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Symmetry operations

In a crystal, some symmetry operations can be classified as intramolecular, meaning that they relate different parts of the same molecule and thus belong to the molecular point-group symmetry. The other symmetry operations, which act as true crystal-packing operators, may be called intermolecular, and these are the ones which relate different molecules in the crystal. This classification implies that molecules be distinguishable in crystals.

The simplest intuitive way of viewing a symmetry operation is that it reproduces in space one, or more if applied repetitively, congruent or enantiomorphic copies of a given object, according to a well-defined convention (Figure 2). The spatial relationship between the parent and the reproduced molecules is strict, so a moment's pondering will convince the reader that some operators are more effective than others towards close packing. For objects of irregular shape, mirror reflection and twofold rotation produce bump-to-bump confrontation, while inversion through a point, screw rotation and glide reflection favour bump-to-hollow, more close-packed arrangements (Figure 3). One must not forget that pure translation ($t$) is always present in a crystal. Except when infinite strings or layers are present, it is an intrinsically intermolecular operator, whose role in close-packing is probably intermediate (Figure 4); space group $P1$ is the eighth most populated one for organic substances.

Figure 2: Sketches of the effect of symmetry operations. Top: inversion through a point. Below, left: twofold screw rotation; below right, glide reflection. The latter two operations give rise to strings in the $y$ direction.

Figure 3: A mirror reflection (mirror plane perpendicular to the page, trace along the solid line) cannot produce close-packing. Translation along some direction is required to allow interlocking of molecular shapes.

Figure 4: A two-dimensional pattern obtained by pure translation: not so bad for interlocking and close packing. For a complete set of two-dimensional space-filling drawings in all the 17 plane groups, see [6].

The clearest proof of the leading role of $\overline{1}$, $2_1$ and $g$ in close packing comes from a statistical analysis of the space-group frequencies of organic compounds, care being taken to distinguish between inter- and intramolecular symmetry operations. Table 2 shows that mirror reflection and twofold rotation appear in organic crystals most often as intramolecular operators: thus $C2/c$ is a favourite for molecules with twofold axes, $Pnma$ for molecules with mirror symmetry, and for these space groups the percentage of structures with molecules in general positions is very low. $C2$ is an apparent exception; in fact, the combination of the centring translation and a twofold rotation results in a twofold screw rotation. Viewing the issue from the other end, Table 3 shows that pairwise combinations of $\overline{1}$, $2_1$ and $g$ produce rows, layers or full structures in all the most populated space groups for organics. A student who cares to work out in detail the results in this Table will understand all the basic principles of geometrical crystallography and crystal symmetry.

Table 3: Arrangements of pairs of common symmetry operations in organic crystals

2, twofold rotation; $m$, mirror reflection; $g$, glide reflection; $2_1$, twofold screw rotation; $\overline{1}$, inversion through a point; $t$, centring translation. The superscript upper-case labels preceding each space-group symbol are as follows: ^C cluster, ^R row, ^LL layer and ^3D full three-dimensional structure. When several possibilities are given for an arrangement, they depend on the relative orientation of the symmetry operations. As the full matrix of these pairs of symmetry operations is symmetric, only the lower triangle is given.

		2		$m$		$g$		$2_1$
$m$		$^CP2/m$
$g$		$^RP2/c$		$^LCm$	$\Biggl\{$	$^LCc$
						$^LPca2_1$
						$^{3D}Pna2_1$
$2_1$	$\Bigl\{$	$^LC2$		$^RP2_1/m$	$\Biggl\{$	$^LP2_1/c$	$\Bigl\{$	$^LP2_12_12$
		$^LP2_12_12$				$^LPca2_1$		$^{3D}P2_12_12_1$
						$^{3D}Pna2_1$
$\overline{1}$	$\Bigl\{$	$^CP2/m$	$\Bigl\{$	$^CP2/m$	$\Bigl\{$	$^RP2/c$	$\Bigl\{$	$^RP2_1/m$
		$^RP2/c$		$^RP2_1/m$		$^LP2_1/c$		$^LP2_1/c$
$t$		$^LC2$		$^LCm$	$\Bigl\{$	$^LCm$		$^LC2$
						$^LCc$

A similar statistic, taking account of the local symmetry of the constituent units, is not available for inorganic compounds, but a similar trend would probably be found. These compounds frequently contain highly symmetrical (tetrahedral, triangular, square-planar) ions or groups, which carry over their symmetry to the crystal. This causes a spread of the space-group frequencies towards the tetragonal, hexagonal or cubic systems (a no-man's land for organics); no space group has more than 10% of the structures for inorganic compounds (Table 4).

Table 4: Space-group frequencies for inorganic crystals from [2]

Rank	Group	Number of crystals	% of total
1	$Pnma$	2863	8.3
2	$P2_1/c$	2827	8.2
3	$Fm\overline{3}m$	1532	4.4
4	$P\overline{1}$	1508	4.4
5	$C2/c$	1326	3.8
6	$P6_3/mmc$	1254	3.6
7	$C2/m$	1180	3.4
8	$I4/mmm$	1176	3.4
9	$Fd\overline{3}m$	1050	3.0
10	$R\overline{3}m$	858	2.5

One can never be careful enough when generalizing on such topics; crystal packing is a subtle, elusive subject. To give just an example of its intricacies, when dealing with the importance of symmetry to crystal packing one should consider that a symmetry operation is relevant only when it relates close-neighbour molecules. Wilson^[8,9] has pointed out that, in some space groups, some symmetry elements⁴ may be silent, or "encumbered"; they are prevented, by their location in space, from acting between first-neighbour molecules. The relative importance of symmetry operators in the most populated space groups has been quantified by packing-energy calculations^[11].

Another reminder: the choice of a space group is to some extent arbitrary; for example one might argue that in some cases the presence or absence of a centre of symmetry is a questionable matter. This may be true for all symmetry operations; a glide reflection can be almost operative, and its assignment can be a matter of sensitivity of the apparatus for the detection of weak reflections, in particular the borderline between a "very weak" and a "systematically absent" reflection can even be a matter of personal taste. In this respect, the sensitivity of single-crystal X-ray diffraction experiments to minor deviations is very high, and the presence of a semi- or pseudo-symmetry operation, violated because of minor molecular details, has the same chemical meaning as that of a fully-observed symmetry operation.

What to say, then, of crystals with more than one molecule in the asymmetric unit, $Z' > 1$? Many are presumably just cases of the accidental overlooking of some symmetry in the crystal-structure determination and refinement, and many more do have pseudo-symmetry operations relating the molecules in the asymmetric unit (see the remarks in the previous paragraph). The conformations of the independent molecules are usually quite similar^[12]. While the overall frequency of structures with $Z' > 1$ is about 8% for organics^[13], they seem to be unevenly distributed among chemical classes. For example, for monofunctional alcohols the frequency rises to 50%; a possible interpretation is in terms of hydrogen-bonded dimers and oligomers which are already present in the liquid state, and are so strongly bound that they are transferred intact to the crystal.

The case is similar for molecules which must pick up solvent molecules to crystallize in the form of solvates, or which can form inclusion compounds with a variety of guest molecules. The reasons for the appearance of these phenomena, and their control, are presently beyond reach but see [14].

Copyright © 2005 International Union of Crystallography