Joint probability distribution functions when a model is available: Fourier syntheses

Abstract

The title of this chapter may seem a little strange; it relates Fourier syntheses, an algebraic method for calculating electron densities, to the joint probability distribution functions of structure factors, which are devoted to the probabilistic estimate of s.i.s and s.s.s. We will see that the two topics are strictly related, and that optimization of the Fourier syntheses requires previous knowledge and the use of joint probability distributions. The distributions used in Chapters 4 to 6 are able to estimate s.i. or s.s. by exploiting the information contained in the experimental diffraction moduli of the target structure (the structure one wants to phase). An important tool for such distributions are the theories of neighbourhoods and of representations, which allow us to arrange, for each invariant or seminvariant Φ, the set of amplitudes in a sequence of shells, each contained within the subsequent shell, with the property that any s.i. or s.s. may be estimated via the magnitudes constituting any shell. The resulting conditional distributions were of the type, . . . P(Φ| {R}), (7.1) . . . where {R} represents the chosen phasing shell for the observed magnitudes. The more information contained within the set of observed moduli {R}, the better will be the Φ estimate. By definition, conditional distributions (7.1) cannot change during the phasing process because prior information (i.e. the observed moduli) does not change; equation (7.1) maintains the same identical algebraic form. However, during any phasing process, various model structures progressively become available, with different degrees of correlation with the target structure. Such models are a source of supplementary information (e.g. the current model phases) which, in principle, can be exploited during the phasing procedure. If this observation is accepted, the method of joint probability distribution, as described so far, should be suitably modified. In a symbolic way, we should look for deriving conditional distributions . . . P (Φ| {R}, {Rp}) , (7.2) . . . rather than (7.1), where {Rp} represents a suitable subset of the amplitudes of the model structure factors. Such an approach modifies the traditional phasing strategy described in the preceding chapters; indeed, the set {Rp} will change during the phasing process in conjunction with the model changes, which will continuously modify the probabilities (7.2).