Polarizability average

For a worm-like chain the difference between the main polarizabilities averaged over all conformations <7i - 72 >s determined by Eqs. (46) - (48) where 71 corresponds to direction h of the end-to-end distance. Hence, the angle e in Eq. (85) is the angle formed by the direction of the dipole moment m of a chain molecule and vector h. 

An axially symmetric molecule is characterized by its linear polarizability in the principal axes and ayy = a" and a" = af/. It is a good approximation to assume that its second- and third-order polarizability tensors each have only one component and respectively, which is parallel to the z-principal axis of the molecule. For linear and nonlinear optical processes, the macroscopic polarization is defined as the dipole moment per unit volume, and it is obtained by the linear sum of the molecular polarizabilities averaged over the stati,stical orientational distribution function G(0). This is done by projecting the optical fields on the molecular axis the obtained dipole is projected on the laboratory axes and orientational averaging is performed. The components of the linear and nonlinear macroscopic polarizabilies are then given by ... 

Results shown for the liquid are averages. Dipole moments are in debye and dipole polarizabilities (average and anisotropic) are in a.u. Rayleigh depolarizations (cr for natural, planar and circular polarizations) are dimensionless. Standard deviations are also shown. 

In the last expression 5n r,cu) stands for the Fourier transform of n r,t) — n r), where n r) is the ground-state density of the system. The quantity that is usually measured in experiments, the photo-absorption cross-section, is essentially proportional to the imaginary part of the djmamical polarizability averaged over the three spatial directions... 

Cartesian electric quadrupole moment operator. The electric dipole - magnetic dipole and electric dipole - electric quadrupole polarizabilities govern the optical rotation of chiral molecules. For samples of randomly oriented molecules, the contribution from the electric dipole - electric quadrupole polarizability averages to zero, and only the trace (the sum of the diagonal elements) of the electric dipole - magnetic dipole polarizability contributes to the optical rotation. The... 

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in temis of an ensemble average over the microscopic nonlmear polarizability tensors of the individual constituents. 

Until now, we have discussed the use of additivity schemes to estimate global properties of a molecule such as its mean molecular polarizability, its heat of formation, or its average binding energy to a protein receptor. 

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

We define the concentration of fluctuation domains at any instant by the symbol N. In addition, we assume that the polarizability associated with one of these domains differs from the macroscopic average value for the substance... 

This approach to separating the different types of interaetions eontributing to a net solvent effeet has elieited much interest. Tests of the tt, a, and p seales on other solvatochromie or related proeesses have been made, an alternative tt seale based on ehemieally different solvatochromie dyes has been proposed, and the contribution of solvent polarizability to ir has been studied. Opinion is not unanimous, however, that the Kamlet-Taft system eonstitutes the best or ultimate extrathermodynamie approaeh to the study of solvent effeets. There are two objections One of these is to the averaging process by which many model phenomena are eombined to yield a single best-fit value. We eneountered this problem in Section 7.2 when we eonsidered alternative definitions of the Hammett substituent eonstant, and similar eomments apply here Reiehardt has diseussed this in the eontext of the Kamlet-Taft parameters. - The seeond objeetion is to the elaim of generality for the parameters and the eorrelation equation we will return to this eontroversy later. 

What is of primary importance chemically is not the ground state, nor the ground configuration, which is some average of valence states, of the free atom but it is the atomic response properties to perturbations by other atoms. That is governed by the energies and spatial extensions and polarizabilities of the upper core and of the compact valence orbitals ([34], p 653). 

In quantum theory as in classical theory the isotropic Raman spectrum is expressed in terms of the average value of the polarizibility tensor a(0) = (1/3) Sp a randomly changing in time due to collisions ... 

The particles of the preceding treatment are here replaced by small elements of volume of the solution. The excess polarizability of one of these volume elements due to the deviation of its concentration from the average may be written... 

Usually, the individual components of the polarizability tensor 5 are not given, but only the average value of its diagonal elements which is defined as = 1/3 (a, + otyy + azz). 

Table 10-3. Compilation of mean absolute deviations for static average polarizabilities [a.u.] of small main group molecules from different sources.

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