Topological Rotational Strengths as Chirality Descriptors for Fullerenes

Abstract

A graph-theoretical procedure is proposed for assigning a chirality descriptor (the topological sign tau(+) or tau(-)) to each enantiomer of a chiral polyhedron, polyhedral molecule or graph, independently of any vertex labelling scheme. Model Cartesian coordinates and rotational strengths are obtained using only adjacency information; a generalised HOMO-LUMO rotational strength is used to associate a sign with a Schlegel diagram and the corresponding three-dimensional structure, polyhedron or molecule. The topological sign gives an unambiguous way of communicating the identity of an enantiomer. The mean-square topological rotational strength is a possible measure of the chirality content of a polyhedral graph or structure.

Chemical crystallography of fullerenes

“…The number of the fullerene-type polyhedra (M) complying with the IPR as a function of the number of atoms in the carbon cage (n) is given below: n 60 62±68 70 72 74 A topological nomenclature of fullerenes taking into account their probable chirality has been suggested. 17 The relative stabilities of different fullerenes have been thoroughly studied theoret-ically by methods of quantum chemistry. 18,19 In particular, the calculations predicted instability of C 72 and C 74 fullerenes (complying with the IPR) as neutral molecules.…”

Chemical Structure Indices in In Silico Molecular Design

“…Randic has introduced graph theorey based two-dimensional chirality descriptors with possible extension to three-dimensional chirality [95]. Rassat et al have proposed topological rotational strengths as chirality descriptors for polyhedral graphs [96]. QSAR models of enantioselective compounds have been developed with chirality sensitive molecular descriptors [97].…”

Geometry of nanostructures and eigenvectors of matrices

“…Based on this topological information we have to determine the Descartes coordinates of the atoms. For fullerenes, nanotubes and nanotori the topological coordinate method supplies the necessary information . This method applies the so‐called bi‐lobal eigenvectors of the adjacency matrix for constructing Descartes coordinates for fullerenes .…”

“…A very similar method was found by mathematicians for embedding graphs in Euclidean spaces (). It was found however, that the construction of nanotori needs four bi‐lobal eigenvectors of adjacency matrix ().This method was successfully applied to propose a procedure, yielding a single configurationally descriptor for any chiral fullerene and its corresponding Schlegel diagram as well (). It fails, however, for nanotube junctions, nanocoils and other nanostructures.…”