Minimally resolution biased electron-density maps

Abstract

Electron-density maps are calculated by Fourier syntheses with coefficients based on structure factors. Diffraction experiments provide intensities up to a limited resolution; as a consequence, the Fourier syntheses always show series-termination errors. The worse the resolution, the less accurate is the Fourier representation of the electron density. In general, each atomic peak is shifted from the correct position, shows a deformed (with respect to the true distribution of the electrons in the atomic domain) profile, and is surrounded by a series of negative and positive ripples of gradually decreasing amplitude. An algorithm is described which is able to reduce the resolution bias by relocating the peaks in more correct positions and by modifying the peak profile to better fit the real atomic electron densities. Some experimental tests are performed showing the usefulness of the procedure.

4. Single crystal and powder XRD techniques: An overview

“…For example, in the EXPO program, procedures have been introduced devoted to -activating, in a completely automatic way, the exploration of all the sets of phases generated by DM; -completing and optimizing the structure model (for a general overview see [41][42][43][44][45][46]). Two of them, the WLSQ-FR and the RBM procedures, are described in the following two paragraphs, respectively.…”

“…Three different kinds of strategies, able to correct the electron density map by typical resolution bias errors, and based on Fourier map calculations, have been implemented in EXPO to complete and optimize the structure models. The first works in the direct space and directly modifies the electron density map [43,44], the second operates in the reciprocal space by modifying the classical atomic scattering factors [45] and the third acts in both direct and reciprocal space [46]. The RBM procedure working in direct space has been chosen as default in the EXPO program.…”

On effective and optical resolutions of diffraction data sets

“…In fact, there is no strict mathematical reason for the minimum distance to be equal to the distance either to the first zero or to the first minimum. Some further discussions about the separation of images of point scatterers and atoms, although at relatively high resolutions only, may be found in Altomare et al (2008). Vaguine et al (1999) proposed another formal procedure to estimate the minimum distance (implicitly, they analyzed atoms with the atomic displacement parameter estimated from the diffraction intensities).…”

The Dual-Space resolution bias correction inEXPO2010

“…Therefore it is combined with the DS approach [3,4] to improve the map further on. It can be summarized in two steps: 1) the ripple contribution is eliminated from each j-th peak in the map; 2) each j-th main peak is best fitted in its own domain by a Gaussian function.…”

“…Therefore Fourier map improvement is fundamental for succeeding in the structure solution. We have developed a new Dual-Space (DUS) procedure [1]: it combines the Reciprocal Space (RS) [2] and the Direct Space (DS) [3,4] approaches: the first one corrects the canonical atomic scattering factors in such a way that the map calculated by using the structure factors obtained by the corrected scattering factors shows reduced truncation errors; the second one modifies directly the electron density map so that ripples are eliminated and peaks are better positioned and resolved. The DUS procedure uses also new computing strategies involving difference electron density maps.…”