Foundations of residual-density analysis

Abstract

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.

Quantifying the Inter- and Intramolecular Interactions in Crystalline Phthalic Acid

“…A plot of the molecule from the final refinement is shown in Figure 1. The quality of the data and refinement model was confirmed with a normal probability plot, a scale factor plot with respect to resolution, and fractal analysis 24 of the residual density. The corresponding plots have been deposited as well as tables of observed and calculated structure factors.…”

Electron densities by the maximum entropy method (MEM) for various types of prior densities: a case study on three amino acids and a tripeptide

“…Accordingly, following the procedure by Hofmann, Netzel & van Smaalen (2007), we find larger optimal values of 2 aim for INV and MP priors than in the case of the IAM priors (Table 2). Bindzus & Iversen (2012) have recently employed the residual density analysis of Meindl & Henn (2008) as a criterion for finding the optimal value of 2 aim . We would like to stress that the method of Hofmann, Netzel & van Smaalen (2007) -by its very principle -leads to smooth MEM electrondensity maps with zero or very few spurious maxima, thus facilitating a meaningful topological analysis of the resulting electron-density maps.…”

Die Bedeutung ionischer Bindungsanteile in Schwefeldioxid – Bindungsordnungen aus Röntgenbeugungsdaten

“…Mit Blick auf eine reine Elektronendichteuntersuchung wird deutlich, dass die Rçntgen-Wellenfunktionsverfeinerung bessere Ergebnisse liefert als ein Multipolmodell, [28] das die neuesten Methoden wie z. Sie ist überlegen hinsichtlich der R-Werte, Deformations-und Restdichteverteilung, einer fraktalen Verteilungsanalyse der Restdichte [31] und topologischer Elektronendichteresultate (Bindungs-und atomare Eigenschaften). Sie ist überlegen hinsichtlich der R-Werte, Deformations-und Restdichteverteilung, einer fraktalen Verteilungsanalyse der Restdichte [31] und topologischer Elektronendichteresultate (Bindungs-und atomare Eigenschaften).…”