Structure factor observed

The first was not the structure of brookite. The second, however, had the same space-group symmetry as brookite (Ft,6), and the predicted dimensions of the unit of structure agreed within 0.5% with those observed. Structure factors calculated for over fifty forms with the use of the predicted values of the nine parameters determining the atomic arrangement accounted satisfactorily for the observed intensities of reflections on rotation photographs. This extensive agreement is so striking as to permit the structure proposed for brookite (shown in Fig. 3) to be accepted with confidence. 

There are therefore four adjustable parameters per atom in the refinement (xy, yy, Zj, By). In the computer experiments we have carried out to test the assumptions of the nucleic acid refinement model we have generated sets of observed structure factors F (Q), from the Z-DNA molecular dynamics trajectories. The thermal averaging implicit in Equation III.3 is accomplished by averaging the atomic structure factors obtained from coordinate sets sampled along the molecular dynamics trajectories at each temperature ... 

A preliminary structural model of a protein is arrived at using one of the methods described above. Calculated structure factors based on the model generally are in poor agreement with the observed structure factors. The agreement is represented by an R-factor defined as found in equation 3.9 where k is a scale factor ... 

Refinement takes place by adjusting the model to find closer agreement between the calculated and observed structure factors. For proteins the refinements can yield R-factors in the range of 10-20%. An example taken from reference 10 is instructive. In a refinement of a papain crystal at 1.65-A resolution, 25,000 independent X-ray reflections were measured. Parameters to be refined were the positional parameters (x, y, and z) and one isotropic temperature factor parameter... 

We have collected a set of observed structure factors - hL for which we have only amplitude information. Denote the required phase angle for... 

Refinement takes place by adjusting the model to find closer agreement between the calculated and observed structure factors. For proteins the... 

Data processing occurs in two stages. Initially, diffraction images are reduced to a tabulation of reflection indices and intensities or, after truncation, structure factors. The second stage involves conversion of the observed structure factors into an experimental electron density map. The choice of how to execute the latter step depends on the method used to determine the phase for each reflection. The software packages enumerated above generally focus on exploitation of anomalous signals to overcome the phase problem for a protein of unknown structure. 

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. 

When observed structure factors are used, the thermally averaged deformation density, often labeled the dynamic deformation density, is obtained. An attractive alternative is to replace the observed structure factors in Eq. (5.8) by those calculated with the multipole model. The resulting dynamic model deformation map is model dependent, but any noise not fitted by the muitipole functions will be eliminated. It is also possible to plot the model density directly using the model functions and the experimental charge density parameters. In that case, thermal motion can be eliminated (subject to the approximations of the thermal motion formalism ), and an image of the static model deformation density is obtained, as discussed further in section 5.2.4. 

The results show only a modest variation when the van der Waals radii are changed within reasonable bounds (Table 6.2). As the data were not refined with the aspherical atom formalism, the scale of the observed structure factors may be biased, an effect estimated on the basis of other studies (Stevens and Coppens 1975) to correspond to a maximal lowering of the scale by 2%. Values corrected for this effect are listed in the last two columns of Table 6.2. Since neutral TTF and TCNQ have, respectively, 72 and 52 valence electrons, the results imply a charge transfer close to 0.60 e. 

An X-ray atomic orbital (XAO) [77] method has also been adopted to refine electronic states directly. The method is applicable mainly to analyse the electron-density distribution in ionic solids of transition or rare earth metals, given that it is based on an atomic orbital assumption, neglecting molecular orbitals. The expansion coefficients of each atomic orbital are calculated with a perturbation theory and the coefficients of each orbital are refined to fit the observed structure factors keeping the orthonormal relationships among them. This model is somewhat similar to the valence orbital model (VOM), earlier introduced by Figgis et al. [78] to study transition metal complexes, within the Ligand field theory approach. The VOM could be applied in such complexes, within the assumption that the metal and the... 

The crystal is placed in an X-ray diffraction apparatus (camera or diffractometer) where the X-ray pattern is recorded photographically or by measuring the intensity of the X-rays electronically. The resulting intensity values are used to obtain the observed structure factors which constitute the fundamental experimental data from which the crystal and molecular structures are derived. The structure derived is used for calculating structure factors that are compared with the experimental structure factors during the period when the derived structure is being modified to fit the experimental data. 

By contrast to the /m3m refinement, the /43m refinement resulted in virtually no outstandingly bad disagreements between observed structure factors (obtained as defined above) and calculated ones. The powder line h2 + k2 + l2 = 34 is illustrative of the improvement of individual comparisons. 

In words, for each reflection, we compute the difference between the observed structure-factor amplitude from the native data set IFobsl and the calculated amplitude from the model in its current trial location IFcalcl and take the absolute value, giving the magnitude of the difference. We add these magnitudes for all reflections. Then we divide by the sum of the observed structure-factor amplitudes (the reflection intensities). 

Translation Search. A translation search was done by using the P2 probe molecule oriented by the rotation function studies and refined by the Patterson correlation method. The translation search employed the standard linear correlation coefficient between the normalized observed structure factors and the normalized calculated structure factors (Funinaga Read, 1987 Brunger, 1990). X-ray diffraction data from 10-3 A resolution were used. Search was made in the range x = 0-0.5, y = 0—0.5, and z = 0-0.5, with the sampling interval 0.0125 of the unit cell length. 

