A Symmetry Adapted Approach to the Dynamic Jahn-Teller Problem

  • Boris TsukerblatEmail author
  • Andrew Palii
  • Juan Modesto Clemente-Juan
  • Eugenio Coronado
Part of the Progress in Theoretical Chemistry and Physics book series (PTCP, volume 23)


In this article we present a symmetry-adapted approach aimed to the accurate solution of the dynamic Jahn-Teller (JT) problem. 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 \({\Gamma }_{1},{\Gamma }_{2}\ldots {\Gamma }_{n}\) labeled by the irreducible representations (irreps) of the actual point group, mixed by the active JT and pseudo JT vibrational modes \({\Gamma }_{1},{\Gamma }_{2}\ldots {\Gamma }_{f}\) (vibrational irreps). The bosonic creation operators b +(Γγ) are transformed as components γ of the vibrational irrep Γ. The first excited vibrational states are obtained by the application of the operators \({b}^{+}(\Gamma \gamma )\) to the vacuum: \({b}^{+}(\Gamma \gamma )\vert n = 0,{A}_{1}\rangle = \vert n = 1,\Gamma \gamma \rangle\) and therefore they belong to the symmetry Γγ. Then the operators b +(Γγ) act on the set \(\vert n = 1,\Gamma \gamma \rangle\) with the subsequent Clebsch-Gordan coupling of the resulting irreps. In this way one obtains the basis set \(\vert n = 2,{\Gamma }^{{\prime}}{\gamma }^{{\prime}}\rangle\) with \({\Gamma }^{{\prime}}\in \Gamma \otimes \Gamma \). In general, the Gram-Schmidt orthogonalization is required at each step of the procedure. Finally, the generated vibrational bases are coupled to the electronic ones to get the symmetry adapted basis in which the full matrix of the JT Hamiltonian is blocked according to the irreps of the point group. The approach is realized as a computer program that generates the blocks and evaluates all required characteristics of the JT systems. The approach is illustrated by the simulation of the vibronic charge transfer (intervalence) optical bands in trimeric mixed valence clusters.


Vibrational Level Creation Operator Vibronic Coupling Energy Pattern Vibrational Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



B.T. acknowledges financial support of the Israel Science Foundation (ISF, grant no. 168/09). A.P. thanks the Paul Scherrer Institute for financial support that made possible his participation in the Jahn-Teller Symposium. The financial support from STCU (project N 5062) and the Supreme Council on Science and Technological Development of Moldova is gratefully acknowledged. J.M.C.J. and E.C. thank Spanish MICINN (CSD2007-00010 CONSOLIDER-INGENIO in Molecular Nanoscience, MAT2007-61584, CTQ-2008-06720 and CTQ-2005-09385), Generalitat Valenciana (PROMETEO program), and the EU (MolSpinQIP project and ERC Advanced Grant SPINMOL) for the financial support. We thank Prof. V. Polinger for the discussion and Dr. O. Reu for his help in the artwork.


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Boris Tsukerblat
    • 1
    Email author
  • Andrew Palii
    • 2
  • Juan Modesto Clemente-Juan
    • 3
  • Eugenio Coronado
    • 3
  1. 1.Chemistry DepartmentBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Institute of Applied PhysicsAcademy of Sciences of MoldovaKishinevMoldova
  3. 3.Instituto de Ciencia MolecularUniversidad de ValenciaPaternaSpain

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