Squares Refinement of the Structure

Preliminary three-dimensional atomic coordinates of atoms in crystal structures are usually derived from electron-density maps by fitting atoms to individual peaks in the map. The chemically reasonable arrangement of atoms so obtained is, however, not very precise. The observed structure amplitudes and their relative phase angles, needed to calculate the electron-density map, each contain errors and these may cause a misinterpretation of the computed electron-density map. Even with the best electron-density maps, the precisions of the atomic coordinates of a preliminary structure are likely to be no better than several hundredths of an A. In order to understand the chemistry one needs to know the atomic positions more precisely so that better values of bond lengths and bond angles will be available. The process of obtaining atomic parameters that are more precise than those obtained from an initial model, referred to as refinement of the crystal structure, is an essential part of any crystal structure analysis. 

Changes in parameters are found that increase the agreement between measured data and those calculated from the revised model that results from these changes. The positions x, y, and z and the atomic displacement (thermal) parameters B derived for each atom in a preliminary crystal structure are adjusted so as to improve the agreement between the observed structure amplitudes, F hkl) o, and those calculated from the determined model F hkl) c. The progress in improving this agreement is usually monitored by a residual index known as the R value, defined as  

The R value is calculated as a measure (although, as we shall see, not a rigorous measure) of the precision of the results of the refinement. When 

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Full-matrix least-squares refinement of the structure model was carried out with programs orfls (9). Since this program in the form we used refines structure factors Fhki rather than the intensities of the powder lines, it was necessary to decompose the intensity of each line having more than one component into contributions from the individual component reflections. This was done by assuming that the F°mz 2 for the several components of a powder line were in the same ratios as the corresponding FcftfcZ 2 obtained from a structure model—the original trial structure or the previous... 

Borkakoti N, Moss DS, Palmer RA (1982) Ribonuclease A least-squares refinement of the structure at 1.45 A resolution. Acta Cryst B38 2210-2217... 

Guss, J.M., Bartunik, H.D., Freeman, H.C. Accuracy and precision in protein structure analysis Restrained least-squares refinement of the structure of poplar plastocyanin at 1.33 A resolution. Acta Crystallogr. B 1992, 48, 790-811. 

During a perfectly straightforward least squares refinement of the structure of Cs H6 one of the hydrogen atoms moved from its initial position to somewhere near the center of the ring.52 The authors, not unnaturally, considered this ridiculous and decided to return it forcibly to its logical position and to remove it from the refinement. No error was thus introduced into the literature. 

Weidenbomer JE (1961) Least squares refinement of the structure of gadolinium-iron garnet, Gd3Fe2Fe30i2. Acta Crystallogr 14 1051-1056... 

Reflection intensity in the SAED negatives was measured with a microdensitometer. The refinement of the structure analysis was performed by the least square method over the intensity data (25 reflections) thus obtained. A PPX single-crystal is a mosaic crystal which gives an "N-pattem". Therefore we used the 1/d hko as the Lorentz correction factor [28], where d hko is the (hkO) spacing of the crystal. In this case, the reliability factor R was 31%, and the isotropic temperature factor B was 0.076nm. The molecular conformation of the P-form took after that of the P-form since R was minimized with this conformation benzene rings are perpendicular to the trans-zigzag plane of -CH2-CH2-. 

When the model used for Fcalc is that obtained by least-squares refinement of the observed structure factors, and the phases of Fca,c are assigned to the observations, the map obtained with Eq. (5.9) is referred to as a residual density map. The residual density is a much-used tool in structure analysis. Its features are a measure for the shortcomings of the least-squares minimization, and the functions which constitute the least-squares model for the scattering density. 

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... 

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. 

The unit cell dimensions determined by least squares refinement of the observed and calculated d-spacings are given in Table I for the four structures considered below. 

