Symmetry-Itemized Enumeration of Cubane Derivatives as Three-Dimensional Entities by the Elementary-Superposition Method of the USCI Approach

Abstract

Abstract Cubane derivatives with a given chemical formula and a given symmetry are enumerated by applying the elementary-superposition method (S. Fujita, Theor. Chim. Acta1992, 82, 473–498) of the unit-subduced-cycle-index (USCI) approach to a cubane skeleton of the point group Oh. A cycle index (CI) for a regular or an irregular case of chiral and achiral ligands is calculated in accord with such a given chemical formula. The CI is superposed elementarily onto respective subduced cycle indices with chirality fittingness (SCI-CFs), which are calculated by starting from unit subduced cycle indices with chirality fittingness (USCI-CFs). Thereby, the numbers of fixed points (derivatives) are obtained with respect to the respective SCI-CFs to give a fixed-point vector (FPV). The resulting FPV is multiplied by an inverse matrix of the mark table of Oh to give an isomer-counting vector (ICV), which contains the numbers of 3D-structural isomers in an itemized fashion with respect to point-group symmetries. The concept of prochirality is examined by using cubane derivatives enumerated by the ICV.