Patterson-function direct methods for structure determination of organic compounds from powder diffraction data. XVI.

A new type of direct methods (DM) called Patterson-function DM are presented that directly explore the Patterson instead of the modulus function. Since they work with the experimental intensities, they are particularly well suited for handling powder diffraction data. These methods are based on the maximization of the sum function S(P) ∝ ∑H(I(H)-<I>)G(-H)(Φ) in terms of the Φ phases of the structure factors. The quantity accessible from the experiment is I(H), the equidistributed multiplet intensity of reflection H, and <I> is the average intensity taken over all non-systematically absent reflections. G(-H)(Φ) is the calculated structure-factor amplitude of the squared structure that includes the positivity and the atomicity of the density function in its definition. The S(P) sum function can be optimized with the Patterson-function tangent formula (TF) using a variant of the S-FFT algorithm [Rius et al. (2007), Acta Cryst. A63, 131-134]. It is important that overlapped reflections also participate in the phase refinement, so that not only the resolved reflections but the whole pattern contribute decisively to the refinement. The increase in effective data resolution minimizes Fourier series termination effects and improves the accuracy of G(Φ). The Patterson-function TF has been applied to synchrotron powder data of various organic compounds. In all cases the molecules were easily identified in the respective Fourier maps. By way of illustration the method is applied to synchrotron powder data of a dimer formed by 30 symmetry-independent non-H atoms. Since single-crystal data may be regarded as overlap-free powder data, it is clear that Patterson-function DM can cope with powder and single-crystal data.