Skip to main content
SpringerLink
Log in

On the cluster structure of linear-chain fermionic wave functions

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

Using the model of cyclic polyenes \(\hbox {C}_N\hbox {H}_N\) with a nondegenerate ground state, \(N = 4 \nu + 2 \; (\nu = 1, 2, \ldots )\), as a prototype of extended linear metallic-like systems we explore the cluster structure of the relevant wave functions. Based on the existing configuration interaction and coupled cluster (CC) results, as obtained with the Hubbard and Pariser–Parr–Pople Hamiltonians in the entire range of the coupling constant extending from the uncorrelated Hückel limit to the fully correlated limit, we recall the breakdown of the CCD or CCSD methods as the size of the system increases and the strongly correlated regime is approached. We introduce the concept of the indecomposable quadruply-excited clusters which arise for \(\nu > 1\) and represent those connected quadruples that do not possess any corresponding disconnected cluster component. It is shown via explicit enumeration that the ratio of the number of these indecomposables relative to that of the decomposables depends linearly on the size of the polyene \(N\), so that the limit of the ratio of the number of indecomposables relative to the total number of quadruples approaches unity as \(N \rightarrow \infty \). We then briefly outline the implications of these results for the applicability of CC approaches to extended systems and provide a qualitative argument for an even more extreme behavior of hexa-excited, octa-excited, etc., clusters as \(N \rightarrow \infty \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Note that in view of the high symmetry of cyclic polyenes all singly-excited clusters vanish so that CCD is equivalent with CCSD.

  2. This approach is often used even when considering spin chains; see, e.g., [13] or [71].

References

  1. R.J. Bartlett, in Modern Electronic Structure Theory, vol. 1, ed. by D.R. Yarkony (World Scientific, Singapore, 1995), pp. 1047–1131.

  2. J. Paldus, X. Li, Adv. Chem. Phys. 110, 1 (1999)

    CAS  Google Scholar 

  3. T.D. Crawford, H.F. Schaefer III, in Reviews of Computational Chemistry, vol. 14, ed. by K.B. Lipkowitz, D.B. Boyd (Wiley, New York, 2000), pp. 33–136

    Chapter  Google Scholar 

  4. J. Paldus, in Handbook of Molecular Physics and Quantum Chemistry, Part 3, Chap. 19, vol. 2, ed. by S. Wilson (Wiley, Chichester, 2003), pp. 272–313.

  5. R.J. Bartlett, M. Musiał, Rev. Mod. Phys. 79, 291 (2007)

    Article  CAS  Google Scholar 

  6. I. Shavitt, R.J. Bartlett, Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory (Cambridge University Press, Cambridge, 2009)

    Book  Google Scholar 

  7. P. Čársky, J. Paldus, J. Pittner (eds.), Recent Progress in Coupled Cluster Methods: Theory and Applications (Springer, Berlin, 2010)

    Google Scholar 

  8. J. Paldus, in Theory and Applications of Computational Chemistry: The First Forty Years, Chap. 7, ed. by C.E. Dykstra, G. Frenking, K.S. Kim, G.E. Scuseria (Elsevier, Amsterdam, 2005), pp. 115–147.

  9. R.J. Bartlett, in Theory and Applications of Computational Chemistry: The First Forty Years, Chap. 42, ed. by C.E. Dykstra, G. Frenking, K.S. Kim, G.E. Scuseria (Elsevier, Amsterdam, 2005), pp. 1191–1221.

  10. J. Paldus, J. Pittner, P. Čársky, in Recent Progress in Coupled Cluster Methods: Theory and Applications, Chap. 17, ed. by P. Čársky, J. Paldus, J. Pittner (Springer, Berlin, 2010), pp. 455–489

  11. H.A. Bethe, Z. Phys. 71, 205 (1931)

    Article  CAS  Google Scholar 

  12. W. Heisenberg, Z. Phys. 49, 619 (1928)

    Article  CAS  Google Scholar 

  13. L. Hulthén, Arkiv. Mat. Astron. Fys. A 26, 1 (1938)

    Google Scholar 

  14. M.T. Batchelor, Phys. Today 60, 36 (2007)

    Article  Google Scholar 

  15. E.H. Lieb, W. Liniger, Phys. Rev. 130, 1605 (1963)

    Article  Google Scholar 

  16. C.N. Yang, Phys. Rev. Lett. 19, 1312 (1967)

    Article  Google Scholar 

  17. E.H. Lieb, F.Y. Wu, Phys. Rev. Lett. 20, 1445 (1968)

    Article  Google Scholar 

  18. E.H. Lieb, F.Y. Wu, Phys. A 321, 1 (2003)

    Article  Google Scholar 

  19. R.J. Baxter, Ann. Phys. (N.Y.) 70, 193 (1972).

  20. I. Bloch, Nat. Phys. 1, 23 (2005)

    Article  CAS  Google Scholar 

  21. Z. Bajnok, L. Šamaj, Acta Phys. Slov. 61, 129 (2011). and references therein

    Google Scholar 

  22. E.H. Lieb, D.C. Mattis (eds.), Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles (Academic Press, New York, 1966)

