Least-squares refinement method

Data analysis procedures have developed substantially over the last few years. In particular, use of least square refinement methods have been developed. Recent progress with theoretical development for the treatment of multiple scattering has resulted in Ugand group refinement such as an imidazole. We can expect further development in this area which ought to lead us to restrained least square refinement procedures for EXAFS data analysis. This type of restrained refinement is commonly used for macromolecular crystallographic structure determination where a similar problem of imderdeterminancy exists... 

Dickerson, R. E., Weinzierl, J. E., and Palmer, R. A. 1968. A least-squares refinement method for isomorphous replacement. Acta Cryst. B24 997. 

The vast majority of crystal structures determined today satisfy stringent quality standards. Nevertheless, it should be stressed that the nominal precision of the resulting atomic positions and derived geometric parameters, estimated by least-squares refinement methods, is quite unrealistic. The stan-... 

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. 

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. 

The problems surrounding Hamilton s test (vide supra), as well as some misconceptions and misuses encountered in the literature, led Rogers to propose an alternative and more reliable method for determining the absolute structure58. A factor >7 is introduced which multiplies the imaginary component Aff of the anomalous scattering terms of the atomic scattering factors of all atoms (equation 11, which replaces equation 9, see Section 4.2.2,1.1), and which is treated as a variable in the least-squares refinement. 

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 intense absorption of water over most of the infrared spectrum restricts the regions where aqueous solutions of carbohydrates can be usefully studied. Absorbance subtraction makes it possible to eliminate water absorbance and magnify the remaining spectral features to the limit of the signal-to-noise ratio. Many other data-processing techniques, such as the ratio method,4 the least-squares refinement,5 and factor analysis,6 should be of benefit in the study of carbohydrate mixtures. 

Quantitative measurements are carried out by use of a single frequency in the conventional i.r. spectrum, but, with digitized spectra and a computer, the entire frequency spectrum of each component in a mixture can be fitted by curve-fitting techniques using such methods as least-squares refinements,5 which can yield an indication as to the precision of the fit. 

The model can be improved in another way by least-squares refinement of the atomic coordinates. This method entails adjusting the atomic coordinates to improve the agreement between amplitudes calculated from the current model and the original measured amplitudes in the native data set. In the latter stages of structure determination, the crystallographer alternates between map interpretation and least-squares refinement. 

As has been commented upon in other papers in this symposium, methods for measurement of intensities in fiber diagrams have received little attention in the last 15 years, during which time considerable progress has been made in handling the atomic coordinates for least squares refinement. Most intensity data has been obtained from linear densitometer traces through the x-ray reflections. We used these procedures for our work on cellulose I and 6-chitin. For cellulose I, (1 ) the area under the peak on a radial scan was determined after subtraction of an estimated Current addresses ... 

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). 

Toraya s Method. The WPPD as implemented by Toraya et al.11 decomposes the peak profiles and background functions to obtain the best fit to the experimental powder pattern of the individual pure phase data by least-squares refinement. The integrated intensities of the pure phases are then stored with the other refined parameters, such as the profile parameters and the unit-cell parameters of the phases to be quantified. During quantification step, the integrated intensity of the phase being quantified is scaled, as defined in Eq. (12.4), such that the total of the scale factors for the component phases sum to unity. The scale factors of the individual components are then refined by least-squares methods until a best fit is observed with respect to the pattern of unknown composition. 

Alternative and much more elegant methods are those using aspherical pseudoatoms least squares refinements. These refinements permit access to the positional and thermal variables of the atoms as well as to the electron density parameters. Several pseudoatoms models of similar quality exist (9-12) and are compared in reference [7J],... 

It should be stressed that, in any experiment for which there is to be a least-squares refinement of parameters, intended to yield not only the best possible parameter values but also respectable estimates of their uncertainties and a test of the validity of the model, it is vital to take the trouble to analyze the methods and the circumstances of the experiment carefully in order to get the best possible values of the a priori weights. 

The structure of a-chitin was refined by using automatic, rigid, subunit, least-squares refinement and the difference-Fourier method. Two distinct types of statistical modification could be present in the structure, both of which would allow complete, intersheet hydrogenbonding between OH-6 groups within the general framework of Carlstrom s structure.56 The R factor is 22%. 

The structure was solved by heavy-atom methods at the U.C. Berkeley CHEXRaY facility using full-matrix least-squares refinement procedures detailed elsewhere. Systematically absent reflections were eliminated from the data set, and those remaining were corrected for absorption by means of the calculated absorption coefficient. A three-dimensional Patterson synthesis gave peaks that were consistent with Xe atoms in Wyckoff position 4c and Ge atoms in 4a in space group Pnmb (see Pnma, No. 62). Three cycles of... 

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