Incommensurate symmetry interactions

Efficient self-assembly occurs when there is a match between the geometric or stereochemical preferences of the components, e.g. the coordination polyhedron of a metal ion and the possible binding site arrangement(s) on a ligand. These pairings are sometimes referred to as incommensurate symmetry interactions or interactional algorithms... 

The incommensurate symmetry interaction approach dissects a target assembly into its symmetry elements and considers the necessary geometric relationships among these elements. For example, a tetrahedral assembly contains four C3 and six C2 axes (Figure 2). These incommensurate symmetry axes must be simultaneously satisfied to form a tetrahedral structure. The structure may be formed from the combination of four Cs-symmetric tris-bidentate chelated... 

Another technique which has been used to prepare 3-D polyhedra is Raymond s symmetry-interaction method.19 This strategy derives from the realization that many natural supramo-lecular assemblies are formed in a symmetry-driven process which relies on incommensurate lock-and-key interactions. By closely examining a desired polyhedral structure, one may establish the relevant symmetry interactions and their associated geometric relationships, and then design metal-ligand systems capable of fulfilling them. 

Each substructure in the inclusion compound can be considered in terms of an incommensurately modulated "basic" structure. The basic structure has 3D lattice periodicity and crystallographic symmetry described by a 3D space group. The incommensurate modulation to a given basic structure represents the structural perturbations that arise from its interaction with the other substructure. Thus, the basic structure is an hypothetical entity and represents the structure to which the real substructure would relax if there were no interaction... 

Composite crystals are crystalline structures that consist of two or more periodic substructures, each one having its own 3-D periodicity to a first approximation. The symmetry of each of these subsystems is characterized by one of the 230 space groups. However, owing to their mutual interaction, the fine structure consists of a collection of incommensurately modulated subsystems. All known composite stractures to date have at least one lattice direction in common and consist of a maximum of three substructures. There are three main classes ... 

The concept of a superlattice can also be applied to regular overlayers of foreign adsorbed atoms (see Section 2.2). If the coupling forces between atoms exceed the atom-surface interaction forces, they can form a structure which is not related to the symmetry of the substrate surface incommensurate structure or incoherent lattice). In such a case det C is an irrational number (Fig. 2.2c). 

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