Solid harmonics symmetries

Here Vr is an appropriate permutational symmetry projection operator for the desired state, T, and YfcM is a product of coupled solid harmonics labeled by the total angular momentum quantum numbers L and M. Permutational symmetry is handled using projection methods in the same manner as described for the potential expansion in the previous section. Again, the reader is referred to the references for details[9,10,12],... 

Solid Harmonic Bases and Their Symmetry Properties. 206... 

We shall consider the recursive evaluation of the scaled regular and irregular solid harmonics (9.13.14) and (9.13.15) in Section 9.13.8. At (ffesenL we only note that these functions exhibit the same inversion and conjugation symmetries (6.4.17) and (6.4.18) as do the standard solid harmonics. 

Note that the real and imaginary components are defined for negative as well as positive indices m. Using (9.13.14), (6.4.22) and (6.4.18), we find that the real regular solid harmonics satisfy the following symmetry properties ... 

Using these relations, it is trivial to set up an explicit expression for their evaluation analogous to (6.4.47) for S/m(r). The recurrence relations may be obtained from (6.4.55), (6.4.56) and (6.4.69). Taking into account the different normalizations (9.13.76) and (9.13.77) and also the symmetry relations (9.13.47) and (9.13.48), we obtain after some algebra the following set of recurrence relations for the scaled regular solid harmonics ... 

Unlike linear optical effects such as absorption, reflection, and scattering, second order non-linear optical effects are inherently specific for surfaces and interfaces. These effects, namely second harmonic generation (SHG) and sum frequency generation (SFG), are dipole-forbidden in the bulk of centrosymmetric media. In the investigation of isotropic phases such as liquids, gases, and amorphous solids, in particular, signals arise exclusively from the surface or interface region, where the symmetry is disrupted. Non-linear optics are applicable in-situ without the need for a vacuum, and the time response is rapid. 

In the derivation of these spin-interaction selection rules the harmonic approximation was made. In taking nuclear vibration into account2,77 these selection rules are often broken. In addition to coupling with the internal vibrational modes of a molecule, coupling with the phonon modes in the solid state may be important.124 Some use of double point group symmetry will be found in Sections IX, XI, and XII. 

Clearly, situations intermediate between perfect order and random distributions occur in arrays of absorbing chromophores, and a treatment is required that allows expression of the orientational distribution of structural units such as crystallites or segments which may be fluorescent in a bulk sample having uniaxial or biaxial symmetry. A complete mathematical approach using a herical harmonic expansion technique has been developed which expresses the distribution as spherical harmonics of various orders in terms of the Euler angles which specify the orientation of the coordinate system in a fixed structural unit with respect to the coordinate system in the bulk sample This is of use in solid systems, where time dependence is not observed. 

Cockroft, J. K., Fitch, A. N., and Simon, A. Powder neutron diffraction studies of orientational order-disorder transitions in molecular and molecular-ionic solids use of symmetry-adapted spherical harmonic functions in the analysis of scattering density distributions arising from orientational disorder. In Collected Papers. Summerschool on Crystallography and its Teaching. Tianjin, China. Sept 15-24, 1988. (Ed., Miao, F.-M.) p. 427. Tianjin Tianjin Normal University (1988). 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

Based on this modeling, molecular dynamics simulations [140] for a complete Vs monolayer are carried out from 10 K to 80 K—that is, ranging from the harmonic herringbone solid through the orientationally disordered solid to the fluid. The herringbone transition occurs at around 22 K, below which enhanced 180° head-to-tail flips of the homonuclear molecules occur. The sixfold symmetry in the orientations persists up to about 50 K,... 

The interfaces in general, and particularly with solid substrates break the head-to-tail symmetry of a liquid crystal phase and induce polar orientational order. The symmetry is reduced to the conical group Coov The latter allows a finite value of the second-order nonlinear susceptibility X2 responsible for the second optical harmonic generation [11]. This phenomenon has been observed in experiments on generation of the second harmonic in a ultrathin nematic layers on a solid substrate as shown in Fig. 10.9. 

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