International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 9.7, pp. 897900

The space group P2_{1}/c accounts for about 1/3 of all known molecular organic structures, whereas the space group P2/m has no certain example. Why? Ultimately, the space group of a crystal of a particular substance is determined by the minimum (or a local minimum) of the thermodynamic potential (Gibbs free energy) of the van der Waals and other forces, but a very simple model goes a long way towards `explaining' the relative frequency of the various space groups within a crystal class or larger grouping. Nowacki (1943) discussed the basic idea that spacegroup frequency is determined by packing considerations, and had earlier (1942) given statistics for the structures known at the time. Nowacki's statistics were used by Kitajgorodskij^{1} (1945), and recent writers tend to cite Kitajgorodskij as having originated the idea. Kitajgorodskij pointed out that the most frequent space groups are those that permit the close packing of triaxial ellipsoids. Later, Kitajgorodskij (1955) showed that the same space groups allowed close packing of objects of any reasonable shape, `close packing' meaning packing with 12point contact. Wilson (1988, 1990, 1993d) used the complementary idea that space groups are rare when they contain symmetry elements – notably mirror planes and rotation axes – that prevent the molecules from freely choosing their positions within the unit cell. A twofold axis excludes molecular centres from a column of diameter equal to some molecular diameter (say M), a mirror plane from a layer of thickness M, and a centre of symmetry from a sphere of diameter M. The volumes excluded by screw axes and glide planes are small in comparison.
In his book^{2} Organicheskaya Kristallokhimiya, Kitajgorodskij (1955) treated the triclinic, monoclinic and orthorhombic space groups in considerable detail, analysing the possibility of (a) forming closepacked layers (sixpoint contact), and (b) close stacking of the layers. On this basis, he divided the layers and the space groups into four categories each. For the layers they are:
The categories of space groups are:
Kitajgorodskij expected the frequency of space groups to decrease in the order (1) > (2) > (3 > (4). In particular, `permissible space groups should be found but rarely, as exceptions'. The categorization is summarized in Table 9.7.1.1, based on Table 8 of Kitajgorodskij (1955).

Kitajgorodskij's categorization proved very successful in broad outline, but Wilson's (1993b,c) detailed statistics revealed about a dozen anomalous spacegroup types. The anomalies were of two kinds. The first was the frequent occurrence of molecules in general positions in space groups in which Kitajgorodskij expected molecules to use inherent symmetry in special positions. Wilson (1993a) pointed out that in such cases structural dimers^{3} can be formed, with two molecules in general positions related by the required symmetry elements – both enantiomers would be required if the element were or m. Such space groups could therefore be added to Kitajgorodskij's table, in the column for `molecular symmetry 1'. The second kind of anomaly was the fairly frequent occurrence of structures with the `impossible' space groups Pc and P2/c. These could be transferred from `impossible' to `permissible', subgroup (a), by the same packing argument that Kitajgorodskij had used for P1. These and a few other reclassifications are indicated in Table 9.7.1.1, the new entries being enclosed in square brackets for distinction. Where the change is a transfer to a higher category, the original position of the space group is indicated in round brackets.
Wilson (1993d) classified the space groups by degree of symmorphism. A fully symmorphic space group contains only the `syntropic' symmetry elements and a fully antimorphic space group contains only the `antitropic' elements The remaining symmetry elements are `atropic'. The two triclinic space groups, P1 and , contain only `atropic' elements, and are thus not classified by these criteria. The rest are divided into five groups, in accordance with the balance of symmetry elements within the unit cell. For the 71 nontriclinic space groups symmorphic in the strict sense (Wilson, 1993d; Subsection 1.4.2.1 ), the classification gives:

The distribution of the 230 space groups (11 enantiomorphic couples merged) by arithmetic crystal class and degree of symmorphism is given in Table 9.7.1.2.

