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Constructing, Solving and Applying the Vibronic Hamiltonian

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The Jahn-Teller Effect

Part of the book series: Springer Series in Chemical Physics ((CHEMICAL,volume 97))

Abstract

The Jahn–Teller effect is shrouded in mysticism and cynicism. To paraphrase a remark that a colleague recently relayed, “For every anomalous spectrum, structural distortion or novel physical property, there is a vibronic Hamiltonian and ensuing explanation that few can appreciate or comprehend.” The aim of this article is to provide a basic introduction to the Jahn–Teller effect, pitched at a level that undergraduates in chemistry can understand, with an emphasis on how to calculate a given experimental quantity. We show that armed with just a little group theory and matrix mechanics, vibronic Hamiltonians can be readily constructed, solved, and the molecular property of interest extracted from the eigenvalues and eigenfunctions. The manifestation of the Jahn–Teller effect does indeed come in many shapes and forms, three signatures of which are briefly discussed. (1) The vibronic energy spectrum is best revealed by spectroscopy and two examples are taken from the literature that elucidate the intricate energy-level pattern of the E ⊗ e vibronic interaction. (2) ‘The Ham effect’, ‘Ham factors’ and ‘Ham quenching’ are now common parlance in spectroscopy and the phenomenon is aptly illustrated by the magnetic and spectroscopic data of the titanium(III) and vanadium(III) aqua ions. (3) The plasticity of the co-ordination sphere is the quintessential feature of transition metals exhibiting strong Jahn–Teller coupling. We show how a concomitant description of structural and spectroscopic data can be obtained employing a model in which the potential energy surface resulting from the cubic Jahn–Teller Hamiltonian is perturbed by anisotropic strain.

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References

  1. S. Sugano, Y. Tanabe, H. Kamimura, Multiplets of Transition Metal Ions In Crystals (Academic, New York, 1970)

    Google Scholar 

  2. M.J. Riley, A. Furlan, Chem. Phys. 210, 389 (1996)

    Article  CAS  Google Scholar 

  3. H.A. Jahn, E. Teller, Proc. Roy. Soc. London, A. 161, 220 (1937)

    Google Scholar 

  4. M.C.M. O’Brien, From her lecture on “The Paramagnetism of Caesium Titanium Alum”, given at the Department of Chemistry, University of Bern in 1997, shortly before her untimely death.

    Google Scholar 

  5. M.A. Hitchman, T.D. Waite, Inorg. Chem. 15(9), 2150 (1976)

    Article  CAS  Google Scholar 

  6. C. Dobe, C. Noble, G. Carver, P.L.W. Tregenna-Piggott, G. McIntyre, A.L. Barra, A. Neels, S. Janssen, F. Juranyi, J. Am. Chem. Soc. 127(10), 3642 (2005)

    Article  CAS  Google Scholar 

  7. P.L.W. Tregenna-Piggott, H. Weihe, A.-L. Barra, Inorg. Chem. 42, 8504–8508 (2003)

    Article  CAS  Google Scholar 

  8. A. Ceulemans, D. Beyens, L.G. Vanquickenborne, J. Am. Chem. Soc. 106, 5824 (1984)

    Article  CAS  Google Scholar 

  9. P. Murray-Rust, H.-B. Bürgi, J.D. Dunitz, Acta. Cryst. A35, 703–713 (1979)

    CAS  Google Scholar 

  10. F.S. Ham, Phys. Rev. 166, 307 (1968)

    Article  CAS  Google Scholar 

  11. M.J. Riley, Top. Curr. Chem. 214, 5 (2001)

    Google Scholar 

  12. I.B. Bersuker, The Jahn–Teller Effect (Cambridge University Press, Cambridge, 2006)

    Google Scholar 

  13. M. Bacci, Chem. Phys. Lett. 58, 537 (1978)

    Article  CAS  Google Scholar 

  14. M. Bacci, Chem. Phys. 40, 237 (1979)

    Article  CAS  Google Scholar 

  15. G. Carver, J. Bendix, P.L.W. Tregenna–Piggott, Chem. Phys. 282, 245–263 (2002)

    Google Scholar 

  16. R. Bruyndonckx, C. Daul, P.T. Manoharan, E. Deiss, Inorg. Chem. 36, 4251 (1997)

    Article  CAS  Google Scholar 

  17. T.K. Kundu, R. Bruyndonckx, C. Daul, P.T. Manoharan, Inorg. Chem. 38, 3931 (1999)

    Article  CAS  Google Scholar 

  18. M. Atanasov, P. Comba, C.A. Daul, A. Hauser, J. Phys. Chem. A 111; 9145–9163 (2007)

    Google Scholar 

  19. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Vol. 1, Chapter V (1977)

