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arXiv:0711.0921 · Physical Review Letters

Finite-Size Correction in Many-Body Electronic Structure Calculations

Hendra Kwee, Shiwei Zhang, Henry Krakauer

Published Mar 28, 2008• 122 citations
arXivHTMLPDF

Abstract

Finite-size (FS) effects are a major source of error in many-body (MB) electronic structure calculations of extended systems. A method is presented to correct for such errors. We show that MB FS effects can be effectively included in a modified local density approximation calculation. A parametrization for the FS exchange-correlation functional is obtained. The method is simple and gives post-processing corrections that can be applied to any MB results. Applications to a model insulator (P2 in a supercell), to semiconducting Si, and to metallic Na show that the method delivers greatly improved FS corrections.

DOI
10.1103/physrevlett.100.126404

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Finite-size (FS) effects are a major source of error in many-body (MB) electronic structure calculations of extended systems. A method is presented to correct for such errors. We show that MB FS effects can be effectively included in a modified local density approximation calculation. A parametrization for the FS exchange-correlation functional is obtained. The method is simple and gives post-processing corrections that can be applied to any MB results. Applications to a model insulator (P2 in a supercell), to semiconducting Si, and to metallic Na show that the method delivers greatly improved FS corrections.
We extend the postprocessing finite-size (FS) correction method, developed by Kwee, Zhang, and Krakauer [Phys. Rev. Lett. 100, 126404 (2008)], to spin-polarized systems. The method estimates the FS effects in many-body electronic structure calculations of extended systems by a modified density functional theory (DFT) calculation, without having to repeat expensive many-body simulations. We construct a unified FS DFT exchange-correlation functional for spin-unpolarized and fully spin-polarized systems, under the local density approximation. The results are then interpolated to arbitrary spin polarizations. Generalization to other functional forms in DFT are discussed. The application of this FS correction method to several typical magnetic systems with varying supercell sizes demonstrates that it consistently removes most of the FS errors, leading to rapid convergence of the many-body results to the infinite size limit.
Quantum Monte Carlo (QMC) calculations are only possible in finite systems and so solids and liquids must be modeled using small simulation cells subject to periodic boundary conditions. The resulting finite-size errors are often corrected using data from local-density functional or Hartree-Fock calculations, but systematic errors remain after these corrections have been applied. The results of our jellium QMC calculations for simulation cells containing more than 600 electrons confirm that the residual errors are significant and decay very slowly as the system size increases. We show that they are sensitive to the form of the model Coulomb interaction used in the simulation cell Hamiltonian and that the usual choice, exemplified by the Ewald summation technique, is not the best. The finite-size errors can be greatly reduced and the speed of the calculations increased by a factor of 20 if a better choice is made. Finite-size effects plague most methods used for extended Coulomb systems and many of the ideas in this paper are quite general: they may be applied to any type of quantum or classical Monte Carlo simulation, to other many-body approaches such as the GW method, and to Hartree-Fock and density-functional calculations.
Further developments are introduced in the theory of finite-size errors in quantum many-body simulations of extended systems using periodic boundary conditions. We show that our recently introduced model periodic Coulomb interaction [A. J. Williamson et al., Phys. Rev. B 55, R4851 (1997)] can be applied consistently to all Coulomb interactions in the system. The model periodic Coulomb interaction greatly reduces the finite-size errors in quantum many-body simulations. We illustrate the practical application of our techniques with Hartree-Fock and variational and diffusion quantum Monte Carlo calculations for ground- and excited-state calculations. We demonstrate that the finite-size effects in electron promotion and electron addition/subtraction excitation energy calculations are very similar.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
This paper presents a general method to describe and analyze electron correlation effects in local-orbital electronic structure calculations using a generalized Hubbard Hamiltonian. In our approach, we first introduce a local density formalism where the total energy of the system is obtained as a function of the orbital occupancies ni{{n}_{i}}ni​ associated with each local orbital; in particular, exchange and correlation local potentials are presented for a multilevel case. In parallel, using the dynamical mean field approximation, a many-body solution is obtained by means of a local self-energy that appropriately interpolates between the low and high correlation limits. We also show that the local density and the many-body solutions are linked through charge consistency conditions. These two solutions are applied to a multilevel Anderson impurity and to a multiband Hubbard lattice, our results showing the high accuracy of the approach presented in this paper. Further on, we discuss how to apply our previous analysis to the case of crystals and molecules and analyze several examples: bulk Si, and HF and H2O{\mathrm{H}}_{2}\mathrm{O}H2​O molecules. The good results obtained for these cases show that our approach for the description of correlation effects offers an interesting alternative to the well-established density functional methods based on the calculation of the electron density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r}).$
We present accurate many-body results of the electronic densities in several solid materials, including Si, NaCl, and Cu. These results are obtained using the ab initio auxiliary-field quantum Monte Carlo (AFQMC) method working in a plane-wave basis with norm-conserving, multiple-projector pseudopotentials. AFQMC has been shown to be an excellent many-body total energy method. Computation of observables and correlation functions other than the ground-state energy requires back-propagation, whose adaption and implementation in the plane-wave basis AFQMC framework are discussed in the present paper. This development allows us to compute correlation functions, electronic densities, and interatomic forces, paving the way for geometry optimizations and calculations of thermodynamic properties in solids. Finite supercell size effects are considerably more subtle in the many-body framework than in independent-electron calculations. We analyze the convergence of the electronic density, and obtain best estimates for the thermodynamic limit. The densities from several typical density functionals are benchmarked against our near-exact results. The electronic densities we obtained can also be used to help construct improved density functionals.