Intermolecular attraction brings molecules together, but there is a priori no implication of order and symmetry. Glasses, in which molecules are oriented at random, are sometimes as stable as crystals, in which molecules are arranged in an ordered fashion. The ordering of irregularly shaped, electrically charged molecules does however imply anisotropy; for mechanical properties, it results in preferential cleavage planes, while the consequences of optical, electrical and magnetic anisotropy lead to a variety of technological applications of crystalline materials. But what is the link between order, symmetry and crystal stability?
Crystal symmetry^{2} has two facets. On one side, in a milestone mathematical development, it was demonstrated that the possible arrangements of symmetry operations (inversion through a point, rotation, mirror reflection, translation, \textit{etc.}) give rise to no less and no more than 230 independent threedimensional space groups. After the advent of Xray crystallography, spacegroup symmetry was determined from the systematic absences in diffraction patterns and used to help in the calculation of structure factors and electrondensity syntheses.
The other side of crystal symmetry has to do with the crystal structure, as resulting from mutual recognition of molecules to form a stable solid. This is a fascinating and essentially chemical subject, which requires an evaluation and a comparison of the attractive forces at work in the crystal. Spacegroup symmetry is needed here to construct a geometrical model of the crystal packing, and it comes into play in judging relative stabilities.
It should be clear that the necessary arrangements of symmetry operations in space bear no immediate relationship to crystal chemistry. The fact that 230 space groups exist does not mean that molecules can freely choose among them when packing in a crystal. Far from it, there are rather strict packing conditions that must be met, and this can be accomplished only by a limited number of arrangements of very few symmetry operations; for organic compounds, these are inversion through a point ( ), the twofold screw rotation (2) and the glide reflection (). Some space groups are mathematically legitimate, but chemically impossible, and the crystal structures of organic compounds so far determined belong to a rather restricted number of space groups^{[1,2,3]} (Table 2).
Rank  Group  No. of crystals  Molecules in general position  Pointgroup symmetry 
1  9056  8032 (89%)  
2  4415  4415 (100%)  1  
3  3285  2779 (85%)  
4  2477  2477 (100%)  1  
5  1371  802 (58%)  
6  1180  1064 (90%)  
7  445  445 (100%)  1  
8  370  370 (100%)  1  
9  275  225 (82%)  2  
10  266  33 (12%)  
12  205  94 (46%)  
14  127  40 (31%)  
16  92  46 (50%)  2  
17  88  51 (58%)  2 
When charge is evenly distributed in a molecule, and there is no possibility of forming hydrogen bonds, no special anchoring points exist. Every region of the molecule has nearly the same potential for intermolecular attraction, and hence it is reasonable to expect that each molecule be surrounded by as many neighbours as possible, forming as many contacts as possible. Empty space is a waste, and molecules will try to interlock and to find good spacefilling arrangements. This closepacking idea appeared very early in its primitive form^{[4]}, but was consciously put forward by Kitaigorodski^{[5]}.
Order and symmetry now come to the fore, since for an array of identical objects a periodic, ordered and symmetrical structure is a necessary (although not sufficient) condition for an efficient close packing. When special interactions (like hydrogen bonds) are present, the closepacking requirement may be a little less stringent (Figure 1), but it turns out that all stable crystals have a packing coefficient^{3} between 0.65 and 0.80.

Copyright © 2005 International Union of Crystallography
IUCr Webmaster