    Google Scholar 

  20. H.C. Longuet-Higgins, U. Opik, M.H.L. Pryce, R.A. Sack, Proc. Roy. Soc. Lond. A244, 1 (1958).

    Google Scholar 

  21. F.S. Ham, Phys. Rev. Lett. 58, 725 (1987)

    Article  CAS  Google Scholar 

  22. J. Ammeter, H.B. Buergi, E. Gamp, V. Meyer-Sandrin, W.P. Jensen, Inorg. Chem. 18(3), 733–750 (1979)

    Article  CAS  Google Scholar 

  23. P.L.W. Tregenna-Piggott, H.P. Andres, G.J. McIntyre, S.P. Best, C.C. Wilson, J.A. Cowan, Inorg. Chem. 42, 1350 (2003)

    Article  CAS  Google Scholar 

  24. D.R. Yarkony, Acc. Chem. Res. 31, 511 (1998)

    Article  CAS  Google Scholar 

  25. P.L.W. Tregenna-Piggott, G. Carver, Inorg. Chem. 43, 8061 (2004)

    Article  CAS  Google Scholar 

  26. A. Dick, E.R. Krausz, K.S. Hadler, C.J. Noble, P.L.W. Tregenna-Piggott, M.J. Riley, J. Phys. Chem. C 112(37), 14555–14562 (2008). References to an abundance of work conducted on copper(II) doped MgO and copper(II) doped CaO may be found in this article

    Google Scholar 

  27. S. Guha, L.L. Chase, Phys. Rev. B 12, 1658 (1975)

    Article  CAS  Google Scholar 

  28. P.L.W. Tregenna-Piggott, Adv. Quantum Chem. 44, 461 (2003)

    Article  CAS  Google Scholar 

  29. M.C.M. O’Brien, Proc. R. Soc., A 281, 323 (1964)

    Google Scholar 

  30. M.C.M. O’Brien, J. Phys. C. Solid State Phys. 16, 85 (1983)

    Article  Google Scholar 

  31. M.J. Riley, C. Noble, P.L.W. Tregenna–Piggott, J. Chem. Phys., in press.

    Google Scholar 

  32. F.S. Ham, in Electron Paramagnetic Resonance, ed. by S. Geshwind (Plenum Press: New York, 1972)

    Google Scholar 

  33. R.W. Reynolds, L.A. Boatner, M.M. Abraham, Y. Chen, Phys. Rev. B. 10, 3802 (1974)

    Article  CAS  Google Scholar 

  34. A. Furlan, M.J. Riley, S. Leutwyler, J. Chem. Phys. 96, 7306–7320 (1992)

    CAS  Google Scholar 

  35. A. Furlan, T. Fischer, P. Fluekiger, H.U. Guedel, S. Leutwyler, H.P. Luethi, M.J. Riley, J. Weber, J. Phys. Chem. 96, 10713–10719 (1992)