    Google Scholar 

  23. R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, New york, 1982)

    Google Scholar 

  24. M. Gaudin, La Fonction d’Onde de Bethe (Masson, Paris, 1983); English translation: The Bethe Wavefunction (Cambridge University Press, Cambridge, 2014)

    Google Scholar 

  25. A. Montorsi (ed.), The Hubbard Model: A Reprint Volume (World Scientific, Singapore, 1992)

    Google Scholar 

  26. V.E. Korepin, N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, 1993)

    Book  Google Scholar 

  27. D. Baeriswyl, D.K. Campbell, J.M.P. Carmelo, F. Guinea, E. Louis, The Hubbard Model: Its Physics and Mathematical Physics (Plenum Press, New York, 1995)

    Book  Google Scholar 

  28. Z.N.C. Ha, Quantum Many-Body Systems in One Dimension (World Scientific, Singapore, 1996)

    Book  Google Scholar 

  29. B. Sutherland, Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems (World Scientific, Singapore, 2004)

    Book  Google Scholar 

  30. F.H.L. Essler, H. Frahm, F. Göhmann, A. Klümper, V.E. Korepin, The One-Dimensionsl Hubbard Model (Cambridge University Press, Cambridge, 2005)

    Book  Google Scholar 

  31. K. Hashimoto, Int. J. Quantum Chem. 28, 581 (1985)

    Article  CAS  Google Scholar 

  32. P. Piecuch, J. Čížek, J. Paldus, Int. J. Quantum Chem. 42, 165 (1992)

    Article  CAS  Google Scholar 

  33. K. Hashimoto, J. Čížek, J. Paldus, Int. J. Quantum Chem. 34, 407 (1988)

    Article  CAS  Google Scholar 

  34. J. Paldus, M. Takahashi, R.W.H. Cho, Phys. Rev. B 30, 4267 (1984)

    Article  CAS  Google Scholar 

  35. R. Pauncz, J. de Heer, P.-O. Löwdin, J. Chem. Phys. 36, 2247 (1962)

    Article  CAS  Google Scholar 

  36. R. Pauncz, J. de Heer, P.-O. Löwdin, J. Chem. Phys. 36, 2257 (1962)

    Article  CAS  Google Scholar 

  37. J. de Heer, R. Pauncz, J. Mol. Spectr. 5, 326 (1960)

    Article  Google Scholar 

  38. R. Pauncz, Alternant Molecular Orbital Method (W. B. Saunders, Philadelphia, 1967)

    Google Scholar 

  39. G.L. Bendazzoli, S. Evangelisti, Chem. Phys. Lett. 185, 125 (1991)

    Article  CAS  Google Scholar 

  40. G.L. Bendazzoli, S. Evangelisti, L. Gagliardi, Int. J. Quantum Chem. 51, 13 (1994)

    Article  CAS  Google Scholar 

  41. S. Evangelisti, G.L. Bendazzoli, Chem. Phys. Lett. 196, 511 (1992)

    Article  CAS  Google Scholar 

  42. G.L. Bendazzoli, S. Evangelisti, Int. J. Quantum Chem. 66, 397 (1998)

    Article  CAS  Google Scholar 

  43. R. Podeszwa, S.A. Kucharski, L.Z. Stolarczyk, J. Chem. Phys. 116, 480 (2002)

  44. J. Paldus, M.J. Boyle, Int. J. Quantum Chem. 22, 1281 (1982)

    Article  CAS  Google Scholar 

  45. J. Paldus, in Theoretical Chemistry: Advances and Perspectives, vol. 2, ed. by H. Eyring, D.J. Henderson (Academic Press, New York, 1976), pp. 131–290.

