Advantages of a Redefinition of Variable‐Space in Direct‐Space Structure Solution from Powder X‐Ray Diffraction Data

Abstract

Techniques for global optimization (which include Monte Carlo and simulated annealing methods, neural networks, genetic algorithms, other evolutionary procedures, and a wide range of other computational strategies) are currently being applied to yield fundamental understanding across diverse areas of the physical sciences, [1] including protein folding, [ 2a] structure prediction of clusters [ 2b-e] and crystals, [ 2f] structure determination from electron diffraction [ 2g] and powder X-ray diffraction [3] data (including applications in the case of nanostructures [ 2h] ), interpretation of spectroscopic data such as NMR spectroscopy [ 2i] and Mçssbauer spectroscopy, [ 2j] understanding protein-substrate interactions, [ 2k] and optimization of laser pulse-shapes for quantum control of chemical reactions, [ 2l, m] to name just a few. Although the details of the computational implementations of global optimization strategies necessarily differs between these different areas of application, there is nonetheless significant commonality in the approaches employed in different fields, and fundamental advances in one field of application are often directly transferable to other fields.Among these areas of application, structure determination of organic molecular solids from powder X-ray diffraction data is nowadays carried out widely, [3] with the recent upsurge of activity in this field coinciding with the development of the "direct-space strategy" for structure solution [4] in which the structure-solution process is transformed into a problem of global optimization. Thus, in the direct-space strategy, trial structures are generated independently of the experimental powder X-ray diffraction pattern, and the quality of each trial structure is assessed by direct comparison between the powder X-ray diffraction pattern calculated for the trial structure and the experimental powder X-ray diffraction pattern. This comparison is quantified using an appropriate figure-ofmerit (in our work, the weighted powder profile R-factor R wp ). Clearly, the aim of the global optimization problem in this case is to locate the trial structure that corresponds to optimal agreement (lowest R wp ) between calculated and experimental powder X-ray diffraction patterns, and is equivalent to exploring a hypersurface R wp (G) to find the global minimum, where G represents the set of variables that define the trial structures. In principle, any technique for global optimization may be used, and our own current work in this field is focused on the use of a genetic algorithm, [5] implemented in the program EAGER.[6] Conventionally, the structural variables in the set G comprise, for each molecule in the asymmetric unit, the position {x, y, z} and orientation {q, f, y} of the whole molecule, and a set of n variable torsion angles {t 1 , t 2 ,.., t n } to define the molecular conformation. An important feature underlying the success of the direct-space strategy is that it incorporates reliable prior knowledge of molecular geometry (i.e....

Crystal structures of 1,8‐bis(dimethylamino)naphthalene (DMAN) and dicarboxylic acids complexes determined from single‐crystal and powder diffraction data

“…In a previous study it was shown that, for torsionally flexible molecules it can be beneficial if a molecule is split into halves, due to torsion angle values being more difficult to determine correctly than positional and orientational coordinates of such fragments. Their results showed that, the increase in the dimensionality of the solution space, by an additional set of positional and orientational coordinates, can be outweighed by removing a torsional angle parameter or parameters.…”

Counteracting stagnation in genetic algorithm calculations by implementation of a micro genetic algorithm strategy

“…Alternative definitions of variable-space, particularly for molecules with considerable conformational flexibility, have been discussed recently. 16 An important feature underlying the success of the direct-space strategy for structure solution is that it incorporates reliable prior knowledge of molecular geometry (i.e. bond lengths, bond angles, geometries of rigid moieties such as aromatic rings, etc.)…”

“…In principle, any technique for global optimization may be used to explore the R wp (C) hypersurface, and our own current work in this field is focused on the use of a genetic algorithm (GA) 14 implemented in the program EAGER. 15 A potential problem in GA calculations is the occurrence of stagnation of the population. When stagnation occurs, a large number of members of the population become very similar to each other, and the resulting reduction of diversity within the population reduces the ability of the GA to search a wide region of the hypersurface and thus decreases the probability of rapidly locating the global minimum.…”

CMR 2005: 12.05: Use of in vivo imaging to follow siRNA‐mediated reporter gene inhibition in a transgenic mouse skin model

“…Five community‐wide blind tests63 of prediction software have been run. The related problem of determining solid‐state molecular structure from powder X‐ray diffraction data has also been treated using GAs and suitably pre‐defined64 molecular fragments.…”