Rigorous treatment of polytopal rearrangements reveal surprising complexity of stereoisomerism configuration landscapes

Previously we posited that a <i>systematic</i> and <i>general</i> description of stereoisomerism could be based upon the principles of the polytopal rearrangement model. The most daunting challenge to this end is to comprehensively describe all possible geometries for arbitrary <i>n</i>-coordinate centres, AB _<i>n</i> , and for this we have developed a physically-inspired rigorous approach. Here we demonstrate the detailed application of this approach to the AB₄ system focussing on <i>e</i>-symmetric distortions of tetrahedral geometry to generate an angular configuration space (the AB₄ <i>T</i>-4 E-mode space). Analytic expressions for the A-B ^<i>i</i> unit vector configurations are presented and the resulting spherical (2D) configuration space is shown to exhibit the symmetries of a disdyakis dodecahedron. Detailed inspection and analysis of the angular configuration space reveals that, in addition to the expected (<i>T</i>-4-<i>R</i>) ⇌ (<i>T</i>-4-<i>S</i>) pseudorotation, it features numerous "orientation permutations" that are also pseudorotations. Through the worked examples of SiF₄, XeF₄, and a chiral silane, we generate the corresponding potential energy surfaces and examine the wider implications. We also outline experimental opportunities for investigating the unexpected configuration space complexity that this work has revealed. This rigorous and mathematically comprehensive approach and framework is part of the Polytope Formalism of isomerism and molecular structure.