Jahn-Teller distortion in polyoligomeric silsesquioxane (POSS) cations

We investigated the symmetry breaking mechanism in cubic octa-tert-butyl silsesquioxane and octachloro silsesquioxane monocations (Si8O12(C(CH3)3)8(+) and Si8O12Cl8(+)) using density functional theory (DFT) and group theory. Under Oh symmetry, these ions possess (2)T2g and (2)Eg electronic states and undergo different symmetry breaking mechanisms. The ground states of Si8O12(C(CH3)3)8(+) and Si8O12Cl8(+) belong to the C3v and D4h point groups and are characterized by Jahn-Teller stabilization energies of 3959 and 1328 cm(-1), respectively, at the B3LYP/def2-SVP level of theory. The symmetry distortion mechanism in Si8O12Cl8(+) is Jahn-Teller type, whereas in Si8O12(C(CH3)3)8(+) the distortion is a combination of both Jahn-Teller and pseudo-Jahn-Teller effects. The distortion force acting in Si8O12(C(CH3)3)8(+) is mainly localized on one Si-(tert-butyl) group, while in Si8O12Cl8(+) it is distributed over the oxygen atoms. The main distortion forces acting on the Si8O12 core arise from the coupling between the electronic state and the vibrational modes, identified as 9t2g + 1eg + 3a2u for the Si8O12(C(CH3)3)8(+) and 1eg + 2eg for Si8O12Cl8(+).

Graph theory and the Jahn–Teller theorem

The Jahn–Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group.