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International Journal of Quantum Chemistry

Analysis of long‐range effects in many‐body correlation approaches for one‐dimensional periodic systems

Marcel Nooijen, Rodney J. Bartlett

Published Jan 1, 1997• 16 citations
Publisher

Abstract

We analyze the divergent contributions to the Hamiltonian for extended, nonmetallic systems in one dimension, to both the ground-state correlation energy and to the correlated band structure. It is shown that the contribution from the long-range divergent part of the Hamiltonian tends to zero as 1/M−2, where M is the extent of the troublesome lattice summations. Therefore, it is well justified to neglect such contributions in higher order many-body perturbation theory or coupled cluster treatments of the electronic structure for polymers. © 1997 John Wiley & Sons, Inc.

DOI
10.1002/(sici)1097-461x(1997)63:3<601::aid-qua1>3.0.co;2-#

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Analysis of long‐range effects in many‐body correlation approaches for one‐dimensional periodic systems

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We analyze the divergent contributions to the Hamiltonian for extended, nonmetallic systems in one dimension, to both the ground-state correlation energy and to the correlated band structure. It is shown that the contribution from the long-range divergent part of the Hamiltonian tends to zero as 1/M−2, where M is the extent of the troublesome lattice summations. Therefore, it is well justified to neglect such contributions in higher order many-body perturbation theory or coupled cluster treatments of the electronic structure for polymers. © 1997 John Wiley & Sons, Inc.

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Coupled-cluster theory with connected single and double excitation operators (CCSD) and related approximations, such as linearized CCSD, quadratic configuration interaction with single and double excitation operators, coupled-cluster with connected double excitation operator (CCD), linearized CCD, approximate CCD, and second- and third-order many-body perturbation theories, are formulated and implemented for infinitely extended one-dimensional systems (polymers), on the basis of the periodic boundary conditions and distance-based screening of integrals, density matrix elements, and excitation amplitudes. The variation of correlation energies with the truncation radii of short- and long-range lattice sums and with the number of wave vector sampling points in the first Brillouin zone is examined for polyethylene, polyacetylene, and polyyne, and is shown to be a function of the degree of pi-electron conjugation or the fundamental band gaps. The t2 and t1 amplitudes in the atomic orbital (AO) basis are obtained by first computing the t amplitudes in the Bloch-orbital basis and subsequently back-transforming them into the AO basis. The plot of these AO-based t amplitudes as a function of unit cells also indicates that the t2 amplitudes of polyacetylene and polyyne exhibit appreciably slower decay than those of polyethylene, although the asymptotic decay behavior is invariably 1/r3. The AO-based t1 amplitudes appear to correlate strongly with the electronic structure, and they decay seemingly exponentially for polyethylene whereas they stay at a constant magnitude across the seventh nearest neighbors of polyacetylene and polyyne, which attests to far reaching effects of nondynamical electron correlation mediated by orbital rotation. Nonetheless, the unit cell contributions to the correlation energies taper below 10(-6) hartree after 15 A for all three polymers. The basis set dependence of the decay behavior of t2 amplitudes is also examined for linear hydrogen fluoride polymer (HF)infinity and linear beryllium polymer (Be)infinity employing the STO-3G, 6-31G, and 6-31G* basis sets, and proves to be rather small.

