Journal of Mathematical Chemistry

, Volume 53, Issue 2, pp 629–650 | Cite as

On the cluster structure of linear-chain fermionic wave functions

  • Josef Paldus
  • Tokuei Sako
  • Geerd H. F. Diercksen
Original Paper


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 \(N \rightarrow \infty \).


Linear chain models Cyclic polyenes Coupled cluster approach (CCA) Cluster analysis Decomposable and indecomposable connected quadruples CCA to extended linear fermionic chains 

Mathematics Subject Classification

81Q05 81Q80 81V55 81V70 92E10 



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.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Josef Paldus
    • 1
    • 2
  • Tokuei Sako
    • 3
  • Geerd H. F. Diercksen
    • 4
  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Chemistry, Guelph-Waterloo Center for Graduate Work in Chemistry (GWC)² - Waterloo CampusUniversity of WaterlooWaterlooCanada
  3. 3.Laboratory of Physics, College of Science and TechnologyNihon UniversityFunabashi, ChibaJapan
  4. 4.Max-Planck-Institut für AstrophysikGarchingGermany

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