Order tensor experimental determination

The operator [157] is a phenomenological spin-orbit operator. In addition to being useful for symmetry considerations, Eq. [157] can be utilized for setting up a connection between theoretically and experimentally determined fine-structure splittings via the so-called spin-orbit parameter Aso (see the later section on first-order spin-orbit splitting). In terms of its tensor components, the phenomenological spin-orbit Hamiltonian reads... 

Since, in HRS, there is no preferred orientation induced by an additional static field, there is the possibility of varying the experimental conditions in order to increase the number of independent observables. The number of theoretically possible independent observations, and hence the number of tensor components that can be obtained by HRS, is at most five. For parametric light scattering, this number is six, due to the possibility of distinguishing between the two optical fundamental fields [20]. The experimental difficulty has precluded the determination of this number of components. What is experimentally realistic in HRS is an additional depolarization measurement, apart from the classical measurement of the intensity of the second-order incoherent scattered light. The two measurements, the total intensity measurement and the depolarization ratio (or two intensity measurements, one with parallel and one with perpendicular polarization for fundamental and second harmonic), represent two independent observables and allow the experimental determination of two tensor components. For molecules of C2 symmetry, these are and P xxy resulting for the total intensity measurement in Eqn. (21),... 

Autschbach and Ziegler performed the relativistic calculations of spin-spin coupling constants (isotropic part) and anisotropies in heavy atom compounds with the two-component zeroth-order regular approximation (ZORA) method. The experimentally determined reduced spin-spin coupling tensor elements Kjk A,B) between two magnetically active nuclei, A and... 

Calculations of nuclear shielding normally provide values for each of the nine components of the second-order shielding tensor. Thus theoretical estimates are available for the principal components of the tensor as well as its anisotropy and the isotropic value of the nuclear shielding, the latter of which is usually available from high-resolution CP/MAS NMR measurements. Methods for experimentally determining the principal tensor components, and its anisotropy, are described in the previous section. 

Fig. 1.4. Rotational Zeeman spectra of the lio - 2xi rotational transition in propene, methyl-enecyclopropene, cyclopentadiene, and fluorobenzene. For better comparison, spectra calculated for the same magnetic field strength are shown. The calculation is based on the experimentally determined -values and susceptibility anisotropies. While the order of magnitude of the M 1 splitting (j -tensor contribution) remains essentially the same, the shifts of the M = 0 satellite and of the M = + 1 doublet due to the jj-tensor contribution increase almost by a factor of ten when going from the small open chain molecule propene to the aromatic ring fluorobenzene. These susceptibility shifts are indicated by the horizontal arrows to the right for M 1 shifts and to the left for M = 0 shifts.

In eq. (4.25), is the orientational order parameter (relative to the external magnetic field) of the vector mn, and < > indicates averaging over internal vibrations. The coefficients sfp in eq. (4.26) are Saupe orientational order tensor elements relative to the external magnetic field in a molecular-fixed frame. Consequently, it is necessary to know the tensor in order to be able to determine experimentally the direct dipolar contribution from... 

Pluta and Sadlej have calculated the dipole moment and static a, P and y and y tensors of urea and thiourea using three high level basis sets of increasing flexibility. Excellent agreement is found with experimental determinations of the dipole moment and linear polarizability. Frequeney-dependent polarizabilities and hyperpolarizabilities are ealeulated in the TDHF approximation and the results are then scaled to allow for electron correlation and the effect of basis set extension. Estimates of the response flmctions for non-linear optical processes are obtained. The introduction of the sulfur atom is found to produce a large increase in the predicted efficiency for third order effects. 

For D D , the reciprocals of these correlation times are raised by an amount [(D /D ) — 1] as indicated in Eq. (7.55). As noted previously, K rriL, ttim) values at a particular temperature are computed from (P2) and (P4). The fourth-rank order parameter P4) cannot be directly measmed from a NMR spectrmn, but may be derived from measurements of the mean square value of a second-rank quantity [7.19-7.22]. In the Raman scattering technique [7.21], the second-rank molecular quantity is the differential polarizability tensor of a localized Raman mode. In fluorescence depolarization [7.19], the average of the product of the absorption and emission tensors is used to determine (P4). Since there is a lack of experimental determination of (P4) in liquid crystals, this may be calculated based on the Maier-Saupe potential... 

In microwave or molecular beam experiments, aU of the three diagonal elements of the rotational g tensor are determined. As for the magnetizabihty, the individual tensor components show some dependence on electron correlation effects, and in order to allow for the calculated results to be within the very tight experimental errors bars for the rotational g tensor elements, electron correlation and the effects of zero-point vibrational corrections need to be included. This is illustrated for the water molecule in O Table 11-5. 

The ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

To fit the experimental results, it is necessary to fix the overall phase. This can be done, for example, by defining h as a real quantity (/ / = 0). The values found for the coefficients /, g, and h can then subsequently be used to calculate the values of the components of the second-order susceptibility, X(2). This is done in detail for a Langmuir-Blodgett film of a poly(isocyanide) in the following section. Note that both phase and magnitude of all tensor components are relative values. The absolute phase cannot be determined... 

One result of studying nonlinear optical phenomena is, for instance, the determination of this susceptibility tensor, which supplies information about the anharmonicity of the potential between atoms in a crystal lattice. A simple electrodynamic model which relates the anharmonic motion of the bond charge to the higher-order nonlinear susceptibilities has been proposed by Levine The application of his theory to calculations of the nonlinearities in a-quarz yields excellent agreement with experimental data. 

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