Least-squares distance refinement

For both structures, all final Si positions were obtained with reasonable accuracy (0.1 -0.2 A) by a 3D reconstruction of HRTEM images followed by a distance least-squares refinement. This kind of accuracy is sufficient for normal property analysis, such as catalysis, adsorption and separation, and as a starting point for structure refinement with X-ray powder diffraction data. The technique demonstrated here is general and can be applied not only to zeolites, but also to other complicated crystal structures. 

As anticipated, lower temperature increases the number of observations from an X-ray diffraction data collection (at constant radiation dose). This is however just one of the advantages that could improve a structure solution or a refinement. In fact, a reduced thermal motion usually implies a more reliable standard model, given that for smaller atomic displacements the harmonic approximation is more appropriate and less correlation is found between variables within a least squares refinement. This returns higher precision of the parameters calculated from those variables (for example bond distances, bond angles, etc.). 

Manceau, A., Chateigner, D., and Gates, W. P. (1998). Polarised EXAFS distance least squares valence modelling (DVLS) and quantitative texture analysis approaches to the structural refinement of Garfield nontronite. Phys. Chem. Miner. 25, 347-65. 

The structures of zinc aspartate tri-hydrate, urea phosphate and cycloserine hydrochloride have been determined by 3-dimensional x-ray methods ami refined by IBM 704 least-squares computations The zinc aspartate analysis [14] reveals H positions and H-bonding which leads to a ring structure for the aspartate ion. Urea phosphate shows strong H-bond throughout the structure [15]. Seven H s are available for nine short bonds. Cycloserine hydrochloride [16, 17] shows seven short mtermolecular distances, suggesting 0—H O,... 

A method for investigating all four types of problems (of which 2a is the most complicated and lb is the least complicated) has been derived by Meier and Villiger (12). The distance least squares (DLS) procedure uses the well-known distances D for Si-O, Al-O, 0-0 (and eventually others) of the framework and refines atomic parameters by a least-squares procedure minimizing... 

In DLS computations, constant atomic distances are used and have been very useful. Therefore, it appears worthwhile to constrain the positional parameters of the framework with respect to the known distances for Si-O, Al-O, and 0-0 for normal structure factor least-squares computations. Constrained refinement essentially reduces the number and/or variability of the parameters and can be helpful for work with limited data sets (e.g., for powder diffraction). Constrained refinement has been discussed by Pawley (26, 27). 

Distance least squares (DLS), a method developed by Meier and Vill-iger (1) for generating model structures (DLS models) of prescribed symmetry and optimum interatomic distances, can supply atomic coordinates which closely approach the values obtained by extensive structure refinement. DLS makes use of the available information on interatomic distances, bond angles, and other geometric features. It is primarily based on the fact that the number of crystallographically non-equivalent interatomic distances exceeds the number of coordinates in framework-type structures. A general DLS program is available (8) which allows any combination of prescribed parameters (interatomic distances, ratios of distances, unit cell constants etc). In addition, subsidiary conditions (as discussed in Refs. 1 and 8) can also be prescribed. 

The shrinkage effect1 is treated in more detail elsewhere in the present article. Due to molecular vibrations interatomic distances observed by electron diffraction do not correspond to a set of distances calculated from a rigid geometrical model. Usually the shrinkage effect is routinely included in electron-diffraction least-squares refinement. In order to do so, it has been found appropriate to introduce a third distance type r defined as the distance between mean positions of atoms at a particular temperature. If the harmonic force field is known, iQ may be calculated from ra according to Eq. (12) ... 

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... 

These maxima in the Fourier transform data, which correspond to the different chromium coordination shells, were isolated using a filter window function. The inverse transform of each peak was generated and fitted using a non-linear least squares program. The amplitude and phase functions were obtained from the theoretical curves reported by Teo and Lee (2 ). The parameters which were refined included a scale factor, the Debye-Waller factor, the interatomic distance, and the threshold energy difference. This process led to refined distances of 1.97(2) and 2.73(2) A which were attributed to Cr-0 and Cr-Cr distances, respectively. Our inability to resolve second nearest neighbor Cr-Cr distances may be a consequence of the limited domain size of the pillars. 

The parameters needed to describe such a model are, for each discrete interaction, the distance, its rms variation and the frequency, and corresponding parameters for describing the emergence of the continuum. The parameter values can be refined either in s space by a least squares procedure minimizing... 

According to a complete X-ray diffraction analysis, Se6 consists of ring molecules with the molecular symmetry of Dzd the crystal and molecular parameters are listed in Table II (17) and the crystal structure is shown in Fig. 2. Refinement by the least squares method resulted in the following atomic parameters of the single atom in the asymmetric unit x = 0.1602 0.00048, y = 0.20227 0.00047, z = 0.12045 0.00120 calculated density, 4.71 g/cm3. An earlier investigation of selenium vapor by electron diffraction led to an internuclear distance of 234 1 pm and an average bond angle of 102 0.5° for the chairlike cyclic Se6 molecule (23). 

The estimated precision in bond lengths obtained by a least-square refinement of a data set measured by X-ray diffraction can be 0.003 A ( 0.3 pm), for a structure with unweighted R-factor less than 3%. If the data set is collected at low temperatures (20 K or 80 K), the decrease in thermal vibration can yield even better bond distances and angles. For H atom coordinates, the precision is one or two orders of magnitude lower, since the electron density around an H atom is relatively low in these cases a neutron diffraction study (which requires very large crystals) can yield better H atom positions. 

Refinement of the framework coordinates was difficult because of the acentricity and pseudosymmetry. A stable least-squares solution was obtained for the framework atoms in which the T-0 distances indicate alternation of A1 and P atoms. However, the 0 atoms showed large displacements from the centroid, particularly 0(2) for which a difference-Fourier map indicated three spearate peaks (Figure 3). Because there is no optical or X-ray evidence for symmetry lower than hexagonal, it is assumed that there are microdomains with tilted tetrahedra, as proposed for high-cristobalite(6) and high-tridymite( 7). For convenience, the displacements of the oxygen atoms are approximated by ellipsoids. 

The structure of the monoclinic form has been refined by the least squares method. The Mo-O distances are therefore known with an accuracy that allows a detailed discussion of the distortions from regular coordination around the molybdenum atoms. The tetrahedral coordination is very regular, while the octahedral coordination is rather irregular, especially in octahedra joined to tetrahedra. 

---