The use of partial observations, partial models and partial residuals to improve the least-squares refinement of crystal structures
Some common misconceptions in least-squares crystal structure refinement can be resolved by recasting the usual equations in terms of partial observations, partial models and partial residuals. An observation has components that are determined by an initial calculated model and each component, including the background, is considered to be a partial observation of the total observation. The use of partial observations, partial models and partial residuals allows various misconceptions to be identified and suggests ways to improve the least-squares methodology. A fixed component of a model of peak-plus-background does not fix its contribution to the observation for each refinement step. A covariance matrix obtained from the least-squares equations enables a standard uncertainty to be estimated for any function of the structural parameters. An oversight in current refinement methods is the failure to estimate the variances of components of the calculated model of an observation and the fraction of each residual associated with the various features of a refinement. A distinction should be made between least-squares equations for model development and least-squares equations for the estimation of a variance–covariance matrix. Methods for detecting systematic errors are discussed. A proposed look-ahead option for model development includes the assessment of the ability to refine parameters. For pseudo-symmetric structures, the use of symmetrized combinations of pseudo-equivalent intensities allows the reliability of minor components of the intensities to be better evaluated. It is also shown how homometric structure solutions can result from the use of powder diffraction data or equally twinned crystals.