The use of Sayre's equation in real space

A method is described for minimizing a least-squares residual to Sayre's equation as a function of electron densities under the constraints that each of the observed structure factors is strictly compatible with the densities, i.e. |F_h| - |F_h^obs| = 0. By evaluation of the residual in real space using fine grid sizes, the method enables one to obtain density maps of atomic resolution even with low-resolution data. Numerical calculations have been made at various resolutions for the crystal structure of a small molecule containing only C, N, O and H atoms. It has been found that the residual can be used as a good figure of merit when the resolution of the observed diffraction data is higher than 1.5-1.8 Å. With 2.0 Å data, however, several unusual structures whose residuals are lower than that of the correct structure have been found. Their existence may indicate a limitation inherent in direct methods based on the principle of `atomicity' in general.