Toward Molecules with Nonstandard Symmetry

Abstract

An algebraic approach to the complex problem of designing molecules with nonstandard (noncrystallographic) symmetry is proposed. Since the five-fold, seven-fold, and higher rotational symmetry are not allowed by classical crystallography, one may use some higher dimensional spaces to construct the cells for structures which are not realizable in the 3-dimensional Euclidean space. This approach makes use of the group theory, geometric, topological, or combinatorial, symmetry, cyclic boundary conditions, k-circulant graph, n-cube, partial cube, etc.. It is shown that planar projections of higher dimensional embeddings can be used to draw any partial cube, in particular, with the generalized diamond structures, the hexagonal tiling, and the diamond crystal, inclusive. Moreover, noncrystal molecules may be constructed using any graph whose automorphism group does not necessary obey the conditions imposed on the automorphism group of a k-circulant, and such a group may induce more than one orbit of atoms (orbitals).