Probabilistic distribution of one-phase structure seminvariants for an isomorphous pair of structures: Theoretical basis and initial applications

Given a special type of triplet of reciprocal-lattice vectors in the monoclinic and orthorhombic systems, there exist eight three-phase structure seminvariants (3PSSs) for a pair of isomorphous structures. The first neighborhood of each of these 3PSSs is defined by the six magnitudes and the joint probability distribution of the corresponding six structure factors is derived according to Hauptman's neighborhood principle. This distribution leads to the conditional probability distribution of each of the 3PSSs, assuming as known the six magnitudes in its first neighborhood. The conditional probability distributions can be directly used to yield the reliable estimates (0 or pi) of the one-phase structure seminvariants (1PSSs) in the favorable case that the variances of the distributions happen to be small [Hauptman (1975). Acta Cryst. A31, 680-687]. The relevant parameters in the formulas for the monoclinic and orthorhombic systems are given in a tabular form. The applications suggest that the method is efficient for estimating the 1PSSs with values of 0 or pi.