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Crystal-structure descriptors

Simple but useful crystal-structure descriptors are the density, the melting point and the packing coefficient; mention of the first is mandatory for papers in Acta Crystallographica, but unfortunately mention of the other two is not.

The intermolecular geometry is another Cinderella in crystallographic papers. Clearly, a long list of intermolecular interatomic distances is generally not useful or significant but, for hydrogen-bonded crystals, the crucial X$\cdots$X or H$\cdots$X contact distances are usually sufficient. As a general rule, the description of intermolecular geometry requires the use of macro-coordinates, like the distances between molecular centres of mass or the angles between mean molecular planes in different molecules or fragments. It can be said that the crystal structure of naphthalene can be described by just two parameters - the angle between the molecular planes of glide-related molecules and the distance between their centres of mass - which contain most if not all of the chemical information on the properties of the crystalline material. It is also unfortunate that such macrogeometry is very seldom highlighted in crystallographic papers, and has to be painstakingly recalculated from the atomic coordinates.

A crystal model suitable for computer use can be built very simply, using the crystal coordinates for a reference molecule (RM) and the space-group matrices and vectors, as given in International Tables for Crystallography Vol. A^[10]. In this respect, finding in the primary literature a set of atomic coordinates representing a completely-connected molecular unit, as near as possible to the origin of the crystallographic reference system, with a reduced cell and in a standard space group, $Z'$ the number of molecules in the asymmetric unit and an indication of the molecular symmetries, helps in saving a substantial amount of time and mistakes (let this be said as an encouragement to experimental X-ray crystallographers to help their theoretician colleagues). The required algebra is as follows. Calling $\mathbf{x}_0$ the original atomic fractional coordinates of the RM, $\mathbf{P}_i$ and $\mathbf{t}_i$ a space-group matrix and (column) translation vector, the atomic coordinates in a given surrounding molecule (SM) are given by:

$$\mathbf{x}_i = \mathbf{P}_i\mathbf{x}_0 + \mathbf{t}_i.$$

From this expression the coordinates of all atoms in the crystal model can be calculated, remembering that translation vectors whose components are an arbitrary combination of integer unit cell translations can always be added to the $\mathbf{t}_i$ vectors.

A most important crystal property that can be calculated by this model is the packing energy. For an ionic crystal, if the coordinates and charges of all ions in the crystal model are known, the interionic distances and hence the Coulombic energy can be calculated. In organic crystals with dispersive--repulsive forces, the packing energy can be approximated by empirical formulae:

$$\mathrm{PE} = {\scriptstyle{1\over 2}} \sum \sum E(R_{ij})$$

$$E(R_{ij}) = A \;\mathrm{exp}(-B R_{ij}) - CR^{-6}_{ij}$$

where $R_{ij}$ is an intermolecular interatomic distance, and $A$, $B$ and $C$ are empirical parameters.

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