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This chapter provides an overview of the methods of structure solution of incommensurately modulated crystals and composite crystals. Assuming the periodic average structure is known, it is shown that modulation functions can often be determined by trial and error, employing structure refinements starting with randomly chosen but small values for the structural parameters. The presentation of systematic methods of structure determination includes Patterson function methods, direct methods, and the method of charge flipping.

The principles of how to solve crystal structures from electron diffraction (ED) data are described. Recording and quantification of ED data is not trivial, considering the sharp diffraction spots that easily saturate the detector. The importance of using thin crystals is stressed. The phase problem in diffraction is presented and how it can be solved by various techniques, such as direct methods using triple relations, the Patterson function, charge flipping and the strong-reflections approach. Origin specification and semi-invariants, normalized structure factors and the Wilson plot are all described in detail.

Direct methods procedures (see Chapter 6) or Patterson techniques (see Chapter 10), primarily the former, have been methods of choice for crystal structure solution of small- to medium-sized molecules from diffraction data. Over the last 30 years, several new phasing algorithms have been proposed, not requiring the use of triplet and quartet invariants, but based only on the properties of Fourier transforms. These were not competitive with direct methods and have never became popular, but they contain a nucleus for further advances. Among these we mention: (i) Bhat (1990) proposed a Metropolis technique (Metropolis et al., 1953; Kirkpatrick et al., 1983; Press et al., 1992), also known as simulated annealing (the reader is referred to Section 12.9 for details on the algorithm). From a random set of phases, an electron density map is calculated, modified, and inverted. The corresponding phases are altered according to the simulated annealing algorithm, and then used to calculate a new electron density map. The procedure is cyclic. (ii) A strictly related simulated annealing procedure has been proposed by Su (1995). The objective function to minimize was . . . R = ∑h (S|Fh|calc − |Fh|obs)2, . . . where S is the scale factor. The scheme is as follows: random atomic positions are generated and in succession shifted; the simulated annealing algorithm is applied to accept or reject atomic shifts. At the end, a new atomic structure is generated, whose positions are shifted in succession, and so on in a cyclic way. (iii) The forced coalescence method (FCP) was proposed by Drendel et al. (1995). Hybrid electron density maps (see Section 7.3.4) were actively used with different values of τ and ω. Even if never popular, the above algorithms opened the way to two other methods which are much more efficient, charge flipping and VLD (vive la difference), to which this chapter is dedicated. Both are based on the properties of the Fourier transform; they do not require the explicit use of structure invariants and seminvariants, or a deep knowledge of their properties. The reader should not, however, conclude that the invariance and seminvariance concepts are not necessary in the handling of these approaches, on the contrary, understanding these basic concepts is essential to the appreciation of these new methods.