Polarizability scalar

The dipole polarizability tensor characterizes the lowest-order dipole moment induced by a unifonu field. The a tensor is syimnetric and has no more than six independent components, less if tire molecule has some synnnetry. The scalar or mean dipole polarizability... 

Formal Theory A small neutral particle at equihbrium in a static elecdric field experiences a net force due to DEP that can be written as F = (p V)E, where p is the dipole moment vecdor and E is the external electric field. If the particle is a simple dielectric and is isotropically, linearly, and homogeneously polarizable, then the dipole moment can be written as p = ai E, where a is the (scalar) polarizability, V is the volume of the particle, and E is the external field. The force can then be written as ... 

The induced dipole moment is Qd and the polarizability (here a scalar) is Qd/E so that... 

Ag+ with 1.2 A ) [99]. Spin-orbit coupling is neglected in our analysis because the results shown in Table 4.2 are from scalar relativistic Douglas-Kroll calculations. Because of the additional shell expansion of the 5ds/2 orbital due to spin-orbit coupling, we expect a further increase of the polarizability of Au. Table 4.3 also... 

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

General properties and definitions of polarizabilities can be introduced without invoking the complete DFT formalism by considering first an elementary model the dipole of an isolated, spherical atom induced by a uniform electric field. The variation of the electronic density is represented by a simple scalar the induced atomic dipole moment. This coarse-grained (CG) model of the electronic density permits to derive a useful explicit energy functional where the functional derivatives are formulated in terms of polarizabilities and dipole hardnesses. 

A variety of properties can be defined and calculated I will restrict attention to the operators involved in the calculation of dipole polarizabilities and NMR parameters, corresponding to the introduction of a uniform electric field E represented by the scalar potential... 

Diffuse augmented basis sets were also developed for both the NR and DK cases by simple even tempered extensions. Valence electrons correlated CCSD(T) dipole polarizabilities calculated with the inclusion of DK scalar relativity are 35.08 and 35.07 a.u. for aug-cc-pVQZ-DK and aug-cc-pV5Z-DK, respectively. These values are about 0.3 a.u. larger than the PP values and therefore somewhat further away from experiment. 

For the P-phase of PVDF the fc-axis is a principal axis of the polarizability tensor of the repeat unit in the absence of an applied field, only the component of parallel to the (>-axis is nonzero. Equation (3) may thus be expressed in scalar form, where Ap is also directed along the b-axis. 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

Orientation angle or scalar polarizability. Polarizability tensor, (4.10). 

Polarizabilities have been determined and compared for a number of azaindole compounds. The data are summarized in Table 33 <90JPC1755>. Overall, the results show that the different structural isomers have very similar values for the polarizability (a) and the second hyperpolarizability (y) while there are dramatic differences in the first hyperpolarizability (/ ). This may result from the fact that a and y are scalar properties while / is a component of a vector. 

Depolarized scattering occurs because of various forms of particle anisotropy. Distinct classes of depolarizing scatterers include nonspherical particles with uniform isotropic (scalar) polarizabilities (sometimes called form anisotropy), inhomogeneous particles with nonuniform distributions of isotropic polarizability, and particles with anisotropic (tensor) polarizabilities. For each of these classes, the intensity of depolarized light scattered by a particle will change as the particle translates, rotates, or manifests internal rearrangement of its scattering elements. DDLS can provide information on the dynamics of each of these processes. 

Molecular polarizability, a, is a measure of the ability of an external electric field, E, to induce a dipole moment, = aE, in the molecule. As such, it can be viewed as contributing to a model for induced dipole (dispersive) interactions in molecules. Because the polarizability is a tensor (matrix) quantity, there is the question of how to represent this in a scalar form. One approach is to use the average of the diagonal components of the polarizability matrix, (a x + otyy + Since the polarizability increases with size (and... 

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