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Physical Review Letters

Convergence Behavior of Many-Body Perturbation Theory with Lattice Summations in Polymers

Jun-Qiang Sun, Rodney J. Bartlett

Published Jan 12, 1998• 22 citations
Publisher

Abstract

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 $N$ as $1/{N}^{3}$ while the second-order MBPT contribution to the band energy converges as $1/{N}^{2}$. These convergence behaviors are demonstrated numerically for polyacetylene.

DOI
10.1103/physrevlett.80.349

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International Journal of Quantum Chemistry•Jan 1, 1997•0 citations
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.
Explicit expressions for electron correlation at the second-order many-body perturbation-theory [MBPT(2)] level are presented and implemented for the total energy per unit cell and for the band structure of extended systems. In the latter case, a formula is presented for a direct evaluation of the band gap rather than obtaining it as a difference of two large numbers. Application is made to alternating trans-polyacetylene. We assess the convergence of MBPT(2) with the number of unit cells (N) included in the lattice summations, the number of k-points (K) taken for the integrals over k in the first Brillouin zone, and the cutoff threshold (10−C) for the two-electron integrals. The MBPT(2) correlation correction to the band structure converges very slowly with N and demands a large K while the MBPT(2) correction to the total energy per unit cell converges much faster with N and needs a much smaller K. Neither MBPT(2) correction is sensitive to the cutoff of the two-electron atomic orbital integrals, 10−C, when C≥5. For polyacetylene, the MBPT(2) band gap is much improved from the SCF result, but does not agree with previous numerical data. Analysis shows that the previous MBPT(2) results were obtained either with too few unit cells such that the convergence with N had not been reached, or that the zeroth-order Hartree–Fock results were inadequately converged. MBPT(2) with a DZP basis improves the Hartree–Fock band gap from 5.57 to 3.22 eV at the experimentally estimated geometry, compared to the measured ∼2 eV peak in the absorption spectrum of the system. We verify our results with three independent programs. We also study the band gap as a function of geometry.
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.
An application of second-order many-body perturbation theory to energies and energy bands of polymers is often hindered by the steep polynomial dependence of its computational cost on the number of wave vector sampling points (K)(K)(K) in the Brillouin zone (BZ). It is shown that a Hartree-Fock (HF) calculation with a large value of KKK (120 in the first BZ) followed by a second-order many-body perturbation calculation with a much smaller value (K=6)(K=6)(K=6) can lead to reliable, interpolated correlated energy bands and density of states of a polymer at less than 1% of the computational cost of the conventional approach. Quantitative simulations on photoelectron spectra of trans- and cis-polyacetylenes and polyethylene show that the correlated energy bands and densities of states thus obtained agree quantitatively with the observed and are significant (sometimes qualitative) improvements over the HF results. The energy bands and photoelectron spectra of polydiacetylene are predicted by this method to assist in the interpretation of future high-resolution measurements.
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.
Second-order many-body perturbation theory [MBPT(2)] is the lowest-ranked member of a systematic series of approximations convergent at the exact solutions of the Schrödinger equations. It has served and continues to serve as the testing ground for new approximations, algorithms, and even theories. This article introduces this basic theory from a variety of viewpoints including the Rayleigh-Schrödinger perturbation theory, the many-body Green's function theory based on the Dyson equation, and the related Feynman-Goldstone diagrams. It also explains the important properties of MBPT(2) such as size consistency, its ability to describe dispersion interactions, and divergence in metals. On this basis, this article surveys three major advances made recently by the authors to this theory. They are a finite-temperature extension of MBPT(2) and the resolution of the Kohn-Luttinger conundrum, a stochastic evaluation of the correlation and self-energies of MBPT(2) using the Monte Carlo integration of their Laplace-transformed expressions, and an extension to anharmonic vibrational zero-point energies and transition frequencies based on the Dyson equation.
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.