skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory

Abstract

Fermi operator expansion (FOE) methods are powerful alternatives to diagonalization type methods for solving Kohn-Sham density functional theory (KSDFT). One example is the pole expansion and selected inversion (PEXSI) method, which approximates the Fermi operator by rational matrix functions and reduces the computational complexity to at most quadratic scaling for solving KSDFT. Unlike diagonalization type methods, the chemical potential often cannot be directly read off from the result of a single step of evaluation of the Fermi operator. Hence multiple evaluations are needed to be sequentially performed to compute the chemical potential to ensure the correct number of electrons within a given tolerance. This hinders the performance of FOE methods in practice. In this paper, we develop an efficient and robust strategy to determine the chemical potential in the context of the PEXSI method. The main idea of the new method is not to find the exact chemical potential at each self-consistent-field (SCF) iteration but to dynamically and rigorously update the upper and lower bounds for the true chemical potential, so that the chemical potential reaches its convergence along the SCF iteration. Instead of evaluating the Fermi operator for multiple times sequentially, our method uses a two-level strategy that evaluatesmore » the Fermi operators in parallel. In the regime of full parallelization, the wall clock time of each SCF iteration is always close to the time for one single evaluation of the Fermi operator, even when the initial guess is far away from the converged solution. We demonstrate the effectiveness of the new method using examples with metallic and insulating characters, as well as results from ab initio molecular dynamics.« less

Authors:
 [1];  [2]
  1. University of California, Berkeley, CA (United States). Dept. of Mathematics
  2. University of California, Berkeley, CA (United States). Dept. of Mathematics; Lawrence Berkeley National Laboratory, Berkeley, CA (United States). Computational Research Division
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1497884
Grant/Contract Number:  
AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 147; Journal Issue: 14; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY

Citation Formats

Jia, Weile, and Lin, Lin. Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory. United States: N. p., 2017. Web. doi:10.1063/1.5000255.
Jia, Weile, & Lin, Lin. Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory. United States. doi:10.1063/1.5000255.
Jia, Weile, and Lin, Lin. Wed . "Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory". United States. doi:10.1063/1.5000255. https://www.osti.gov/servlets/purl/1497884.
@article{osti_1497884,
title = {Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory},
author = {Jia, Weile and Lin, Lin},
abstractNote = {Fermi operator expansion (FOE) methods are powerful alternatives to diagonalization type methods for solving Kohn-Sham density functional theory (KSDFT). One example is the pole expansion and selected inversion (PEXSI) method, which approximates the Fermi operator by rational matrix functions and reduces the computational complexity to at most quadratic scaling for solving KSDFT. Unlike diagonalization type methods, the chemical potential often cannot be directly read off from the result of a single step of evaluation of the Fermi operator. Hence multiple evaluations are needed to be sequentially performed to compute the chemical potential to ensure the correct number of electrons within a given tolerance. This hinders the performance of FOE methods in practice. In this paper, we develop an efficient and robust strategy to determine the chemical potential in the context of the PEXSI method. The main idea of the new method is not to find the exact chemical potential at each self-consistent-field (SCF) iteration but to dynamically and rigorously update the upper and lower bounds for the true chemical potential, so that the chemical potential reaches its convergence along the SCF iteration. Instead of evaluating the Fermi operator for multiple times sequentially, our method uses a two-level strategy that evaluates the Fermi operators in parallel. In the regime of full parallelization, the wall clock time of each SCF iteration is always close to the time for one single evaluation of the Fermi operator, even when the initial guess is far away from the converged solution. We demonstrate the effectiveness of the new method using examples with metallic and insulating characters, as well as results from ab initio molecular dynamics.},
doi = {10.1063/1.5000255},
journal = {Journal of Chemical Physics},
number = 14,
volume = 147,
place = {United States},
year = {2017},
month = {10}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 1 work
Citation information provided by
Web of Science

Figures / Tables:

Figure 1 Figure 1: Fermi-Dirac distribution for phospherene nanoribbon (PNR) 180 atoms (left) and graphene nanoribbon (GRN) 180 atoms (right). The blue dashed line shows the exact number of electrons (900 for PNR and 720 for GRN).

