The construction of crystal structures by computer

The procedure for determining the structure of a real crystal goes through chemical synthesis of the constituting compound, followed by purification and recrystallization 

Assuming a known and fixed molecular conformation - either the molecule is rigid, or structure optimization has been carried out by some computational method - and that there is no more than one molecule in the asymmetric unit, the variables describing the crystal structure are three coordinates of the center of mass, three orientation angles, and the cell parameters, which range in number from one for a hypothetical cubic crystal to six for a triclinic crystal. These variables define the multidimensional space that must be scanned by the computational procedure. It is a relatively easy matter to instruct a computer to search this space and to provide computational polymorphs for a given molecular structure. 

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On the basis of these structural features, it is easy to understand why zeolites constructed by Si and A1 cannot have extra-large pores. Nevertheless, pure-silica zeolites with 14-membered rings, i.e. CIT-5 and UTD-1, have been synthesized recently, and further investigation into crystallization mechanisms in combination with the vast experimental data available and with theoretical simulation and computation may help us to rationally design and synthesize extra-large microporous aluminosilicate molecular sieves with special channels such as multidimensionally interconnected and chiral ones. 

X-ray erystallographic studies are an experimental technique that allows a three-dimensional reconstruction of the structure of a chemical compound. It is based on the property of crystals to dilfract X-rays. The diffraetion pattern is dependent on the atoms present in the molecule and can be read by an X-ray crystallographer who computationally constructs the structure and shape of a molecule. 

Colloidal crystals . At the end of Section 2.1.4, there is a brief account of regular, crystal-like structures formed spontaneously by two differently sized populations of hard (polymeric) spheres, typically near 0.5 nm in diameter, depositing out of a colloidal solution. Binary superlattices of composition AB2 and ABn are found. Experiment has allowed phase diagrams to be constructed, showing the crystal structures formed for a fixed radius ratio of the two populations but for variable volume fractions in solution of the two populations, and a computer simulation (Eldridge et al. 1995) has been used to examine how nearly theory and experiment match up. The agreement is not bad, but there are some unexpected differences from which lessons were learned. 

Crystal structure of a protein molecule can also be determined by x-ray crystallography. Purified protein is crystallized either by batch methods or vapor diffusion. X-rays are directed at a crystal of protein. The rays are scattered depending on the electron densities in different positions of a protein. Images are translated onto electron density maps and then analyzed computationally to construct a model of the protein. It is especially important for structure-based drug designs. 

The Duffing Equation 14.4 seems to be a model in order to describe the nonlinear behavior of the resonant system. A better agreement between experimentally recorded and calculated phase portraits can be obtained by consideration of nonlinear effects of higher order in the dielectric properties and of nonlinear losses (e.g. [6], [7]). In order to construct the effective thermodynamic potential near the structural phase transition the phase portraits were recorded at different temperatures above and below the phase transition. The coefficients in the Duffing Equation 14.4 were derived by the fitted computer simulation. Figure 14.6 shows the effective thermodynamic potential of a TGS-crystal with the transition from a one minimum potential to a double-well potential. So the tools of the nonlinear dynamics provide a new approach to the study of structural phase transitions. 

In modern crystallography virtually all structure solutions are obtained by direct methods. These procedures are based on the fact that each set of hkl planes in a crystal extends over all atomic sites. The phases of all diffraction maxima must therefore be related in a unique, but not obvious, way. Limited success towards establishing this pattern has been achieved by the use of mathematical inequalities and statistical methods to identify groups of reflections in fixed phase relationship. On incorporating these into multisolution numerical trial-and-error procedures tree structures of sufficient size to solve the complete phase problem can be constructed computationally. Software to solve even macromolecular crystal structures are now available. 

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