  46. J. Paldus, in Mathematical Frontiers in Computational Chemical Physics, IMA Series, vol. 15, ed. by D.G. Truhlar (Springer, Berlin, 1988), pp. 262–299

  47. R.G. Parr, The Quantum Theory of Molecular Electronic Structure (Benjamin, New York, 1963)

    Google Scholar 

  48. N. Mataga, K. Nishimoto, Z. Phys, Chem. 13, 140 (1957)

    CAS  Google Scholar 

  49. J. Čížek, J. Paldus, I. Hubač, Int. J. Quantum Chem. Symp. 8, 293 (1974)

    Article  Google Scholar 

  50. Y. Ooshika, J. Phys. Soc. Jpn. 12, 1246 (1957)

    Article  CAS  Google Scholar 

  51. T. Murai, Progr. Theor. Phys. 27, 899 (1962)

    Article  CAS  Google Scholar 

  52. D. Cazes, L. Salem, C. Tric, J. Polymer Sci.:Part C, No. 29, pp. 109–118 (1970).

  53. J. Hubbard, Proc. R. Soc. A 244, 199 (1958)

    Article  Google Scholar 

  54. M. Urban, J. Noga, S.J. Cole, R.J. Bartlett, J. Chem. Phys. 83, 4041 (1985)

    Article  CAS  Google Scholar 

  55. K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett. 157, 479 (1989)

    Article  CAS  Google Scholar 

  56. J. Čížek, J. Chem. Phys. 45, 4256 (1966)

    Article  Google Scholar 

  57. J. Čížek, Adv. Chem. Phys. 14, 35 (1969)

    Google Scholar 

  58. J. Čížek, J. Paldus, Int. J. Quantum Chem. 5, 359 (1971)

    Article  Google Scholar 

  59. J. Paldus, J. Čížek, I. Shavitt, Phys. Rev. A 5, 50 (1972)

    Article  Google Scholar 

  60. J. Paldus, J. Čížek, Adv. Quantum Chem. 9, 105 (1975)

    Article  CAS  Google Scholar 

  61. P.E.S. Wormer, J. Paldus, Adv. Quantum Chem. 51, 59 (2006)

    Article  CAS  Google Scholar 

  62. J. Paldus, J. Chem. Phys. 67, 303 (1977)

    Article  CAS  Google Scholar 

  63. P. Piecuch, J. Paldus, Theor. Chim. Acta 78, 65 (1990)

    Article  CAS  Google Scholar 

  64. P. Piecuch, R. Toboła, J. Paldus, Phys. Rev. A 54, 1210 (1996)

    Article  CAS  Google Scholar 

  65. P. Piecuch, J. Paldus, Theor. Chim. Acta 83, 69 (1992)

    Article  Google Scholar 

  66. P. Piecuch, R. Toboła, J. Paldus, Int. J. Quantum Chem. 55, 133 (1995)

    Article  CAS  Google Scholar 

  67. A.E. Kondo, P. Piecuch, J. Paldus, J. Chem. Phys. 104, 8566 (1996)

    Article  CAS  Google Scholar 

  68. X. Li, J. Paldus, J. Chem. Phys. 101, 8812 (1994)

    Article  CAS  Google Scholar 

  69. B. Jeziorski, J. Paldus, P. Jankowski, Int. J. Quantum Chem. 56, 129 (1995)

    Article  CAS  Google Scholar 

  70. D.J. Thouless, in The Quantum Mechanics of Many-Body Systems, 2nd edn. (Academic, New York, 1972), p. 57 (p. 35 in the 1961, 1st edn.)

  71. J.-M. Maillet, in Quantum Spaces: Poincaré Seminar 2007, Progress in Mathematical Physics, vol. 53, ed. by B. Duplantier, V. Rivasseau (Birkhäuser Verlag, Basel, 2007), pp. 161–201

Download references

Acknowledgments

Two of the authors (J.P. and T.S.) are greatly indebted to the Alexander von Humboldt Foundation for its kind support that enabled their stay at the Max-Planck-Institute for Astrophysics at Garching bei München in Germany and they thank the latter Institute for its hospitality during their stay. Their heartfelt thanks are also due to their host, Prof. Dr. Geerd H. F. Diercksen, for his kind advice and collaboration and for making their stay as pleasant and productive as possible.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Josef Paldus

  2. Department of Chemistry, Guelph-Waterloo Center for Graduate Work in Chemistry (GWC)² - Waterloo Campus, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Josef Paldus

  3. Laboratory of Physics, College of Science and Technology, Nihon University, 7-24-1 Narashinodai, Funabashi, Chiba, 274-8501, Japan

    Tokuei Sako

  4. Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, 85741, Garching, Germany

    Geerd H. F. Diercksen

Authors
  1. Josef Paldus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tokuei Sako
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Geerd H. F. Diercksen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josef Paldus.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paldus, J., Sako, T. & Diercksen, G.H.F. On the cluster structure of linear-chain fermionic wave functions. J Math Chem 53, 629–650 (2015). https://doi.org/10.1007/s10910-014-0445-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-014-0445-7

Keywords

Mathematics Subject Classification

Search

Navigation