The direct solution of crystal structures by a Boolean programming optimization methodology

Abstract

In studying the structure of matter, X-ray diffraction techniques are limited by a phenomenon known as the "Phase Problem". Previous contributions to the direct solution of crystal structures have made use of statistical relationships between measured diffraction intensities. This dissertation describes the application of a novel approach to three-dimensional centrosymmetric cases using an optimization methodology. The method involves casting the "Phase Problem" in terms of a classical integer linear programming problem. The variables, which represent the phases, are restricted to take on the values of zero or one. The assignment of zero to a phase angle, in effect, causes it to take on the value of 180°, just as the assignment of one corresponds to a phase angle of 0°. When a priori knowledge of atomic positions exists, the coefficients in the objective function may be biased accordingly to speed up the convergence process. The inequality constraints required by the problem formulation are based on a non-negativity restriction that applies to all points in the unit cell. Upper bound constraints may also be generated for points in real space that correspond to pronounced valleys in Patterson (vector) space. The operation of the method is described in detail and the results of the successful solution of two test cases are presented. The two structures involved are spirodienone, C��H₈O₂, and potassium lead hexanitrocuprate II, K₂PbCu(NO₂)₆. This direct method is significant in that it is a nonstatistical approach that has been shown to work with real structures by using measured intensities. A priori knowledge of, atomic positions Is not required and algorithm efficiency for small structures is comparable to other existing direct methods.