The thermal rearrangements of 1,1,2,2,4,4-hexafluorospiropentane and perfluorospiropentane

The title compounds were synthesized $\text{via}$ CF₂: addition to the respective methylenecyclopropanes. The hexafluorospiropentane thermally rearranges to 2,2,4,4-tetrafluoro-1-(difluoromethylene)cyclobutane in competition with CF₂: extrusion, whereas perfluorospiropentane decomposes exclusively by CF₂: extrusion.     

Vibrational analyses employing cartesian coordinates

Geometry-optimised ab initio (STO-3G minimal basis set) calculations have been carried out for S-trans and orthogonal forms of 1,3-butadiene. The central C-C bond is shorter (1.482 Å) in the S-trans form than the orthogonal form (1.504 Å). Calculations on the orthogonal form indicate that the additional bond length decrease destabilizes the σ bond. the stabilization of the S-trans form being associated with conjugative (or π delocalization) effects.     

Variations on a Result of Bressoud

The well-known Rogers-Ramanujan identities have been a rich source of mathematical study over the last fifty years. In particular, Gordon’s generalization in the early 1960s led to additional work by Andrews and Bressoud in subsequent years. Unfortunately, these results lacked a certain amount of uniformity in terms of combinatorial interpretation. In this work, we provide a single combinatorial interpretation of the series sides of these generating function results by using the concept of cluster parities. This unifies the aforementioned results of Andrews and Bressoud and also allows for a strikingly broader family of q-series results to be obtained. We close the paper by proving congruences for a “degenerate case” of Bressoud’s theorem.