General quadratic functions in real and reciprocal space and their application to likelihood phasing

A general multivariate quadratic function of the structure factors is constructed and transformed to obtain a quadratic function of the continuous electron density. Two special cases, where structure factors are independent and where electron-density values are independent, are examined. These results are related to the new likelihood-based framework of Terwilliger [Terwilliger (1999), Acta Cryst. D55, pp. 1863-1871] for employing structural information which was previously exploited by means of conventional density-modification calculations. The treatment here involves different assumptions and highlights new features of Terwilliger's calculation. The generalization quadratic construction allows the generation of cross terms relating all reflections and electron densities. Other applications of this approach are considered.