Refinement least squares

In general, the least-squares refinement procedure works well in simations in which we can make reasonable guesses at starting values for the parameter set. It does not work so well when we have many unknown variables, e.g. low-quality powder diffraction data that have been collected under non-standard conditions of temperature and/or pressure. Then the process becomes a lot messier, and it may be possible to fit multiple structures to the same data set with equal levels of confidence. In cases like these, it is important to have additional information from other techniques, with computational methods playing an increasingly important role. We continue our discussions on least-squares refinement in Sections 3.9, 10.3.2 and 10.7.6. 

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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. 

The reliability factor B was 0276 after the first refinement and 0-211 after the fourth refinement. The parameters from the third and fourth refinements differed very little from one another. The final values are given in Table 1. As large systematic errors were introduced in the refinement process by the unavoidable use of very poor atomic form factors, the probable errors in the parameters as obtained in the refinement were considered to be of questionable significance. For this reason they are not given in the table. The average error was, however, estimated to be 0-001 for the positional parameters and 5% for the compositional parameters. The scattering power of the two atoms of type A was given by the least-squares refinement as only 0-8 times that of aluminum (the fraction... 

The structure was refined with block diagonal least squares. In cases of pseudo-symmetry, least squares refinement is usually troublesome due to the high correlations between atoms related by false symmetry operations. Because of the poor quality of the data, only those reflections not suffering from the effects of decomposition were used in the refinement. With all non-hydrogen atoms refined with isotropic thermal parameters and hydrogen atoms included at fixed positions, the final R and R values were 0.142 and 0.190, respectively. Refinement with anisotropic thermal parameters resulted in slightly more attractive R values, but the much lower data to parameter ratio did not justify it. 

The difference electron density map following the last cycle of least squares refinement did not show evidence for a simple disorder model to explain the anomalously high B for the hydroxyl oxygen. Attempts to refine residual peaks with partial oxygen occupancies did not significantly improve the agreement index. 

The most effective spectrophotometric procedures for pKa determination are based on the processing of whole absorption curves over a broad range of wavelengths, with data collected over a suitable range of pH. Most of the approaches are based on mass balance equations incorporating absorbance data (of solutions adjusted to various pH values) as dependent variables and equilibrium constants as parameters, refined by nonlinear least-squares refinement, using Gauss-Newton, Marquardt, or Simplex procedures [120-126,226],... 

Lingane, P. J. Hugus, Z. Z. Jr., Normal equations forthe gaussian least-squares refinement of formation constants with simultaneous adjustment of the spectra of the absorbing species, Inorg. Chem. 9, 757-762 (1970). 

A preliminary least-squares refinement with the conventional, spherical-atom model indicated no disorder in the low-temperature structure, unlike what had been observed in a previous room-temperature study [4], which showed disorder in the butylic chain at Cl. The intensities were then analysed with various multipole models [12], using the VALRAY [13] set of programs, modified to allow the treatment of a structure as large as LR-B/081 the original maximum number of atoms and variables have been increased from 50 to 70 and from 349 to 1200, respectively. The final multipole model adopted to analyse the X-ray diffraction data is described here. 

The weighted residuals plots in Fig. 3.4 are the results of the analyses, where the shape of the reference spectra are matched to those of acceptor and donor spectra by least-squares refinement. Poor shape-analysis leads to high weighted residuals, which can reveal impurities, decomposition, or other artifacts. In the present cases, no difficulties were encountered. 

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. 

Least squares refinement of coordinates together with anisotropic temperature factors, in the final stages, gave an R factor of 0.11. 

Table 5.12. The first step of non-linear least square refinement of the parameters, ft, and e for selected depths in the water column. The last three columns represent the elements of the matrix...

For the 2-cyanoguanidine molecule6 the static deformation density has been mapped by least-square refinement against low-temperature X-ray data in order to explain the fact that the C—N bonds around the C atom are almost identical and the fact that a large negative charge (—0.2 e) is on the N(3) atom. Hence one must take all the resonance forms (2) into consideration. 

Mathematically, all this iirformation is used to calculate the best fit of the model to the experimental data. Two techniques are currently used, least squares and maximum hkelihood. Least-squares refinement is the same mathematical approach that is used to fit the best line through a number of points, so that the sum of the squares of the deviations from the line is at a minimum. Maximum likelihood is a more general approach that is the more common approach currently used. This method is based on the probability function that a certain model is correct for a given set of observations. This is done for each reflection, and the probabilities are then combined into a joint probability for the entire set of reflections. Both these approaches are performed over a number of cycles until the changes in the parameters become small. The refinement has then converged to a final set of parameters. 

The modeling of electron diffraction by the pattern decomposition method, for which no structural information is required, can be successfully applied for extraction of the diffraction information from the pattern. Several parameters can be refined during the procedure of decomposition, including the tilt angle of the specimen the unit cell parameters peak-shape parameters intensities. The procedure consists of fitting, usually with a least-squares refinement, a calculated model to the whole observed diffraction pattern. 

We have developed a software package MSLS [1], in which multi-slice calculation software is combined with least squares refinement software used in X-ray crystallography. With multi-slice calculations which are standardly used for image calculations of HREM images, dynamic diffraction is taken into account explicitly. 

Recent microwave data for the potential interstellar molecule Sis is used together with high-level coupled-cluster calculations to extract an accurate equilibrium structure. Observed rotational constants for several isotopomers have been corrected for effects of vibration-rotation interaction subsequent least-squares refinements of structural parameters provide the equilibrium structure. This combined experimental-theoretical approach yields the following parameters for this C2v molecule re(SiSi) = 2.173 0.002A and 0e(SiSiSi) = 78.1 O.2 ... 

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