Increasing the power of classical direct methods of solving crystal structures. The χ2 tangent formula

Abstract

The conventional tangent formula can be derived from the fact that Z(Φ), i.e. the integral ∫vρ3dV expressed as a function of the collectivity Φ of phases of the largest E's, is a positive maximum for the correct Φ. In practice, however, refinement of phases with the tangent formula can also lead, for certain space groups and atomic arrangements, to false maxima of Z(Φ). To reduce the number of such false maxima, Z(Φ) is multiplied by the penalty function P(Φ) = [1 + λ(1 − χ2(Φ)/n)], with λ and n being, respectively, a suitable weighting factor and the number of terms in the sum of a χ2 statistic that takes into account the conditional probability distributions of all the triplets involving two largest E's plus a largest, a medium-large or a weakest one. It will be shown how the combined function Z(Φ) × P(Φ) can be maximized by a tangent formula. The application of this new restrained tangent formula will be illustrated on the basis of a representative example.