Save / Share:

Works referenced in this record:

XIX. A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares
journal, August 1852

  • Sylvester, J. J.
  • The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 4, Issue 23
  • DOI: 10.1080/14786445208647087

SIESTA-PEXSI: massively parallel method for efficient and accurate ab initio materials simulation without matrix diagonalization
journal, July 2014


Accelerating atomic orbital-based electronic structure calculation via pole expansion and selected inversion
journal, June 2013


SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix
journal, February 2011

  • Lin, Lin; Yang, Chao; Meza, Juan C.
  • ACM Transactions on Mathematical Software, Vol. 37, Issue 4
  • DOI: 10.1145/1916461.1916464

DGDFT: A massively parallel method for large scale density functional theory calculations
journal, September 2015

  • Hu, Wei; Lin, Lin; Yang, Chao
  • The Journal of Chemical Physics, Vol. 143, Issue 12
  • DOI: 10.1063/1.4931732

Linear-scaling density-functional theory with tens of thousands of atoms: Expanding the scope and scale of calculations with ONETEP
journal, July 2009

  • Hine, N. D. M.; Haynes, P. D.; Mostofi, A. A.
  • Computer Physics Communications, Vol. 180, Issue 7
  • DOI: 10.1016/j.cpc.2008.12.023

SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems
journal, June 2003

  • Li, Xiaoye S.; Demmel, James W.
  • ACM Transactions on Mathematical Software, Vol. 29, Issue 2
  • DOI: 10.1145/779359.779361

Pole-Based approximation of the Fermi-Dirac function
journal, August 2009

  • Lin, Lin; Lu, Jianfeng; Ying, Lexing
  • Chinese Annals of Mathematics, Series B, Vol. 30, Issue 6
  • DOI: 10.1007/s11401-009-0201-7

The SIESTA method for ab initio order- N materials simulation
journal, March 2002

  • Soler, José M.; Artacho, Emilio; Gale, Julian D.
  • Journal of Physics: Condensed Matter, Vol. 14, Issue 11
  • DOI: 10.1088/0953-8984/14/11/302

Recent progress in linear scaling ab initio electronic structure techniques
journal, March 2002


Minimax rational approximation of the Fermi-Dirac distribution
journal, October 2016

  • Moussa, Jonathan E.
  • The Journal of Chemical Physics, Vol. 145, Issue 16
  • DOI: 10.1063/1.4965886

\mathcal{O}(N) methods in electronic structure calculations
journal, February 2012


Ab initio molecular simulations with numeric atom-centered orbitals
journal, November 2009

  • Blum, Volker; Gehrke, Ralf; Hanke, Felix
  • Computer Physics Communications, Vol. 180, Issue 11
  • DOI: 10.1016/j.cpc.2009.06.022

Nearsightedness of electronic matter
journal, August 2005

  • Prodan, E.; Kohn, W.
  • Proceedings of the National Academy of Sciences, Vol. 102, Issue 33
  • DOI: 10.1073/pnas.0505436102

Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach
journal, April 2005

  • VandeVondele, Joost; Krack, Matthias; Mohamed, Fawzi
  • Computer Physics Communications, Vol. 167, Issue 2
  • DOI: 10.1016/j.cpc.2004.12.014

Edge-Modified Phosphorene Nanoflake Heterojunctions as Highly Efficient Solar Cells
journal, February 2016


Electronic structure and aromaticity of large-scale hexagonal graphene nanoflakes
journal, December 2014

  • Hu, Wei; Lin, Lin; Yang, Chao
  • The Journal of Chemical Physics, Vol. 141, Issue 21
  • DOI: 10.1063/1.4902806

Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation
journal, February 2012


PSelInv—A Distributed Memory Parallel Algorithm for Selected Inversion: The Symmetric Case
journal, December 2016

  • Jacquelin, Mathias; Lin, Lin; Yang, Chao
  • ACM Transactions on Mathematical Software, Vol. 43, Issue 3
  • DOI: 10.1145/2786977

Daubechies wavelets for linear scaling density functional theory
journal, May 2014

  • Mohr, Stephan; Ratcliff, Laura E.; Boulanger, Paul
  • The Journal of Chemical Physics, Vol. 140, Issue 20
  • DOI: 10.1063/1.4871876

    Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.