Orientational polarizability, relation

When Stokes shifts are plotted as a function of the orientation polarizability A f (Lippert s plot, see Section 7.2.2), solvents are distributed in a rather complex manner. A linear relationship is found only in the case of aprotic solvents of relatively low polarity. The very large Stokes shifts observed in protic solvents (methanol, ethanol, water) are related to their ability to form hydrogen bonds. 

Molecular distortion polarizability is a measure of the ease with which atomic nuclei within molecules tend to be displaced from their zero-field positions by the applied electric field. (3) Orientation polarizability is a measure of the ease with which dipolar molecules tend to align against the applied electric field. The electron polarizability of an individual molecule is related to the -> permittivity (relative) of a dielectric medium by the -> Clausius-Mossotti relation. 

The dielectric constant is a natural choice of order parameter to study freezing of dipolar liquids, because of the large change in the orientational polarizability between the liquid and solid phases. The dielectric relaxation time was calculated by fitting the dispersion spectrum of the complex permittivity near resonance to the Debye model of orientational relaxation. In the Debye dispersion relation (equation (3)), ij is the frequency of the applied potential and t is the orientational (rotational) relaxation time of a dipolar molecule. The subscript s refers to static permittivity (low frequency limit, when the dipoles have sufficient time to be in phase with the applied field). The subscript oo refers to the optical permittivity (high frequency limit) and is a measure of the induced component of the permittivity. 

The Polarization of Polar Molecules. Equation (4.7) gives an expression for the electronic polarizability of a spherically symmetric atom or molecule. If a molecule has a freely rotating permanent dipole, which has a time-averaged dipole moment of zero, in the presence of an external field E, there may develop an induced orientational dipole. This would then be related to the orientational polarizability of the molecule. If at some instant, the permanent dipole II of the molecule is at an angle 6 to the applied field, its energy in the field will be given by... 

We now consider this issue in a more rigorous fashion. The inference of molecular orientation can be explamed most readily from the following relation between the surface nonlinear susceptibility tensor and the molecular nonlinear polarizability... 

The Raman measurements provide values directly for P)mn, the coefficients of the Legendre expansion related to coordinates axes chosen with respect to the principal axes of the differential polarizability tensor, hence the superscript r. The coefficients Pimn for the orientation of the units of structure must then be obtained by further calculation from the P)mn. 

The EM method has been tested on the inert gases and a range of small molecules and gives good agreement with experimental results in almost all cases.17 This method will be discussed further in relation to the orientation dependence of the electron impact ionization cross section in a later section. The semiempirical polarizability method described below was developed to calculate and to use it with the amax values obtained from this method in order to calculate the energy dependence of the cross section. 

Dipole-Dipole Interaction. The first of the four terms in the total electrostatic energy depends on the permanent dipole moment of the solute molecule of radius a (assuming a spherical shape) immersed in a liquid solvent of static dielectric constant D. The function f(D) = 2(D - l)/(2D + 1) is known as the Onsager polarity function. The function used here is [f(D) — f(n2)] so that it is restricted to the orientational polarity of the solvent molecules to the exclusion of the induction polarity which depends on the polarizability as of the solvent molecules, related to the slightly different Debye polarity function q>(n2) according to... 

In this section, a simple description of the dielectric polarization process is provided, and later to describe dielectric relaxation processes, the polarization mechanisms of materials produced by macroscopic static electric fields are analyzed. The relation between the macroscopic electric response and microscopic properties such as electronic, ionic, orientational, and hopping charge polarizabilities is very complex and is out of the scope of this book. This problem was successfully treated by Lorentz. He established that a remarkable improvement of the obtained results can be obtained at all frequencies by proposing the existence of a local field, which diverges from the macroscopic electric field by a correction factor, the Lorentz local-field factor [27],... 

The orientation of the polarizability tensor related to the planar molecule axes (i.e., to the crystal structure) is investigated by measuring the orientation and dispersion of the indicatrix. Usually, the principal axes of polarizability coincide with the directions of the three principal axes of the indicatrix X, Y, and Z. Perpendicular to each of the two circular sections of radius np is an optic axis. The maximum absorption is always parallel to one of the three optical directions. 

In order to link An to the orientational order parameter 5, one needs to develop a molecular model to connect n, a macroscopic property, to the related molecular property, the polarizability a. One widely used model is that described by Vuks and others, which leads to a quantity Kv(hdX is directly proportional to 5(7) ... 

As to infrared spectroscopy - and the same holds good for other spectral ranges -the orientational order is readily observable in form of dichroism Being related to the molecular shape, the molecular polarizability is anisotropic as well. By the alignment of the molecules this anisotropy is transferred to the sample, however damped due to the imperfect order as described by the order parameters. As a consequence, the dielectric function and furthermore the (complex) refractive index are anisotropic, so that eventually (linear) dichroism and birefringence occur. 

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