Local similarity in organic crystals and the non-uniqueness of X-ray powder patterns

• Published in: Acta crystallographica. Section B, Structural science Vol. 55; no. 6; pp. 1075 - 1089
• Main Authors: Karfunkel, Heinrich, Wilts, Heike, Hao, Zhimin, Iqbal, Abul, Mizuguchi, Jin, Wu, Zhijian
• Format: Journal Article
• Language: English
• Published: 5 Abbey Square, Chester, Cheshire CH1 2HU, England International Union of Crystallography 01.12.1999; Blackwell
• Subjects: Boundary-preserving isometry; Condensed matter: structure, mechanical and thermal properties; Crystal structure prediction; Crystalline state (including molecular motions in solids); Exact sciences and technology; Local similarity; Physics; Single-crystal and powder diffraction; Structure of solids and liquids; crystallography; Synthomorphism; Theory of crystal structure, crystal symmetry; calculations and modeling; X-ray diffraction and scattering; X-ray powder patterns; Isometry; Powder pattern; Nitrogen heterocycle; Transforms; Bicyclic compound; Molecular crystals; Theoretical study; Lactams; XRD; Organic compounds; Crystal structure
• ISSN: 0108-7681; 1600-5740
• DOI: 10.1107/S0108768199006114

Two new concepts for molecular solids, `local similarity' and `boundary‐preserving isometry', are defined mathematically and a theorem which relates these concepts is formulated. `Locally similar' solids possess an identical short‐range structure and a `boundary‐preserving isometry' is a new mathematical operation on a finite region of a solid that transforms mathematically a given solid to a locally similar one. It is shown further that the existence of such a `boundary‐preserving isometry' in a given solid has infinitely many `locally similar' solids as a consequence. Chemical implications, referring to the similarity of X‐ray powder patterns and patent registration, are discussed as well. These theoretical concepts, which are first introduced in a schematic manner, are proved to exist in nature by the elucidation of the crystal structure of some diketopyrrolopyrrole (DPP) derivatives with surprisingly similar powder patterns. Although the available powder patterns were not indexable, the underlying crystals could be elucidated by using the new technique of abinitio prediction of possible polymorphs and a subsequent Rietveld refinement. Further ab initio packing calculations on other molecules reveal that `local crystal similarity' is not restricted to DPP derivatives and should also be exhibited by other molecules such as quinacridones. The `boundary‐preserving isometry' is presented as a predictive tool for crystal engineering purposes and attempts to detect it in crystals of the Cambridge Structural Database (CSD) are reported.