21-25 August 2012
Portorož, Slovenia
UTC timezone
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The Jahn-Teller theorem for graphs

Presented by Prof. Arnout CEULEMANS
Type: Keynote Lecture

Content

Molecular stability and electronic degeneracy do not form an evident combination. In degenerate states of highly symmetric molecules, there is a mismatch between the symmetries of electronic and nuclear charge distributions. This imbalance gives rise to a spontaneous distortion of the nuclear frame to a lower symmetry, which lifts the degeneracy. The effect is known as the Jahn-Teller (JT) effect. Based on the analogy between molecules and graphs, the following conjecture was made: 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.1 In this lecture we will establish the strict analogy between JT instabilities in molecules and graphs for the special case of the simplex structures, represented by the complete graphs, Kn. The proof is based on the irreducible representations of the symmetric group, Sn, and consists of two parts: a molecular part in which the simplex is treated as a geometric structure, consisting of equivalent sites, which can undergo distortions, and a graph-theoretical part, which derives the symmetries of changes of the edge weigths. The complete graph is found to represent the ideal case, where the Jahn-Teller effect is 'symmetry-precise'. Deviations from the ideal case may be 'symmetry-deficient'. This is the case when not all of the coupling operators that give rise to the lifting of the degeneracy, have a counterpart in the space of edge distortions. Symmetry-deficiency raises intriguing questions about the distortivity of graphs with degenerate eigenvalues. 1 Graph theory and the Jahn-Teller theorem. Ceulemans, A.; Lijnen, E.; Fowler, P. W.; Mallion, R. B.; Pisanski, T. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 468, issue 2140, pp. 971-989 (2012)

Place

Location: Portorož, Slovenia
Address: University of Primorska, Faculty of Tourism Studies, Obala 11a, SI-6320 Portorož - Portorose, Slovenia
Room: VP1

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