A few points about the symmorphic groups are worth noting. The 14 `fully symmorphic' space groups are those that (i) have primitive cells and (ii) have no secondary or tertiary axes (three each monoclinic, orthorhombic, tetragonal, hexagonal; two trigonal; no cubic). Secondary axes, even though syntropic in the conventional spacegroup notation, generate additional antitropic axes in accordance with the principles set out by Bertaut (2005, Chap. 4.1 ). These additional axes are not indicated in the `full' Hermann–Mauguin spacegroup symbol, but should appear in the `extended' symbol (Bertaut, 2005, Table 4.3.2.1 ^{4}). As a result of the additional axes, 21 symmorphic space groups with primitive cells are shifted to the `tending to symmorphism' column (five tetragonal, six trigonal, five hexagonal, five cubic). Lattice centring has a similar or greater effect; the 36 centred symmorphic space groups are spread over the three columns `tending to symmorphism' (seven), `equally balanced' (20) and `tending to antimorphism' (nine).
From the nature of the definitions, no symmorphic space group can be `fully antimorphic'. The 14 groups under the latter heading consist of (i) 12 space groups with no special positions (Subsection 9.7.4.1), and (ii) two space groups whose only special positions have symmetry (Subsection 9.7.4.2). On these criteria, the two triclinic groups excluded from discussion would fall naturally into the column `fully antimorphic'. The remaining space groups have no obvious outstanding characteristics. Most of them fall under the heading `tending to antimorphism', though there are some in each of the columns `tending to symmorphism' and `equally balanced'.
Since both Kitajgorodskij's and Wilson's classifications were made with a view to `explaining' the empirically observed frequencies of space groups – though neither makes any use of empirical frequencies – it would be expected that there should be considerable correlation between them. All `closestpacked' space groups are also `fully antimorphic', and most of the `limitingly close packed' and `permissible' are `tending to antimorphism'; a few requiring high molecular symmetry (222, mm2, mmm) and a couple of others are `equally balanced'. Two `fully antimorphic' groups, Pc and Cc, are merely `permissible'. All `fully symmorphic' space groups are `impossible'.
Structural classes (Belsky & Zorky, 1977, and papers cited there and below) are not an a priori classification of space groups but are a classification of structures within a spacegroup type in accordance with the number and kind of Wyckoff positions occupied by the molecules. As a considerable knowledge of the structures is required before their structural classes can be assigned, they form an a posteriori classification, and will be described (Section 9.7.5 below) after the empirical frequencies of space groups have been discussed.
References
Belsky, V. K., Zorkaya, O. N. & Zorky, P. M. (1995). Structural classes and space groups of organic homomolecular crystals: new statistical data. Acta Cryst. A51, 473–481.Belsky, V. K. & Zorky, P. M. (1977). Distribution of homomolecular crystals by chiral types and structural classes. Acta Cryst. A33, 1004–1006.
Bertaut, E. F. (2005). International tables for crystallography, Vol. A, fifth edition, Chap. 4.1. Heidelberg: Springer.
Kitaigorodskii, A. I. (1961). Organic chemical crystallography. New York: Consultants Bureau.
Kitaigorodsky, A. I. (1945). The closepacking of molecules in crystals of organic compounds. J. Phys. (Moscow), 9, 351–352.
Kitajgorodskij, A. I. (1955). Organicheskaya Kristallokhimiya. Moscow: Academy of Science.
Nowacki, W. (1942). Symmetrie und physikalischchemische Eigenschaften krystallisierter Verbindungen. I. Die Verteilung der Kristallstrukturen über die 219 Raumgruppen. Helv. Chim. Acta, 25, 863–878.
Nowacki, W. (1943). Symmetrie und physikalischchemische Eigenschaften kristallisierter Verbindungen. II. Die allgemeinen Bauprinzipien organischer Verbindungen. Helv. Chim. Acta, 26, 459–462.
Wilson, A. J. C. (1988). Space groups rare for organic structures. I. Triclinic, monoclinic and orthorhombic crystal classes. Acta Cryst. A44, 715–724.
Wilson, A. J. C. (1990). Space groups rare for organic structures. II. Analysis by arithmetic crystal class. Acta Cryst. A46, 742–754.
Wilson, A. J. C. (1993a). Kitajgorodskij's categories. Acta Cryst. A49, 210–212.
Wilson, A. J. C. (1993b). Kitajgorodskij and spacegroup popularity. Acta Chim. Acad. Sci. Hung. 130, 183–196.
Wilson, A. J. C. (1993c). Symmetry of organic crystalline compounds in the works of Kitajgorodskij. Kristallografiya, 38, 153–163. [In Russian.]
Wilson, A. J. C. (1993d). Space groups rare for organic structures. III. Symmorphism and inherent molecular symmetry. Acta Cryst. A49, 795–806.