Distance least-square program

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. 

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. 

One of the more powerful manipulation commands in MAECIS is COMPARE which allows the current molecule to be superimposed upon the alternate molecule. This is accomplished in the following steps. First, the user specifies the atom pairs in the two structures which are to be overlapped. At least three pairs of atoms must be specified. The program then performs a nonlinear least-squares calculation to minimize the distance between these atom pairs. Finally, the user specifies that the molecules are to be drawn superimposed upon each other. The superimposed molecules can be drawn with any of the options available with the DRAW command. An illustration of this is contained in Figure 2 where two musk odorants are compared. Thus, a chemist can obtain an idea of the structural similarity of any two molecules. 

We will come to the form of the interaction potential employed in the next section given the appropriate form, the coupled equations (10.165) were solved by a numerical method due to Johnson [218] and described by Hutson [217]. For most states the coupled equations were solved for intemuclear distances up to 25 A, and the energies of the levels converged to 2 MHz. A computer program for interactive nonlinear least squares fitting of the parameters to the observed transition frequencies has been described by Law and Hutson [219]. 

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