Positional Isomorphous Phases

The relative stability of different cation sites in a zeolite framework is one more interesting example of energy difference calculation, in this case 

In this case, the level of theory is almost irrelevant for evaluating the relative stability of the different sites (Table 13). The proton is preferably predicted at the Sin site. Li and Na cations are more stable when coordinated on the hexagonal prism (SII), whereas K definitely prefers the eight-member ring site (SnP) for steric reasons. 

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X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

Jensen et al. have very successfully used difference Fourier maps using phases calculated from the atomic co-ordinates, in the refinement of the structure of the small protein, rubredoxin. Shifts derived from the first difference Fourier synthesis at 1.5 A resolution reduced the R factor (the agreement factor) from 0.372 to 0.321. Eventually difference syntheses revealed the positions of 127 water molecules oxygen atoms representing water were included if peaks were present in the 2 A isomorphous phased Fourier as well as the difference Fourier map. Occupancy factors were made proportional to peak height. This led to an appreciable decrease in R. After calculation of four difference Fourier syntheses, the method of least squares was used to refine the positional and thermal parameters despite the fact that their number was exceeded by the number of structure factors by... 

There seem to be many binary metallic systems in which there are phases of this sort. In the sodium-lead system there are two such phases. One of them, based on the ideal structure Na3Pb, extends from 27 to 30 atomic percent lead, with its maximum at about 28 atomic percent lead and the other, corresponding to the ideal composition NaPb3, extends from 68 to 72 atomic percent lead, with maximum at about 70 atomic percent. The intensities of X-ray reflection have verified that in the second of these phases sodium atoms occupy the positions 0, 0, 0, and the other three positions in the unit cell are occupied by lead atoms isomorphously replaced to some extent by sodium atoms (Zintl Harder, 1931). These two phases are interesting in that the ranges of stability do not include the pure compounds Na8Pb and NaPb3. 

Multiple isomorphous replacement allows the ab initio determination of the phases for a new protein structure. Diffraction data are collected for crystals soaked with different heavy atoms. The scattering from these atoms dominates the diffraction pattern, and a direct calculation of the relative position of the heavy atoms is possible by a direct method known as the Patterson synthesis. If a number of heavy atom derivatives are available, and... 

The impurities can become incorporated in the solid phase in three modes i) interstitially, i.e. between regular lattice positions, ii) by coprecipitation as a separate insoluble phase or iii) by isomorphous substitution of... 

The addition of one or more heavy atoms to a macromolecule introduces differences in the diffraction pattern of the derivative relative to that of the native. If this addition is truly isomorphous, these differences will represent the contribution from the heavy atoms only thus the problem of determining atomic positions is initially reduced to locating the position of a few heavy atoms. Once the positions of these atoms are accurately determined, they are used to calculate a set of phases for data measured from the native crystal. Although, theoretically, one needs only two isomorphous derivatives to determine the three-dimensional structure of a biological macromolecule, in practice more than two are needed. This is due to errors in data measurement and scaling and in heavy-atom positions, as well as lack of isomorphism. 

In most cases, however, the protein molecules are larger or the resolution of the data is lower, and phasing becomes a two-stage process. If two or more intensity measurements are available for each reflection with differences arising only from some property of a small substructure, the positions of the substructure atoms can be found first, and then the substructure can serve as a bootstrap to initiate the phasing of the complete structure. Suitable substructures may consist of heavy atoms soaked into a crystal in an isomorphous replacement experiment, or they may consist of the set of atoms that exhibit... 

If the phasing model is not isomorphous with the desired structure, the problem is more difficult. The phases of atomic structure factors, and hence of molecular structure factors, depend upon the location of atoms in the unit cell. In order to use a known protein as a phasing model, we must superimpose the structure of the model on the structure of the new protein in its unit cell and then calculate phases for the properly oriented model. In other words, we must find the position and orientation of the phasing model in the new unit cell that would give phases most like those of the new protein. Then we can calculate the structure factors of the properly positioned model and use the phases of these computed structure factors as initial estimates of the desired phases. 

Crystals of the material are grown, and isomorphous derivatives are prepared. (The derivatives differ from the parent structure by the addition of a small number of heavy atoms at fixed positions in each — or at least most — unit cells. The size and shape of the unit cells of the parent crystal and the derivatives must be the same, and the derivatization must not appreciably disturb the structure of the protein.) The relationship between the X-ray diffraction patterns of the native crystal and its derivatives provides information used to solve the phase problem. 

