Least-squares refinement crystal structures

Traditionally, least-squares methods have been used to refine protein crystal structures. In this method, a set of simultaneous equations is set up whose solutions correspond to a minimum of the R factor with respect to each of the atomic coordinates. Least-squares refinement requires an N x N matrix to be inverted, where N is the number of parameters. It is usually necessary to examine an evolving model visually every few cycles of the refinement to check that the structure looks reasonable. During visual examination it may be necessary to alter a model to give a better fit to the electron density and prevent the refinement falling into an incorrect local minimum. X-ray refinement is time consuming, requires substantial human involvement and is a skill which usually takes several years to acquire. 

For both structures, all final Si positions were obtained with reasonable accuracy (0.1 -0.2 A) by a 3D reconstruction of HRTEM images followed by a distance least-squares refinement. This kind of accuracy is sufficient for normal property analysis, such as catalysis, adsorption and separation, and as a starting point for structure refinement with X-ray powder diffraction data. The technique demonstrated here is general and can be applied not only to zeolites, but also to other complicated crystal structures. 

In crystal structure analyses of proteins, the presence of Ca ions is usually determined indirectly. Ions, being larger and more electron dense, are differentiated from water molecules based on their peak heights in electron density maps, and their high occupancies and low thermal motion parameters during least-squares refinement of the... 

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. 

The measurable range is greatly extended (6500 1 for the KBr-amylose data) and errors can be accounted for more accurately in the process of obtaining intensity data. These errors may even be used in weighting of least-squares refinement calculations as is presently done in single crystal structure studies. 

The crystal structure of chrysene has been accurately analysed by Bums and Iball (1960). The coordinates of the carbon and hydrogen atoms and the anisotropic thermal vibrations of the carbon atoms were determined by three-dimensional least-squares refinement, as well as by three-dimensional Fourier and difference Fourier syntheses, a total of 1037 planes being employed. The value of the residual, R, is 0-076. 

Two advanced techniques have been proposed and applied to some crystal structures (Section IV,C), in which aspherical distributions of valence electrons around an atom are directly taken into account in the least-squares calculations. Aspherical atomic form factors are introduced in the least-squares refinement in the first method (29, 38, 80) and multipole parameters describing the aspherical valence distributions are used in the second method (31, 34, 46). 

Magnetic anisotropies xlz (l/3)Tr/ for R = Ce-Yb except Pm, Gd (0.002 < AFj < 0.06, table 9) have been computed with eq. (58) and using five contact contributions Sfj (i = H9, H11-H14) and the geometrical G factors obtained from the crystal structures of (HHH)-[/ Co(L5)3]6+ (R = La, Lu). A qualitative good agreement (AF = 0.23) is obtained between the experimental magnetic anisotropies (scaled to -100 for Dy(III) and corrected for the variation of the crystal-field parameter near the middle of the series (vide supra), table 9) and Bleaney s factors (table 3). Further non-linear least-squares refinements of the molecular... 

Analyses of electron density distributions have enabled the positions of major elements of high atomic weights such as iron and other transition elements to be located relative to lighter elements such as magnesium and aluminium in mineral crystal structures. The widespread availability of automated X-ray dif-fractometry and least squares refinement programs have increased the availability of site occupancy data for transition metal ions in most contemporary crystal structure refinements. 

The estimated precision in bond lengths obtained by a least-square refinement of a data set measured by X-ray diffraction can be 0.003 A ( 0.3 pm), for a structure with unweighted R-factor less than 3%. If the data set is collected at low temperatures (20 K or 80 K), the decrease in thermal vibration can yield even better bond distances and angles. For H atom coordinates, the precision is one or two orders of magnitude lower, since the electron density around an H atom is relatively low in these cases a neutron diffraction study (which requires very large crystals) can yield better H atom positions. 

In the X-ray analysis of a protein crystal structure, solvent molecules appear as spheres of electron density in difference Fourier maps calculated at the end of a refinement. In a strict sense, the electron density map exhibits preferred.s/tes of hydration which are occupied by freely interchanging solvent molecules. This electron density is well defined for the tightly bound solvent molecules and can be as spurious as just above background for ill-defined molecules which exhibit large temperature factors and/or only partly occupied atomic positions. Since these two parameters are correlated in least-squares refinement, this gives rise to methodological problems. 

