Graph theory and the Jahn–Teller theorem

Abstract

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 edgeweighted 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.

Nanomaterials design for super-degenerate electronic state beyond the limit of geometrical symmetry

“…To interpret these results theoretically, simple tight-binding models 2 , 30 , 31 were constructed and analyzed as follows (Fig. 2 ).…”

Some Observations on Triplet Ground-States in the Context of ‘Topological’ (HLPM) Ring-Currents in Conjugated Systems

“…In the course of performing calculations on individual molecules by means of this HLPM approach a problem has, from time-to-time, been encountered whenever, on application of the Aufbau Principle, [11,[20][21][22][23][24] the π-electronic ground-state configuration of the conjugated hydrocarbon in question is found to be a triplet ground-state. We have occasionally noted specific examples of this phenomenon as we have serendipitously come across them in our previously reported work.…”

“…approach [9,10] is frequently cast, these are the eigenvalues and eigenvectors, [11,21] respectively, of the vertex-adjacency matrix [8] of that associated labelled molecular-graph. [8] (b) Then the Aufbau process [20][21][22][23][24] is applied, during the course of which the available π-electrons are distributed amongst these calculated energy-levels in order to determine a π-electron ground-state configuration for the hydrocarbon in question. If the Aufbau procedure has been successful -which is not always the case, as is discussed in Refs.…”

The clone cover

“…Along these lines, Clone(K n ) was considered in [7], where it was shown that with few exceptions, Clone(K n ) is the only m-sheeted covering graph of K n , for m ≤ n − 1, such that the full automorphism group of K n has a lift. On the other hand, Clone(K 5 ) was studied in connection with a graph-theoretical interpretation of the Jahn-Teller effect [2]. In order to clarify the role of degenerate eigenvalues (that is, eigenvalues having higher multiplicities) in the Jahn-Teller distortion, graphs with symmetry group S 5 were sought [3].…”