The earliest attempts at model analysis of polysaccharides -typified by the x-ray crystal structure analysis of amylose triacetate - were usually conducted in three steps ( L). In the first step, a model of the chain was established which was in agreement with the fiber repeat and the lattice geometry, as obtained from diffraction data. In the second step, the invariant chain model was packed into the unit cell, subject to constraints imposed by nonbonded contacts. This was followed, in the third step, by efforts to reconcile calculated and observed structure factor amplitudes. It was quickly realized that helical models of polysaccharide chains could be easily generated and varied using the virtual bond method. Figure 1 illustrates the generation of a two-fold helical model of a (l- U)-linked polysaccharide chain. 

R-factors" Calculated between Pairs of Observed Structure Factor Sets Determined by Different Investigators... 

We derive here the governing equations necessary to describe the structure factor, S (q, t), for a complex liquid mixture subject to flow. Since this observable is the Fourier transformation of the spatial correlation of concentration functions, it is first required to develop an equation of motion for 5c (r, t). The approach described here employs a modified Cahn-Hilliard equation and is described in greater detail in the book by Goldenfeld [91]. To describe the physical system, an order parameter, q/ (r, ), is introduced. In a complex mixture, this parameter would simply be /(r, 0 = c(r, r)-(c), where c(r, t) is the local concentration of one of the constituents and mean concentration. The order parameter has the property of being zero in a disordered, or on phase region, and non-zero in the ordered or two-phase region. The observed structure factor, which is the object of this calculation, is simply... 

In 1995, an elaborated method was developed for accurate structure analysis using X-ray powder diffraction data, that is, the MEM/Rietveld method [1,9]. The method enables us to construct the fine structural model up to charge density level, and is a self-consistent analysis with MEM charge density reconstruction of powder diffraction data. It also includes the Rietveld powder pattern fitting based on the model derived from the MEM charge density. To start the methods, it is necessary to have a primitive (or preliminary) structural model. The Rietveld method using this primitive structural model is called the pre-Rietveld analysis. It is well known that the MEM can provide useful information purely from observed structure factor data beyond a presumed crystal structure model used in the pre-Rietveld analysis. The flow chart of the method is shown in Fig. 2. 

The process of observed intensity estimation for overlapping reflections in the Rietveld analysis is shown schematically in Fig. 3. Suppose we have partially overlapped 1- and 2-reflections as shown in Fig. 3a. Observed structure factors were evaluated by dividing the observed intensities at each data point, Yobs(d)y into the Tlobs(0) and the y2obs(0) based on the calculated contributions of individual reflections, Yicai(0) and Y2ca (6). This evaluation is done by the following Eq. 1 in a modified Rietveld-refinement program as shown in Fig. 3b,... 

In the MEM/Rietveld analysis, each of the observed structure factors of intrinsically overlapped reflections (for instance, 333 and 511 in a cubic system) can be deduced by the structure model based on a free atom model in the Rietveld refinement. In such a case, the obtained MEM charge density will be partially affected by the free atom model used. In order to reduce such a bias, the observed structure factors should be refined based on the deduced structure factors from the obtained MEM charge density. The detail of the process is described in the review article [9,22-24]. In addition, the phased values of structure factors based on the structure model used in Rietveld analysis are used in the MEM analysis. Thus, the phase refinement is also done for the noncentrosymmetric case as P2, of Sc C82 crystal by the iteration of MEM analysis. The detail of the process is also described elsewhere [25]. All of the charge densities shown in this article are obtained through these procedures. 

The cell parameters are determined as a= 18.2872(8) A, fe=l 1.2120(4) A, c=l 1.1748(4) A,and /J=107.91(l)° (monoclinic P21),by the Rietveld analysis. The reliability factor (R-factor) based on the Bragg intensities, R(, and weighted profile R-factor, Rwp, were 6.3 and 2.6%, respectively. The MEM charge density based on Rietveld analysis was derived with 502 observed structure factors by dividing the unit cell into 128x72x72 pixels. The reliability factor of the final MEM charge density was 3.8%. 

X-ray crystallography seeks to obtain the best model to describe the periodic electron density in a crystal by a least squares fit of the parameters of the model (used to calculate structure factors) against the observed structure factors derived from the diffraction experiment. All models used are atomic in nature, but vary in the complexity of the description of the atomic electron density. 

In normal, routine structure determination experiments, the atomic scattering factors (fj) used are derived from spherically averaged ground state electronic configurations of neutral atoms. The positions of these scattering centers convoluted with thermal motion are then used to calculate structure factors (FH), which are compared with observed structure factors derived from the observed Bragg intensities [3]. 

Although it is possible to determine the complete electron density distribution using the Fourier transform of the observed structure factors, Eq. (1), the errors inherent in the structure factor amplitudes and, in the case of non-centrosymmetric structures, the errors in their phases introduce significant noise and bias into the result. Because of this, it has become normal practice to model the electron density by a series of pseudo-atoms consisting of a frozen, spherical core and an atom centered multipole expansion to represent the valence electron density [2,17]. 

DM uses the normalized structure factors Em obtained from the observed structure factors Fhtd and the individual theoretical atomic scattering factors /, ... 

The summation was made on all the 0.65 A-1 resolution observed structure factors. 

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