Thermally Distinguishable Polyhedral Shapes in Chemistry: 6- and 7‑Coordination.

Coordination polyhedra in metal complexes occasionally exhibit marked deviations from ideal geometric shapes, complicating their accurate characterization and symmetry assignment. We introduce a rigorous mathematical approach to establish a complete yet chemically relevant set of thermally distinguishable polyhedral shapes, TDPSs, suitable for describing coordination geometries, specifically focusing on coordination numbers 6 and 7. Anchored in Steinitz's theorem, we constructed all combinatorially distinct convex polyhedra and then optimized their spatial arrangements using a novel repulsive-type crowding potential, respecting maximum symmetry and minimal repulsion. Owing to thermal smearing, several of these polyhedral forms became experimentally indistinguishable within crystallographic uncertainties. Therefore, we classified these geometries into subsets of thermally interconvertible polyhedra, from which we established a concise set of thermally distinguishable polyhedral shapes. While some of these are already well-established, others represent previously unrecognized shapes in the coordination chemistry literature. Our comprehensive analysis of over 42,000 structures deposited in the Cambridge Structural Database confirmed the not infrequent occurrence of these new shapes. Remarkably, among the five thermally distinguishable shapes identified for hexacoordinated complexes, we identified the digonal anticupola (DAC-6) as a prevalent structure previously overlooked in the coordination chemistry literature. For heptacoordinated complexes, 14 thermally distinguishable polyhedral shapes emerged, including chiral and previously unrecognized shapes that we predominantly identified in lanthanide and alkali-metal complexes. This study thus establishes a robust theoretical foundation, enabling refined structural characterization, more precise symmetry assignments, and facilitating quantitative links to lattice-dynamical and thermal-transport properties (e.g., heat capacity and thermal conductivity) in molecular coordination solids.