8. Bonding in Rings and Clusters

The chapter presents a theory with a qualitative flavor, Tensor Surface Harmonics (TSH) , that demonstrated its elegant power for explaining the bonding in clusters in the era preceding the computer revolution and the wider availability of the quantum chemical computer programs. It uses symmetry reasons, based on the parentage of delocalized molecular orbitals from spherical harmonics functions and their derivatives, when a cluster is more or less approximated with a pseudo-globular pattern. The derivative term, in the above phrase, literally meant the mathematical operation of differentiation, applied on spherical harmonics, as function of polar coordinates, diversifying the basis of elements in which the structural description can be conceived. Within TSH, quasi-spherical molecules can be regarded as giant atoms. The realization of stable occupation schemes in quasi-degenerate orbital patterns draws rationales for stereochemistry and compositional “magic numbers” in cluster chemistry. The axial symmetry, explaining the electronic structure of rings, in a manner similar to Hückel’s crude and clear approximation, can be regarded as a particularization of TSH. To be distinguished from the classical qualitative use of TSH, we revisit elements of this theory with the support of computational methods exposed in the previous chapters, proving and enlarging the illuminating power of this paradigm. One may note that we met and exploited the spherical harmonics in several instances, throughout this book, starting with the well-known encounter as the angular part of atomic orbitals, continued with the lesser known use in the parameterization of two-electron integrals (that enabled a hands-on lucrative approach of many electron atoms) and culminating with Ligand Field Theory, where the spherical harmonics are cornerstones of phenomenological Hamiltonian and effective bases. The Tensor Surface Harmonics is approached here as a welcome completion of the conceptual heuristics and phenomenological modeling leverage emerging from spherical symmetry. Finally, the aromaticity concept may be linked, in poly-aromatic hydrocarbon (PAH) molecules, with the isomers constitution. This connection appears on 16 PAHs molecules and their isomers, by combining the topo-reactivity method of specific-bond-by-adjacency procedure with Kekulé, Clar, and Fries benzenoid descriptions. The resulting approach determines a new method of classification for aromatic molecules, in general, and may be used to predict which molecules are most likely to adopt the most aromatic (Kekulé + Clar + Fries) conformation in chemical reactions.