Exact Conditional Probability Density Function of Three-Phase Invariant in P1: Some Theoretical and Practical Considerations

Abstract

The applicability of probabilistic approaches to crystallography was first realized by Wilson (1949) in his study of intensity statistics, and found its place as the fundamental mathematical avenue to direct methods of phase determination (e.g. Hauptman & Karle, 1953; Cochran & Woolfson, 1955; Cochran, 1955; see also the Chapters by M. M. Woolfson on “The Cochran Distribution” and by S. Fortier and I. R. Castleden on “Some Applications of Probability Theory in Direct Methods”, in this book). The actual probabilistic formalisms, used until 1984, have been based either on low-order approximations afforded by the central limit theorem, as invoked, e.g., by Wilson (1949) and Cochran (1955), or constitute higher approximations (e.g. Hauptman & Karle, 1953; Klug, 1958; Naya, Nitta & Oda, 1964,1965 - to mention a few early ones) which can be cast into the formalism of Gram-Charlier or Edgeworth expansions. These expansions are rather lengthy and have poor convergence properties. In fact, most structure-solving packages are still based on the central limit theorem approximations, which perform well for nearly equal-atom structures, with a not too small number of atoms in the asymmetric unit and with no significant population of special positions or other sites imparting rational dependence (Karle & Hauptman, 1959). There remain, of course, small- and intermediate-molecule cases in which these approximations are not adequate. This presentation includes a description of some of our work on the development of exact probabilistic approaches, which properly account for the space-group symmetry and (arbitrarily heterogeneous) atomic composition.