A multisolution method of phase determination by combined maximization of entropy and likelihood. II. Application to small molecules

The approach described by Bricogne & Gilmore [Acta Cryst. (1990). A46, 284-297] (I) is applied to three small organic molecules. Phase extension for sucrose octaacetate (C₂₈H₃₈O₁₉) from a basis set of 300 correctly phased U magnitudes confirms the stability of the exponential modelling and plane-search algorithms under very demanding conditions; the extrapolated phases are of comparable quality with those produced by the tangent formula, although it is possible, by overfitting the observed and calculated U magnitudes, to obtain results that are better than tangent refinement. The ab initio phasing of two small molecules, one (diamantan-4-ol, C₁₄H₂₀O) centrosymmetric and the other [(-)-platynecine, C₈H₁₅NO₂] non-centrosymmetric, shows that the likelihood function is a more powerful discriminator between phase choices than any figure of merit hitherto available in conventional direct methods; correct discrimination of phase sets arising from phase-angle permutation is readily achieved even in situations where less than ten phased reflexions are in the basis set. In the non-centrosymmetric case, the conditional probability criterion P(δq) ∝ ∫ δq(x)²/q^ME(x) d³x [I, equation (1.4)] plays a vital rôle in exploring the multimodality of statistical phase indications and is used as a filter in building the phasing tree. Finally, the likelihood function is successfully used in phase refinement in (-)-platy-necine where the mean absolute phase error is reduced by 6.1° for the acentric reflexions using a basis set of only 27 reflexions. Such a calculation would be impossible in traditional direct methods.