A Multisolution Phase Determination Method in X-Ray Crystallography

Abstract

The maximum entropy method offers several advantages over traditional direct methods in the a-priori solution of crystal structures. These include the full use of all invariants at every point without their explicit generation; the use of a non-uniform distribution of atoms, qME(x), which is constantly updated guaranteeing that the aproximate joint probability distribution of structure factors remains valid even for large deviations from uniformity; the natural incorporation of the variances of the structure factors, a tolerance towards errors in the intensity data, and a stability that is independent of data resolution. In ab-initio studies on small organic molecules, starting with origin defining reflections only, the maximum entropy method is used to generate qME(x) which can then be used as a source of new phase information via extrapolation, and this new information is used to update qME(x) in a cyclic fashion. When the extrapolation process has exhausted the currently assumed phase information, strong unphased structure factors are given permuted phases; this recenters the asymptotic expansion for the joint probability distribution of the phased structure factors, and a likelihood criterion is used to select the most probable phases for these reflections and to carry out phase refinement.