Structure Solution I. Direct Methods

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Direct Methods

Structure solution methods can be divided into two broad categories: those based on reciprocal-space methods and those based on real-space methods. Typically, the former involve direct use of the measured reciprocal space data, i.e. the |F|² values, while the latter are concerned with the position and conformation of the molecules (or individual atoms/ions) within the unit cell. This page presents a brief overview of the so-called direct methods.

Structure factors can be considered as a vector quantity in the sense that they have an amplitude, F, (i.e. magnitude) and a phase, ƒ, (i.e. direction). As discussed earlier in the section on Fourier maps, the vectors F may be used to calculate a map of real space showing electron or nuclear density for the case of X-ray or neutron structure factors, respectively. A simple grid search can then be used to locate the peaks in the map, which correspond to atomic positions. Assignment of the peaks to particular atoms provides a convenient description of the crystal structure.

The only problem with this method is that the diffraction experiment measures only the magnitude of F and not its phase, and without phases a Fourier map cannot be constructed. Worse still, it can be shown that the phases actually have a higher "information" content than amplitudes, so that over 50% of the information is lost in the experiment itself. At first sight the problem may seem a little intractable: despite the fast number of observations made in most crystallographic experiments, the number of unknowns, ƒ, is always equal to the number of knowns, F. However, the scattering density giving rise to F is "lumpy" due to the existence of atoms and, in addition, it is normally positive definite (except for a few nuclei in neutron diffraction that exhibit negative b values). The consequence is that the phases are not random, but have well-defined relationships that relate to the magnitude of the structure factors, F.

Direct methods programs exploit these relationships and attempt to determine the phases directly from the observed amplitudes, hence their name. It is beyond the scope of this course to explain in detail how these programs actually work, but it is important to be aware of the assumptions they make since effect the success or failure of this approach. The main assumption is that atoms are found as peaks in the density map and that, due to chemical bonding, the peaks are separated, typically about 1-2 Å apart in covalently bonded structures. Therefore, one requirement for success in direct methods is that the data is measured to atomic resolution, and experience shows that the minimum d spacing required is typically in the range 0.9-1.0 Å. This condition is readily adhered to in most single-crystal experiments, but is hard to achieve by powder diffraction given the extensive overlap due to unit cells of any reasonable size, say ones over 800 Å³.

Despite this limitation a significant number of structures have been solved by direct methods from powder diffraction data. Indeed, some of the single-crystal programs have been specifically modified to take into account the intrinsically lower quality of powder data in much the same way that modifications have been made for protein crystallography, the latter having similar problems even for single-crystal diffraction.

Patterson Maps

While the electron (or nuclear) density map results from the Fourier transform of F, another type of map, known as a Patterson map, can be produced by the Fourier transform of the measured quantity |F|². The peaks in this map correspond not to atoms, but to the vector distances between atoms. Patterson maps are very useful in single-crystal diffraction for the case where a few scatterers dominate the scattering process, e.g. heavy atoms in X-ray diffraction. This approach has been tried with powder diffraction, but with limited success,

Given the limitations imposed by direct methods, the alternative approach has been to use real-space methods, and these are discussed on the next page.

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