Point groups of high symmetry

Character tables of point groups of high symmetry (n > 2) have entries other than 1, but we need not deal with such groups here. 

The III character table is given in Table A.46 in Appendix A. The very high symmetry of this point group results in symmetry species with degeneracies of up to five, as in and... 

I. Croups with very high symmetry. These point groups may be defined by the large number of characteristic symmetry elements, but most readers will recognize them immediately as Platonic solids of high symmetry, a. Icosahedrd, Ik.—The icosahedron (Fig. 3.10a), typified by the B12H 2 ion (Fig. 3.10b), has six C3 axes, ten C3 axes, fifteen C2 axes, fifteen mirror... 

Therefore, k+bm and k label the same representation and are said to be equivalent (=). By definition, no two interior points can be equivalent but every point on the surface of the BZ has at least one equivalent point. The k = 0 point at the center of the zone is denoted by T. All other internal high-symmetry points are denoted by capital Greek letters. Surface symmetry points are denoted by capital Roman letters. The elements of the point group which transform a particular k point into itself or into an equivalent point constitute the point group of the wave vector (or little co-group of k) P(k) C P, for that k point. 

If the molecule is not linear, it may belong to a point group of extremely high symmetry such as Td, Oh, or Ih. 

When a molecule is relatively small and/or belongs to a point group of relatively high symmetry, it is possible to elucidate the molecular structure by using the symmetry selection rules discussed in Section 1.14. Molecules of XY2 (linear or bent C2v), XY3 (planar D3h or pyramidal C3v), XY4 (square-planar D4h or tetrahedral Td) and XY5 (trigonal-bipyramidal D3h or tetragonal-pyramidal C4v) types may take one of the structures indicated in parentheses. Since the number of IR/Raman-active vibrations is different for each structure, the most probable structure can be chosen by comparing the number of observed IR/Raman bands with that predicted for each structure by symmetry selection rules. 

In the first step, the vibrations of the hypothetical isolated species are analyzed with respect to its point group. Secondly, the symmetry of the species in the static lattice (the site symmetry) is deteiTnined. The site is normally of lower symmetry than the isolated species. If the species are highly symmetrical, splitting of the degenerate vibrations is observed. In the last step, the correlations between the different groups in the unit cell are analyzed (factor group analysis, Sec. 2.7.6). Since the unit cell of the scheelite crystal contains two WO4 species, twice as many vibrations are observed as expected from site group analysis. 

Fig. 3.10 Point groups and molecules of high symmetry (a) icosahedron, (b) the BijHi ion. (c) octahedron, (d) sulfur hexaflucride, (e) bexacyanocoballatc(lll) amon. (f) tetrahedron, (g) ammonium cation, and (h> teirafhioroborate anion.

We use calligraphic symbols such as C when referring to the abstract symmetry group. We will begin with point groups that have the lowest symmetry and work up to those of high symmetry., ... 

The symmetry properties of spontaneous strains are most conveniently understood by referring to the irreducible representations and basis functions for the point group of the high symmetry phase of a crystal of interest. These are given in Table 2 for the point group Almmm as an example. Basis functions x + y ) and are associated with the identity representation and are equivalent to (ei + ei) and C3 respectively. This is the same as saying that both strains are consistent with Almmm symmetry e = ei). The shear strain e - ei) is equivalent to the basis function (x - y ) which is associated with the Big representation, the shear strain e is equivalent to xy (B2g) and shear strains e and e to xz, yz respectively (Eg). The combinations (ci + 62) and (ci - ei) are referred to as symmetry-adapted strains because they have the form of specific basis functions of the... 

---