Crystal structure prediction stoichiometry

The crystal structure and stoichiometry of these materials is determined from two contributions, geometric and electronic. The geometric factor is an empirical one (8) simple interstitial carbides, nitrides, borides, and hydrides are formed for small ratios of nonmetal to metal radii, eg, rx / rM < 0.59. When this ratio is larger than 0.59, as in the Group 7—10 metals, the structure becomes more complex to compensate for the loss of metal—metal interactions. Although there are minor exceptions, the H gg rule provides a useful basis for predicting structure. 

The lattice energies of the single-molecule crystals and co-crystal are assumed to be those for the most stable forms of each. To predict the stability of a given co-crystal it is therefore necessary to perform crystal structure prediction on each molecule independently and on the co-crystal itself. If it is necessary to predict which stoichiometry of co-crystal is the most stable, it is also necessary to perform crystal structure prediction on crystals with each possible stoichiometry. Since the cost and difficulty of a crystal structure prediction calculation increases considerably with the number of independent molecules in the asymmetric unit, this becomes a very hard problem which few have ventured to tackle. The role of crystal structure prediction is to identify the lowest energy structures that are possible for both co-crystal and its single-molecule crystalline components. The prediction of stoichiometry for a co-crystal requires consideration of all dissociation processes available for a given number of molecules in the asymmetric unit, for example ... 

To assess the relative stabilities of the various stoichiometries, global minima were identified by crystal structure prediction and compared to the energy of 2 moles of crystalline caffeine and 2 moles of crystalline HBA. As before for caffeine, the lattice energy of the lowest energy crystal was used, ignoring the observed experimental disorder. 2-HBA has one known polymorph and 3-HBA and 4-HBA have two known polymorphs each. However, because of uncertainties in their structure determinations only one polymorph for each of 3-HBA and 4-HBA was considered in the analysis. 

Novel techniques for the creation of co-crystals and solvates such as neat and liquid assisted grinding have challenged the ability of crystal structure prediction to predict stoichiometry from first principles. Recent work has addressed the problem of predicting solvate stoichiometry of acetic acid (the solvent) with various molecules including carbamazepine (CBZ) and its 10,11-dihydro derivative (DHCBZ), urea and theobromine(see Figure 4.8). 

Chapter 6 therefore deals in detail with this issue, including the latest attempts to obtain a resolution for a long-standing controversy between the values obtained by thermochemical and first-principle routes for so-called lattice stabilities . This chapter also examines (i) the role of the pressure variable on lattice stability, (ii) the prediction of the values of interaction coefficients for solid phases, (iii) the relative stability of compounds of the same stoichiometry but different crystal structures and (iv) the relative merits of empirical and first-principles routes. 

Fluorite, CaF2, shows a radius ratio of 0.96, which predicts that the calcium ions will occupy cubic holes formed by the fiuoride anions. Note in Figure 7.22a that the calcium ions do indeed occupy such sites. However, as required by stoichiometry (see Problem 7.39), half of the cubic holes must be unoccupied. (Note that the center of the unit cell is an unoccupied cubic hole formed by the fluoride ions.) The unit cell of the lattice therefore cannot be the simple cubic of fluorides with one calcium in the body. Rather, a larger unit cell of fee calcium ions with fluorides filling the tetrahedral holes is the more appropriate description. Note that the coordination number of the fluorides is 4, which is consistent with Equation (7.6). Table 7.11 indicates that there is a 90% correlation between the known crystal structure and the calculated radius ratio for compounds that assume the fluorite structure. 

Even though crystal forms have been and are the subject of intense investigations, polymorphism as a phenomenon still represents a substantial scientific challenge. Indeed it is hard to predict whether a given molecule will crystallize in one or several crystal forms, whether it will form solvates with different stoichiometries or will ever be happy to link up with other molecules and form stable co-crystals. Such variability and unpredictability have been taken by some scientists as an intrinsic drawback of being able to construct desired crystal structures (and obtain relevant properties) from a purposeful choice of the molecular components, which is the paradigm of molecular crystal engineering [15-19]. 

Kubik and co-workers designed amide-based cyclopeptides 18-19 from two amino acid building blocks, Boc-L-prohne and 6-aminopicolinic acid benzyl ester [49, 50]. Cyclopeptide 18 formed sandwich-type 2 1 anion complexes with halide and sulfate anion with ranging from 10" to 10 in 80 % D2O/CD3OD. Crystal structure of 182 I showed that one iodide ion was located in a cavity between two cyclopeptide molecules, which interlocked perfectly the anion with six N I H-bonds ranging from 3.723 to 3.935 A (Fig. 5.16). As predicted, dicyclopeptide 19 which has more amide NH sites also displayed excellent binding constants (up to 10 M ) to anion with 1 1 stoichiometry. 

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