Additional refinement parameters

We seek a set of parameters that minimize the function E . These parameters include the atom positions, of course, because the atom positions in the model 

The factor IFc I in Eq. (7.7) can be expanded to show all the parameters included in refinement, as follows  

Although this equation is rather forbidding, it is actually a familiar equation (5.15) with the new parameters included. Equation (7.8) says that structure factor Fhk[ can be calculated (Fc) as a Fourier series containing one term for each atom j in the current model. G is an overall scale factor to put all Fcs on a convenient numerical scale. In the /th term, which describes the diffractive contribution of atom j to this particular structure factor, n- is the occupancy of atom j f- is its scattering factor, just as in Eq. (5.16) Xj,yjt and zf are its coordinates and Bj is its temperature factor. The first exponential term is the familiar Fourier description of a simple three-dimensional wave with frequencies h, k, and / in the directions x, y, and 7. The second exponential shows that the effect of Bj on the structure factor depends on the angle of the reflection [(sin 0)/X]. 

At the end of successful refinement, the 2F0 Fc map almost looks like a space-filling model of the protein. (Refer to Plate 2 b, which is the final model 

Fourier transform, the crystallographer moves back and forth between real and reciprocal space to nurse the model into congruence with the data. 

---

If the distances satisfy the triangle inequalities, they are embeddable in some dimension. One possible solution is therefore to try to start refinement in four dimensions and use the allowed deviation into the fourth dimension as an additional annealing parameter [43,54]. The advantages of refinement in higher dimensions are similar to those of soft atoms discussed below. 

It is easily possible to introduce refinements into the dilated van Laar model which would further increase its accuracy for correlating activity coefficient data. However, such refinements unavoidably introduce additional adjustable parameters. Since typical experimental results of high-pressure vapor-liquid equilibria at any one temperature seldom justify more than two adjustable parameters (in addition to Henry s constant), it is probably not useful for engineering purposes to refine Chueh s model further, at least not for nonpolar or slightly polar systems. 

Using the refinement parameters, electron density maps were calculated. Figure 1 shows an example derived from CU96. According to all refinement results the intensities of the strong reflections were too low. Therefore, they were omitted for the electron density studies. The problem is currently under study. Additionally, low-temperature measurements are planned for the near future. 

Despite these modifications there remain a number of well-documented problems with the AM1/PM3 core-repulsion function [37] which has resulted in further refinements. For example, Jorgensen and co-workers have developed the PDDG (pair-wise distance directed Gaussian) PM3 and MNDO methods which display improved accuracy over standard NDDO parameterisations [38], However, for methods which include d-orbitals (e.g. MNDO/d [23,24], AMl/d [25] and AMI [39,40]) it has been found that to obtain the correct balance between attractive and repulsive Coulomb interactions requires an additional adjustable parameter p (previously evaluated using the one-centre two-electron integral Gss, Eq. 5-7), which is used in the evaluation of the two-centre two-electron integrals (Eq. 5-8). 

Qi and Qj are the net charges of atoms i and j Nvai(i) and Nvai(j) their number of valence electrons. Cexch and pexCh are empirical parameters. Some additional refinements exist within SIBFA as explicit addition of lone pairs for the exchange term [50],... 

Carbon-13 magnetic resonance (CMR) can play a useful role. Since carbon magnetic resonance deals with analyzing the carbon distribution types, the obvious structural parameter to be determined is the aromaticity, fa Direct determination from the various types of carbon environments is one of the better methods for the determination of aromaticity. Thus, through a combination of proton and carbon magnetic resonance techniques, refinements can be made on the structural parameters, and for the solid-state high-resolution carbon magnetic resonance technique, additional structural parameters can be obtained. 

In short the approach based upon the concept of a limiting distribution offers a viable alternative to that based upon tolerance sets. The stated objective of reducing the number of samples required for making correct decisions has been achieved. Additional refinements in the selection of parameters for the limiting distribution should further enhance its applicability in evaluating acute exposures. 

The parameters of the JT distortions were calculated by the X -method for a series of crystals in good agreement with experimental melting temperatures [14]. The details of the theory and specific calculations seemingly require additional refinements, but the main idea of the JT origin of the liquid-crystal phase transition seems to be quite reasonable. This work thus makes an important next step toward a better understanding of the relation between the macroscopic property of SB and the microscopic electronic structure, the JT effect parameters. 

Toraya s WPPD approach is quite similar to the Rietveld method it requires knowledge of the chemical composition of the individual phases (mass absorption coefficients of phases of the sample), and their unit cell parameters from indexing. The benefit of this method is that it does not require the structural model required by the Rietveld method. Furthermore, if the quality of the crystallographic structure is poor and contains disordered pharmaceutical or poorly refined solvent molecules, quantification by the WPPD approach will be unbiased by an inadequate structural model, in contrast to the Rietveld method. If an appropriate internal standard of known quantity is introduced to the sample, the method can be applied to determine the amorphous phase composition as well as the crystalline components.9 The Rietveld method uses structural-based parameters such as atomic coordinates and atomic site occupancies are required for the calculation of the structure factor, in addition to the parameters refined by the WPPD method of Toraya. The additional complexity of the Rietveld method affords a greater amount of information to be extracted from the data set, due to the increased number of refinable parameters. Furthermore, the method is commonly referred to as a standardless method, since the structural model serves the role of a standard crystalline phase. It is generally best to minimize the effect of preferred orientation through sample preparation. In certain instances models of its influence on the powder pattern can be used to improve the refinement.12... 

