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

## TL;DR

This paper introduces a new mathematical method to identify distinct polyhedral shapes in metal complexes, revealing previously unknown coordination geometries for 6- and 7-coordinated structures.

## Contribution

A novel mathematical framework for identifying thermally distinguishable polyhedral shapes in coordination chemistry, revealing new geometries overlooked in prior literature.

## Key findings

- The digonal anticupola (DAC-6) is a prevalent but previously overlooked hexacoordinated structure.
- 14 thermally distinguishable polyhedral shapes were identified for heptacoordinated complexes, including chiral and novel forms.
- New coordination geometries were confirmed to occur frequently in over 42,000 structures from the Cambridge Structural Database.

## Abstract

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.