On the application of phase relationships to complex structures. XVI. A random approach to structure determination

With triple-phase relationships treated as linear equations it is possible to refine a set of phases from given initial values. Phases so obtained are better than those found by refining to self-consistency with the tangent formula. An investigation of the radius of convergence of the least-squares refinement process showed that a substantially correct solution may often be found even starting with random phases. Systems containing up to 300 phases have been investigated and the results and their implications are discussed. It is concluded that the random approach can, at the very least, be used to obtain 70--100 phases as a good starting point for phase development. There is also the possibility of obtaining a sufficient number of phases directly to define a reasonably complex structure, especially with a computer augmented by an array processor. A problem which can arise with linear equations, as with the tangent formula, is that the phases obtained do not adequately define the enantiomorph and give an E map with a pseudo centre of symmetry. Two methods of overcoming this problem are described.