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This chapter demonstrates that the experimental observable in diffraction analysis is the intensity of each diffraction spot which provides only the amplitude of the corresponding structure factor and not its phase — the fundamental phase problem in crystallography. To obtain an image of the molecule forming a crystal we need to calculate a Fourier transform, which requires that we know both the amplitude and phase of each structure factor. Nevertheless, very important information can be derived by calculating a ‘phase-less’ Fourier transform of the intensities alone, which is known as the Patterson function. Although this function is inherently more complex than an electron density map since it displays all inter-atomic vectors, certain sections of the Patterson function, known as Harker sections, can yield information on the positions of the most electron-rich atoms within the crystal. The Patterson function is exploited in most methods of the solving the phase problem for proteins and simple rules for the interpretation of a Patterson function are derived.

The Patterson synthesis is a reverse Fourier transform in which the amplitudes are replaced by their squares and the phases are omitted. Peaks in a Patterson map represent vectors between pairs of atoms; the Patterson function is the convolution of the electron density with itself. Patterson space groups are the combinations of Laue classes with unit cell centring arrangements. The large number of broad Patterson peaks leads to much overlap, with peaks due to pairs of heavy atoms (many electrons) dominating. Heavy atoms may thus often be found by Patterson map analysis, especially when they are related by symmetry, giving peaks in special positions (Harker sections and lines). Examples are shown in several space groups. Care is needed in some cases, when ambiguous solutions may be found. Geometrically rigid groups of atoms may also be located from a Patterson map by combined rotational and translational search methods.

According to the basic principles of structural crystallography, stated in Section 1.6: (i) it is logically possible to recover the structure from experimental diffraction moduli; (ii) the necessary information lies in the diffraction amplitudes themselves, because they depend on interatomic vectors. The first systematic approach to structure determination based on the above principle was developed by Patterson (1934a,b). In the small molecule field related techniques, even if computerized (Mighell and Jacobson, 1963; Nordman and Nakatsu, 1963), were relegated to niche by the advent of direct methods. Conversely, in macromolecular crystallography, they survived and are still widely used today. Nowadays, Patterson techniques have been reborn as a general phasing approach, valid for small-, medium-, and large-sized molecules. The bases of Patterson methods are described in Section 10.2; in Section 10.3 some methods for Patterson deconvolution (i.e. for passing from the Patterson map to the correct electron density map) are described, and in Section 10.4 some applications to ab initio phasing are summarized. The use of Patterson methods in non-ab initio approaches like MR, SAD-MAD, or SIR-MIR are deferred to Chapters 13 to 15. We do not want to leave this chapter without mentioning some fundamental relations between direct space properties and reciprocal space phase relationships. Patterson, unlike direct methods, seek their phasing way in direct space; conversely, DM are the counterpart, in reciprocal space, of some direct space properties (positivity, atomicity, etc.). One may wonder if, by Fourier transform, it is possible to immediately derive phase information from such properties, without the heavy probabilistic machinery. In Appendix 10.A, we show some of many relations between electron density properties and phase relationships, and in Appendix 10.B, we summarize some relations between Patterson space and phase relationships. Patterson (1949) defined a second synthesis, known as the Patterson synthesis of the second kind. Even if theoretically interesting, it is of limited use in practice. We provide information on this in Appendix 10.C.