In the attempt to synthesize molecular sieves with isomorphous substitutions of A1 and/or Si by the divalent calcium element in the tetrahedral positions, we obtained a new calcium silicate phase by inclusion of heteroatom calcium into silicate sols. The characterization results showed that as-synthesized calcium silicate, named CAS-1 (Calcium silicate No. 1), was a novel zeolite-like crystal material with the cation reversibly exchangeable and selectively adsorptive properties. In this paper, the effects of composition of raw materials, reaction temperature and the different alkali ion on the hydrothermal synthesis of calcosilicate crystal material CAS-1 were investigated and the uptake of different cation on the thermal stability of CAS-1 structure was also examined. The sample was characterized by XRD, TEM, SEM, DT-TGA, BET, AAS and chemical analysis. 

Once diffraction data have been gathered the next stage in the structure analysis is usually focused on seeking a definition of the orientation of the particle and, often, its position in the crystal cell. This information is essential for both phase refinement and the analysis of isomorphous replacement experiments. Because of the inevitable noncrystallographic redundancy (a minimum of 5-fold) and the fixed relationship between the various icosahedral symmetry axes it is often possible to solve the orientation problem by analysis of the diffraction data in the absence of a model, usually by use of a self-rotation function. 

It is clear that in the case of MFI, the zeolite pore entrances should not be considered as rigid apertures. Instead, zeolite framework topologies can show flexibility. While the O-Si-0 angle in the tetrahedral unit is rigid (109 + 1 °), the Si-O-Si angle between the units can vary between 145 and 180°. Based on isomorphous substitution of Si by other T-atoms in the framework [18], framework defects [19], cation positions, changes in the water content [16], external forces on the crystalline material [20] and upon adsorption of guest molecules [21] phase transitions can occur that have a dramatic influence in particular cases on the framework atom positions. 

Heavy-atom derivative of a protein The product of soaking a solution of the salt of a metal of high atomic number into a crystal of a protein. If the heavy-atom derivative is to be of use in structure determination, the heavy atom must be substituted in only one or two ordered positions per asymmetric unit. Then the method of isomorphous replacement can be used to determine the relative phase angles of the Bragg reflections. 

FIGURE 8.2 The unknown phase of a wave of measurable amplitude can be determined by beating it against a reference wave (created experimentally) whose phase and amplitude are both known. The unknown wave at top and the reference wave below are allowed to interfere with one another to produce the resultant wave at bottom. Although the phase of the resultant wave cannot be measured, its amplitude can. That resultant amplitude is a function of the amplitudes of the unknown and reference waves (both known or measurable), the phase of the reference wave (experimentally set to zero), and the phase of the unknown wave, which is in question. In isomorphous replacement the unknown wave is the native structure factor (measured), the reference wave is the structure factor of the heavy atom alone (calculated from its position), and the resultant wave is the structure factor of the heavy atom derivatized crystal (measured). 

In order to exploit the heavy atom method with crystals of conventional molecules, or to utilize the isomorphous replacement method or anomalous dispersion technique for macro-molecular structure determination, it is necessary to identify the positions, the x, y, z coordinates of the heavy atoms, or anomalously scattering substituents in the crystallographic unit cell. Only in this way can their contribution to the diffraction pattern of the crystal be calculated and employed to generate phase information. Heavy atom coordinates cannot be obtained by biochemical or physical means, but they can be deduced by a rather enigmatic procedure from the observed structure amplitudes, from differences between native and derivative structure amplitudes, or in the case of anomalous scattering, from differences between Friedel mates. 

Unless direct methods are used to locate heavy atom positions, an understanding of the Patterson function is usually essential to a full three-dimensional structure analysis. Interpretation of a Patterson map has been one of two points in a structure determination where the investigator must intervene with skill and experience, judge, and interpret the results. The other has been the interpretation of the electron density map in terms of the molecule. Interpretation of a Patterson function, which is a kind of three-dimensional puzzle, has in most instances been the crucial make or break step in a structure determination. Although it need not be performed for every isomorphous or anomalous derivative used (a difference Fourier synthesis using approximate phases will later substitute see Chapter 10), a successful application is demanded for at least the first one or two heavy atom derivatives. 

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