Reinvestigation of the Use of Patterson Maps to Extrapolate Data to Higher Resolution

Abstract

The idea of using suitably modified Patterson maps for the purpose of extrapolating X-ray diffraction amplitudes to higher resolution is not new. It was first proposed by Karle and Hauptman [1] in 1964 as a means of refining E-values to obtain values which were more consistent with the point atom approximation upon which direct methods probability relationships are based. What K&H proposed is that one could compute a sharpened, origin-removed Patterson function using |E|²-1 as coefficients, and then modify the density to remove unexpected negative regions, as well as peaks which represented vectors which would place molecules closer than a van der Waals contact to proper symmetry elements such as rotation axes, inversion centers and mirror planes. When one back transformed the modified map, one not only obtained refined estimates for the observed |E|²-1 amplitudes, but also obtained values for terms which were not included in the original map. Each cycle of refinement included calculation, modification, and back transformation of the map, and renormalization of the extrapolated ||E|²-1|> values to fit the expected distribution, and insure that negative IEI² values were reset to zero as their smallest possible value. One must also eliminate reflections corresponding to space group extinctions, as the modified Patterson will not respect this constraint. Only one subsequent publication [2] appears to have used this Patterson refinement scheme to improve the ‘accuracy’ of the measured E-values to solve a structure.