    CAS  Google Scholar 

  36. G. Delacrétaz, E.R. Grant, R.L. Whetten, L. Wöste, J.W. Zwanziger, Phys. Rev. Lett. 56, 2598 (1986)

    Article  Google Scholar 

  37. M.J. Riley, A. Furlan, H.U. Guedel, S. Leutwyler, J. Chem. Phys. 98, 3803–3815 (1993)

    CAS  Google Scholar 

  38. A. Furlan, S. Leutwyler, M.J. Riley, J. Chem. Phys. 100, 840–855 (1994)

    CAS  Google Scholar 

  39. F.S. Ham, Phys. Rev. 138, A1727 (1965)

    Article  Google Scholar 

  40. M.C.M. O’Brien, C.C. Chancey, Am. J. Phys. 61, 688–697 (1993)

    Article  Google Scholar 

  41. A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Metal Ions (Oxford University Press, Oxford, 1970), p. 815

    Google Scholar 

  42. F.I.B. Williams, D.C. Krupa, D.P. Breen, Phys. Rev. 179, 255 (1969)

    Article  CAS  Google Scholar 

  43. J.H. Van Vleck, Electric and Magnetic Susceptibilities (Oxford University Press, New York, 1932)

    Google Scholar 

  44. J.H. Van Vleck, J. Chem. Phys. 7, 61–71 (1939)

    Google Scholar 

  45. J.K. Beattie, S.P. Best, B.W. Skelton, A.H. White, J. Chem. Soc., Dalton Trans. 2105 (1981)

    Google Scholar 

  46. C. Daul, A. Goursot, Inorg. Chem. 24, 3554–3558 (1985)

    Article  CAS  Google Scholar 

  47. S.P. Best, J.B. Forsyth, J. Chem. Soc., Dalton Trans. 1721 (1991)

    Google Scholar 

  48. B. Bleaney, G.S. Bogle, A.H. Cooke, R.J. Duffus, M.C.M. O’Brien, K.W.H. Stevens, Proc. Phys. Soc. London, Ser. A 68, 57 (1955)

    Google Scholar 

  49. P.L.W. Tregenna-Piggott, S.P. Best, M.C.M. O’Brien, K.S. Knight, J.B. Forsyth, J.R. Pilbrow, J. Am. Chem. Soc. 119, 3324–3332 (1997)

    Article  CAS  Google Scholar 

  50. P.L.W. Tregenna-Piggott, M.C.M. O’Brien, J.R. Pilbrow, H.U. Güdel, S.P. Best, C. Noble, J. Chem. Phys. 107, 8275 (1997)

    Article  CAS  Google Scholar 

  51. L. Dubicki, M.J. Riley, J. Chem. Phys. 106, (1997) 1669

    Article  CAS  Google Scholar 

  52. P.L.W. Tregenna-Piggott, C.J. Noble, J.R. Pilbrow, J. Chem. Phys. 113, 3289–3301 (2000)

    CAS  Google Scholar 

  53. G. Carver, C. Dobe, T.B. Jensen, P.L.W. Tregenna-Piggott, S. Janssen, E. Bill, G.J. McIntyre, A.L. Barra, Inorg. Chem. 45, 4695 (2006)

    Article  CAS  Google Scholar 

  54. P.L.W. Tregenna-Piggott, H.U. Güdel, Inorg. Chem.40, 5497–5506 (2001)

    Article  CAS  Google Scholar 

  55. P.L.W. Tregenna-Piggott, M.C.M. O’Brien, H. Weihe, H.U. Güdel, J. Chem. Phys. 109, 2967 (1998)

    Article  CAS  Google Scholar 

  56. G. Carver, S. Spichiger P.L.W. Tregenna-Piggott, J. Chem. Phys. 122, 124511 (2005)

    Article  Google Scholar 

  57. D. Spichiger, G. Carver, C. Dobe, J. Bendix, P.L.W. Tregenna-Piggott, R. Meier, G. Zahn, Chem. Phys. Lett. 337, 391 (2001)

    Article  CAS  Google Scholar 

  58. P.L.W. Tregenna-Piggott, D. Spichiger, G. Carver, B. Frey, R. Meier, H. Weihe, J. Cowan, G. McIntyre, G. Zahn, A.-L. Barra, Inorg. Chem. 43, 8049–8060 (2004)

