Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Englman R (1972) The Jahn–Teller effect in molecules and crystals. Wiley, London
Bersuker IB, Polinger VZ (1989) Vibronic interactions in molecules and crystals. Springer, Berlin
Bersuker IB (2006) The Jahn-Teller effect. Cambridge University Press, Cambridge
Ding X-Q, Bominaar EL, Bill E, Winkler H, Trautwein AX, Drüeke S, Chaudhuri P, Weighardt K (1990) J Chem Phys 92:178; Gamelin DR, Bominaar EL, Kirk ML, Wieghardt K, Solomon EI (1996) J Am Chem Soc 118:8085
Marks AJ, Prassides K (1993) New J Chem 17:59; Marks AJ, Prassides K (1993) J Chem Phys 98:4805
(a) Borshch SA, Bominaar EL, Blondin G, Girerd G (1993) J Am Chem Soc 115:5155; (b) Bominaar EL, Borshch SA, Girerd JJ (1994) J Am Chem Soc 116:5362
Clemente-Juan JM, Coronado E (1999) Coord Chem Rev 361:193
(a) Borras-Almenar JJ, Clemente-Juan JM, Coronado E, Tsukerblat BS (1995) Chem Phys 195:1; (b) Borras-Almenar JJ, Clemente-Juan JM, Coronado E, Tsukerblat BS (1995) Chem Phys 195:17; (c) Borras-Almenar JJ, Clemente-Juan JM, Coronado E, Tsukerblat BS (1995) Chem Phys 195:29
(a) Polinger VZ, Boldirev SI (1986) Phys Stat Sol (b) 137:241; (b) Boldyrev SI, Polinger VZ, Bersuker IB (1981) Fiz Tverdogo Tela (Russ) 23:746
Sakamoto N, Muramatsu S (1978) Phys Rev B 17:868
Perlin YuE, Tsukerblat BS (1984) In: Perlin YuE, Wagner M (eds) The dynamical Jahn-Teller effect in localized systems, vol 7. Elsevier, Amsterdam, pp 251–346
Gütlich P, Hauser A, Spiering H (1994) Angew Chem Int Ed Engl 33:2024
Kaplan MD, Vekhter BG (1995) Cooperative phenomena in Jahn-Teller crystals Plenum, New York
Köppel H, Domcke W, Cederbaum LS (1984) Adv Chem Phys 57:59
Faraji S, Gindensperger E, Köppel H (2009) In: Köppel H, Yarkony DR, Barentzen (eds) The Jahn-Teller effect. Fundamentals and implications for physics and chemistry, Series of Chemical Physics, vol 97. Springer, Heidelberg, pp 239–276
Grosso G, Martinelli L, Parravicini GP (1995) Phys Rev B 51:13033
Pooler DR (1978) J Phys A 11:1045; Pooler DR (1980) J Phys A 13:1029
O’Brien MCM (1969) Phys Rev 187:329; O’Brien MCM (1971) J Phys C 4:2524
Tsukerblat B (2006) Group theory in chemistry and spectroscopy. Dover, Mineola/New York
Koster GF, Dimmok JO, Wheeler RG, Statz H (1963) Properties of the thirty-two point groups. MIT Press, Cambridge
Clemente-Juan JM, Palii A, Coronado E, Tsukerblat B (2011) J Comp Chem, submitted
Borras-Almenar JJ, Clemente-Juan JM, Coronado E, Palii AV, Tsukerblat BS (2001) In: Miller J, Drillon M (eds) Magnetoscience-from molecules to materials. Willey-VCH, New York, pp 155–210
(a) Wong KY, Schatz PN (1981) Prog Inorg Chem 28:369; (b) Piepho SB, Krausz ER, Shatz PN (1978) J Am Chem Soc 100:2996
(a) Piepho SB (1988) J Am Chem Soc 110:6319; (b) Piepho SB (1990) J Am Chem Soc 112:4197
Borrás-Almenar JJ, Coronado E, Ovstrosvsky SM, Palii AV, Tsukerblat BS (1999) Chem Phys 240:149
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Tsukerblat, B., Palii, A., Clemente-Juan, J.M., Coronado, E. (2011). A Symmetry Adapted Approach to the Dynamic Jahn-Teller Problem. In: Atanasov, M., Daul, C., Tregenna-Piggott, P. (eds) Vibronic Interactions and the Jahn-Teller Effect. Progress in Theoretical Chemistry and Physics, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2384-9_3
Download citation
DOI: https://doi.org/10.1007/978-94-007-2384-9_3
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2383-2
Online ISBN: 978-94-007-2384-9
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)