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International Journal of Quantum Chemistry

On the Brillouin‐zone integrations in second‐order many‐body perturbation calculations for extended systems of one‐dimensional periodicity

Tomomi Shimazaki, So Hirata

Published Apr 30, 2009• 26 citations
Publisher

Abstract

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

DOI
10.1002/qua.22176

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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
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 propose progressive downsampling of wave vectors in the Brillouin zone integrations occurring in the second-order many-body or Møller-Plesset perturbation (MP2) method for extended systems of one-dimensional periodicity. Higher-lying unoccupied and lower-lying occupied Bloch orbitals are subject to downsampling by an exponentially increasing factor (with base n), making the total number of Bloch orbitals included in the MP2 lattice sums grow only logarithmically with respect to the number of basis functions per unit cell. Unlike the mod n downsampling scheme proposed earlier, this log n scheme reduces the scaling of the computational cost and thus achieves a greater speedup as the unit cell size increases. Correct band indexing is essential for accuracy. Two-electron integrals entering the MP2 energy and quasiparticle energy expressions must be multiplied by quadrature weights that are a function of the energy bands involved, and an algorithm to compute the weights is proposed. A combined use of the log n and mod n schemes can speedup the MP2/6-31G** calculation of polyacetylene typically by a factor of 20 with an error in the correlation energy within a few percent relative to the conventional calculation. Similar combinations can reproduce the MP2 quasiparticle energy bands accurately at a fraction of the usual computational cost.
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.
Abstract The multireference, state‐specific, second‐order, Brillouin–Wigner perturbation theory is presented as a robust approach to the electron correlation problem for systems demanding the use of a multireference function. A posteriori modifications are made which, in the case of a single reference function, recover the well known formula of second‐order many‐body perturbation theory (MBPT2), i.e. Møller–Plesset (MP2) theory, and in the multireference case can be shown to be equivalent to state‐specific multireference Rayleigh–Schrödinger‐like perturbation theory. It is shown that multireference many‐body Brillouin–Wigner perturbation theory when restricted to second‐order is a true many‐body theory, that is, it is a theory, which has the property of extensivity. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
A quantum Monte Carlo method that combines the second-order many-body perturbation theory and Monte Carlo (MC) integration has been developed for correlation and correlation-corrected (quasiparticle) energy bands of one-dimensional solids. The sum-of-product expressions of correlation energy and self-energy are transformed, with the aid of a Laplace transform, into high-dimensional integrals, which are subject to a highly scalable MC integration with the Metropolis algorithm for importance sampling. The method can compute correlation energies of polyacetylene and polyethylene within a few mEh\text{m}{E}_{\text{h}}mEh​ and quasiparticle energy bands within a few tenths of an eV. It does not suffer from the fermion sign problem and its description can be systematically improved by raising the perturbation order.
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.