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 O_h octahedra to elongated D_4h 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 D_2h 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 D_4h tetragonal bipyramid isomers, the 8 face midpoints represent distortions to D_3d trigonal antiprism isomers, and the 12 edge midpoints represent lower symmetry rhombic D_2h intermediates. In the case of Jahn-Teller T ? h distortions of regular I_h icosahedra, the corresponding top-rep becomes a regular icosahedron in which the 12 vertices represent distortions to pentagonal D_5d isomers, the 20 face midpoints represent distortions to trigonal D_3d isomers, and the 30 edge midpoints represent D_2h 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 T_h isomers and D_3d intermediates, respectively.