The periodic table epitomizes chemistry and evolving representations of chemical periodicity should reflect the ongoing advances in chemistry. In this respect, the traditional Mendeleev-style table appears distinctly suboptimal for describing a variety of important "higher-order" periodicity patterns that have become apparent in the post-Mendeleevian quantal era. In this article we analyze the rigorous mathematical origins of chemical periodicity in terms of the quantal nodal features of atomic valence orbitals. We propose a variety of alternative 2D and 3D display symbols, tables, and models that are better able to exhibit the chemical richness of generalized isodonor, isoacceptor, and isovalency kinships, while also drawing closer connection to the instrinsic spectroscopic radial (n) and angular (l ) descriptors of atomic valence shells. These improved periodicity displays are also able to exhibit the recently discovered "Lewis-like" bonding analogies between main-group and transition-group elements, including the remarkably extended possibilities for high-order metal–metal bonding (e.g., the recently synthesized Cr–Cr quintuple bond). We argue that alternative display topologies such as a 2D "left-step" or "step-pyramid" table or a novel 3D "periodic towers" model can fruitfully supplement or supplant the standard IUPAC-approved table by better emphasizing such higher-order patterns of chemical association and reactivity, rather than the physical resemblances of standard-state elemental substances (as emphasized in the time-honored Mendeleev-style groupings).
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