International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.7, pp. 904-906

Section 9.7.7. Molecular packing

A. J. C. Wilson,a V. L. Karenb and A. Mighellb

aSt John's College, Cambridge CB2 1TP, England, and bNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

9.7.7. Molecular packing

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9.7.7.1. Relation to sphere packing

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The effect of molecular symmetry cannot be ignored in overall statistical surveys as well as in structure prediction. However, in most structures, the molecular symmetry is low or it is not used in the packing. In about 90% of crystalline compounds, the molecules crystallize in low-symmetry space groups, so that a given molecule has a 12-point contact with neighbouring molecules. As 12 corresponds to the number of nearest neighbours in cubic and hexagonal closest packing of spheres, the periodic assembly of most molecular structures can be regarded as the closest packing of distorted spheres, where symmetry ensures the interlocking of complex shapes (Gavezzotti, 1994[link]).

For the relatively infrequent cases where high molecular symmetry is reflected in high crystal symmetry, the packing of molecules can be derived from the appropriate, though not necessarily the densest, packing of spheres. For example, ten-point, eight-point and six-point molecular contacts can be achieved, respectively, by tetragonal close packing (I4/mmm), by I-centred cubic packing ([Im\overline {3}m]), and by primitive cubic packing ([Pm\overline {3}m]). For a review and some derivations of the densest packing of equal spheres, see Chapter 9.1[link] and Patterson & Kasper (1959[link]), Coutanceau Clarke (1972[link]), and Smith (1973[link]); and for packing of clusters of unequal spheres, see Williams (1987[link]).

With spheres having infinite point symmetry [{\cal K}_{\infty h}], every sphere can be located on syntropic symmetry elements at special positions with high symmetry up to the symmetry of the lattice. The lattice translations, pertinent to the fully symmorphic space group, are then able to generate the entire crystal structure. When spheres are deformed, symmetry is removed and the non-lattice translations involved with antimorphic space groups (which must be subgroups of the sphere-packing groups) become necessary to ensure space filling with the repetitive patterns of complex molecular shapes. In a similar manner, other objects with infinite elements of symmetry (e.g. rods; Lidin, Jacob & Andersson, 1995[link]) can be subjected to a rigorous analysis of close packing.

9.7.7.2. The hydrogen bond and the definition of the packing units

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The variously shaped molecular packing units of organic crystal structures are not necessarily identical with the individual molecule. The molecule (of a shape defined by chemical bonds on the inside and van der Waals forces on the outside) can be subjected to clustering under formation of intermolecular hydrogen bonds. Although far weaker than the chemical bond, hydrogen bonds are strong enough to alter the shape of the packing units of the crystal structure significantly. This may have far-reaching consequences for the adopted packing and symmetry. An extreme example is represented by the clustering of H2O molecules, where two hydrogen bonds and two regular O—H bonds create a [\overline {4}3m] point symmetry at each O atom, and a highly symmetrical structure emerges with an infinite bond network, similar to that in quartz, SiO2. From the point of view of an individual H2O molecule, the structure is very open. In contrast, a pseudo-close-packed structure of crystalline water, assuming an effective H2O radius of 1.38 Å, would have specific density of 1.8 g cm−3.

Analogous principles apply to organic structures with hydrogen bonding. CH3OH, for example, forms hydrogen-bonded zig-zag chains in its crystal structure. Obviously, the shape of the hydrogen-bonded cluster of molecules depends on the number and orientations of the hydrogen bonds relative to the size and shape of the molecule, causing three-dimensional, planar and linear `polymers', or the formation of dimers and trimers. As in the example of water, this introduces additional symmetry elements and decreases the degree of space filling.

There is a general rule that ensures that this phenomenon is widespread. The principle of maximum hydrogen bonding states that all the H atoms in the active (polar) groups of a molecule are employed in hydrogen-bond formation (Evans, 1964[link]). Therefore, as the [{\rm O}\cdots{\rm HO}] and [{\rm O}\cdots{\rm HN}] hydrogen bonds are both energetic and common, they are also of the greatest importance in this respect. Although most pronounced in smaller molecules, the symmetry-altering influence of hydrogen bonding also applies to relatively large molecules with a lower proportion of hydrogen bonding as, for example, in long-chain carboxylic acids that are linked in pairs. In large molecules with many active groups, however, the hydrogen bonds merely become the new delimiters of the shape of the individual molecule. The perils of the symmetry-statistical treatments of the hydrogen-bonded structures are well recognized and, for some purposes, the strategy adopted is to exclude such systems from the statistical pool (Filippini & Gavezzotti, 1992[link]).

References

Coutanceau Clarke, J. A. R. (1972). New periodic close packings of identical spheres. Nature (London), 240, 408–410.
Evans, R. C. (1964). An introduction to crystal chemistry. Cambridge University Press.
Filippini, G. & Gavezzotti, A. (1992). A quantitative analysis of the relative importance of symmetry operators in organic molecular crystals. Acta Cryst. B48, 230–234.
Gavezzotti, A. (1994). Molecular packing and correlations between molecular and crystal properties. Structure correlation, Vol. 2, edited by H.-B. Bürgi & J. D. Dunitz, Chap. 12, pp. 509–542. Weinheim/New York/Basel/Cambridge/Tokyo: VCH Publishers.
Lidin, S., Jacob, M. & Andersson, S. (1995). A mathematical analysis of rod packings. J. Solid State Chem. 114, 36–41.
Patterson, A. L. & Kasper, J. S. (1959). Close packing. International tables for X-ray crystallography, Vol. II, Mathematical tables, pp. 342–354. Birmingham: Kynoch Press.
Smith, A. J. (1973). Periodic close packings of identical spheres. Nature (London) Phys. Sci. 246(149), 10–11.
Williams, D. E. G. (1987). Close packing of spheres. J. Chem. Phys. 87, 4207–4210.








































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