Structure solution by minimal-function phase refinement and Fourier filtering. I. Theoretical basis

Eliminating the N atomic position vectors r_j, j = 1, 2, ..., N, from the system of equations defining the normalized structure factors E_H yields a system of identities that the E_H's must satisfy, provided that the set of E_H's is sufficiently large. Clearly, for fixed N and specified space group, this system of identities depends only on the set {H}, consisting of n reciprocal-lattice vectors H, and is independent of the crystal structure, which is assumed for simplicity to consist of N identical atoms per unit cell. However, for a fixed crystal structure, the magnitudes |E_H| are uniquely determined so that a system of identities is obtained among the corresponding phases φ_H alone, which depends on the presumed known magnitudes |E_H| and which must of necessity be satisfied. The known conditional probability distributions of triplets and quartets, given the values of certain magnitudes |E|, lead to a function R(φ) of phases, uniquely determined by magnitudes |E| and having the property that R_T < ½ < R_R, where R_T is the value of R(φ) when the phases are equal to their true values, no matter what the choice of origin and enantiomorph, and R_R is the value of R(φ) when the phases are chosen at random. The following conjecture is therefore plausible: the global minimum of R(φ), where the phases are constrained to satisfy all identities among them that are known to exist, is attained when the phases are equal to their true values and is thus equal to R_T. This `minimal principle' replaces the problem of phase determination by that of finding the global minimum of the function R(φ) constrained by the identities that the phases must satisfy and suggests strategies for determining the values of the phases in terms of N and the known magnitudes |E|. Equivalently, the minimal principle leads to the solution of the (in general redundant) system of equations satisfied by the phases φ_H.