The structure of the closely related molecule, 1,2-cyclopentenophen-anthrene, has been determined and refined with partial three-dimensional data by least-squares methods by Entwhistle and Iball (1961). Independent confirmation of the correctness of this structure has been provided by Basak and Basak (1959) who did not, however, carry out any refinement of the structure. Entwhistle and Iball s results show that the molecule is not planar the deviations of the carbon atoms from the mean molecular plane are shown in Fig. 9 (the standard deviations of the atomic coordinates lie between 0-009 and 0-015 A). The three aromatic rings appear to be linked in a slightly twisted arrangement. Atoms H and K, which are bonded to the overcrowded hydrogen atoms, are displaced almost the same distance on opposite sides of the mean plane. In the five-membered ring, atoms C and E are below the molecular plane by about 0-10 A while atom D lies 0-18 A... 

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... 

When bond lengths with different chemical sorrounding are compared to each other, the experimental error limits for the individual values are of great importance. It has become common practice in ED studies to report estimated uncertainties, which usually are 2 to 3 times the standard deviation of the least-squares refinement. It is assumed that such uncertainties account for possible systematic errors. Most MW studies also report estimated uncertainties, which may be considerably larger than the standard deviations derived from fitting the rotational constants. In X-ray crystallography it is common use to cite standard deviations from the least-squares analysis of the structure factors. 

Baker EN, Dodson EN (1980) Crystallographic refinement of the structure of actinidin at 1.7 A resolution by fast-Fourier least-squares methods. Acta Cryst A 36 559-572... 

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. 

Other well-established methods of analysis include modeling the experimental data using geometrical solid-body representations of the scattering species. This method allows for the construction of many kinds of geometrical models (e.g., hollow shells, core-shell particles, lamellar structures, etc. and provides for the interactive least-squares refinement of the dimensions of the models to fit the data. This approach has been widely used to analyze SANS data obtained from colloidal and polymeric systems. ... 

Disorder in a crystal structure is frequently revealed by the shapes of the thermal ellipsoids obtained from the least-squares refinement of the anisotropic displacement parameters. An example is provided by the crystal structure determination of potassium dihydrogen isocitrate. One carboxyl oxygen atom is very anisotropic as a result of two possible hydrogen bonding schemes in which it can take part (Figure 13.10). 

The set of anisotropic displacement parameters, obtained from the least-squares refinement of the crystal structure (as described by Chapter 10) can be analyzed to obtain T, L and S. It has been assumed that there is no correlation between the motion of different atoms. Values of Uij are analyzed (again by an additional least-squares analysis) in such a way that good agreement is obtained between the refined values and those predicted when constants have been obtained for the T, L, and S tensors. The total number of anisotropic displacement parameters (6 per atom) is the input, and a total of 12 parameters for a centrosymmetric structure, or 20 parameters for a noncentrosymmetric structure, is the output of this least-squares analysis. The results consist of the molecular translational (T), librational (L), and screw (S) tensors. This treatment leads to estimates of corrections that should be made to bond distances. On the other hand, this type of analysis cannot be used for intermolec-ular distances because the correlation between the motion of different molecules is not known. 

Figure 7.39. Relative shifts of individual atoms (left-hand scale) and average shift (A) to standard deviation (cr) ratio during the first 12 cycles of the least squares refinement of the coordinates of all atoms in the model of the crystal structure of the anhydrous monoclinic FeP04. Both the P-0 distances and O-P-0 angles were soft restrained with the weight of 4 during the first five cycles. The weight was set to 10 beginning from cycle No. 6. The first five cycles indicate erratic shifts of P and O atoms. Beginning from the sixth cycle, the shifts of all atoms steadily decrease and approach zero after cycle No. 12.

Single-crystal x-ray studies of the contact-ion pair complexes shown in Table II have been completed. The data were measured with a Picker automated diffractometer, and the structures were solved by direct methods. Hydrogen atom positions were included in all the structures but usually not refined. Refinements of the structures were made using a full-matrix, least-squares technique with neutral-atom scattering factors. 

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