Infinite point groups

The two infinite point groups [Pg.133]

The reduction formula can only be applied to finite point groups. For the infinite point groups, D h and C h, the usual practice is to reduce the representations by inspection of the character table. 

The reduction formula cannot be applied to the infinite point groups (Chapter 4). Here inspection of the character table may help. Since 2 cosd> at appears with the I u irreducible representation, it is worth a try to subtract this one from rvib ... 

Since this equation is not applicable to the infinite point groups (C approaches have been proposed (see Refs. 75 and 76). 

For infinite point groups, Greek letters are used, where Z is used for IRRs having a character of I for their identity operation and II, A, O, or F are used if the character for the identity operation is larger than I. 

Solution. The degrees of freedom basis set for the CO2 molecule was shown in Example 8.21. Using the D21, subgroup of the infinite point group ... 

Partial correlation tables for infinite point groups ... 

The Kfj point group contains an infinite number of axes and a centre of inversion i. It also contains elements generated from these. 

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

A point group is the symmetry group of art object of finite extent, such as an atom or molecule. (Infinite lattices, occruring in the theory of crystalline solids, have translational symmetry in addition.) Specifying the point group to which a molecule belongs defines its symmetry completely. 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

A There is a Cm axis passing through the ball along its largest dimension and an infinite number of vertical planes containing that axis. There are also an infinite number of C2 axes that are perpendicular to the major axis and which are contained by the horizontal plane. This passes the nC2 JL Cn test with n = so the ball belongs to the point group. 

The number of elements in a group is called the order of the group and is usually given the symbol g e.g. for the point group for the symmetric tripod g is 6 and for an infinite group g is oo. 

In the second place, we restrict our discussion to those representations which are composed of matrices which cannot be simultaneously broken down (reduced) by a similarity transformation into block form (e.g. the matrices in 5-9). It will be for these irreducible representations that (in the next chapter) we will be able to prove a number of far reaching theorems, one of the most important of which is the theorem that the number of non-equivalent irreducible representations is equal to the number of classes in the point group. So that, for example, for the point group which has three classes, rather than dealing with an infinite number of representations we will have only the three which are non-equivalent and irreducible to worry about. Also in the next chapter we will show how to obtain, with the least amount of work, the essential information concerning the non-equivalent irreducible representations which exist for any point group. 

The point group Cwv is a special case for linear molecules such as IC1 and HCN (Fig. 3.13e, f), because it is possible to rotate the molecule about its principal axis to any desired degree and to draw an infinite number of vertical planes. 

The list of point groups is split into two classes seven infinite families and seven sporadic cases. Every point group contains a normal subgroup formed by its rotations. 

A finite object that exhibits the highest possible symmetry is the sphere, whose point group symbol Kb is derived form the German word Kugel. It has an infinite number of rotation axes of any order in every direction, all passing through the center, as well as an infinite number of ah planes each perpendicular to a Coo axis. Obviously, K is merely of theoretical interest as no chemical molecule can possess such symmetry. 

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