Non-linear polarizability

A surface is illuminated with a high-intensity laser, and photons are generated at the second-harmonic frequency through non-linear optical process. For many materials only the surface region has the appropriate symmetry to produce a SHG signal. The non-linear polarizability tensor depends on the nature and geometry of adsorbed atoms and molecules. 

Non-linear optical techniques, such as second harmonic generation (SHG), have recently been used as surface probes. Bulk materials with inversion symmetry do not generate second harmonic signals, while surfaces and interfaces cannot have inversion symmetry, so the total SHG signal will come from the surface region for many systems. The components of the non-linear polarizability tensor have been used to determine the orientation of chemisorbed molecules. 

The linear and non-linear polarizabilities of a molecule in solution differ from those of the isolated molecule in the gas phase since the molecular properties are modified by solute-solvent interactions. Some of these interactions are present even in the absence of externally applied static or optical fields. For molecules with a non-zero dipole moment fj in the electronic ground state the dominant interaction is usually due to the reaction field contribution The molecular dipole moment polarizes the solvent environment and thus generates a polarization field which interacts with the solute. This field is given by (88) (Boettcher, 1973 Wortmann and Bishop, 1998). 

The linear and non-linear polarizabilities of organic molecules are usually determined from measurements of macroscopic susceptibilities of liquid solutions. Classical examples are the measurements of the refractive index, n, or the relative permittivity of pure organic liquids and their interpretation by the well-known Lorentz-Lorenz and Clausius-Mosotti equations. These... 

Heavy elements have many core electrons of high potential energy that are easy to polarize. Hence, high refractive indices (n") and therefore high linear polarizabilities, a, are associated with their presence in molecules. In comparable structures, an increased linear polarizability may be intuitively suspected to lead to an increased non-linear polarizability. The homologous series of haloforms shows this to be the case (Table 1) (Kama and Dupuis, 1990 Samoc et al., 1992 Kohler et al., 1993). Unfortunately, the maximum values of /3 achievable with tr systems even of very high molar refractivities are too low for practical applications. In addition, the incorporation of rr-bonded heavy elements tends to impart chemical (thermal, photochemical) instability on the molecule. 

Electro-optical properties molar r raction and molecular correlation the Kerr effect its component terms and molecular theory molecular angular correlation non-linear polarizability generalized Kerr effect translational and molecular field fluctuations electric saturation. 

Concluding remarks and recapitulation. Rdated electric field-induced phenomena non-linear polarizabilities of atoms and molecules radial and angular correlations fluctuations of molecular fields. 

Solution of a dipolar liquid in a quadrupolar solvent. Let us assume the solvent (component 1) to be non-dipolar, and the solute (component 2) as consisting of dipolar molecules. Moreover, for the sake of simplicity, let the molecides of both components be axially symmetric. Equation (227) now yields, on ne ecting fluctuations and non-linear polarizability ... 

The Role of Non-linear Polarizability. We have just proved that the anisotropic molar saturation constants are non-zero even in liquids consisting of atoms and non-polar molecules. Beside the previous statistical-fluctuational processes, simple liquids reveal mechanisms that are due to the non-linear polarizabilities of atoms and such molecules. This is so because, in general, in addition to linear electric polarizability in the form (288), if a suffidently strong electric field is applied, we have to deal with the non-linear polarizability ... 

In the absence of electron dispersion and absorption, the tensor of second-order non-linear polarizability ft(— cog coi, tog) can be dealt with as totally symmetric, and the numbers of its non-zero and independent elements are to be found in Table 11. Static values of the nonJinear polarizability = i(0 0,0) have been calculated theoretically for some molecules. Tensor elements A" = fi(— Kerr effect or molecular light scattering in liquids. ... 

If the atoms or molecules undergo a non-linear polarizability in the fluctuating field of near neighbourhood, the change in their polarizability tensor is ... 

Numerical values of linear and non-linear polarizabilities of atoms and molecules calculated theoretically and determined from experimental data. Only mean values are given a = (ou + a%i + 033)/s, b = (6u3 + 6333 +... 

The first non-linear polarizability is a tensor of rank 3 and possesses non-vanishing elements only in the case of molecules without a centre of symmetry. This Born distortion tensor was resorted to by Piekara for explaining the non-linear dielectric behaviour of nitrobenzene. In the case of axially symmetric molecules, the tensor of the first non-linear distortion (hyperpolarizability ) can be written as follows ... 

The second non-linear polarizability y is a symmetric tensor of rank 4 having non-vanishing elements even in the case of spherical symmetry, when one has ... 

Table 11 Elements of the symmetric tensor p of non-linear polarizability... 

By equation (193), the Kerr constant raises the value of Ae but lowers Ae". The general relation (194) states that in order to have the value — 2 resulting from Langevin s reorientation theory one has to neglect the quantity Cm due to statistical fluctuations and non-linear polarizabilities. On neglecting Cm and and taking into account that 2 = 5K, equation (194) yields + 3, in conformity with Voigt s non-linear deformation theory. 

It would be highly desirable to measure A 2nAn, and Ass cs 2 A j, independently of one another and to use the result in conjunction with (161) and (194) for deciding which of the mechanisms predominates in the liquid under consideration isotropic statistical fluctuations (Cm)> non-linear polarizability (, K ), or anisotropic statistical processes (KS). The last two can be separated, with some degree of accuracy, also by measurements as a function of temperature and pressure or of analysing-light frequenQT. 

General Treatment of Fluctuational Processes. The previous treatment is good only as long as we deal with strongly dipolar substances and all other polarizational effects remain negligible. In the majority of substances, besides reorientation of permanent dipoles, one has to consider reorientation of the polarizability ellipsoids as well as statistical-fluctuational processes. In calculating the electric polarization (277), one has to include the term accounting for linear distortional polarizability of the dielectric (non-linear polarizabilities are dealt with below) ... 

The coefficient Pp is called the second-order non-linear polarizability or the hyperpolarizability. The corresponding equation in a condensed medium is... 

Sdva et alP have made a comparative study of the non-linear polarizabilities of sohtonie polyenic chains using AMI semi-empirical and ab initio 6-3IG with MP2 methods. 

B. Kirtman and M. Hasan, Chem. Phys. Lett., 57, 123 (1989). Ab Initio Longitudinal Polarizabilities and Hypeipolarizabilities for Polydiacetylene and Polybutatriene Oligomers. For the long-chain semiempirical calculations, see B. Kirtman, Int.. Quantum Chem., to be published. Finite Chain Approximation for Linear and Non-Linear Polarizabilities of Polydiacetylene and Polybutatriene. 

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