A symmetry adapted approach to the dynamic Jahn-Teller problem: Application to mixed-valence polyoxometalate clusters with keggin structure

In this article, we present a symmetry-adapted approach aimed to the accurate solution of the dynamic vibronic problem in large scale Jahn-Teller (JT) systems. The algorithm for the solution of the eigen-problem takes full advantage of the point symmetry arguments. The system under consideration is supposed to consist of a set of electronic levels mixed by the active JT and pseudo JT vibrational modes. Applying the successive coupling of the bosonic creation operators, we introduce the irreducible tensors that are called multivibronic operators. Action of the irreducible multivibronic operators on the vacuum state creates the vibrational symmetry adapted basis that is subjected to the Gram-Schmidt orthogonalization at each step of evaluation. Finally, the generated vibrational basis is coupled to the electronic one to get the symmetry adapted electron-vibrational (vibronic) basis within which the full matrix of the JT Hamiltonian is blocked according to the irreducible representations (irreps) of the point group. The proposed approach is a part of our study of the nanosized mixed valence (MV) clusters with large number of delocalized electrons that are at the border line between quantum and classical objects. Here, we illustrate in detail the developed technique by the application to the 2e-reduced MV dodecanuclear Keggin anion in which the electronic pair is delocalized over 12 sites (overall symmetry T _d) giving rise to the ( ¹T ₂+ ¹E+ ¹A ₁)O× - (e+t ₂) ( ³T ₁+ ³T ₂)O× - (e+t ₂) combined JT/pseudo JT problems for the spin-singlet and spin-triplet states. © 2012 Wiley Periodicals, Inc. Large scale Jahn-Teller systems, especially high-nuclearity mixed-valence clusters, represent an important class of objects in molecular nanoscience. While the quantum-mechanical solution for the eigen-problem has been proposed earlier, the vibronic properties of these systems remained unknown due to the lack of efficient computational tools. Here, a symmetry-adapted approach to the dynamic vibronic problem is developed and applied to a reduced mixed-valence Keggin polyoxometalate. In this molecular metal-oxide an electronic pair is delocalized over twelve sites giving rise to a complicated vibronic problem for the spin-singlet and spin-triplet states.