Point groups conjugate elements

Define the following terms symmetry operation, symmetry element, principal axis, identity operation, improper rotation, inversion, symmetry group, point group, conjugate elements, similarity transformation and class. 

The elements of a group which are conjugate to each other are said to form a doss and the number of elements in the tth class is given the symbol g. For the symmetric tripod point group the elements fall into three classes E o, a" C C 1 with gx = 1, g% = 3 and g% — 2,... 

The elements of a group which are conjugate to each other are said to form a class and the number of elements in the tth class is given the symbol gt. For the symmetric tripod point group the elements fall into three classes E o, o, o" Ct, C 1 with gx = 1, g% = 3 and g9 = 2. For this and all groups E is conjugate with itself X lEX = E (all AT) and henoe E is always in a class of its own we will always consider E to be the first class. The reader, with the aid of Table 3-4.1, can confirm for himself that if Q = or o or o" then with any of the six elements... 

If Eq. (E.14) is satisfied for all elements of some point group G, A will be an invariant operator [13] (the Hermitian conjugate as well as the sum and/or product of two invariant operators are also invariant operators). Such an operator can be expanded in the form... 

The conjugated atoms of porphin have a C4 axis (z), four C2 axes in the xy plane, four planes of reflection symmetry containing the z axis (<7v), a center of inversion symmetry, and identity (Fig. 4.7B). These elements form the point group. 

It is less easy to find examples for the C h point groups there are several C2h molecules, such as ( )- ,2-diphenylethene (common name stilbene), shown in Figure 3.18. This molecule is planar due to conjugation between the ethene double bond and phenyl substituents. Any planar molecule must have the plane of the molecule as a symmetry element, and in this case this is the only symmetry plane present. The principal axis is C2 and is perpendicular to the plane, which must, therefore, be labelled Since C2 and <7h are present we must have an S2 axis, but the only resulting operation is equivalent to an inversion (i = 2 ) and so only i is quoted. 

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