Schonflies notation

Magnetic ordering, 746 Magnetic point groups, 738, 739 international notation, 739 properties of, 740 Schonflies notation, 739 Shubnikov notation, 739 Magnetic point symmetry, determination of, 744... 

Scattering processes, 586 Schiff, L. J., 437,444 Schiffer, M.% 363 SchUchting, H., 24 Schmidt orthogonalization, 65 Schonflies notation for magnetic point groups, 739... 

The SchOnflies notation for this operation is i, although the circumflex is... 

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. 

Table 2.5. Abbreviated International symbols and Schonflies notation.

In each column, the symbol for the point group is given in International notation on the left and in Schonflies notation on the right. When n = 2, the International symbol for D2h is mmm. When n is odd, the International symbol for C v is nm, and when n is even it is nmm. Note that n = n/2. In addition to these groups, which are either a proper point group P, or formed from P, there are the three cyclic groups 1 or Ci = E, 1 or Q = EI, and morCs = E a. ... 

Table 2.9. The thirty-two crystallographic point groups in both International and Schonflies notation.

Underlines in the International notation for G show which operators are complementary ones. Alternatively, these may be identified from the classes of G H by multiplying each operator by 0 G is the ordinary crystallographic point group from which G was constructed by eq. (14.1.2) H is given first in International notation and then in Schonflies notation, in square brackets. Subscript a denotes the unit vector along [1 1 0]. 

Figure 6. Complete subgroup lattice of continuous point groups. Solid circles represent point goups that can be represented by geometrical figures Ki, (sphere), (cylinder), Cw (cone). Open circles represent point goups that cannot be represented by geometrical figures. Schonflies notations are accompanied by Hermann-Mauguin (international) notations in brackets.

Fig. 23. Structures of the tetrasubstituted TEEs. Below the molecules the symmetry is given in international and Schonflies notation. In the upper right corner the different conjugation paths and the coordinate system are depicted as used in the derivation of the symmetry relations...

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 Schonflies notation rather than the Hermann Mauguin notation favoured in crystallography. 

Table 7.3 shows a few point groups of interest to molecules (and to crystals). The Schonflies notation is being replaced in the crystallographic literature by the Herrmann11-Mauguin12 or international notation. 

Scheme for determining molecular point groups (Schonflies notation). 

The point group, which is D3 in the Schonflies notation used for example in molecular spectroscopy, is called 32 in the International (or Hermann-iMauguin) notation used by crystallographers. 

Chemists use the Schonflies notation for molecular point groups, the labels used being listed in the flow chart shown in Fig. 2 and explained below. For a non-linear molecule with at least one rotation axis, the first important question is whether there is a principal axis, a unique CQ axis with... 

A word about notation. For the points groups we shall generally use the Schonflies notation. In this notation basically the same symbols are used for the point groups as for the symmetry operations. Whenever there is any doubt as to the meaning we shall use a caret (example Cn) to indicate a symmetry operation. 

In the Schonflies notation, the symbols have the following meanings ... 

Symmetry operations Order Schonflies notation International notation Symmetry notation Isomorphisms... 

This corresponds in the "Schonflies notation" (Schonflies 1891) to the crystal classes Ci (single rotation axis, tricline), Cs (single mirror plane, monocline), C3 (triple rotation axis, rhombohedric) and 3 ... 

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