    Article  CAS  Google Scholar 

  59. S.P. Best, B.N. Figgis, J.B. Forsyth, P.A. Reynolds, P.L.W. Tregenna–Piggott, Inorg. Chem. 34, 4605–4610 (1995)

    Google Scholar 

  60. M.J. Riley, M.A. Hitchman, A.W. Mohammed, J. Chem. Phys. 87, 3766 (1987)

    Article  CAS  Google Scholar 

  61. K.S. Hadler, J.R. Kilmartin, G.R. Hanson M.A. Hitchman, C.J. Simmons, M.J. Riley, Inorg. Chem. 47, 8188 (2008)

    Article  CAS  Google Scholar 

  62. V.M. Masters, M.J. Riley, M.A. Hitchman, Inorg. Chem. 40(5), 843–849 (2001)

    Article  CAS  Google Scholar 

  63. C.J. Simmons, M.A. Hitchman, H. Stratemeier, A.J. Schultz, J. Am. Chem. Soc. 115, 11304 (1993)

    Article  CAS  Google Scholar 

  64. C. Dobe, H.P. Andres, P.L.W. Tregenna-Piggott, S. Mossin, H. Weihe, S. Janssen, Chem. Phys. Lett. 362, 387 (2002)

    Article  CAS  Google Scholar 

  65. C. Dobe, T. Straessle, F. Juranyi, P.L.W. Tregenna-Piggott, Inorg. Chem. 45, 5066 (2006)

    Article  CAS  Google Scholar 

  66. G. Carver, M. Thut, C. Noble, P.L.W. Tregenna-Piggott, J. Chem. Theory Comput. 4, 603–613 (2008)

    Article  CAS  Google Scholar 

  67. Weatherburn D.C., in Handbook on Metalloproteins, ed. by I. Bertini, A. Sigel, H. Sigel (Marcel Dekker, New York, 2001), pp. 193–268. P.E.M. Siegbahn, Curr. Opin. Chem. Biol. 6, 227 (2002)

    Google Scholar 

  68. M.U. Triller, D. Pursche, W. Hsieh, V.L. Pecoraro, A. Rompel, B. Krebs, Inorg. Chem. 42, 6274 (2003).

    Article  CAS  Google Scholar 

  69. Q. Scheifele, C. Riplinger, F. Neese, H. Weihe, A.L. Barra, F. Juranyi, A. Podlesnyak, P.L.W. Tregenna-Piggott, Inorg. Chem. 47, 439–447 (2008)

    Article  CAS  Google Scholar 

  70. R.J. Deeth, M.A. Hitchman, Inorg. Chem. 25, 1225 (1986)

    Article  CAS  Google Scholar 

  71. B.N. Figgis, M.A. Hitchman, Ligand-Field Theory and Its Applications (Wiley-VCH, New York, 2000)

    Google Scholar 

  72. M.J. Riley, Inorg. Chim. Acta 268, 55 (1998)

    Article  CAS  Google Scholar 

  73. I. Krivokapic, C. Noble S. Klitgaard, P.L.W. Tregenna-Piggott, H. Weihe, A.L. Barra, Angew. Chem. Int. Ed. 44, 3613 (2005)

    Article  CAS  Google Scholar 

  74. Q. Scheifele, T. Birk, J. Bendix, P.L.W. Tregenna-Piggott H. Weihe, Angew. Chem. Int. Ed. 47, 148 (2007)

    Article  Google Scholar 

  75. D. Reinen, M. Atanasov, W. Massa, Z. Anorg. Allg. Chem. 632, 1375 (2006)

    Article  CAS  Google Scholar 

  76. P.L.W. Tregenna-Piggott, Inorg. Chem. 47, 448 (2008)

    Article  CAS  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratory for Neutron Scattering, ETH Zürich and Paul Scherrer Institut, CH-5232, Villigen PSI, Switzerland

    Philip L. W. Tregenna-Piggott

  2. School of Chemistry and Molecular Biosciences, University of Queensland, St. Lucia, QLD, 4072, Australia

    Mark J. Riley

Authors
  1. Philip L. W. Tregenna-Piggott
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  2. Mark J. Riley
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Correspondence to Philip L. W. Tregenna-Piggott .

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Editors and Affiliations

  1. LS für Theoretische Chemie, Universität Heidelberg, Im Neuenheimer Feld 229, Heidelberg, 69120, Germany

    Horst Köppel

  2. Dept. Chemistry, Johns Hopkins University, Baltimore, 21218, U.S.A.

    David R. Yarkony

  3. MPI für Festkörperforschung, Heisenbergstr. 1, Stuttgart, 70569, Germany

    Heinz Barentzen

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Tregenna-Piggott, P.L.W., Riley, M.J. (2009). Constructing, Solving and Applying the Vibronic Hamiltonian. In: Köppel, H., Yarkony, D., Barentzen, H. (eds) The Jahn-Teller Effect. Springer Series in Chemical Physics, vol 97. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03432-9_13

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