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Physical Review A

Coupled-cluster approaches with an approximate account of triexcitations and the optimized-inner-projection technique. III. Lower bounds to the ground-state correlation energy of cyclic-polyene model systems

Piotr Piecuch B, Sohrab Zarrabian, Josef Paldus, Jiřı́ Čı́žek

Published Nov 1, 1990• 23 citations
Publisher

Abstract

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 ${\mathrm{C}}_{\mathit{N}}$${\mathrm{H}}_{\mathit{N}}$, 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.

DOI
10.1103/physreva.42.5155

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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 {\mathrm{C}}_{\mathit{N}}$${\mathrm{H}}_{\mathit{N}}, 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.
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.
The authors tested several approximations commonly used in molecular energy calculations by applying them to the six-site half-filled Hubbard model and comparing the results with the exact solution. Three main conclusions follow from this study: (i) for $\frac{\ensuremath{\beta}}{U}$ in the range of interest for molecules, perturbations to the molecular-orbital ground state produce better results than the other more elaborate methods tested; (ii) Nesbet's pair theory is only slightly worse in this range but has the advantage of working well for all values of the parameters, and (iii) perturbative corrections to the unrestricted Hartree-Fock ground state give very poor results when this state has lower symmetry than the Hamiltonian.
In single-reference coupled-cluster (CC) methods, one has to solve a set of non-linear polynomial equations in order to determine the so-called amplitudes that are then used to compute the energy and other properties. Although it is of common practice to converge to the (lowest-energy) ground-state solution, it is also possible, thanks to tailored algorithms, to access higher-energy roots of these equations that may or may not correspond to genuine excited states. Here, we explore the structure of the energy landscape of variational CC and we compare it with its (projected) traditional version in the case where the excitation operator is restricted to paired double excitations (pCCD). By investigating two model systems (the symmetric stretching of the linear H4 molecule and the continuous deformation of the square H4 molecule into a rectangular arrangement) in the presence of weak and strong correlations, the performance of variational pCCD (VpCCD) and traditional pCCD is gauged against their configuration interaction (CI) equivalent, known as doubly occupied CI, for reference Slater determinants made of ground- or excited-state Hartree–Fock orbitals or state-specific orbitals optimized directly at the VpCCD level. The influence of spatial symmetry breaking is also investigated.
We have studied electron correlations in the doped two-dimensional (2D) Hubbard model by using the coupled-cluster method (CCM) to investigate whether or not the method can be applied to correct the independent particle approximations actually used in ab initio band calculations. The double excitation version of the CCM, implemented using the approximate coupled pair (ACP) method, accounts for most of the correlation energies of the 2D Hubbard model in the weak $(U/t\ensuremath{\simeq}1)$ and the intermediate U/tU/tU/t regions $(U/t\ensuremath{\simeq}4).$ The error in this case is always less than 11%.1 The ACP approximation gets less accurate for large U/tU/tU/t $(U/t\ensuremath{\simeq}8)$ and near half-filling. Further incorporation of electron correlation effects is necessary in this region. The accuracy depends neither on the system size, nor the gap between the lowest unoccupied level and the highest occupied level in the finite size effect. Hence, the CCM may be favorably applied to ab initio band calculations on metals as well as semiconductors and insulators.
Methods which aim at universal applicability must be able to describe both weak and strong electronic correlation with equal facility. Such methods are in short supply. The combination of symmetry projection for strong correlation and coupled cluster theory for weak correlation offers tantalizing promise to account for both on an equal footing. In order to do so, however, the coupled cluster portion of the wave function must be optimized in the presence of the symmetry projection. This paper discusses how this may be accomplished, and shows the importance of doing so for both the Hubbard model Hamiltonian and the molecular Hamiltonian, all with a computational scaling comparable to that of traditional coupled cluster theory.