Point groups characteristic symmetry elements

Point group Characteristic symmetry elements Comments... 

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... 

The thirty-two crystal classes (crystallographic point groups) described in Section 9.1.4 can also be classified into the same seven crystal systems, depending on the most convenient coordinate system used to indicate the location and orientation of their characteristic symmetry elements, as shown in Table 9.2.1. 

In this section, actual molecular structures are shown for the various point groups. The Schoenflies notation is used and the characteristic symmetry elements are enumerated. 

Table 3.1 Characteristic symmetry elements of some important classes of point groups. The characteristic symmetry elements of the T, Oij and 4 are omitted because the point groups are readily identified (see Figure 3.8). No distinction is made in this table between CTy and <7d planes of symmetry. For complete lists of symmetry elements, character tables should be consulted.

Table 3.1 Characteristic symmetry elements of some important classes of point groups. The characteristic symmetry elements of the Oj,...

Polyhedral crystals bounded by flat crystal feces usually take characteristic forms controlled by the symmetry elements of the crystal (point) group to which the crystal belongs and the form and size of the unit cell (see Appendix A.5). When a unit cell is of equal or nearly equal size along the three axes, crystals usually take an isometric form, such as a tetrahedron, cube, octahedron, or dodec-... 

Figure 8. In this Venn diagram and in those of the following Figures the ellipses contain the representative points of all the structures with at least all of the symmetry elements characteristic of the indicated space groups. As an example, the ellipse indicated by P-1 contains all the structures with at least one inversion center, even if, commonly only the structures in the dashed area are described in P-1. In this sense the diagram represents the distribution of symmetry elements among the more frequent space groups.

Table 3.1 summarizes the most important classes of point group and gives their characteristic types of symmetry elements E is, of course, common to every group. Some particular features of significance are given below. 

One of the noncentrosymmetric point groups (cubic 432) has symmetry elements which prevent polar characteristics. 

The following symmetry elements are present in the NH3 molecule a C3 (z) axis and three planes 0 (1) that contain the N—H, bonds and bisect the opposite H—N—H angle (6-21). These symmetry elements are characteristic of the C3V point group. 

Consider a complex in which the metallic atom is surrounded by four ligands that are placed at the corners of a square (6-23). The symmetry elements of this system are characteristic of the Dau point group. The axes are shown in 6-24. The planes of symmetry are xy (oj,), xz and yz (cfdb), respectively, together with the planes that bisect xz and yz and each contain two M—L bonds (Oya and respectively). The inversion centre is of course at the origin, coincident with the central atom. 

Consider a complex ML4 in which the four ligands are situated at the apices of a tetrahedron. Each ligand has a a orbital which points towards the metallic centre (6-25). The symmetry elements, which are characteristic of the point group, are ... 

Point groups describe the symmetry characteristics of molecules in terms of their symmetry elements. 

A fundamental characteristic of spatially periodic systems is the existence of a group of translational symmetry operations, by means of which the repeating pattern may be brought into self-coincidence. The translational symmetry of the array, expressing its invariance with respect to parallel displacements in different directions is represented by a lattice. This lattice consists of an array of evenly spaced points (Fig. 3-13), such that the structural elements appear the same and in the same orientation when viewed from each and every one of the lattice points. Another important property of spatially periodic arrays is the existence of two characteristic length scales, corresponding to the average microscopic distance between lattice... 

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