Inorganic Frontier Orbital Theory: How Penetration, Shielding and Relativity Reshape the Periodic Table

For many decades, persistent irregularities in atomic and molecular electronic structure have resistedany unifying explanation. Classical and modern work by Mulliken, Coulson, Roothaan, Fukui, Hoffmann,Pyykkö, Schwerdtfeger and others mapped emblematic cases: the 4d/5s inversion in Pd, the 6s/5d frontierreordering in Pt and Au, the collapse of the 4f-derived HOMO in Eu/Yb, the 5f/6d competition in U/Pu,and the 𝜋/𝜎 anomalies in CO, NO+ and O2. What remained unclear was whether these “exceptions” sharea common microscopic mechanism or are unrelated curiosities. Here these phenomena are interpreted as manifestations of a single inorganic-chemical mechanism:competition between frontier orbitals with very different penetration, shielding and relativistic response,leading to a curvature-driven reordering of HOMO and LUMO. Instead of the full operator calculus,the analysis focuses on descriptors that can be extracted from modern relativistic electronic-structurecalculations: (i) energy branches of candidate frontier orbitals as functions of effective nuclear charge orchemical environment; (ii) their curvature and near-degeneracy patterns; and (iii) a reduced 2 × 2 mixingproblem that encodes the degree of 𝑠/𝑝–𝑑/ 𝑓 or 𝜎–𝜋 rehybridisation. Within this perspective, anomalies arise whenever two frontier orbitals approach closely in energywhile possessing strongly contrasting radial penetration and angular character. At such points, the morepenetrating or more relativistically stabilised orbital gains control of the frontier region, enforcing anabrupt change of HOMO/LUMO identity. The 4d/5s collapse in Pd, the 6s/5d interplay in Pt/Au, the 4f/6sbalance in Eu/Yb, the 5f/6d covalency in U/Pu, and the 𝜋/𝜎 inversions in CO, NO+ and O2 emerge aschemically transparent instances of the same mechanism. Periodic trends, oxidation-state flexibility, unusual covalencies and small-molecule 𝜋/𝜎 anomaliesthus appear as local projections of a single frontier-orbital manifold. The geometry of competing orbitalchannels provides a spectrally informed, inorganic theory of frontier orbitals across the periodic table.