TWIN VARIABLES AND DETERMINANTS IN DIRECT-METHODS

In most algorithms of direct methods, the variables are the normalized structure factors (SF) E(H). An alternative set of variables is propo sed which provides more flexibility for handling, in a single algorith m, phase relationships and direct-space constraints, as well as the co mplete set of diffraction data. This set of variables Psi(H) consists of SF associated with a complex periodic function psi(r) such that rho (r) = \psi(r)\(2). The pair of variables {E(H), Psi(H)}, called twin v ariables, play a crucial role in the subsequent theory. The phase rela tions are enhanced by using pairs of non-negative 'twin determinants' {D-m,D-m+1'}; D-m is a classical Karle-Hauptman (K-H) determinant invo lving E and D'(m+1) is generated by bordering D-m with an (m+1)th row and column containing Psi. The associated regression equation establis hes a relation between E and Psi. Furthermore, a remarkable expression is obtained for the gradient of the phase given by the classical tang ent formula, as well as for the gradients involved in the related form ulae pertaining to the Psi set. The flexibility of the algorithm is il lustrated by the ab initio transferring to the Psi set of the a priori known information (such as the whole set of the observed moduli), bef ore starting the sequential phase determination of the unknown phases. All constraints are included in a global minimization function. Analy tical formulae are given for the gradient of this function with respec t to the Psi set of variables. In the final result, the Psi set is sim ultaneously compatible in the least-squares sense with the whole set o f observed SF and with various other constraints and phase relations. Application to two known structures permitted testing the different pa rts of the algorithm.