Structural point groups, crystallography

It is the arrangement and symmetry of the ensemble of the atomic nuclei in the molecule that is considered to be the geometry and the symmetry of the molecule. The molecules are finite structures with at least one singular point in their symmetry description and, accordingly, point groups are applicable to them. There is no inherent limitation on the available symmetries for molecules whereas severe restrictions apply to the symmetries of crystals, at least in classical crystallography. 

Space groups (like point groups) constitute a very pretty branch of mathematics, but that (presumably) is not why chemists study them. Space groups are important to a chemist because they are essential to solving and interpreting crystal structures. Therefore, we conclude this chapter with three topics that relate space group theory directly to the use of X-ray crystallography to obtain chemically useful information. 

Symmetry is the fundamental basis for descriptions and classification of crystal structures. The use of symmetry made it possible for early investigators to derive the classification of crystals in the seven systems, 14 Bravais lattices, 32 crystal classes, and the 230 space groups before the discovery of X-ray crystallography. Here we examine symmetry elements needed for the point groups used for discrete molecules or objects. Then we examine additional operations needed for space groups used for crystal structures. 

Simplifications can be brought about whenever the surface structure has symmetries. Point-group symmetries help moderately to reduce the matrix dimensions. On the other hand, two-dimensional periodicity can help drastically by reducing the number N to the number of atoms within a single two-dimensional unit cell with a depth perpendicular to the surface of a few times the electron mean free path. For surface crystallography this is, however, not yet sufficient, because surface structural determination requires repeating such calculations for hundreds of different geometrical models of the surface structure. 

Table 1.2 Results of a query from CSDSymmetry, showing the ten molecular point groups (with more than 30 occurrences in the CSD) that give rise most often to noncentrosymmetric crystal structures. Reproduced from [28] by permission of the International Union of Crystallography...

Deeper structural studies by crystallography, molecular modeling, and SAR were performed until recently to modulate the potency and the selectivity of synthetic harmine (1) derivatives (see e.g., [37]). These smdies showed that lipophilic substituents, replacing the methyl group of the methoxy moiety at C-7, increased the potency for MAO-A inhibition in comparison with 1. Additionally, it was found that synthetic compounds containing a cyclohexyl group as substituent at C-7 were more potent MAO-B inhibitors than 1. Docking simulations demonstrated that this cyclohexyl chain points to a lipophihc pocket in the entrance cavity of the human MAO-B active site. 

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