A theory of phase determination for the four types of non-centrosymmetric space groups 1P222, 2P22, 3P₁2, 3P₂2

Abstract

Joint probability distributions and relevant expected values and variances are obtained for selected (but typical) non-centrosymmetric space groups belonging to the four types 1P222, 2P22, 3P12, 3P22. These lead to formulas for phase determination the analysis and interpretation of which constitute the major goal of this paper. The analysis is strongly dependent on the theory of invariants and seminvariants, and the agreement between this theory and certain consequences of the probability theory is noteworthy.

Electron Crystallography of Small Organic Molecules: Criteria for Data Collection and Strategies for Structure Solution

“…To re-determine the X-ray crystal structure of triphenylene by direct methods, a combined package of tangent-formula-based (Karle & Hauptman, 1956) software [e.g. SIR88 (Burla et al, 1989)] was able to find a fragment containing all but one atomic position (as well as an additional incorrect position).…”

“…An initial attempt to solve the crystal structure of triphenylene from measured three-dimensional electron diffraction data was made by a combination of symbolic addition (Karle & Karle, 1966) (for centrosymmetric zones) and the Sayre (1952) equation or the tangent formula (Karle & Hauptman, 1956) (for extension into the noncentric set). When this was shown to be unsatisfactory, another approach, searching for an optimal molecular packing that would minimize the lattice energy, was employed (Gavezzotti, 1991;Voigt-Martin et al, 1992;Voigt-Martin, Simon et al, 1995).…”

The crystal structure of 2-dimethylsulfuranylidene-1,3-indanedione

“…The position of the sulfur atom was determined from a sharpened Patterson function, and phase angles computed from the sulfur atom parameters were refined by the tangent formula (Karle & Hauptman, 1956). An E map calculated from the refined phases revealed the positions of all atoms other than hydrogen.…”

Conformation of the tRNA minor constituent dihydrouridine

“…We collected about 2000 intensity data on an automatic Siemens diffractometer with N&filtered Cu radiation and converted them to normalized structure factors (E's). The structure was determined by direct methods involving a cyclic application of the tangent formula [4] combined with multiple solution techniques [S] . A three-dimensional E-map computed with the phases derived from the …”