Sphericities of Cycles. What Pólya’s Theorem is Deficient in for Stereoisomer Enumeration

Three methods and their extended versions for enumerating stereoisomers, which have been developed by modifying or simplifying Fujita’s USCI (unit-subduced-cycle-index) approach based on the concept of sphericities of orbits in order not to take account of symmetry itemization, are applied to the enumeration problem of ethane and propane derivatives. The proligand method and its extended version based on the concept of sphericities of cycles are also applied to the same enumeration problems. These results are compared with the results based on Pólya’s theorem (and Pólya’s corona). Thereby, it is shown that Pólya’s theorem enumerates chemical compounds as graphs, not as stereoisomers (3D chemical structures) if all of the permutations corresponding to proper and improper rotations are adopted. Moreover, if the permutations corresponding to proper rotations are adopted, Pólya’s theorem enumerates chemical compounds as chiral ones, where enantiomeric relationship and achiral nature (i.e., self-enantiomeric relationship) are not characterized properly. The two types of applications of Pólya’s theorem do not take account of improper rotations properly. Thereby, what Pólya’s theorem is deficient in is concluded to be the concept of sphericity