On integrating the techniques of direct methods and isomorphous replacement. I. The theoretical basis

Recent advances in direct methods are applied to structurally isomorphous pairs; in particular the probabilistic theory of the three-phase structure invariant in P1 is worked out. The neighborhood principle plays the central role. Specifically, the six-magnitude first neighborhood of each of the four kinds of three-phase structure invariant is defined and the joint probability distribution of the corresponding six structure factors is derived. This distribution leads to the conditional probability distribution of each kind of three-phase structure invariant, assuming as known the six magnitudes in its first neighborhood. In the favorable case that the variance of the distribution happens to be small, one obtains a reliable estimate (0 or π of the structure invariant [the neighborhood principle: Hauptman (1975). Acta Cryst.. A31, 680-687].