Results of similar experiments by Gault and his co-workers (93) with a 10-wt % Pt/Al203 catalyst (mean crystallite size 150-200 A) required the assumption that several successive rearrangements took place in the adsorbed phase before desorption. A model was developed in which either a dehydrocy-clization-hydrogenolysis event or a methyl or ethyl shift involving a tertiary atom competed with desorption. By assuming that the isomeric hexanes had the same desorption probability (d) and the different bond-shift processes proceeded with the same chance (r), it was found possible to reproduce the observed initial product distributions with these two independent parameters. In general, values of d 0.5 and t = 0.10-0.20 fitted the results best. As an additional refinement, the ratio of the C2—C3 and C3—C4 bond scission probabilities for methylcyclo-pentane (0) was taken to be 3.3, rather than the statistical value of 2, to improve further the fit. 

X-ray powder diffraction has been the primary tool used in zeolite structure research. With new high-flux sources, the size requirement of useful single-crystals for structure determination studies has decreased significantly. In addition, refinements of atomic coordinates of known structures using Rietveld powder techniques have become common (24). The solution of a dozen or more new zeolite structure types within the last several years has added to our knowledge base for looking at unknowns (for examples, see references 25-31), and has made us better able to characterize catalyst materials and to correlate synthesis, sorptive, catalytic, and process parameters to their structures (32,33). 

An electronic measure, the NMR chemical shift values of the amide proton in coil conformations (Table 3) also show a high degree of correlation (r = 0.70-0.89) with hydrophihcity scales and with strand versus coil conformations (Table 4). NMR studies reveal that the amide proton is shielded to a greater extent in coil conformations as compared with extended ( 3) structures (37) increased electron density exists at this atom in the coil conformation. Taken together, the data suggest strong interactions between hydrophihcity and electronic parameters in folding and provide support for additional refinement of the Hp index. 

Now that fairly precise measures of electron density can be made, atomic displacement parameters can be refined so that the best possible fit to the experimental electron-density profiles of each atom is obtained. This is done by the introduction of additional atomic parameters, one parameter if the displacements are isotropic, six if they are anisotropic. When this least-squares refinement of displacement parameters is completed, the crystallographer is then left with the problem of explaining the atomic displacement parameters so obtained in terms of vibration, static disorder, dynamic disorder, or a combination of these. 

A difference Fourier map, calculated at this point, reveals an additional small electron density maximum in the tetrahedral cavity next to the partially occupied V2. Thus, it is reasonable to assume that the V2 site splits into two independent partially occupied positions with the coordinates, which distribute V atoms in a random fashion in two adjacent tetrahedral positions rather than being simply vanadium-deficient. We label these two sites as V2a (corresponding to the former V2) and V2b (corresponding to the Fourier peak). Refinement of this model slightly improves the fit. Subsequently, additional profile parameters (F, F , and sample displacement) were included in the refinement, followed by a typical procedure of refining the porosity in the Suortti approximation with fixed atomic coordinates and Ui o, and then fixing the porosity parameters for the remainder of the refinement. 

If the selected cell has M 20 (= M20) > 20 for triclinic crystal systems, or M20 > 30 i.e., M20 > 10) for monoclinic or higher symmetry crystal systems, it will be automatically refined by PI RUM, originally an interactive program, suitably modified to perform the automatic refinement of the unit-cell parameters. If more than 25 observed lines are available, the first 25 lines will be used for finding the cell, while all the lines will be involved in the refinement step. At the end of the PI RUM refinement a statistical study of the index parity of the assigned reflections is performed to detect the presence of doubled axes or of additional lattice points (A-, B-, C-, I-, R- or F-centred cell). If one of the index parity conditions is verified, an additional refinement is performed taking into account this information. 

The evolution of one or another of the parameters affects the entire calculated pattern. Therefore, the scope of variations in the values of the various refinement parameters is small and, most importantly, interdependent. The parameters are said to be constrained. During the refinement, the person conducting the experiment can include additional constraints, for example, imposing that the sum of two occupation... 

This method is particularly efficient [HIL 00] and has been widely implemented in the past few years. It is however necessary to have a good knowledge of the stmctures of the phases in the sample. Additionally, in this type of study, the number of refinement parameters can be very large, which means that the refinement strategy has to be carefully defined [CON 00]. 

In the second line, the number of refined parameters for a given cycle is specified and, regardless of the pattern s quality, it is always comprised of a continuous scattering background in addition to the diffraction peaks. There are... 

Although FDS is able to produce a shortlist of equipment, it is possible to further refine the equipment choice. This requires consideration of additional selection parameters (indicated through Tables 5.7-5.11) and provides a basis for deciding which types of solid/liquid separation device merit the time and cost of executing detailed testing and/or simulation programmes. 

When the final least-squares parameters Xi with standard deviations af) have been obtained with some fixed parameters, additional refinements are carried out changing one of the assumed parameters [X at a time by a small amount. In this way, dXildXl may be obtained. A standard deviation for the assumed parameter (crj) is estimated, and the standard deviation for Xi calculated from... 

---