The electron density distribution for solvent molecules can be improved if the contribution from bulk water to the X-ray scattering is included in the model. This affects the low-angle j X-ray intensity data which are omitted in early stages of the least-squares refinement of protein crystal structures. If they are included in refinement and properly accounted for, the signal-to-noise ratio in the electron density maps is significantly improved and the interpretation of solvent sites is less ambiguous. 

Crystals of the compound of empirical formula FiiPtXe are orthorhombic with unit-cell dimensions a = 8-16, h = 16-81. c = 5-73 K, V = 785-4 A . The unit cell volume is consistent with Z = 4, since with 44 fluorine atoms in the unit cell the volume per fluorine atom has its usual value of 18 A. Successful refinement of the structure is proceeding in space group Pmnb (No. 62). Three-dimensional intensity data were collected with Mo-radiation on a G.E. spectrogoniometer equipped with a scintillation counter. For the subsequent structure analysis 565 observed reflexions were used. The platinum and xenon positions were determined from a three-dimensional Patterson map, and the fluorine atom positions from subsequent electron-density maps. Block diagonal least-squares refinement has led to an f -value of 0-15. Further refinements which take account of imaginary terms in the anomalous dispersion corrections are in progress. 

Structure Refinement. Initially, the structure was solved using the data from crystal no. 4 only (20 < 40 ). This set of data yielded an averaged set of 633 unique reflections, of which529 satisfying the condition I > 3o(/) were used in the least-squares refinement. 

In the pharmaceutical community, quantitative analyses has conventionally been based on the intensity of a characteristic peak of the analyte. It is now recognized that phase quantification will be more accurate if it is based on the entire powder pattern.This forms the basis for the whole-powder-pattern analyses method developed in the last few decades. Of the available methods, the Rietveld method is deemed the most powerful since it is based on structural parameters. This is a whole-pattern fitting least-squares refinement technique that has also been extensively used for crystal structure refinement and to determine the size and strain of crystallites. 

It is important for the reader to understand that in a least-squares refinement of a crystal structure it is the shifts in parameters that are calculated in order to improve the structure, not the parameters themselves. The preliminary parameters that are shifted to more appropriate values come from the trial structures (see Chapters 8 and 9). 

The least-squares method is a very powerful tool, provided the model is sufficiently close to the true structure. If the initial model is basically correct, the shifts in parameters indicated by the least-squares refinement will drive the energy of the structure to a global minimum. Unfortunately, if the model is not quite correct, the least-squares refinement will produce a structure that is trapped in a local minimum of energy which is not the true structure (see Figure 10.12). This problem manifests itself by monitoring the R value. Often there is a hint of trouble in that the R value is higher than expected and will not decrease to acceptable values. Several crystal structures have been reported with this type of problem, but they are generally corrected in the subsequent literature. 

Correlation between parameters A correlation is a measure of the extent to which two mathematical variables are dependent on each other. In the least-squares refinement of a crystal structure, parameters related by symmetry are completely correlated, and temperature factors and occupancy factors are often highly correlated. 

Atoms in crystals seldom have isotropic environments, and a better approximation (but still an approximation) is to describe the atomic motion in terms of an ellipsoid, with larger amplitudes of vibration in some directions than in others. Six parameters, the anisotropic vibration or displacement parameters, are introduced for each atom. Three of these parameters per atom give the orientations of the principal axes of the ellipsoid with respect to the unit cell axes. One of these principal axes is the direction of maximum displacement and the other two are perpendicular to this and also to each other. The other three parameters per atom represent the amounts of displacement along these three ellipsoidal axes. Some equations used to express anisotropic displacement parameters, which may be reported as 71, Uij, or jdjj, axe listed in Table 13.1. Most crystal structure determinations of all but the largest molecules include anisotropic temperature parameters for all atoms, except hydrogen, in the least-squares refinement. Usually, for brevity, the equivalent isotropic displacement factor Ueq, is published. This is expressed as ... 

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