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Topological representations of Jahn-Teller distortions

Topological representations (top-reps), which originally were developed to model molecular polyhedral isomerization processes, can be extended to depict the relationships between the polygons and polyhedra involved in Jahn-Teller (JT) distortions. Using this approach, the top-rep of the E ⊗ (b 1g + b 2g) distortion of square planar molecules to rectangle and rhombus isomers becomes a rhombus in which the vertices alternately represent distortions to the rectangle and rhombus isomers. Similarly the top-rep of the E ⊗ e distortion of regular Oh octahedra to elongated D4h tetragonal bipyramids becomes a triangle in which the vertices represent the three distinct tetragonal bipyramids from a given octahedron and the edge midpoints represent lower symmetry rhombic D2h intermediates. A regular octahedron can be used as a top-rep for the T ⊗ (e + t 2) distortions of regular octahedra if the 6 vertices represent distortions to D4h tetragonal bipyramid isomers, the 8 face midpoints represent distortions to D3d trigonal antiprism isomers, and the 12 edge midpoints represent lower symmetry rhombic D2h intermediates. In the case of Jahn-Teller T ⊗ h distortions of regular Ih icosahedra, the corresponding top-rep becomes a regular icosahedron in which the 12 vertices represent distortions to pentagonal D5d isomers, the 20 face midpoints represent distortions to trigonal D3d isomers, and the 30 edge midpoints represent D2h intermediates. A 4-dimensional analogue of the tetrahedron (i.e. the 4-simplex) can be used as a top-rep for the G ⊗ g problem and the H ⊗ g component of the H ⊗ (g + 2h) problem for JT distortions of regular icosahedra. In this case the 5 vertices and the 10 edge midpoints correspond to Th isomers and D3d intermediates, respectively.