Polarization tensor properties

Lazzeretti and coworkers175 calculated nuclear electric and electromagnetic shielding tensors for 1 and oxirane. These properties are related to atomic polar tensors and atomic axial tensors used by infrared and VCD spectroscopists. The authors demonstrated that they could obtain fairly accurate sum rules for atomic polar tensors and atomic axial tensors with relatively little computational effort. 

Piezoelectric materials are materials that exhibit a linear relationship between electric and mechanical variables. Electric polarization is proportional to mechanical stress. The direct piezoelectric effect can be described as the ability of materials to convert mechanical stress into an electric field, and the reverse, to convert an electric field into a mechanical stress. The use of the piezoelectric effect in sensors is based on the latter property. For materials to exhibit the piezoelectric effect, the materials must be anisotropic and electrically poled ie, there must be a spontaneous electric field maintained in a particular direction throughout the material. A key feature of a piezoelectric material involves this spontaneous electric field and its disappearance above the Curie point. Only solids without a center of symmetry show this piezoelectric effect, a third-rank tensor property (14,15). 

From tensor algebra, the tensor property relating two associated tensor quantities, of rank / and rank g, is of rank (/-b g). Hence, the physical property connecting /, and aj is the third-rank tensor known as the piezoelectric effect, and it contains 3 = 27 piezoelectric strain coefficients, dyk. The piezoelectric coefficients are products of electrostriction constants, the electric polarization, and components of the dielectric tensor. 

The signals in 2D-IR experiments are fourth-rank tensor properties with indices corresponding to the four polar vector components of the incident and detected electric fields. In an isotropic medium there are only three independent fourth... 

It is clear that the above results are contrary to the simple and common model of fixed atomic point charges that predicts an isotropic response to an external field. The source of the problem is that the atomic polar tensor is not isotropic. This tensor has nine elements (for each nucleus), corresponding to the derivatives of each component of the dipole moment with respect to the three Cartesian coordinates of the nucleus. The properties of these tensors and their interpretation as atomic charges have been extensively discussed in the literature in connection with infrared intensities.It is well known that the anisotropy of these tensors can be explained as follows. If we write the total molecular dipole moment as formed from point charges, then a typical element of the atomic polar tensor is, for example. 

Of rows in the next group for electric properties, the fourth shows the total electric dipolar moment. The next five rows present an analysis of net electronic populations associated with each atomic centre, according to an atomic polar tensor [13] each value listed represents the net alteration of electronic population associated with a particular atomic centre through its participation as a constituent of the particular molecule in a specific isomeric form. To distinguish the two hydrogenic atomic centres if they lie in chemically inequivalent positions, is nearer a carbon atom than H likewise if the two nitrogens have inequivalent positions N, is nearer a carbon atom than Ni,. The next six rows present elements of a symmetric... 

From infrared and Raman spectra of bicyclobutane and appropriate deuterated isomers a new vibrational assignment was made with the help of the spectrum calculated using the 6-3IG basis set. A normal coordinate analysis furnished atomic polar tensors and related properties. The results were compared with similar data for cyclopropane and [l.l.l]propellane. 

Flexoelectricity is a basic mechano-electric phenomenon in liquid crystal physics. The first hints for the existence of a curvature-polarization tensor in liquid crystals, although with different sjmunetry transformation properties, can be foimd in a manuscript by Freedericksz (1940) that made use of the tensorial analysis method of general relativity and was published only... 

Transformation properties of polar tensors are independent on sign of det(o/,). An axial tensor (or a pseudotensor) of p-order is defined by transformation equation... 

In practice, the electric and magnetic dipole transition moments are usually expressed as summations of atomic properties, namely the atomic polar tensor (APT), Pgp, and atomic axial tensor (AAT), respectively. In the... 

The second rank-order parameter S can be derived from measurements of the macroscopic tensor properties such as birefringence and diamagnetic susceptibility. It varies typically between 0.4 at the clearing temperature to 0.7 at T ni- r= 20K in nematic phase. The fourth rank-order parameter (P ) may play an important role for a subtle analysis of the orientational distribution function and can be determined using polarized Raman spectroscopy. ... 

