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International Journal of Quantum Chemistry

Interpolant polynomial technique applied to the PPP model. I. Asymptotics for excited states of cyclic polyenes in the finite cyclic Hubbard model

Paul Bracken, J. � ek

Published Mar 15, 1996• 7 citations
Publisher

Abstract

For theoretical reasons, and on account of the development of a new interpolation technique, it is useful and important to examine the asymptotic behavior of the solution to the one-dimensional Hubbard model. In this article, it is shown how perturbative expansions for the energy can be developed in the asymptotic region of the relevant coupling for all the relevant excited states of cyclic polyene rings. © 1996 John Wiley & Sons, Inc.

DOI
10.1002/(sici)1097-461x(1996)57:6<1019::aid-qua1>3.0.co;2-w

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Physical review. B, Condensed matter•Oct 15, 1984•138 citations
For theoretical reasons, and on account of the development of a new interpolation technique, it is useful and important to examine the asymptotic behavior of the solution to the one-dimensional Hubbard model. In this article, it is shown how perturbative expansions for the energy can be developed in the asymptotic region of the relevant coupling for all the relevant excited states of cyclic polyene rings. © 1996 John Wiley & Sons, Inc.
An interpolation technique which is based on the strongly and weakly correlated cases of the one-dimensional Hubbard model is proposed. The input information consists of the perturbation expansions which are obtained from the Lieb–Wu equations in both limits. The Hubbard model is used to describe cyclic polyene rings, and for the case of N=6 sites, this would correspond to benzene. The technique has been applied to several symmetries of the model. It has been shown that the exact secular problem can be reconstructed for these symmetries, and the results for N=6 agree exactly with the results which have been obtained in a different way, that is, by using a full configuration interaction calculation.
The optimized-inner-projection (OIP) method is implemented and applied to determine the lower bounds to the ground-state correlation energy for the Pariser-Parr-Pople and Hubbard models of cyclic polyenes CN{\mathrm{C}}_{\mathit{N}}CN​HN{\mathrm{H}}_{\mathit{N}}HN​, N=4\ensuremath{\nu}+2, \ensuremath{\nu}=3--5, in the whole range of the coupling constant. In actual calculations, the orthogonally spin-adapted form of the OIP equations is used, and full advantage is taken of the high symmetry of the cyclic polyene model, so that systems with N\ensuremath{\ge}14 can be explored. Comparison is made with the results of both full and limited configuration-interaction calculations and with the correlation energies obtained with certain approximate coupled-cluster schemes. Contrary to the six- and ten-membered rings, the OIP lower bounds to the ground-state correlation energy of cyclic polyenes with larger number of sites are very poor and cannot serve as a meaningful source of information. Since the eigenvalues of the intermediate Hamiltonian, in terms of which the OIP technique is defined, are highly degenerate even in the weakly correlated region, and in view of the random behavior of the OIP bracketing function, the OIP method becomes computationally very demanding for N\ensuremath{\ge}14 rings. Moreover, simple iterative schemes usually diverge and sophisticated root searching procedures have to be applied. The usefulness of the OIP technique for the determination of lower bounds to energy eigenvalues in larger systems is thus questioned.
Abstract The applicability of the finite‐order many‐body perturbation theory to the electron correlation problem in extended one‐dimensional systems is examined. The cyclic polyenes CNHN, N = 4ν + 2, ν = 1, 2, …, with the DNh geometry as described by both the Pariser–Parr–Pople and Hubbard Hamiltonians, are employed to model the metallic‐like one‐dimensional systems. The second‐order perturbation theory contributions to the correlation energy are obtained with three different partitionings of the Hamiltonian (Hückel, M⊘ller–Plesset, and Epstein–Nesbet). The third‐ and fourth‐order contributions are also calculated in special cases. A comparison with other methods is given wherever available. For the Hubbard Hamiltonian the asymptotic behavior of the perturbation theory expansion is examined numerically. It is shown that the finite‐order perturbation expansion can provide reliable results for the correlation energy of one‐dimensional systems even in the correlation region which corresponds to the spectroscopically determined physical value of the coupling constant.
The radius of convergence of a perturbation expansion is estimated using Pad\'e approximants. Location of the branch point which is nearest to the origin is evaluated directly from the pole of the denominator of an [M,M] Pad\'e approximant and indirectly from the logarithmic derivative of this approximant. Both methods are applied to the perturbation expansion for the ground-state energy of the finite cyclic Hubbard systems. The method using the logarithmic derivative yields numerical results which are in good agreement with those obtained by other methods. Kato's formula is also applied.
The many-electron correlation problem for one-dimensional metalliclike systems with Born--von K\'arm\'an boundary conditions, represented by the Pariser-Parr-Pople and the Hubbard Hamiltonian cyclic polyene models, CN{\mathrm{C}}_{\mathrm{N}}CN​HN{\mathrm{H}}_{\mathrm{N}}HN​, N=2n=4\ensuremath{\nu}+2, \ensuremath{\nu}=1,2,..., is studied using the coupled-cluster approach in the localized Wannier basis representation. Various truncation schemes for the pair clusters are examined. It is shown that already the intracell pair-cluster approximation, which can be handled analytically and yields the same expression for the correlation energy as the variational approach of Ukrainskii, provides a reasonable approximation in the entire range of the coupling constant. Using all doubly excited clusters composed of locally excited particle-hole pairs, one obtains the exact correlation energy in the fully correlated limit assuming that the coupled-pair equations are corrected for the connected quadruply excited cluster contributions. This is achieved by using the recently developed approximate coupled-pair approach with corrections for the quadruply excited clusters (ACPQ), which is almost identical, except for a numerical factor of certain diagrams, with the standard approximate coupled-pair approach. The ACPQ approach removes the singularities which otherwise plague the standard coupled-pair approach and yields very good correlation energies in the entire range of the coupling constant even when the n3{n}^{3}n3 doubly excited pair clusters are truncated to only n+10 locally and quasilocally excited pair clusters in the localized Wannier basis representation.
The many-electron correlation problem for metalliclike systems with Born---von K\'arm\'an boundary conditions, as modeled by the cyclic polyene Pariser-Parr-Pople and Hubbard Hamiltonians, is examined over the entire range of the coupling constant using the coupled-pair (CP) many-electron theory based on the exponential-cluster ansatz for the exact wave function. It is shown that the standard CP theory breaks down not only in the highly correlated, but even in the intermediately correlated, regions, and the nature of its singular behavior in these regions is examined. This breakdown is linked with the increasingly important role played by the connected quadruply excited clusters, which invalidate the basic assumption of the CP theory, as the highly correlated limit and/or the extended character of the model are approached. The contribution from the quadruply excited clusters is then taken into account using the new version of the approximate coupled-pair theory corrected for connected quadruply excited clusters, called ACPQ. This approximation is almost identical with the standard approximate coupled-pair (ACP) -theory approach, in which only factorizable (with respect to hole pairs) nonlinear terms are retained, and differs from it only by a numerical factor associated with one of the nonlinear diagrams. It is shown that the ACPQ not only removes the singularities and associated convergency problems of the standard CP approaches, but, in fact, provides excellent quantitative results over the entire range of the coupling constant, yielding the exact correlation energy in the strongly correlated limit.