STRUCTURE SOLUTION BY MINIMAL-FUNCTION PHASE REFINEMENT AND FOURIER FILTERING .1. 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 an d specified space group, this system of identities depends only on the set {H}, consisting of n reciprocal-lattice vectors H, and is indepen dent of the crystal structure, which is assumed for simplicity to cons ist of N identical atoms per unit cell. However, for a fixed crystal s tructure, the magnitudes \E(H)\ are uniquely determined so that a syst em of identities is obtained among the corresponding phaseS phi(H) alo ne, which depends on the presumed known magnitudes \E(H)\ and which mu st of necessity be satisfied. The known conditional probability distri butions of triplets and quartets, given the values of certain magnitud es Absolute value of Absolute value of E lead to a function R(phi) of phases, uniquely determined by R(T) < 1/2 magnitudes Absolute value of E and having the property that R(T) < 1/2 < R(R), where R(T) is the v alue of R(phi)) when the phases are equal to their true values, no mat ter what the choice of origin and enantiomorph, and R(R) is the value of R(phi) when the phases are chosen at random. The following conjectu re is therefore plausible: the global minimum of R(phi), where the pha ses are constrained to satisfy all identities among them that are know n 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 prob lem of phase determination by that of finding the global minimum of th e function R(phi) constrained by the identities that the phases must s atisfy and suggests strategies for determining the values of the phase s in terms of N and the known magnitudes Absolute value of Absolute va lue of E. Equivalently, the minimal principle leads to the solution of the (in general redundant) system of equations satisfied by the phase s phi(H).