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

Two new concepts for molecular solids, 'local similarity' and 'boundary-pre serving isometry', are defined mathematically and a theorem which relates t hese concepts is formulated. 'Locally similar' solids possess an identical short-range structure and a 'boundary-preserving isometry' is a new mathema tical operation on a finite region of a solid that transforms mathematicall y a given solid to a locally similar one. It is shown further that the exis tence of such a 'boundary-preserving isometry' in a given solid has infinit ely many 'locally similar' solids as a consequence. Chemical implications, referring to the similarity of X-ray powder patterns and patent registratio n, are discussed as well. These theoretical concepts, which are first intro duced in a schematic manner, are proved to exist in nature by the elucidati on of the crystal structure of some diketopyrrolopyrrole (DPP) derivatives with surprisingly similar powder patterns. Although the available powder pa tterns were not indexable, the underlying crystals could be elucidated by u sing the new technique of ab initio 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 qu inacridones. The 'boundary-preserving isometry' is presented as a predictiv e tool for crystal engineering purposes and attempts to detect it in crysta ls of the Cambridge Structural Database (CSD) are reported.