Topology and the Chemical Elements: A Spectral–Topological Approach to Quantum Chemical Anomalies

Periodic anomalies in the electronic configurations of transition and related metals have long re- sisted straightforward explanation. Here it is shown that these so-called exceptions are not accidental but arise as quantized, topologically protected features of the atomic spectrum. By formulating the Hamiltonian in terms of spectral branches over parameter space, each anomalous configuration is identified with a singularity—indexed by invariants such as Berry phase, Chern number, or winding number—between distinct symmetry-adapted states. This spectral–topological approach unifies ob- served and predicted anomalies within a single mathematical structure, recasting the periodic table as a landscape of topological transitions rather than empirical rules. The results provide a predictive ex- planation for electronic anomalies and suggest new avenues for the design of materials and quantum systems with tailored properties.