material would result from the simultaneous maximization of the molecular second order coefficients (first hyperpolarizabilities, p) and the polar order parameters of the assembly of subunits. (1)... 

In the limit of the oriented gas model with a one-dimensional dipolar molecule and a two state model for the polarizability (30). the second order susceptibility X33(2) of a polymer film poled with field E is given by Equation 4 where N/V is the number density of dye molecules, the fs are the appropriate local field factors, i is the dipole moment, p is the molecular second order hyperpolarizability, and L3 is the third-order Langevin function describing the electric field induced polar order at poling temperature Tp - Tg. 

The polarization can now be expanded in several different ways. For example, if the expansion is in terms of powers of the first two terms on the right of eqn (8.2) which together comprise the effective fields as defined by the conventional field factors then effective polarizabilities and hyperpolarizabilities will be defined which incorporate the effects of the discrete terms that have been omitted from the fields in the power expansion. If the expansion is in terms of all contributions except then the polarizabilities and hyperpolarizabilities so defined will... 

X specifies the experimental angle between the external electric field and the light polarization at frequency p, and h is Planck s constant. The scalars and S, and the vectors R and R are functions of the transition moment polarizability and hyperpolarizability tensors, m is a unit vector oriented along the transition dipole moment and F is the internal electric field at the molecule, which depends on the externally applied field such that... 

When evaluated, the summation in Eq. (11) is a rank-one polytensor that represents the potential experienced at molecule A in terms of field components, field gradient components, and so on. This can be used with response properties such as shielding polarizabilities to find property changes dues to electrical influence. The evaluation is analogous to Eq. (11). The incorporation of mutual or back polarization/hyperpolarization requires a self-consistent solution for the induced moments, and this can be done iteratively [170] or if there are no hyperpolarizabilities, it can be done by matrix inversion. 

The interaction of long-chain molecules such as polymers is a problem area where the nature of polarization response can be a significant concern on its own. An example is from a study of parallel hexatriene molecules carried out to represent a truncated form of solid-state polyacetylene [192]. This smdy included both ab initio calculations and an electrostatic model using polarizability, a, and second hyperpolarizability, y, tensors distributed to the carbon centers. The ab initio calculations on a single hexatriene molecule were used to find the distributed tensors for the electrical analysis. The objective in this smdy was not the interaction energy, but the effect on each molecule s polarizability and hyperpolarizability due to intermolecular interaction. The ab initio evaluations benchmarked the electrostatic model calculations both for... 

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