Anisotropic polarizability tensors

All of the EFP interaction terms are pairwise additive, except the induction (polarization) energy, which is modeled with asymmetric anisotropic polarizability tensors located at the centroids of the localized molecular orbitals. The analytic energy gradients (both forces and torques) for anisotropic polarizability tensors are derived via a direct differentiation approach in the form of matrix equations. 

Using this relation for each component of the anisotropic polarizability tensor a ap, which is much smaller than the isotropic part, one gets... 

The field induced orientation of poly-4BCMU in toluene in the red phase (<72 C) indicates an anisotropic polarizability tensor characteristic of a rodlike conformation. The field-induced signal from the polymer in its coil conformation (yellow phase) is extremely weak (smaller by a factor of at least 10 ) confirming the rod-coil transition. From studies of the transient response of the anisotropic light scattering from the red solution, we find a rotational relaxation time of 0.1 sec. This value is consistent with the rotational diffusion constant expected for rodlike polydiacetylene macromolecules. 

The dipole and quadrupole polarizability tensor components of LiH were calculated by MCSCF linear response theory with the basis set of Roos and Sadlej [57] which consists of 13s-, 8p-, 6d-, and 2f-type sets of uncontracted Gaussian functions on Li and 12s-, 8p-, and 5d-type sets of uncontracted Gaussians on H. Due to the small size of the molecule we could perform MCSCF calculations over the whole range of internuclear distances with a very large CAS 0000 520,10,10,4 p g present the tensor components, isotropic, and anisotropic values of the dipole polarizability tensor a as function... 

We noted in the preceding section that the polarizability of an ellipsoid is anisotropic the dipole moment induced by an applied uniform field is not, in general, parallel to that field. This anisotropy originates in the shape anisotropy of the ellipsoid. However, ellipsoids are not the only particles with an anisotropic polarizability in fact, all the expressions above for cross sections are valid regardless of the origin of the anisotropy provided that there exists a coordinate system in which the polarizability tensor is diagonal. 

It is not difficult to generalize the results of this section to an anisotropic ellipsoid the axes of which coincide with the principal axes of its permittivity tensor. The principal values of the polarizability tensor of such a particle are... 

The stress-optical behaviour of an elastomeric network of PDET is measured over a wide range of elongation ratios and temperatures. Theoretical calculations are carried out with the RIS model. For Act, no reasonable modification of the conformational energies or contributions to the anisotropic part of the polarizability tensor would achieve agreement between theory and experiments. The discrepancy between theoretical and experimental results may be qualitatively explained by intermolecular interactions. Agreement between theory and experiment is only obtained assuming the unlikely value of about + 4.2 kJ mol-1 for E(on). 

Besides the isotropic overlap-induced dipole component familiar from the rare gas pairs, we will now in general have other significant induced dipole components if molecules are present, namely multipole-induced and distorted frame-induced dipole components, see Chapter 4 for details. Moreover, these anisotropic dipole components couple with the polarizability tensor and thus give rise to simultaneous transitions in two (or perhaps more) molecules. Furthermore, molecules in general interact with more or less anisotropic forces which to some extent does also affect the spectra of molecular systems. 

The polarizability tensor, a, introduced in section 4.1.2, is a measure of the facility of the electron distribution to distortion by an imposed electric field. The structure of the electron distribution will generally be anisotropic, giving rise to intrinsic birefringence. This optical anisotropy reflects the average electron distribution whereas vibrational and rotational modes of the molecules making up a sample will cause the polarizability to fluctuate in time. These modes are discrete, and considering a particular vibrational frequency, vk, the oscillating polarizability can be modeled as... 

To see why this is the case, we first consider the portion of the response that arises from llsm. According to Equation (10), we can express (nsm(t) nsm(0)> in terms of derivatives of llsm with respect to the molecular coordinates. Since in the absence of intermolecular interactions the polarizability tensor of an individual molecule is translationally invariant, FIsm is sensitive only to orientational motions. Since the trace is a linear function of the elements of n, the trace of the derivative of a tensor is equal to the derivative of the trace of a tensor. Note, however, that the trace of a tensor is rotationally invariant. Thus, the trace of any derivative of with respect to an orientational coordinate must be zero. As a result, nsm cannot contribute to isotropic scattering, either on its own or in combination with flDID. On the other hand, although the anisotropy is also rotationally invariant, it is not a linear function of the elements of 11. The anisotropy of the derivative of a tensor therefore need not be zero, and nsm can contribute to anisotropic scattering. 

If the polarizability tensors a and a have elements only in the directions of the three principal axes 1,2,3, equation (170) yields for the anisotropic part of the Kerr constant ... 

Many complex fluids contain orientable molecules, particles, and microstmctures that rotate underflow, and under electric and magnetic fields. If these molecules or microstructures have anisotropic polarizabilities, then the index of refraction of the sample will be orientation-dependent, and thus the sample will be birefringent. In general, the anisotropic part of the index of refraction is a tensor n that is related to the polarizability a of the sample. The polarizability is the tendency of the sample to become polarized when an electric field is applied thus P = a E, where P is the polarization and E is the imposed electric field. When the anisotropic part of the index of refraction is much smaller than the isotropic part (the usual case), the index-of-refraction tensor n can be related to a by the Lorentz-Lorenz formula ... 

The terms isotropic and anisotropic are meant to refer to the invariants of the polarizability tensor, not to scattering per se we prefer generally terms such as trace and anisotropy scattering. 

So far only the minimal multipole moments and their contribution to the j, a., and (t2) values have been considered. However, they are not the only manifestations of the degeneracy in the characteristics of the absorption and Raman spectra under consideration. For instance, an important additional contribution to the torque can arise because of the anisotropy of the dispersion interaction between the molecules owing to the nonzero matrix elements of the anisotropic components of the polarizability tensor, i.e., the anisotropy of the cubic symmetry molecules in degenerate states. 

In the general case, polarizability is anisotropic it depends on the position of the molecnle with respect to the orientation of the external electric field. To consider the orientation, a is expressed by a fnnction — atom polarizability tensor — that defines the indnced dipole moment for each possible direction of the electric field. This atom polarizability tensor a describes the distortion in the nuclear arrangement in a molecnle (i.e., the tendency of polarization in three dimensions). The polarizability determined in an experiment is an average polarizability it is the sum of polarizabilities in three principal directions that are collinear with the external field [59],... 

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