On the solution of the phase problem. III. Distributions fitted by comparing their moments

The proposed method of determination of the correct set of phases of structure factors enables in principle full benefit to be taken of a priori structure information contained in the probability distributions of seminvariants. Unlike the direct comparison of the probability distributions discussed in the preceding paper, the method discussed here, by neglecting the moments of higher orders, allows concentration on the main characteristics of the distributions taken for the test. The basic principle of the method for determination of the correct set of phases using the fit between moments of the theoretical and trial distributions has been widely used in different modifications. However, most of these figures of merit compare only first distribution moments. In many cases this results in insufficient discriminating ability. The comparison of the second moments raises the effectiveness of these methods and may be useful in the last stage of the phase-problem solution. The utilization of moments of higher orders may be dangerous, especially using the global coefficient of moments fitting and in the case of a small number of seminvariants (unreliable determination of higher moments). The method of successive comparison of moments of different orders seems to be more reliable and economical. It permits the survey of a large number of potential solutions, thus increasing the likelihood that a correct solution is included. From the economic point of view, it is convenient to include only those regions of magnitudes and those distribution types which have not been used in the preceding step of the search of the trial solutions. It explains the excellent results obtained using figures of merit based on the special seminvariant types, e.g. NQEST, NQC [De Titta, Edmonds, Langs & Hauptman (1975). Acta Cryst. A31, 472-479; Schenk (1974). Acta Cryst. A30, 477- 481].