Hypersymmetry

A typical characteristic of hypersymmetry operations is that they exercise their influence in well-defined discrete domains. These domains do not overlap—they do not even touch each other. The usual hypersymmetry elements lead to point-group properties. This means that no infinite molecular chains could be selected, for example, to which these hypersymmetry operations would apply. They affect, instead, pairs of molecules or very small groups of molecules. Thus, they can really be considered as local point-group operations. These hypersymmetry elements, accordingly, divide the whole crystalline system into numerous small groups of molecules, or transform the crystal space into a layered structure. 

A prerequisite for hypersymmetry is that there should be chemically identical (having the same structural formula), but symmetrically independent, molecules in the crystal structure—symmetrically independent, that is, in the sense of the three-dimensional space group to which the crystal belongs. The question then arises as to whether these symmetrically independent but chemically identical molecules will have the same structure or not. Only if they do have the same structure, conformation as well as bond configuration, can we talk about the validity of the hypersymmetry operations. Here, preferably, quantitative criteria should be introduced, which is the more difficult since, for example, with increasing accuracy, structures that could be considered identical before, may no longer be considered so later when more accurate data become available. 

A special case of hypersymmetry is when the otherwise symmetrically independent molecules in the crystal are related by hypersymmetry operations making them enantiomorphous pairs. 

Hypersymmetry is a rather widely observed, and sometimes ignored, phenomenon which is not restricted to any special class of compounds. It may be supposed, however, that certain types of molecules are more apt to have this kind of additional symmetry in their crystal structures than others. 

There are hypersymmetry phenomena in some crystal structures that are characterized by extra symmetry operations applicable to infinite chains of molecules. This kind of hypersymmetry has proved to be more easily detectable and has been reported often in the literature [104], 

A prerequisite for hypersymmetry is that there should be chemically identical (having the same structural formula), but symmetrically indepen- 

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There are some crystal structures in which further symmetries are present in addition to those prescribed by their three-dimensional space groups. The phenomenon is called hypersymmetry [102], Thus, it refers to symmetry features not included in the system of the 230 three-dimensional space groups. For example, phenol molecules, connected by hydrogen bonds, form spirals with threefold screw axes as indicated in Figure 9-55. This screw axis does not extend, however, to the whole crystal, and it does not occur in the three-dimensional space group characterizing the phenol crystal. 

Hypersymmetry may be interpreted on the basis of the symmetry of the potential energy functions describing the conditions of the formation of the molecular crystal. The molecules around a certain... 

There is obvious that this kind of symmetry, i.e., the presence of the axis of 2 type, can not be found in none of the S5nnmetries of the space groups, being this the reason for which it is introduced as a hypersymmetry, but still being a symmetry. 

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