Polarizability displacement

Let us assume internal equilibrium in Zf, which corresponds to the mutually open subsystems, Zfe=(Z/> Zf 2), with equalized chemical potentials, nf = nf = P.r = dE/dN, at the global chemical potential level. The internal stability refers to intra-5 (hypothetical) charge displacements, dA/y(A) = (A, — A), that preserve N. The corresponding quadratic energy change due to this polarizational displacement from the initial internal equilibrium state ... 

The first term represents the contribution of the polarizability (displacement polarization) and is always positive whereas the second term originates from the permanent dipole moment (orientation polarization) and is positive or negative depending on whether P is smaller or larger than 55°. 

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... 

The most recently introduced optical teclmique is based on the retardation of light guided in an optical waveguide when biomolecules of a polarizability different from that of the solvent they displace are adsorbed at the waveguide surface (optical waveguide lightmode spectroscopy, OWLS) [H]. It is even more sensitive than ellipsometry, and the mode... 

Here X is tire reorganization energy associated witli the curvature of tire reactant and product free energy wells and tlieir displacement witli respect to one another. Assuming a stmctureless polarizable medium, Marcus computed the solvent or outer-sphere component of tire reorganization energy to be... 

As for the change of dipole moment, the change of polarizability with vibrational displacement x can be expressed as a Taylor series... 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

For a given a the force constant ko can be chosen in a way that the displacement d of the Drude particle remains much smaller than the interatomic distance. This guarantees that the resulting induced dipole jl, is almost equivalent to a point dipole. In the Drude polarizable model the only relevant parameter is the combination q /ko which defines the atomic polarizability, a. It is... 

When the Drude particles are treated adiabatically, a SCF method must be used to solve for the displacements of the Drude particle, d, similarly to the dipoles Jtj in the induced dipole model. The implementation of the SCF condition corresponding to the Born-Oppenheimer approximation is straightforward and the real forces acting on the nuclei must be determined after the Drude particles have attained the energy minimum for a particular nuclear configuration. In the case of N polarizable atoms with positions r, the relaxed Drude particle positions r + d5CF are found by solving... 

Isotope superlattices of nonpolar semiconductors gave an insight on how the coherent optical phonon wavepackets are created [49]. High-order coherent confined optical phonons were observed in 70Ge/74Ge isotope superlattices. Comparison with the calculated spectrum based on a planar force-constant model and a bond polarizability approach indicated that the coherent phonon amplitudes are determined solely by the degree of the atomic displacement, and that only the Raman active odd-number-order modes are observable. 

Figure 2.8 Shell model of ionic polarizability (a) unpolarized ion (no displacement of shell) (b) polarized (displaced shell) (c) interactions 1, core-core 2, shell-shell 3, core-shell.

For larger displacement UA-, the variation of i(r, r ) relative to UK can be computed by using the nonlinear polarizability kernels defined below [26] (see Section 24.4). Forces and nonlocal polarizabilities are thus intimately related. 

Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Refer to Fig. 12.1 and its caption for more detail. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. Any such property, P, can be expressed in terms of a Taylor series expansion over the displacements of the coordinates from their equilibrium positions,... 

Orientation effects in benzene derivatives operate in two ways. If the substituent is inductive there are large first order charge displacements at the ortho and para positions, and these can be estimated approximately using the atom polarizabilities (which is very small at the meta position). The changes of bond order, however, and consequently of free valence, vanish in first order and hence depend on Sa. The charge g g at position s therefore increases or decreases from the value unity in the... 

The electron cloud of an ion subjected to an electric field undergoes deformations that may be translated into displacement of the baricenters of negative charges from the positions held in the absence of external perturbation, which are normally coincident with the centers of nuclear charges (positive). The noncoincidence of the two centers causes a dipole moment, determined by the product of the displaced charge (Z ) and the displacement d. The displacement is also proportional to the intensity of the electrical field (F). The proportionality factor (a) is known as ionic polarizability ... 

The terms in equation 1.166 represent total ionic polarizability, composed of electronic polarizability a plus an additional factor a , defined as a displacement term, due to the fact that the charges are not influenced by an oscillating electric field (as in the case of experimental optical measurements) but are in a static field (Lasaga, 1980) ... 

A molecular polarizability effect occurring by the inductive mechanism of electron displacement. Substituent polarizability is a factor governing reactivity. 

Conductivity is a measure of the number of ions per unit volume and their average velocity in the direction of the applied field. Polarizability is a measure of the number of bound charged particles per cubic unit and their average displacement in the direction of the applied field. 

There are two types of charging currents and condenser charges, which may be described as rapidly forming or instantaneous polarizations and slowly forming or absorptive polarizations. The total polarizability of the dielectric is the sum of contributions due to several types of charge displacement in the materials caused by the applied field. The relaxation time is the time required for polarization to form or disappear. The magnitude of the polarizability, k, of a dielectric is related to the dielectric constant, s, as follows ... 

For the uncoupled system (C = 0) two transition temperatures are obtained. One corresponds to the ordering of the protons and is of first order and pure order/disorder type, whereas the other is second order and displacive related to the polarizable sublattice (Fig. 5). 

Abstract This chapter describes the experimentai compiement of theoretical models of the microscopic mechanism of ferroelectric transitions. We use the hydrogen-bonded compounds as examples, and attempt to show that the new experimental data obtained via recently developed high resolution nuclear magnetic resonance techniques for solids clearly support the hypothesis that the transition mechanism must involve lattice polarizability (i.e. a displacive component), in addition to the order/disorder behaviour of the lattices. 

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