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.
σ2
R
, a reciprocal-space measure of the quality of macromolecular electron-density maps