Optimization Techniques for Phase Retrieval Based on Single-Crystal X-Ray Diffraction Data
In this thesis we develop innovative optimization models for phasing crystal structures from X-ray diffraction data. First, an integer minimal principle for quartets and triplets is proposed for initially setting the phases of structure factors that compose a Karle-Hauptman matrix. Initialization in this manner is motivated by a proof of the relation between invariants and the Karle-Hauptman matrix determinant generated from a phase set. Phase initialization by the quartet and triplet model is shown to benefit CRUNCH for a variety of test structures. Next a reciprocal space integer minimal principle model and polynomial-time Sieve method are developed. The Sieve method is shown to phase one order-of-magnitude faster on average than SnB for a variety of test structures. A shift in emphasis to direct space algorithms is then precipitated by the applicability of the Sieve method to only centrosymmetric structures and reliance on invariant subsets completely free of odd triplets. Four direct space methods are introduced, a completely general density assignment MINLP, a NLP relaxation of the MINLP, a MILP for density assignment for restricted reflections, and finally, a MILP relaxation of the density assignment for restricted reflections. The potential for these models to provide accurate phasing is verified for a variety of test structures. Success of these models is limited by the ability to prepare a small grid from Patterson information, with greater than 25% of the atom positions present. This is prohibitive in the sense that the majority of structures must be solved utilizing periodic grid definition in the absence of a resolvable Patterson map. Finally, given the individual limitations of the reciprocal space and direct space methods, three direct-reciprocal space formulations are developed. Consideration of direct and reciprocal space simultaneously is demonstrated, for a variety of test structures, to enable detection of odd triplets and work in the context of periodic grids, which require no prior structural information.
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign