Polarizability units

We consider first the polarizability of a molecule consisting of two or more polarizable parts which may be atoms, bonds, or other units. When the molecule is placed in an electric field the effective field which induces dipole moments in various parts is not just the external field but rather the local field which is influenced by the induced dipoles of the other parts. The classical theory of this interaction of polarizable units was presented by Silberstein36 and others and is summarized by Stuart in his monograph.40 The writer has examined the problem in quantum theory and finds that the same results are obtained to the order of approximation being considered. 

On the other hand, electrostatic models regard the ligands or the whole crystal as polarizable units and thereby lead to weaker Coulomb and spin-orbit interactions. In a dielectric screening model (DSM) from Morrison et al. (1967) the f element is placed within an empty sphere with radius Rs which is embedded into an infinite medium with dielectric constant e. This leads to a reduction AFk of the Slater parameters (Newman, 1973) ... 

First, as with spheres, the strength of interaction is proportional to the z product of volumes, this time the product of 7rR2, the volume per unit length. Superficially at least, the masses of the two rods act as polarizable units. Figure L1.53... 

Number densities of the set of ions of valence v in region i (with v subscript to distinguish from index in frequency summation and index of refraction). Number densities of suspended particles, molecules, atoms, polarizable units. 

The title of this chapter merits the question, what defines Junctionality For the purposes of this discussion, we have hmited our definition of JunctiomUty to a group, or groups, of polarizable units with an orientation suitable for cooperative substrate-directed interactions. In most cases, directed interactions should be considered to be toward a metal-bound substrate, but in some instances (i.e., supramolecular self-assemblies and sensors) metal ions may be absent from our discussion. 

For molecules with a small size, a, qa< ) the fields scattered by aU the polarizable units are in phase. In this case a single quantity, the polarizability of the molecule determines the Raman spectra. The symmetry of the molecnle and of the tensor is fundamental for the calcnlation of the symmetry of the vibrational modes and of the intensity of the relative Raman an IR absorption bands (Long, 1977). 

In harmonic crystals, the vibrations are phonons, i.e. plane waves involving the motion of all atoms in the sample. Phonons are described by a frequency illuminated volume seen by the detector, and fields irradiated by aU polarizable units interfere. As a consequence, one phonon scattering processes (first-order Raman scattering) are subjected to the selection rules ... 

The symmetry properties of in any medium can be established considering that it transforms as a fourth-rank polar tensor under the macroscopic symmetry operations. The symmetry of the microscopic polarizable units can be used to simplify the microscopic expression... 

The microscopic origin of the nonlinear response is the distortion induced in the molecular charge distribution due to the electrical field. The presence of a microscopic dipole produces a macroscopic polarization in the unit volume P = N r) where N is the number density of polarizable units and (er) the expectation value of the dipole moment induced in each unit. In order to evaluate (sr) we will use the density matrix formalism, because it is the easiest way to relate microscopic properties to macroscopic ones and to cope with macroscopic coherence effects. In the absence of fields, the medium is supposed to be described by an unperturbed Hamiltonian Hq and to be at equilibrium. When the fields are applied, the field-matter interaction contributes a time-dependent term V(t) =-E(t)P(t) to the global energy. The evolution of the system under this perturbation can be described through the equation of motion of the density operator ... 

Before the development of molecular orbital approaches to the calculation of ROA, a number of simple models of ROA were advanced to provide a conceptual basis for understanding ROA spectra. In various ways, these models arise from considering the polarizability of the molecule as the sum of local polarizability units, such as those associated with bonds or atoms. While the polarizability and its individual local components are independent of the location of the origin of the molecule, the magnetic dipole and electric quadrupole ROA tensors do depend on this origin. In the FFR approximation, we can express these tensors in terms of local contributions as... 

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