Point group symmetry, effects

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

Podolsky method, Renner-Teller effect, triatomic molecules, Hamiltonian equations, 612—615 Poincare sphere, phase properties, 206 Point group symmetry ... 

The nature of the relation which exists between the symmetry of diffraction effects and that of the crystal may be gathered by consideration of the hkO intensities of two simple tetragonal crystals—urea (0---C(NH2)2), whose point-group symmetry is 42m, and penta-erythritol (C(CH2OH)4), belonging to class 4. For urea (see Fig. 144), the intensity of 310 is the same as that of 310, as is obvious from the relation of the molecules to the traces of these planes the equality... 

Effect of screw axes and glide planes on X-ray diffraction patterns 251 Diffraction symmetry in relation to point-group symmetry 258... 

The true point group symmetry is Can hut the effective potential can be approximated to that of Us. The difference between these two symmetries is that terms such as A S [a (x + iy)a + b (x — iy)8] enter into the expression of the potential Ve in a complete calculation. 

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

We next comment on the effect of sparsity in the integral list (apart from that caused by point group symmetry) on the relative performance. It is obvious that sparsity considerations are... 

Another possible source of chemical MIF is self-reaction of the SO radical, which either yields S and SO2 as disproportionation products [41], or the SO dimer as an association product [42]. Two of the four lowesf energy singlef sfafes of the SO dimer [43] have a reduced point group symmetry upon sulfur isotope substitution. If sulfur atom exchange is another possible outcome in the self reaction of SO, then a symmetry-derived mass-independent effect analogous to O3 formation may be possible. A similar conclusion may be drawn for the triplet states of the SO dimer. 

Going from the molecular picture to that of collective properties in a solid means adding translational symmetry to the point group symmetry. The theoretical description does this by introducing a phase of the distortion throughout the material, which is determined by the spatial variation of the variously distorted molecules. If, as is usual in a classical crystal, the phase of the distortion shows the translational symmetry of the solid, the so-called cooperative Jahn-Teller effect appears where the shape of one molecule and the space group determines the shape of all the others. If the distortions are not correlated, however, the phase is random and the situation is not different from that of isolated molecules. This is the dynamic Jahn-Teller effect where the distortions cannot be detected but the solid-state consequences still appear in the electronic structure [16]. 

A number of reviews have recently appeared in the literature concerning the Raman spectroscopy of catalytic systems. The purpose of this chapter is not to review the literature, but to demonstrate what is currently achievable in Raman spectroscopy of heterogeneous catalysis. The theory of the Raman effect and its relationship to signal intensity are presented in Section 2. Point-group symmetry and the Frequency/Length/... 

It is apparent that the methods discussed here are particularly effective for molecules whose MO s are largely determined by their point group symmetry because, in these cases, central AO s are good MO templates. Clearly, there are other types of molecules in which central AO s are less suitable and other simple orbitals are more effective templates. For general pi-electron systems, for example, free-elec-tron MO s would be more useful GO s (14). In its broadest sense, then, the generator orbital approach is based on the principle that major characteristics of complicated molecular orbitals can be semi-quantitatively anticipated by conceptual analogy to simpler orbitals with which they share essential physical features. We feel that there is an elucidating quality in this reduction of the more complex to the less complex, which is of value to theorists and non-theorists alike. 

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