Type-Itemized Enumeration of Five Types of Stereoisograms and Their Simplified Diagrams for Characterizing Cubane Derivatives

Abstract

Abstract For discussing stereochemistry of cubane derivatives, the concept of stereoisograms controlled by the RS-stereoisomeric group $\boldsymbol{{O}}_{\boldsymbol{{h}}\tilde{\boldsymbol{{\sigma}}}\skew2\hat{\boldsymbol{{I}}}}$ (order 98) is formulated by the extension of the point group Oh (order 48). The vertical directions of a stereoisogram are concerned with chirality as the first kind of handedness, which is controlled by rotations and reflections of Oh. The horizontal directions of a stereoisogram are concerned with RS-stereogenicity as the second kind of handedness, which is controlled by an RS-stereogenic group $\boldsymbol{{O}}_{{\tilde{\boldsymbol{{\sigma}}}}}$ (order 48). The diagonal directions of a stereoisogram are concerned with sclerality/asclerality, which is controlled by an LR-permutation group $\boldsymbol{{O}}_{{\skew2\hat{\boldsymbol{{I}}}}}$ (order 48). These groups are characterized by combined-permutation representations (CPRs), which are used to calculate respective cycle indices with chirality fittingness (CI-CFs) for enumeration under the GAP system. Enumerations are conducted under O, Oh, $\boldsymbol{{O}}_{{\tilde{\boldsymbol{{\sigma}}}}}$, $\boldsymbol{{O}}_{{\skew2\hat{\boldsymbol{{I}}}}}$, and $\boldsymbol{{O}}_{\boldsymbol{{h}}\tilde{\boldsymbol{{\sigma}}}\skew2\hat{\boldsymbol{{I}}}}$. Then, the enumeration results are discussed in terms of simplified diagrams. It has been proven that there are two main categories for characterizing the appearance of five types of stereoisograms. The first category is an ascleral case which is characterized by the presence of type-I and type-IV stereoisograms. In contrast, the second category is a scleral case which is characterized by the presence of type-II, type-III, and type V stereoisograms. There also exist minor cases in which the coexistence of the first and second categories is observed.