Frequency statistical method for evaluating cosine invariants of three-phase relationships

Direct methods: a paradox with regard to the convergence of random phase trials toward solutions

A frustrating observation, based on an R(min) variance analysis within the `shake and bake' framework of direct methods phasing, is described. The variance of R(min) can on occasion identify large subsets of phases that have a significantly lower mean phase error than the entire direct methods phase set of otherwise unsuccessful phasing trials for which the overall phase error occasionally dips below 75 or 80°. This is the first time, other than for a handful of Σ1 phase indications in optimal situations, that a priori phase estimates have been attained for large numbers of E values, prior to solving the structure. Although the a priori variance of R(min) is a useful tool for identifying such phases, the a posteriori phase refinement shifts indicated by its minimum often prevent a successful convergence to the solution. Similar efforts to encourage solution convergences in the realm of real space have also been discouraging.