Crystal packing coefficients

Fig. 8.2. Scatterplots of the crystal packing coefficient (equation 8.2) and the self-occupation coefficient (equation 1.11) for a very large number of organic molecules. Both distributions show a slight decline on increasing molecular volume.

Table 11. Crystal data and packing coefficients for the alcohol inclusions of 1...

Figure 11 Packing coefficients in various states organic crystals, protein cores, inclusion crystals, encapsulated hosts and the liquid state.

This remarkable finding is accounted for by the open nature of the crystal structures, resulting from the steric effect of substituents, as attested by packing coefficients and molecular modelling. 

In protein crystals, due to the large size of the molecule, the empty space can have cross sections of 10-15 A or greater. The empty space between the protein molecules is occupied by mother liquor. This property of protein crystals, shared by nucleic acids and viruses, is otherwise unique among the crystal structures. In fact, the values of the packing coefficient of protein crystals range from 0.7 to 0.2, but the solvent molecules occupy the empty space so that the total packing coefficient is close to 1 [37]. Nevertheless, a detailed theoretical study has been carried out to examine the models of DNA-DNA molecular interactions on the basis of hard-sphere contact criteria. The hard-sphere computations are insufficient for qualitative interpretation of the packing of DNA helices in the solid state, but... 

Nakano, K., Sada, K., Kurozumi, Y. and Miyata, M. (2001) Importance of packing coefficients of host cavities in the isomerization of open host frameworks guest-size-dependent isomerization in cholic acid inclusion crystals with monosubstituted benzenes. 

Simple descriptors of crystal compactness include the crystal density, and the Kitaigorodski packing coefficient, Cr [5] ... 

Indices for measuring the extent of isostructurality of two or more organic crystal structures can be applied not only to homomolecular crystals but also to molecular associates. such as inclusion compounds. A qualitative summary of these descriptors follows, and the reader is referred to a recent account for explicit mathematical definitions. Earlier descriptors of isostructurality included the " d-gree of isostructurahty" Ii(n) (based on the distance differences ARj between the crystal coordinates of identical nonhydrogen atoms within the same section of the asymmetric units of two or more related structures) the packing coefficient increment. A(pc) and the unit cell similarity index... 

The coefficients can be drawn from experimental values (but this is limited to the cases in which there is one site only for both A and B), from ad hoc calculations, or by a fitting of AE values. In the past, large use has been made of experimental AE values, derived, e. g., from crystal packing energies, but now the main sources are the variationally computed... 

The packing coefficient p should not be confused with the space-filling coefficient k in molecular crystals, which is calculated on the assumption that atoms contact each other at van der Waals radii, much larger than bond radii (see Sect. 4.3). In fact, in crystals comprising separate molecules, chains or layers (e.g. graphite), each atom participates in two different types of contacts, described by metallic/covalent and van der Waals radii, respectively. In this case the term radius should not be taken literally. Thus, the crystal structures of P, As, Sb and Bi (No = 3) have p = 0.23 but k 0.7. 

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