Twin variables and derminants in direct methods

Abstract

In most algorithms of direct methods, the variables are the normalized structure factors (SF) EH. An alternative set of variables is proposed which provides more flexibility for handling, in a single algorithm, phase relationships and direct-space constraints, as well as the complete set of diffraction data. This set of variables ΨH consists of SF associated with a complex periodic function ψ(r) such that ρ(r) = | ψ(r)|2. The pair of variables {EH, ΨH}, called twin variables, play a crucial role in the subsequent theory. The phase relations are enhanced by using pairs of non-negative `twin determinants' {Dm, D′m+1}, Dm is a classical Karle–Hauptman (K-H) determinant involving E and D′m+1 is generated by bordering Dm with an (m + 1)th row and column containing Ψ. The associated regression equation establishes a relation between E and Ψ. Furthermore, a remarkable expression is obtained for the gradient of the phase given by the classical tangent formula, as well as for the gradients involved in the related formulae pertaining to the Ψ set. The flexibility of the algorithm is illustrated by the ab initio transferring to the Ψ set of the a priori known information (such as the whole set of the observed moduli), before starting the sequential phase determination of the unknown phases. All constraints are included in a global minimization function. Analytical formulae are given for the gradient of this function with respect to the Ψ set of variables. In the final result, the Ψ set is simultaneously compatible in the least-squares sense with the whole set of observed SF and with various other constraints and phase relations. Application to two known structures permitted testing the different parts of the algorithm.