Hermann-Mauguin notations

These simple examples serve to show that instinctive ideas about symmetry are not going to get us very far. We must put symmetry classification on a much firmer footing if it is to be useful. In order to do this we need to define only five types of elements of symmetry - and one of these is almost trivial. In discussing these we refer only to the free molecule, realized in the gas phase at low pressure, and not, for example, to crystals which have additional elements of symmetry relating the positions of different molecules within the unit cell. We shall use, therefore, the Schdnflies notation rather than the Hermann-Mauguin notation favoured in crystallography. 

A rotoreflection is a coupled symmetry operation of a rotation and a reflection at a plane perpendicular to the axis. Rotoreflection axes are identical with inversion axes, but the multiplicities do not coincide if they are not divisible by 4 (Fig. 3.3). In the Hermann-Mauguin notation only inversion axes are used, and in the Schoenflies notation only rotoreflection axes are used, the symbol for the latter being SN. 

Schonflies notation is widely used to describe molecules or assemblages of atoms (polyhedron) such as the local environment of an atom. Thus, it is widely used to describe the symmetry of structural sites. It is a more compact notation but less complete than the Hermann-Mauguin notation. It consists generally of one capital letter, followed by one subscript number and one final letter. 

Two forms of symmetry notation are commonly used. As chemists, you will come across both. The Schoenflies notation is useful for describing the point symmetry of individual molecules and is used by spectroscopists. The Hermann-Mauguin notation can be used to describe the point symmetry of individual molecules but in addition can also describe the relationship of different molecules to one another in space—their so-called space-symmetry—and so is the form most commonly met in crystallography and the solid state. We give here the Schoenflies notation in parentheses after the Hermann-Mauguin notation. 

The Hermann-Mauguin notation for the description of point group symmetry (in contrast to the Schonflies system used in Chapter 6) is widely adopted in crystallography. An n-fold rotation axis is simply designated as n. An object is said to possess an n-fold inversion axis h if it can be brought into an equivalent configuration by a rotation of 360°/n in combination with inversion through a... 

The Hermann- Mauguin notation is generally used by crystallographers to describe the space group. Tables exist to convert this notation to the Schoen-flies notation. The first symbol is a capital letter and indicates whether the lattice is primitive. The next symbol refers to the principal axis, whether it is rotation, inversion, or screw, e.g.,... 

Table 7.3 Point Groups of Interest to Chemistry (in Schonflies and also Hermann-Mauguin Notation), with Examples of Molecules that Belong to Them3...

The periodicity of a lattice limits the number of compatible rotation operations to onefold, twofold, threefold, fourfold, and sixfold. This, in turn, limits the number of point groups to thirty-two. Point groups are used to describe individual molecules. Table 14.1 shows the thirty-two point groups in both the Hermann-Mauguin notation and the Schoenflies notation divided into seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The correspondence between the Schoenflies and Hermann-Mauguin notation for the 32 crystallographic point groups is given in Table A3.4. 

The complete designation of the symmetry of a crystal requires the correct assignment of axes and identification of (he symmetry elements. There are a total of 32 different combinations of symmetry elements. Each of these has a unique Hermann-Mauguin notation or point group and is called a crystal class. The 32 crystal classes can be divided into six crystal systems. We will (ry to give you an appreciation of point groups and crystal classes, but our main emphasis will be on the more general crystal systems. 

Table 5.2 lists the Hermann-Mauguin notation for expressing the symmetry operators. Some combinations... 

TABLE 5.2 Symmetry Operators (Hermann-Mauguin Notation) ... 

Molecular point group Hermann-Mauguin notation (Schoenflies notation) Percentage of crystal structures in noncentrosymmetric space groups Number of structures considered... 

I 1.9. Determine the crystallographic point group for each of the following crystals, where the rotational axes and mirror planes are indicated. Use both the Schoenflies and Hermann-Mauguin notations. 

Further, some elements of crystalline stmctures will be presented and analyzed from symmetry operations transcriptions point of view in symmetry compact international (Hermann-Mauguin) notation. These compact S5mibols follow some simple writing conventions ... 

FIGURE 2.31 Symmetry operations and international (Hermann-Mauguin) notation determinations as for elasses 2/m2/m2/m (on left) and 2mm (on right), after Chiriac-Putz-Chiriae (2005). 

---