On integrating the techniques of direct methods with anomalous dispersion. I. The theoretical basis

The recently secured mathematical formalism of direct methods is here generalized to the case that the atomic scattering factors are arbitrary complex numbers, thus including the special case that one or more anomalous scatterers are present. Once again the neighborhood concept plays the central role. Final results from the probabilistic theory of the two- and three-phase structure invariants are briefly summarized. In particular, the conditional probability distribution of the three-phase structure invariant, given the six magnitudes |E| in its first neighborhood, is described. The distribution yields an estimate for the three-phase structure invariant which is particularly good in the favorable case that the variance of the distribution happens to be small (the neighborhood principle). Particularly noteworthy is the fact that, in sharp contrast to all earlier work, the estimate is unique in the whole range 0 to 2π. An example shows that the method is capable of yielding unique estimates for tens of thousands of three-phase structure invariants with unprecedented accuracy, even in the macromolecular case. The clear implication is that the fusion of the traditional techniques of direct methods with anomalous dispersion, which is described here, will facilitate the solution of those crystal structures which contain one or more anomalous scatterers.