Point groups isometrics

Polyhedral crystals bounded by flat crystal feces usually take characteristic forms controlled by the symmetry elements of the crystal (point) group to which the crystal belongs and the form and size of the unit cell (see Appendix A.5). When a unit cell is of equal or nearly equal size along the three axes, crystals usually take an isometric form, such as a tetrahedron, cube, octahedron, or dodec-... 

Similar problems arise when we deal with possible reference structures that can be interconverted by internal rotations. Consider, for example, the diphenylmethane molecule (Scheme 2.2), where conformations with the planes of both phenyl groups either parallel or perpendicular to the C-CH2-C plane have C2V symmetry. Rotation of a phenyl group through 71 produces a conformation that is isometric with the initial one, but such a rotation is not a symmetry operation of the C2V point group. However, it is clear that such operations have to be included in the group of all operations that leave the reference structure invariant. We shall not attempt to review here the work that has been done on the classification of such symmetry operations on non-rigid molecules [23], nor the controversies that have ensued from it. Instead we present four representative examples of such molecules. 

Fig. 2.10. Isometric structures of the ethane molecule, obtained by symmetry operations of the point group Dj and by rotation of one methyl group with respect to the other...

Fig. 2.11. Isometric conformations of the diphenylmethane molecule. The conformations 1, 2, 3 and 4 can be transformed into one another by point-group symmetry operations that leave the central frame unaltered or by appropriate rotations of the phenyl groups...

Fig. 2.12. Conformational map of diphenylmethane. The 16 equivalent positions (open circles) are images of the 16 isometric conformations with different values of the two torsion angles. The unit cell shown is non-primitive, the primitive iattice having translation distances of n along Wa d along Wg. The plane group is cmm, with translation vectors y, = coa +Wb> 2 = Wa-Wb- The general positions of this plane group are images of arbitrary conformations, the special positions images of conformations with point-group symmetry...

Now let us try an experiment with the (111) face in an isometric crystal in the point group 4/m32/m. The object of our experiment will be to see what set of faces is formed by operating on the (111) face with the symmetry elements of this point group. 

All of the isometric point groups form the cube and dodecahedron, but only two of the five classes form the tetrahedron, two form the pyritohedron, and three form the octahedron. As we have implied, many crystals also exist as combinations of more than one form. This is shown in Figure 98 where the cube/octahedron and cube/dodecahedron combinations are seen. 

Crick and Watson were the first to suggest that small viruses were built up of small protein subunits packed together symmetrically to form a protective shell for the nucleic acid. They reasoned that formation of small identical molecules was an efficient use of the limited information contained in the virus nucleic acid. They also realized that, of the types of symmetry possible for a three-dimensional structure enclosing space, only the cubic point groups could lead to an isometric particle, which was the known symmetry of many viruses at the time. Three types of cubic symmetry exist namely,... 

AMYL. The 5-carbon aliphatic group CsHji, also known as pentyl. Eight isometric arrangements (exclusive of optical isomers) are possible. The amyl compounds occur (as in fusel oil) or are formed (as from the petroleum pentanes) as mixtures of several isomers, and, sinee llieir boiling points are close and their other properties similar, it is neither easy nor usually necessary to purify them. 

It should be pointed out that the fixed points play an important role in geometrical application of isometric groups, e.g. stereochemistry, cf. Sects. 3.4 and 3.5. 

In Sect. 2.2.2 we have shown that if a SRM admits primitive period isometric transformations, representations of two groups and may be derived. Extension of a representation of J7 by leads to the corresponding representation of St, whereas extension of the representations of St by g gives those of St. The use of St or depends on the problem to which the isometric group is to be applied, as has been pointed out in Section 2.2.2. In order to simplify the notation we shall for general discussions not distinguish between the representations of St and St. 

As may be derived from Table 11, the isometric substitution group J has fixed points ... 

It should be pointed out that the polarizability tensor of a SRM may exhibit a more complicated transformation behavior than expressed by Eq. (3.47). This goes back to the fact, that the polarizability tensor involves all electronic states and the latter do not necessarily all have the same isometric group. 

It should be pointed out that the chirality problem is based entirely on the concept of RNCs. This immediately implies that for its treatment the isometric group (l)( ( )) is sufficient and the primitive period isometries may be omitted. 

The isometric group of SRMs has been used for enumeration of the conformational isomers of NRMs49). From the point of view of permutational symmetry, this problem has been treated by Mislow et al.s°). The problem of enumeration of permutational isomers of rigid molecules has been studied by Polya5 and more generally by Ruch et al.52). The determination of classes and number of permutational isomers of molecules with a nonrigid skeleton has been attacked by Leonard53,54 ... 

To complete the discussion on the symmetry of QRMs and NRMs it is interesting to point out the analogy between the covering group < e of the restructure of a QRM and the isometric group St %) of the SRM associated with a NRM... 

The hyperbolic description implies that to a reasonable approximation, tetrahedral water, silicate, silicon and germanium frameworks are characterised by a preferred area per vertex group and a preferred Gaussian curvature. Thus, identical tessellations of isometric surfaces, with equal areas and curvatures at corresponding points on the surface, should offer alternative possibilities for stable frameworks. Indeed this is the case for the zeolite frameworks, faujasite and analcime, which are related to each other through the Bonnet transformation. Within an intrinsic two-dimensional description, these two frameworks are indistinguishable. We have seen in section 2.6 that the Bonnet transformation describes well a number of characteristics of the fee -> bcc martensitic phase transformation in metals and alloys. The success of this model suggests that the hyperbolic picture, intuitive and obvious for zeolites, is also valid for other atomic structures. 

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