Tensor of polarizability

Table 2 Ground-State Polarizability (ai) and Trace of the Tensor of Polarizability Variation (l/3)Tr[Aa] for Several Optical Dyes and Charge Transfer Complexes...

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

Let us illustrate this conclusion by the example of spherical-top molecules. In the degenerate states E, T , T2, and G3/2 there is the E representation in the decomposition of the appropriate symmetric product, and hence in these cases the matrix elements are nonzero for anisotropic components of the tensors of polarizability and quadrupole moment, since E is first met in the decomposition of the spherical representation D2. Moreover, in tetrahedral systems in states of the type T and G3/2 the matrix elements of the dipole moment are nonzero, since [T2] and G3/2 contain the representation T2. 

Here the operators with the indicies Ty belong to the line y of representations T T and rM are representations of the tensor of polarizability and the moment p, respectively bf and hv[ are numerical coefficients dependent on the choice of the form of the irreducible tensor operators in the point group of each concrete molecule. The irreducible tensor operators (p (g) p)ry are determined by the Clebsh-Gordan coefficients of the point group of the molecule (Koster et al., 1963) as follows ... 

Ability to transit to the localized orbitals basis set [17,18] and use the additive scheme to obtain the physicochemical properties of molecules (electrical dipole moment, tensor of polarizability), electrooptical characteristics of IR- and Raman spectra (derivative from the first on normal vibration coordinates) [19], and also bond length [20,21]. 

X2-Y2 complex. Analogous approach can also be applied to the X2-X2 and X2-Y2 complexes. In the modified model the tensors of polarizability and quadmpole moment of effective atoms X(Y) are equal each to other and their total polarizability and quadrupole moment are the same as that of the molecule. So, retaining the leading term, the dipole moment of the complex X2-X2 may be written as... 

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. 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

The fundamental scattering mechanism responsible for ROA was discovered by Atkins and Barron (1969), who showed that interference between the waves scattered via the polarizability and optical activity tensors of the molecule yields a dependence of the scattered intensity on the degree of circular polarization of the incident light and to a circular component in the scattered light. Barron and Buckingham (1971) subsequently developed a more definitive version of the theory and introduced a definition of the dimensionless circular intensity difference (CID),... 

It was proven [96] that the molecular polarizability can be written as a sum of intrinsic atomic polarizabilities of the atoms in the molecule and a charge delocalization term. Thus, the xy element of the molecular polarizability tensor of molecule A can be decomposed as... 

A computation of Raman intensities can be done precisely in the same way as for infrared intensities. One needs here, in addition to the wave functions of the initial and final state, the polarizability tensor a(r,0, < )). This is a symmetric tensor of rank 2 that in Cartesian coordinates can be written as... 

FIGURE 32. Diagonal elements of the polarizability tensors of ethylene and the aUyUc compounds (of. Table 7). Reprinted with permission from Reference 32. Copyright 1998 American Chemical Society... 

The polarizability tensor of a molecule related the components of the induced dipole moment of the molecule to the components of the electric field doing the inducing. It therefore has 9 components, axx, ctxy, etc., only 6 of which are independent. The theory of the Raman effect shows that a vibrational transition, from the totally symmetric ground state to an excited state of symmetry species F, will he Raman active if at least one of the following direct products contains the totally symmetric representation ... 

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... 

Measurements of dipole moments, Kerr constants, and dielectric absorption have been employed (81RCR336) widely to obtain information on the conformational equilibrium in acyl heterocycles. Details on conformer structures and populations depend on the choice of additive scheme, group moments, or polarizability tensor in the case of Kerr constants. Several early conclusions, especially for furan- and thiophene-2-carboxaldehyde, appeared contradictory, owing to the choice of these quantities. A more precise definition of polarizability tensors for several heterocycles and a choice of group moments and additive schemes tested on a large amount of available experimental results and supported by accurate theoretical calculations have led to more confidence in the use of experimental dipole moments and Kerr constants in conformational analysis. A limitation of the method is that the... 

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. 

Here a designates the trace of the polarizability tensor of one molecule (l/47i o) times the factor of a represents the electric fieldstrength of the quadrupole moment q2. Other non-vanishing multipole moments, for example, octopoles (e.g., of tetrahedral molecules), hexadecapoles (of linear molecules), etc., will similarly interact with the trace or anisotropy of the polarizability of the collisional partner and give rise to further multipole-induced dipole components. 

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. 

A fundamental transition will be Raman active (i.e., will give rise to a Raman shift) if the normal mode involved belongs to the same representation as one or more of the components of the polarizability tensor of the molecule. 

Of particular interest in connection with the derivation of the stress-optical law is, that an analogous equation exists for the average polarizability tensor of the single chain 62). This expression is derived from the work of Kuhn and Grun (64) and reads ... 

In inelastic Raman scattering a photon loses (or gains) one quantum of rotational or vibrational energy to (or from) the molecule. The process involves the electric field of the radiation inducing an electric dipole in the molecule and so depends on the polarizability tensor of the molecule. (A (second-order) tensor is a physical quantity with nine components.) The induced electric dipole D is proportional to the electric field E ... 

It should be pointed out that the polarizability tensor of a SRM may exhibit a more complicated transformation behavior than expressed by Eq. (3.47). This goes back to the fact, that the polarizability tensor involves all electronic states and the latter do not necessarily all have the same isometric group. 

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