Symmetry element, definition

From the definition of, it follows that (7 = 51 i = 0082, since a and i are taken as separate symmetry elements the symbols 5i and 82 are never used. 

Note The symmetry elements always bear a special relationship to the chain axis (see also the note, Definition 2.9). 

In Fig. 2-3.1 we illustrate these definitions for a square-based pyramid by labelling the four comers of the base. This labelling is merely to enable us to see that an operation has taken place and it has no physical significance the whole point of the symmetry operation is that the final orientation is indistinguishable from the original one. Of the operations shown in Fig. 2-3.1, only three, excluding E, are distinct Cit Cti and Cj. It is conventional when choosing the symbol for a rotational operation to do so in such a way that is as small as possible, e.g. Ct is used in preference to Cj. Finally, it is apparent that quite often symmetry elements will coincide and in such cases we will link the symmetry elements e.g. the C Cti and C axes in Fig. 2-3.1 will be written as CV-Cj-Cj. 

Translation simply means movement by a specified distance. For example, by the definition of unit cell, movement of its contents along one of the unit-cell axes by a distance equal to the length of that axis superimposes the atoms of the cell on corresponding atoms in the neighboring cell. This translation by one axial length is called a unit translation. Unit cells often exhibit symmetry elements that entail translations by a simple fraction of axial length, such as a 4. 

The symmetry classification of wavefunctions is based on the symmetry properties of molecules. Most small molecules possess certain symmetry elements such as a plane (a), or an w-fold axis (OJ, or a centre of symmetry (i), or perhaps a variety of these elements in combination. In order to be as definite as possible we shall develop the argument in terms of a specific example. The ground state of... 

The structure of Rhi2(CO)25(C2), reported in Fig. 10, is a further example of carbide stabilization (13). This is the most irregular cluster as yet characterized like Rh8(CO)i9C, it has no symmetry element. This cluster is also derived chemically from the oxidation of [Rh COVC]2-. The 2 central carbide atoms are definitely bonded together (1.47 A) and lead to... 

To determine the symmetry of a molecule, we first need to identify the symmetry elements it may possess and the symmetry operations generated by these elements. The twin concepts of symmetry operation and symmetry element are intricately connected and it is easy to confuse one with the other. In the following discussion, we first give definitions and then use examples to illustrate their distinction. 

In crystallography one is accustomed to the idea that a structure either has a particular symmetry element or it does not. The membership is thus either 1 or zero. In morphological analysis the symmetry has a value of zero through 1 depending upon how closely the profile approaches the symmetry being considered. The definitions of the symmetry operations are shown below ... 

Note that according to the foregoing definition, chirality occurs only in molecules that do not have a rotation/reflection axis. However, if the molecule has only ( ) an axis of rotation, it is chiral. For example, both trans-1,2-dibromocyclohexane (D in Figure 3.3) and the dibromosuccinic acid E have a two-fold axis of rotation (C2) as the only symmetry element. In spite of that, these compounds are chiral because the presence of an axis of rotation, in contrast to the presence of a rotation/reflection axis, is not a criterion for achirality. 

The application of a symmetry element is a symmetry operation and the symmetry elements are the symmetry operators. The consequence of a symmetry operation is a symmetry transformation. Strict definitions refer to geometrical symmetry, and will serve us as guidelines only. They will be followed qualitatively in our discussion of primarily non-geometric symmetries, according to the ideas of the Introduction. 

This can be done only if a molecule has no symmetry elements. But in most cases the number of different reactions of permutation isomerism is less than n. And there are definite reasons for that. 

In order to find the character of the representation of rotational motions and librations, the transformation properties of an axial vector (Fig. 2.1-lb) have to be taken into account. This vector is defined not only by its length and orientation, but also by a definite sense of rotation inherent to it. For linear molecules with only two degrees of rotational freedom, the Xr R) values for Cj and a depend on the orientation of the molecular axis to the symmetry elements. All values of XriR) tire also included in Table 2.7-1. 

The rigorous group theoretical requirement for the existence of chirality in a crystal or a molecule is that no improper rotation elements be present. This definition is often trivialized to require the absence of either a reflection plane or a center of inversion in an object, but these two operations are actually the two simplest improper rotation symmetry elements. It is important to note that a chiral object need not be totally devoid of symmetry (i.e., be asymmetric), but that it merely be diss)nn-metric (i.e., containing no improper rotation symmetry elements). The tetrahedral carbon atom bound to four different substituents may be asymmetric, but the reason it represents a site of chirality is by virtue of dissymmetry. 

Chirality, an important shape property of molecules, can be regarded as the lack of certain symmetry elements. Chirality measures are in fact measures of symmetry deficiency. These principles, originally used for crisp sets, also apply for fuzzy sets. Considering the case of three-dimensional chirality, the lacking point symmetry elements are reflection planes a and rotation-reflections 82 of even indices. Whereas the lacking symmetry elements can be of different nature in different dimensions, nevertheless, all the concepts, definitions, and procedures discussed in this section have straightforward generalizations for any finite dimension n. 

By analogy with chirality and various chirality measures, more general symmetry deficiencies and various measures for such symmetry deficiencies can be defined with reference to an arbitrary collection of point symmetry elements. We shall discuss in some detail only the three-dimensional cases of symmetry deficiencies, however, as it has been pointed out in reference [240], all the concepts, definitions, and procedures listed have straightforward generalizations for any finite dimension n. 

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