The symmetry properties of in any medium can be established considering that it transforms as a fourth-rank polar tensor under the macroscopic symmetry operations. The symmetry of the microscopic polarizable units can be used to simplify the microscopic expression... 

Second-order response properties, such as electric polarizabilities, magnetic susceptibilities, and atomic polar tensors, can be readily partitioned into either atomic or atom-pair contributions with the help of the theory of AIMs. The former partitioning is accomplished by taking derivatives of the pertinent first-order properties with respect to strengths of external perturbations, whereas the latter involves a somewhat more complicated (albeit more theoretically consistent) formalism. In general, the atomic and atom-pair contributions to the second-order response properties are the sums of the atomic basin and surface relaxation terms. ... 

When calculating the elements of the RPH, it does not require much additional effort to calculate also other properties of the reaction complex that depend on s. An example can be found in the investigation of the isomerization reaction of the methoxy radical to the hydroxymethyl radical by Colwell and Handy, where the authors also determined the dipole moment and the components of the polarization tensor as a function of s. A systematic step in this reaction has been made by Konkoli, Kraka, and Cremer, who analyzed the electron density distribution p(r.s) of the reaction complex along the RP, calculating difference density distribution Ap(r,s) (equation 39), Laplace concentrations V-plr.s),... 

Pyroelectricity is a first-rank tensor property that relates the change in temperature to a change in electrical displacement D (or polarization P since no field is applied) ... 

The values of the elements of atomic polar tensors as evaluated from q. (4.12) depend on die choice of a reference Cartesian coordinate system. Applications in describing structural properties of molecules require to tabulate atomic polar tensor elements for different atoms in various molecular environments. With values depending... 

The elements of atomic polar tensors are in units of electric charge. The quantity ) is, therefore, also in units of electric charge. Tensor properties require that the sum of squares of all elements are independent on the particular choice of a reference... 

Atomic polar tensors, defined widi respect to an aibitraty Cartesian system, nn be transformed into quantities referring to a bond axis system using Eq. (4.22). If a molecule has sets of equivalent atmns the transfonnatiatomic polar tensors, provided the local refermce systems are chosen in a consistent way. With such representations it is clear that the number of independent atomic polar tensor elements is smaller for molecules with higher symmetiy. Decius and Mast [117] analyzed in detail the site symmetiy properties of atomic polar tensors exjnessed in tenns of bond axis system. The treatment covers molecules with sufficient symmetiy so that the directions of transitional dipole moments are uniquely determined. Molecules with symmetiy point grotq> G = C2h> Ci, C, Q and Ci are excluded from fire analysis. As already discussed, all elements of the Pq matrix for such molecules cannot be determined from experiment. 

In the present section a theoretical framework for analysis of vibrational intensities recendy developed by Galabov et al. [146] is presented. Fully corrected for rotational contributions atomic polar tensors are transformed into quantities termed effective bond charges. The effective bond charges are expected to reflect in a generalized manner, polar properties of the valence bonds in molecules. Aside from die usual harmonic approximation no other constraints are imposed on the dipole moment functirm. 

The semiclassical theories described so far are aimed mostly at interpreting the experimentally determined vibrational absorption intensities of molecules in terms of quantities associated with the charge distribution and dynamics. Fewer attempts have been made for quantitative predictions of intensities based on transferable intensity parameters. Successful predictions are difficult to achieve because transferability properties are not so well expressed as in the case of force constants. This is determined by a number of factors (1) the high sensitivity of vibrational intensities associated with particular modes to changes in molecular environment (2) the physical limitations of the approximate point-charge models and (3) mathematical difficulties in applying non-approximate models such as polar tensors or bond polar parameters for larger molecules. 

The Raman scattered light emanating from even a random sample is polarized to a greater or lesser extent. For randomly oriented systems, the polarization properties are determined by the two tensor invariants of the polarization tensor, that is, the trace and anisotropy tensors. The depolarization ratio is always less or equal 3/4. For a specific scattering geometry, this polarization is dependent upon the symmetry of the molecular vibration giving rise to the line. As shown in Fig. 2.15, the line of CCU at 457 cm is completely polarized. 

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