Figures of merit based on PSIO and on strong triplets are often unreliable for structures with superstructure effects. Prior information on pseudotranslational symmetry is used in order to estimate one-phase seminvariants. These are used, together with quartet invariants, for finding the correct solution in a multisolution process.

The mathematical model proposed in paper III of this series [Cascarano, Giacovazzo & Luić (1988). Acta Cryst. A44, 176-183] for describing structures with superstructure effects has been used to derive probabilistic formulas for estimating triplet invariants. The formulas obtained proved sufficiently robust to be applied successfully to a wide range of structures with superstructure effects, in which deviations both of replacive and of displacive type from ideal pseudotranslational symmetry occur.

A general mathematical model is presented which can describe a large variety of structures showing superstructure effects. In particular the model can take into account deviations, both of displacive and of replacive type, of the substructural part from ideal pseudotranslational symmetry. The formulation is used to predict statistical effects of deviations on diffraction data. It is shown that the scattering power of the substructural part may be estimated via a statistical analysis of diffraction data for ideal pseudotranslational symmetry or for displacive deviation from it, while it is not estimable in the case of replacive deviation.

A probabilistic theory of triplet invariants is provided which may be used when pseudotranslations occur in the crystal structure. The final formula for estimating a triplet phase invariant in centrosymmetric space groups is of Cochran-Woolfson type, in non-centrosymmetric space groups of von Mises type with maximum at 2π single phases are determined via a special tangent formula. Thus the usual algorithms for phase expansion and refinement can be employed with few modifications. Parameters occur in the von Mises and tangent formulae which are markedly different from the usual ones. In particular, the reliability of each triplet depends not only on |Eh|, |Ek|, |Eh - k|, but also on the actual h, k, h - k indices and on the nature of the pseudotranslations. An automatic phasing procedure and some applications are also described for the solution of superstructures and other structures showing pseudotranslations.

The effects on the reciprocal space of one or more pseudotranslations occurring in a crystal structure are studied. A quantitative theory is described, which gives full account of the subsets of pseudonormalized structure factors whose mean intensity significantly deviates from unity. Conversely, statistical criteria are suggested aiming at facilitating the recognition of the nature of the superstructure. The theory has been implemented into a computer program that, from 72 different pseudotranslational symmetries, chooses the most probable one, estimates the number of atoms suffering pseudosymmetry and renormalizes structure factors.

The algebraic relations between one-phase seminvariants and Harker sections are described. One-phase seminvariants of the first rank can be estimated via the Fourier transform of single Harker sections, and one-phase seminvariants of the second rank can be estimated via the Fourier transform of pairs of Harker sections.