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International Journal of Quantum Chemistry

Perturbation theory and electron correlation in extended systems: Cyclic polyene model

M. Takahashi, Josef Paldus, Jakub Čı́žek

Published Dec 1, 1983• 43 citations
Publisher

Abstract

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.

DOI
10.1002/qua.560240614

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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 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.
Abstract The perturbation expansion coefficients for the ground‐state energy of the half‐filled one‐dimensional Hubbard model with N = 4ν + 2, (ν = 1, 2,…) sites and satisfying cyclic boundary conditions were calculated in the Hückel limit (U/β → 0), where β designates the one‐electron hopping or resonance integral, and U, the two‐electron on‐site Coulomb integral. This was achieved by solving—order by order—the Lieb–Wu equations, a system of transcendental equations that determines the set of quasi‐momenta {ki} and spin variables {τα} for this model. The exact values for these quantities were found for the N = 6 member ring up to the 20th order in terms of the coupling constant B = U/2β, as well as numerically for 10 ⩽ N ⩽ 50, and the N = 6 Lieb–Wu system was reduced to a system of sextic algebraic equations. These results are compared with those of the infinite system derived analytically by Misurkin and Ovchinnikov [Teor. Mat. Fiz. 11, 127 (1972)]. It is further shown how the results of this article can be used for the interpolation by the root of a polynomial. It is demonstrated that a polynomial of relatively small degree provides a very good approximation for the energy in the intermediate coupling region. © 1995 John Wiley & Sons, Inc.
The Pariser-Parr-Pople (PPP) model of a single-band one-dimensional (1D) metal is studied at the Hartree-Fock level, and by using the second-order perturbation theory of the electronic correlation. The PPP model provides an extension of the Hubbard model by properly accounting for the long-range character of the electron-electron repulsion. Both finite and infinite version of the 1D-metal model are considered within the PPP and Hubbard approximations. Calculated are the second-order electronic-correlation corrections to the total energy, and to the electronic-energy bands. Our results for the PPP model of 1D metal show qualitative similarity to the coupled-cluster results for the 3D electron-gas model. The picture of the 1D-metal model that emerges from the present study provides a support for the hypothesis that the normal metallic state of the 1D metal is different from the ground state.
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.