Patterson function and δ recycling: derivation of the phasing equations.

Two phasing equations based on the Fourier syntheses δ(P) = T(-1)[(E(2) - <E(2)>)exp(iφ)] and δ(M) = T(-1)[(E - <E>)exp(iφ)] were recently described [Rius (2012). Acta Cryst. A 68, 77-81] (E is the quasi-normalized structure factor and <E> is the average over all reflections). These equations were found by comparison with the direct methods origin-free modulus sum function and constitute the core of the `δ recycling' phasing procedure. The derivation of these phasing equations from the minimization of a residual (R(P)) between two differently calculated density functions (one of them including the positivity constraint) is shown.