Symbols point groups

System and lattice symbol Point group Hermann-Mauguin symbol Short Full Plane group number Symmetry elements additional to discrete tranlations... 

A. Term Symbols for Non-Degenerate Point Group Symmetries... 

Examples are rare except for the S2 point group. This point group has only an S2 axis but, since S2 = i, it has only a centre of inversion, and the symbol generally used for this point group is C,. The isomer of the molecule ClFHC-CHFCl in which all pairs of identical FI, F or Cl atoms are trans to each other, shown in Figure 4.11(b), belongs to the C, point group. 

The point group Cj , contains only a plane of symmetry, in addition to I. It is therefore the same as Ci and is usually given the symbol Q. 

Harnung SE, Schaffer CE (1972) Phase-fixed 3-G Symbols and Coupling Coefficients for the Point Groups. 12 201-255... 

A Hermann-Mauguin point-group symbol consists of a listing of the symmetry elements that are present according to certain rules in such a way that their relative orientations can... 

Cubic point groups have four threefold axes (3 or 3) that mutually intersect at angles of 109.47°. They correspond to the four body diagonals of a cube (directions x+y+z, -x+y-z, -x-y+z and x-y-z, added vectorially). In the directions x, y, and z there are axes 4, 4 or 2, and there can be reflection planes perpendicular to them. In the six directions x+y, x-y, x+z,. .. twofold axes and reflection planes may be present. The sequence of the reference directions in the Hermann-Mauguin symbols is z, x+y+z, x+y. The occurrence of a 3 in the second position of the symbol (direction x+y+z) gives evidence of a cubic point group. See Fig. 3.8. 

Symmetrical geometric figures and their point group symbols in each case, the short Hermann-Mauguin symbol is given to the left, and the Schoenflies symbol to the right... 

Figures 3.8 and 3.9 list point group symbols and illustrate them by geometric figures. In addition to the short Hermann-Mauguin symbols the Schoenflies symbols are also listed. Full Hermann-Mauguin symbols for some point groups are ...

As indicated above there may be many equivalent matrix representations for a given operation in a point group. Although the form depends on the choice of basis coordinates, the character is Independent of such a choice. However, for each application there exists a particular set of basis coordinates in terms of which the representation matrix is reduced to block-diagonal form. This result is shown symbolically in Fig. 4. ft can be expressed mathematically by the relation... 

Note that dj2 is the short notation for d7l2 2 2, as it appears in the cubic point groups (Appendix VII). Similarly, fti, fxtj and are the abbreviated symbols for... 

Collectively, the symmetry elements present in a regular tetrahedral molecule consist of three S4 axes, four C3 axes, three C2 axes (coincident with the S4 axes), and six mirror planes. These symmetry elements define a point group known by the special symbol Td. 

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

All the possible combinations of these symmetry elements result in 32 crystallographic point-group symmetries or crystal classes their symbols are listed in Table 3.3. Notice that in putting together the symbols to denote the symmetries of any crystal classes the convention is to give the symmetry of the principal axis first for instance 4 or 4, for tetragonal classes. If there is a plane of symmetry perpendicular to the principal axis, the two symbols are associated as in 4 m or Aim (4 over m), then the symbols for the secondary axes, if any, follow, and then any other symmetry planes. In a symbol such as Almmm, the second and third m refer to planes parallel to the four-fold axis. 

Notice that the symmetry operations of each point group by continued repetition always bring us back to the point from which we started. Considering, however, a space crystalline pattern, additional symmetry operations can be observed. These involve translation and therefore do not occur in point groups (or crystal classes). These additional operations are glide planes which correspond to a simultaneous reflection and translation and screw axis involving simultaneous rotation and translation. With subsequent application of these operations we do not obtain the point from which we started but another, equivalent, point of the lattice. The symbols used for such operations are exemplified as follows ... 

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

The examples used above to illustrate the features of the software were kept deliberately simple. The utility of the symbolic software becomes appreciated when larger problems are attacked. For example, the direct product of S3 (order 6) and S4 (isomorphic to the tetrahedral point group) is of order 144, and has 15 classes and representations. The list of classes and the character table each require nearly a full page of lineprinter printout. When asked for, the correlation tables and decomposition of products of representations are evaluated and displayed on the screen within one or two seconds. Table VII shows the results of decomposing the products of two pairs of representations in this product group. 

Catalogue raisonee of the common point groups symbols,... 

Fig. 5.5. Geometrical structure of a close-packed metal surface. Left, the second-layer atoms (circles) and third-layer atoms (small dots) have little influence on the surface charge density, which is dominated by the top-layer atoms (large dots). The top layer exhibits sixfold symmetry, which is invariant with respect to the plane group p6mm (that is, point group Q, together with the translational symmetry.). Right, the corresponding surface Brillouin zone. The lowest nontrivial Fourier components of the LDOS arise from Bloch functions near the T and K points. (The symbols for plane groups are explained in Appendix E.)...

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