Estimation of quartet phase sums from a new joint probability distribution of normalized structure factors

Abstract

A new joint probability distribution of normalized structure factors is derived for equal-atom structures in space group P1. From this general distribution, a series expansion, the conditional joint probability distribution of the quartet phase sum is obtained, when restrictive conditions among the reciprocal vectors are imposed. The main difference from existing quartet distributions is the possibility of enclosing higher-order terms to any given order of N, although an approximation employed in the derivation limits the number of them considerably. The higher-order terms present are easily employed in the series since the determination of their explicit appearance has been automated: a computer program derives the terms up to a desired order and stores them in a coded form. In general, the incorporation of selective terms up to order N-3 appears to yield sufficient convergence. Only high |E| values or a low N value may necessitate the use of higher-order terms. Test results show that, in contrast to results from the quartet distributions of Hauptman and Giacovazzo, systematic estimation errors are hardly present, while absolute estimation errors are reduced as well.