The direct determination of centrosymmetric electron density distributions by maximizing the integrated cube of the electron density

The two characteristics of all fully refined electron density distributions are that they do not go negative and the electron density is concentrated into a small number of peaks. The correct set of structure factor signs is defined as a set which gives rise to an electron density distribution which has these characteristics. A simple function, sensitive to these characteristics, is the integrated cube of the electron density; it can distinguish between positive and negative peaks and between a 'peaky' and a dispersed distribution. An electron density map is calculated using only the known signs. Then each structure factor's contribution to the map is added with both alternative signs, in order of magnitude. The correct sign is that giving rise to the larger value of the integrated cube of the electron density. The term is added with the correct sign and the next structure factor similarly dealt with. A recycling procedure is also described. The method of refinement has been tried on several known two- and three- dimensional structures with great success enabling the electron density to be fully refined without recourse to any model. The same criterion can be used to determine the signs of the initial set of structure factors. Electron density distributions are calculated for all permutations of signs of the initial set. The particular combination chosen as correct is that combination giving the largest value of the integrated cube of the electron density. In all the cases tried the method gave the correct signs.