The difference electron density: a probabilistic reformulation

The joint probability distribution function P(E, E-p), where E and E-p are the normalized structure factors of the target and of a model structure, respectively, is a fundamental tool in crystallographic methods devoted to crystal structure solution. It plays a central role in any attempt for improving phase estimates from a given structure model. More recently the difference electron density P-q = rho - rho(p) has been revisited and methods based on its modifications have started to play an important role in combination with electron density modification approaches. In this paper new coefficients for the difference electron density have been obtained by using the joint probability distribution function P(E, E-p, E-q) and by taking into account both errors in the model and in measurements. The first applications show the correctness of our theoretical approach and the superiority of the new difference Fourier synthesis, particularly when the model is a rough approximation of the target structure. The new and the classic difference syntheses coincide when the model represents the target structure well.