The (Fo-Fc) Fourier synthesis: a probabilistic study

(F(o)-F(c)) and (2F(o)-F(c)) Fourier syntheses are considered the most powerful tools for recovering the remainder of a structure and for correcting crystal structure models. A probabilistic approach has been applied to derive the formula for the variance for the expected value of the coefficient (F(o)-F(c)). This has allowed a better understanding of the features of the difference Fourier synthesis; in particular, a subset of well phased reflections has been separated from the subset of reflections best phased by the standard F(o) Fourier synthesis. An iterative procedure, based on the electron-density modification of the difference Fourier map, has been devised which aims to improve phase and modulus estimates of the reflections with higher variance value, by using as lever arm the set of reflections with lower variance value. The new procedure (DEDM) has been implemented and verified on a wide set of test structures, the partial models of which were obtained by molecular replacement or by automatic model-building routines applied to experimental electron-density maps. Phase and modulus estimates of the difference Fourier syntheses improve in all the test cases; as a consequence, the quality of the difference Fourier maps also improves in the region where the target structure deviates from the partial model. A new procedure is suggested, combining DEDM with standard electron-density modification techniques, which leads to significant reduction of the phase errors. The procedure may be considered a starting point for further developments.