Crystal symmetry

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 symmetry² 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 three-dimensional space groups. After the advent of X-ray crystallography, space-group symmetry was determined from the systematic absences in diffraction patterns and used to help in the calculation of structure factors and electron-density 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. Space-group 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 ( ${}\overline{1}$ ), 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).

Table 2: Space-group frequencies from [1] for a sample of organic crystals

Rank	Group	No. of crystals	Molecules in general position	Point-group symmetry
1		9056	8032 (89%)	$\overline{1}$
2		4415	4415 (100%)	1
3	$P\overline{1}$	3285	2779 (85%)	$\overline{1}$
4		2477	2477 (100%)	1
5		1371	802 (58%)	$2,\overline{1}$
6		1180	1064 (90%)	$\overline{1}$
7		445	445 (100%)	1
8		370	370 (100%)	1
9		275	225 (82%)	2
10		266	33 (12%)	$m,\overline{1}$
12		205	94 (46%)	$\overline{1},2$
14		127	40 (31%)	$\overline{1},m$
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 space-filling arrangements. This close-packing 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 close-packing requirement may be a little less stringent (Figure 1), but it turns out that all stable crystals have a packing coefficient³ between 0.65 and 0.80.

**Figure 1:** A molecule without strongly polar sites or hydrogen-bonding capability chooses to close-pack in the crystal, bump into hollows, in order to maximize dispersive interactions. When strong forces are present, the close-packing requirement may be less compelling (water is an extreme and almost unique example).
$\includegraphics[width=8cm]{fig1.eps}$