Point Group 23 7 Materials

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

It is assumed that the reader has previously learned, in undergraduate inorganie or physieal ehemistry elasses, how symmetry arises in moleeular shapes and struetures and what symmetry elements are (e.g., planes, axes of rotation, eenters of inversion, ete.). For the reader who feels, after reading this appendix, that additional baekground is needed, the texts by Cotton and EWK, as well as most physieal ehemistry texts ean be eonsulted. We review and teaeh here only that material that is of direet applieation to symmetry analysis of moleeular orbitals and vibrations and rotations of moleeules. We use a speeifie example, the ammonia moleeule, to introduee and illustrate the important aspeets of point group symmetry. 

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],... 

The structure factor itself is expressed as the sum of energy diffracted, over one unit-cell, of the individual scattering factors, fi, for atoms located at X, y and z. Having done this, we can then identify the exact locations of the atoms (ions) within the unit-cell, its point-group sjmimetiy, and crystal system. This then completes our picture of the structure of the material. 

If g is an element of the point group of the material meaning that p(r) and p(gr) are indistinguishable for all elements g in that group, corresponding Fourier components can differ only by a phase factor ... 

Symmetry The word symmetry means the same measure, which denotes harmony and beauty of the parts. It also plays a very important role in molecular architecture and material properties. The study of molecular symmetry is through a branch of group theory, that is, the point groups of rotations that leave one point... 

Table 1.8. Material, P and of selected molecular organic superconductors classified according to the point group of the donor...

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

Table I lists the properties of several organic materials which have been studied as single crystals. The materials are listed alphabetically according to the acronym applied to them in order to avoid the appearance of prejudice with respect to any one material. Listed are the molecular P (if known), space group and point group of the crystal, SHG powder intensity relative to urea, NLO coefficients for SHG, its transparency cutoff, and the figure of merit for SHG if known. Other aspects are...

By simply extending the methods used for the point group selection rules, one can obtain selection rules for molecules involving rotation-translation and reflection-translation. Two approaches are available. The older method is the Bhagavantum-Ventkatarayudu (BV) method (50), and necessitates the availability of the structure of the material being studied. The other method is that of Halford-Hornig (HH) (51-53) and considers only the local symmetry of a solid and the number of molecules in the unit cell and is simpler to work with. This method is also called the correlation method and depends on the proper selection of the site symmetry in the unit cell. 

Upon appropriate reduction in the number and nature of the independent tensorial components of i/i(s) (= j/, ), resulting from the common point-group symmetry elements of the sphere and cube (applied to fourth-rank tensors), the material tensor can be shown quite generally to be of the form (Zuzovsky et al., 1983)... 

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