Patterson function and δ recycling: derivation of the phasing equations

Two phasing equations based on the Fourier syntheses δ_P = T⁻¹[(E² − 〈E²〉)exp(iφ)] and δ_M = T⁻¹[(E − 〈E〉)exp(iφ)] were recently described [Rius (2012). Acta Cryst. A68, 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.