Schoenflies notation

The symbols used to designate the elements of moleciflar point groups in the Schoenflies notation and their descriptions are as follows  

C — rotation about an axis through Injn radians. The principal axis is the axis of highest n 

Gh — reflection in a horizontal plane, that is, the plane through the origin perpendicular to the axis of highest n 

A group G is a set of elements related by an operation which we will call 

The product of any two elements is in the set that is, the set is closed under group multiplication. 

A molecular point group is a set of symmetry elements. Each symmetry element describes an operation which when carried out on the molecular skeleton leaves the molecular skeleton unchanged. Elements of point groups may represent any of the following operations  

Reflections in planes containing the origin (center of mass)  

Improper rotations—a rotation about an axis through the origin followed by a reflection in a plane containing the origin and perpendicular to the axis of rotation  

A group G =. .., gt. is a set of elements related by an operation which we will call group multiply for convenience and which has the following properties  

There is an inverse, c/f. to each element, such that e. An 

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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. 

Note 5 From a crystallographic point of view, the uniaxial nematic structure is characterised by the symbol Dooh in the Schoenflies notation (Wmm in the International System). 

Note 3 The smectic C structure corresponds to monoclinic symmetry characterised by the symbol C2h, in the Schoenflies notation and the space group t Hm in the International System. 

Note 4 Locally, the structure of the chiral smectic C mesophase is essentially the same as that of the achiral smectic C mesophase except that there is a precession of the tilt direction about a single axis. It has the symmetry C2 in the Schoenflies notation. 

Note 4 The structure of a smectic B mesophase is characterised by a D6h point group symmetry, in the Schoenflies notation, by virtue of the bond orientational order. 

Note 4 The point-group symmetry is C2h (2/m) in the Schoenflies notation, and the space group, 121m in the International System. 

Note 3 The relevant space group of a Colh mesophase is P 6lmmm (equivalent to P 6/m 2 m in the International System and point group Dhh in the Schoenflies notation). 

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 equivalent symmetry element in the Schoenflies notation is the improper axis of symmetry, S which is a combination of rotation and reflection. The symmetry element consists of a rotation by n of a revolution about the axis, followed by reflection through a plane at right angles to the axis. Figure 1.14 thus presents an S4 axis, where the Fi rotates to the dotted position and then reflects to F2. The equivalent inversion axes and improper symmetry axes for the two systems are shown in Table 1.1. 

S = rotation through 2n/n radians followed by crA (see below) SCHOENFLIES NOTATION... 

Chemists use the Schoenflies Notation and rotation-reflection for improper rotation in assigning point groups. Crystallographers... 

To make matters even more complicated, it is much more convenient in crystallography to use an entirely different system of symmetry symbols, termed the Herman-Maugin (as opposed to Schoenflies) notation (Table 8.2). 

Table 7.8 The 32 Crystallographic Point Groups, Listed by Main Symmetry Axes or Plane, Using Both the Schoenflies Notation (S, e.g., C2v) and the Hermann-Mauguin or International Notation (HM, e.g., mm2)3...

The Schoenflies notation for rotation axes is C , and for mirror-rotation axes the notation is S2 , where n is the order of the rotation. The symbol i refers to the center of symmetry (cf. Section 2.4). Symmetry planes are labeled cr crv is a vertical plane, which always coincides with the rotation axis with an order of two or higher, and... 

Point-group symmetries not listed in Table 3-1 may easily be assigned the appropriate Schoenflies notation by analogy. Thus, e.g., C5v, C5h, C7, Cg, etc. can be established. Such symmetries may well occur among real molecules. 

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. 

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. 

A full description of the Schoenflies notation is contained in the references given in the Bibliography. 

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