Symmetry-Itemized Enumeration of Cubane Derivatives as Three-Dimensional Entities by the Partial-Cycle-Index Method of the USCI Approach

Abstract

Abstract By placing chiral and achiral proligands on the eight positions of a cubane skeleton of Oh, resulting cubane derivatives are counted as three-dimensional (3D) structural isomers by the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, 1991; S. Fujita, Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers, University of Kragujevac, Kragujevac, 2007), where the numbers of 3D structural isomers with given constitutional formulas are itemized with the subsymmetries of Oh. For this purpose, partial cycle indices with chirality fittingness (PCI-CFs) for respective subgroups of Oh are calculated by using unit subduced cycle indices with chirality fittingness (USCI-CFs) and an inverse matrix of the mark table of Oh. After introducing ligand-inventory functions into the PCI-CFs, expansions of the resulting equations provide us with generating functions for giving the numbers of 3D structural isomers. A Maple program source for counting cubane derivatives as 3D structural isomers is given as an example of practical calculation for listing the resulting data in tabular forms.