1-Phase seminvariants and Harker sections

Abstract

The algebraic relations between 1-phase seminvariants of first rank and Harker sections recently described by Ardito, Cascarano, Giacovazzo and Luic [Z. Kristallogr. 172 (1985) 25 -34)] are further on discussed. A special least-squares procedure is proposed for discriminating spurious peaks among those lying on Harker sections and for improving positional and thermal parameters of heavy atoms. The first applications show that 1-phase seminvariants of first rank may be reliably estimated also for crystal structures with medium weight atoms.Symbols m: order of the space group (it equals the number of symmetry operators) Ν: number of atoms in the unit cell t : number of atoms in the asymmetric unit Γ; ξ (xj, yj, zj): positional vector of the yth atom Cp = (R p , T p ): pth symmetry operator. R p is the rotational part, T p the translational part CpTj = RpT; + T p : positional vector of the atom symmetry equivalent to r, I: identity 3x3 operator u/Cj, C q ) = (C s -C,) τ/, interatomic vector between two symmetry equivalent atoms HS(C s , C 4 ): Harker section containing interatomic vectors u/C s , C q ) for .7 = 1,2,...

Integration of Patterson information into direct methods. I. The theoretical background

“…It is, however, anticipated that Harker sections can provide additional information for triplet estimation. In particular, Harker sections will prove highly efficient for the estimation of onephase seminvariants of first rank; this is in accordance with some recent algebraic results (Ardito, Cascarano, Giacovazzo & Lull, 1985;Cascarano, Giacovazzo, Lui~, Pifferi & Spagna, 1987).…”

SIR2004: an improved tool for crystal structure determination and refinement

“…In summary, the combination procedure does not lead to any substantial improvement since (i) when a very good Patterson trial is present (say with MPE < 40 ) the Patterson deconvolution method itself is able to solve the protein; (ii) when all the Patterson trials have MPE > 80 , the structure solution relies on the ST procedure; The combination deteriorates the quality of the original ST trial; (iii) when the best Patterson and tangent trials have comparable MPE, the combination produces negligible improvements An alternative procedure has been explored, which combines only one-phase seminvariants. However, we have verified that (a) the seminvariant phase estimates through Harker sections (Cascarano et al, 1987;Ardito et al, 1985) are systematically worse than those derived by the Patterson deconvolution techniques used in SIR2004; (b) the combination procedure is able to improve the ST seminvariant phase subset, but it has a negligible effect on the whole phase set.…”