The application of phase relationships to complex structures. III. The optimum use of phase relationships

TLDR

The study improves the automatic crystal‑structure solution procedure in the LSAM program. The authors enhance LSAM by developing symbol relationships, initiating fresh symbolic‑addition cycles, permitting controlled failures for multisolution, and implementing the MULTAN phase‑permutation program with a weighted tangent formula to accept phases immediately. These modifications yield a fourfold speed increase in generating complete phase sets and introduce an absolute figure of merit to assess the likelihood of correct multisolution phase sets.

Abstract

An improvement is described in the automatic procedure for solving crystal structures incorporated in the computer program LSAM . The development of signs from an initial set containing symbols is carried only as far as is necessary to establish strong relationships between the symbols. The information so gained is used in a fresh beginning of the symbolic-addition process. Some failure of relationships between symbols is allowed to give a multisolution method. A phase-permutation computer program for non-centrosymmetric structures, MULTAN , incorporates a weighted tangent formula. This is of the form {\rm tan}\varphi_{\bf h} = {{\sum_{\bf h'}w_{\bf h'},w_{\bf h-h'}|E_{\bf h'}E_{\bf h-h'}| \sin (\varphi_{\bf h'} + \varphi_{\bf h-h'})}\over{\sum_{\bf h'}w_{\bf h'},w_{\bf h-h'}|E_{\bf h'}E_{\bf h-h'}| \cos (\varphi_{\bf h'} + \varphi_{\bf h-h'})}} = {{T_{\bf h}}\over{B_{\bf h}}} and w_{\bf h} = {\rm tanh}\{ \sigma_3\sigma_2^{-3/2}|E_{\bf h}|(T_{\bf h}^2 + B_{\bf h}^2)^{1/2}\}.All phases are accepted as soon as they are found with the associated weight. This gives a fourfold increase in speed in development of the complete phase set. An absolute figure of merit is described to indicate probably correct phase sets for multisolution methods.