Least squares procedures density maps

After the crystal structure of the compound has been solved, or deduced, from the X-ray data, the initial parameters (atomic positions, bond lengths, and bond angles) are only approximate and have to be improved. The usual method employed is that of least-squares refinement, although electron-density difference-maps and trial-and-error procedures are also used. Electron-density difference-maps give the approximate difference between the actual structure and the trial structure. 

Often the least-squares refinement is combined with an EDM (electron density modification) procedure the advantage is that it is not necessary to interpret electron density peaks in terms of atomic species. In practice the map is modified by a suitable function g to obtain a better representation p of the structure ... 

The ease with which a molecular model can be fitted to the observed density depends on the resolution at which the map is calculated (compare figures 2.1(c), 2.1 (d), 10.4 and 10.5) and its quality. The resolution limit of the calculation is set by the phase determination method. For the method of isomorphous replacement, phasing is successful usually to =2.5-3.0 A. In the case where a related structure is already known the method of molecular replacement (Rossmann (1972)) can be used whereby rotation and translation matrices are determined and then calculated phases used. Clearly, these two procedures are both approximate methods. The model is usually improved by using least-squares methods of refinement (for a collection of papers see Machin, Campbell and Elder (1980)) and higher resolution data (better than 2 A or so). Refinement methods involve the determination of shifts to the atomic parameters (coordinates and thermal parameters) so as to agree better with the observed diffraction data whilst preserving the known stereochemical features of proteins and nucleic acids. This is achieved by minimising a composite observational function ... 

The experimental diffraction data were analyzed by a combined technique involving Rietveld analysis, the maximum entropy method (MEM), and MEM-based pattern fitting (MPF) [10-15]. Rietveld analysis, which is used to refine the crystal structure from the powder diffraction data by a least squares method, was carried out using the RIETAN-2000 program [27], which yields structure factors and their errors after structural refinement. It is known that MEM can be used to obtain a nuclear density distribution map based on neutron structure factors and their errors [5, 6, 8, 10-15, 26-29] any type of complicated nuclear density distribution is allowed so long as it satisfies the symmetry requirements. MEM calculations were carried out using the PRIMA program [29]. To reduce the bias imposed by the simple structural model in the Rietveld refinement, an iterative procedure known as the REMEDY cycle [29] was applied after MEM analysis (Fig. 6.3). In this procedure, structure factors... 

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