The Journal of Chemical Physics•Feb 4, 2004•162 citations
We analyze the divergent contributions to the Hamiltonian for extended, nonmetallic systems in one dimension, to both the ground-state correlation energy and to the correlated band structure. It is shown that the contribution from the long-range divergent part of the Hamiltonian tends to zero as 1/M−2, where M is the extent of the troublesome lattice summations. Therefore, it is well justified to neglect such contributions in higher order many-body perturbation theory or coupled cluster treatments of the electronic structure for polymers. © 1997 John Wiley & Sons, Inc.
We analyze the divergent contributions to the Hamiltonian for extended, nonmetallic systems in one dimension, to both the ground-state correlation energy and to the correlated band structure. It is shown that the contribution from the long-range divergent part of the Hamiltonian tends to zero as 1/M−2, where M is the extent of the troublesome lattice summations. Therefore, it is well justified to neglect such contributions in higher order many-body perturbation theory or coupled cluster treatments of the electronic structure for polymers. © 1997 John Wiley & Sons, Inc.
Recently the possible divergence of many-body perturbation theory or many-body green function (propagator) methods with lattice summations in extended systems has been raised. The convergence of these methods with lattice summations is not only the key to establishing their correct scaling properties in inhomogeneous systems, but is also a necessity if numerical calculations are to be meaningful. In this report, it is rigorously shown that many-body perturbation theory (MBPT), coupled cluster theory (CC), and many-body green function (MBGF) methods all converge uniformly with lattice summations, although the integrand for the integration over the reciprocal lattice vector, k, could become infinite at special k values. Our proof is given not only for infinite polymers but also for crystals. In our proof, only the continuity of the zeroth-order band structure and Bloch orbitals with k is used. We show that MBPT, CC, and MBGF methods converge with the radius of the lattice summation range R at least as fast as 1/R1/n where n is the dimensionality of the system. It could be much faster if the zeroth-order band structure and Bloch orbitals have continuous first- or even higher-order derivatives with k. In practical numerical calculations, one should not pursue the convergence of the integrand, but the convergence of the correction considered with lattice summations.
The convergence behavior of many-body perturbation theory (MBPT) with lattice summations directly reflects the spatial property of electron correlation in extended systems. A knowledge of such properties is essential for both theory and numerical calculations. It is shown solely from theory that the second-order MBPT contribution to the total energy per unit cell in polymers converges with lattice size NNN as 1/N31/{N}^{3}1/N3 while the second-order MBPT contribution to the band energy converges as 1/N21/{N}^{2}1/N2. These convergence behaviors are demonstrated numerically for polyacetylene.
Abstract The validity and accuracy of various ways of drastically reducing the number of k‐points in the Brillouin zone integrations occurring in second‐order many‐body perturbation calculations of one‐dimensional solids has been investigated. The most promising approximation can recover correlation energies of polyethylene and polyacetylene within 1% of converged values at less than a tenth of usual computational cost. The quasi‐particle energy bands have also been reproduced quantitatively with the same approximation. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
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.
Coupled-cluster theory with connected single and double excitation operators (CCSD) and related approximations, such as linearized CCSD, quadratic configuration interaction with single and double excitation operators, coupled-cluster with connected double excitation operator (CCD), linearized CCD, approximate CCD, and second- and third-order many-body perturbation theories, are formulated and implemented for infinitely extended one-dimensional systems (polymers), on the basis of the periodic boundary conditions and distance-based screening of integrals, density matrix elements, and excitation amplitudes. The variation of correlation energies with the truncation radii of short- and long-range lattice sums and with the number of wave vector sampling points in the first Brillouin zone is examined for polyethylene, polyacetylene, and polyyne, and is shown to be a function of the degree of pi-electron conjugation or the fundamental band gaps. The t2 and t1 amplitudes in the atomic orbital (AO) basis are obtained by first computing the t amplitudes in the Bloch-orbital basis and subsequently back-transforming them into the AO basis. The plot of these AO-based t amplitudes as a function of unit cells also indicates that the t2 amplitudes of polyacetylene and polyyne exhibit appreciably slower decay than those of polyethylene, although the asymptotic decay behavior is invariably 1/r3. The AO-based t1 amplitudes appear to correlate strongly with the electronic structure, and they decay seemingly exponentially for polyethylene whereas they stay at a constant magnitude across the seventh nearest neighbors of polyacetylene and polyyne, which attests to far reaching effects of nondynamical electron correlation mediated by orbital rotation. Nonetheless, the unit cell contributions to the correlation energies taper below 10(-6) hartree after 15 A for all three polymers. The basis set dependence of the decay behavior of t2 amplitudes is also examined for linear hydrogen fluoride polymer (HF)infinity and linear beryllium polymer (Be)infinity employing the STO-3G, 6-31G, and 6-31G* basis sets, and proves to be rather small.