Least-square constraints plane

The first and second moment conditions can be very easily introduced into the r5-fit method as least-squares constraints [7,54] if the number of isotopomers is sufficient for a complete restructure. The effect on the coordinates is not expected to be particularly unbalanced unless the moment conditions are required for the sole purpose of locating atoms that could not be substituted (e.g., fluorine or phosphorus) or that have a near-zero coordinate. While all coordinates may change, the small coordinates will, of course, change more. In the cases tested, the coordinate values of the rs-fit with constraints and those of the corresponding r/e-fit (not of the r0-fit), including errors and correlations, differed by only a small fraction of the respective errors, i.e., much less than reported above. This was true under the provision that all atoms could be substituted and that the planar moments that were excluded from the r -fit because of substitution on a principal plane or axis, were also omitted from the r/E-fit. With these modifications, the basic physical considerations and the input data are the same in both cases, and the results should be identical in the limit where the number of observations equals that of the variables. 

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