Foundations of residual-density analysis

New and concise descriptors of the residual density are presented, namely the gross residual electrons, the net residual electrons and the fractal dimension distribution. These descriptors indicate how much residual density is present and in what way it is distributed, i.e. the extent to which the distribution is featureless. The amount of residual density present accounts for noise in the experimental data as well as for modeling inadequacies. Therefore, the minimization of the gross residual electrons during refinement serves as a quality criterion. In the case where only Gaussian noise is present in the residual density, the fractal distribution is parabolic in shape. Deviations from this shape therefore serve as an indicator for systematic errors. The new measures have been applied to simulated and experimental data in order to study the effects of noise, model inadequacies and truncation in the experimental resolution. These measures, although designed and examined with particular regard to applications of space residual density, are very general and can in principle also be applied to space and momentum residual densities in a one-, two-, three- or higher-dimensional Euclidean space.