Section 16.2.3.2. Conventional direct methods and their limitations

The traditional theory of direct methods assumes a uniform distribution q(x) of random atoms and proceeds to derive joint distributions of structure factors belonging to an N-atom random structure, using the asymptotic expansions of Gram–Charlier and Edgeworth. These methods have been described in Section 1.3.4.5.2.2 of IT B (Bricogne, 2001) as examples of applications of Fourier transforms. The reader is invited to consult this section for terminology and notation. These joint distributions of complex structure factors are subsequently used to derive conditional distributions of phases when the amplitudes are assigned their observed values, or of a subset of complex structure factors when the others are assigned certain values. In both cases, the largest structure-factor amplitudes are used as the conditioning information.

It was pointed out by the author (Bricogne, 1984) that this procedure can be problematic, as the Gram–Charlier and Edgeworth expansions have good convergence properties only in the vicinity of the expectation values of each structure factor: as the atoms are assumed to be uniformly distributed, these series afford an adequate approximation for the joint distribution only near the origin of structure-factor space, i.e. for small values of all the structure amplitudes. It is therefore incorrect to use these local approximations to near as if they were the global functional form for that function `in the large' when forming conditional probability distributions involving large amplitudes.