Electric moment transformation properties

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

If we draw an arrow to the coordinate axes to which the symmetry of a given molecule is referred, then the transformation properties of these translation vectors under the symmetry operation of the group are the same as the electric dipole moment Vector induced in the molecule by absorption of light (Figure 3.10). 

The induced magnetic dipole moment has transformation properties similar to rotations Rx, Rt, and Rz about the coordinate axes. These transformations are important in deducing the intensity of electronic transitions (selection rules) and the optical rotatory strength of electronic transitions respectively. If P and /fare the probabilities of electric and magnetic transitions respectively, then... 

The relation between the spherical components AJ0( ) of a general tensor A of rank 2 and the cartesian components A, ( ) are given in Appendix 4. Equations (3.36) will form the basis for derivation of selection rules for rotation-internal motion transitions of SRMs presented in the next section. They also may serve for derivation of the transformation properties of the electric and magnetic dipole moment operators referred to the laboratory system (VH G... 

For example the dipole polarizability given in (2.33) has spherical tensor components OQQdl) and a2K(11) The dipole-quadrupole polarizability (A .gY in Cartesian notation), which describes the quadrupole moment induced by an electric field or the dipole moment induced by an electric field gradient, has components 0 (12), 021(02) and 02k(12). The polarizabilities are even (g) or odd (u) under inversion according as 1+1 is even or odd. This information is then sufficient, with the help of Table 3, to determine the transformation properties in the molecular symmetry group. Any component which transforms according to the totally symmetric representation may have a non-zero value. 

The analysis of the preceding results leads to several important conclusions about the electric properties of vibronic systems. First, in accordance with the results obtained in the preceding, nonpolar molecules may have both types of behavior of the mean dipole moment—that for rigid dipole molecules and that for nondipolar ones. Only in the cases of limit values of temperatures or vibronic coupling constants can they be related to either the former or the latter. This statement can be illustrated by the case of a molecule with two dipolar-type minima [Eq. (19)]. Consider the two limit cases A kT. In the former case the function a(T) transforms into the classical linear dependence on 11 kT inherent to rigid dipole molecules. In the limit case of low temperatures, a(T) is reduced to a constant value equal to the static polarizability of molecules that have no proper dipole moment. 

A natural way to introduce equations for excited states into a quantum chemical approach is to consider stimulating the molecule by a time-varying electric field to which the molecule can respond by excitation, and derive solutions from the time-dependent Schroedinger equation. Analysis then leads to equations for the excitation energies and properties of the excited state eigensolutions like transition moments. In particular, such an approach, after a Fourier transformation from time to frequency, will yield the dynamic polarizability whose spectral expansion is... 

LC elastomers based on the side-chain and main-chain LC polymers containing rather small concentrations of the chiral mesogens can form SmC mesophase-possessing domains with permanent electric dipole moment, which exhibit piezoelectric properties. The application of an uniaxial mechanical field (shear) produces a centrosymmetric morphology, where the piezoelectric effeas are observed. The piezoelectric coefficient reaches its maximum at a certain shear angle that corresponds to the completion of polydomain to monodomain transformation. The piezoelectric module of different types of such LC elastomers can be higher than those... 

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