Distance least squares

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. 

Table 4.1 Distance least-squares optimized atomic coordinates for UZM-5.

Kroll, H., Maurer, H., Stockelmann, D., Becker, W., Fulst, J., Kriisemann, R., et al. (1992). Simulation of crystal structures by a combined distance-least-squares valence-rule method. Zeit. Kristallogr. 199, 49 66. 

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. 

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... 

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. 

Figure 8. Stereoview of the BSS framework drawn as straight lines connecting adjacent tetrahedral vertices. Atomic co-ordinates are based on distance least squares optimized values.

Banfield JF, Veblen DR, Snuth DJ (1991) The identification of naturally occurring Ti02(B) by structure determination nsing high-resolntion eleetron nueroseopy, image simnlation, and distance-least-squares refinement. Am Mineral 76 343-353... 

Examine visually the representations (projections down the three principal crystallographic directions) of the full unit-cell contents for each of the saved models, followed by addition of framework oxygen atoms between neighboring T-atoms, and DLS (Distance Least Squares) optimization. 

The distance least squares refinement of liottite (LIO), given in the third revised edition, has been replaced by a single crystal x-ray refinement. 

If the chemical connectivity of a given system can be anticipated, as for example in the case of framework structured solids, reasonable bounds for first and second neighbour bond distances can be readily defined. Distance targets can, therefore, be set up for such a system and the resulting distance least squares procedure (Meier and Villiger, 1969) (DLS) takes as input a trial or random set of coordinates, the crystal symmetry and possibly site symmetry constraints, which, when combined with defined connectivity and target bond... 

A published crystal structure has been refined by least-squares based on about 8600 reflections measured with an automated three-circle single-crystal diffractometer. A distance least-squares computation (DLS) revealed the space group Am (No. 8) to be the most probable one. The published structure was found to be essentially correct. The weighted R-factor for all reflections used is 9.4%. Further detailed work is in progress. 

The distance least-squares (DLS) computations (11) were initiated on the atomic parameters of Bartl and Fischer (2). The result of this computation favors the space group Am. This contradicts the assumption of Amirov et al. (1) who adopted the space group A2. Further details of DLS-computations will be published elsewhere. 

Distance Least Squares Methods.- Perhaps the earliest computational method in this field was that due to Meier and Villinger, who devised the method of distance least squares (DLS) to provide a suitable method for the refinement of zeolite crystal structures. 

A number of zeolite structures have been refined to reasonably precise levels based on synchrotron powder X-ray diffiaction data, including those of ZSM-11 [57,58] and silica-ZSM-12 [18] where, in both cases, the previously reported atontic coordinates had been c timized solely by distance least squares. Complete stracture determinations of... 

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