Pole-Based approximation of the Fermi-Dirac function

  • Lin LinEmail author
  • Jianfeng Lu
  • Lexing Ying
  • E. Weinan


Two approaches for the efficient rational approximation of the Fermi-Dirac function are discussed: one uses the contour integral representation and conformal mapping, and the other is based on a version of the multipole representation of the Fermi-Dirac function that uses only simple poles. Both representations have logarithmic computational complexity. They are of great interest for electronic structure calculations.


Contour integral Fermi-Dirac function Rational approximation 

2000 MR Subject Classification

41A20 65F30 


  1. [1]
    Baroni, S. and Giannozzi, P., Towards very large-scale electronic-structure calculations, Europhys. Lett., 17(6), 1992, 547–552.CrossRefGoogle Scholar
  2. [2]
    Ceriotti, M., Kühne, T. D. and Parrinello, M., A hybrid approach to Fermi operator expansion, 2008, preprint. arXiv:0809.2232v1Google Scholar
  3. [3]
    Ceriotti, M., Kühne, T. D. and Parrinello, M., An effcient and accurate decomposition of the Fermi operator, J. Chem. Phys., 129(2), 2008, 024707.CrossRefGoogle Scholar
  4. [4]
    Goedecker, S. and Colombo, L., Efficient linear scaling algorithm for tight-binding molecular dynamics, Phys. Rev. Lett., 73(1), 1994, 122–125.CrossRefGoogle Scholar
  5. [5]
    Goedecker, S. and Teter, M., Tight-binding electronic-structure calculations and tight-binding molecular dynamics with localized orbitals, Phys. Rev. B, 51(15), 1995, 9455–9464.CrossRefGoogle Scholar
  6. [6]
    Goedecker, S., Linear scaling electronic structure methods, Rev. Mod. Phys., 71(4), 1999, 1085–1123.CrossRefGoogle Scholar
  7. [7]
    Greengard, L. and Rokhlin, V., A fast algorithm for particle simulations, J. Comput. Phys., 73(2), 1987, 325–348.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Hale, N., Higham, N. J. and Trefethen, L. N., Computing A α, log(A), and related matrix functions by contour integrals, SIAM J. Numer. Anal., 46(5), 2008, 2505–2523.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Krajewski, F. R. and Parrinello, M., Stochastic linear scaling for metals and nonmetals, Phys. Rev. B, 71(23), 2005, 233105.CrossRefGoogle Scholar
  10. [10]
    Liang, W. Z., Baer, R., Saravanan, C., et al, Fast methods for resumming matrix polynomials and Chebyshev matrix polynomials, J. Comput. Phys., 194(2), 2004, 575–587.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Liang, W. Z., Saravanan, C., Shao, Y. H., et al, Improved Fermi operator expansion methods for fast electronic structure calculations, J. Chem. Phys., 119(8), 2003, 4117–4125.CrossRefGoogle Scholar
  12. [12]
    Lin, L., Lu, J., Car, R., et al, Multipole representation of the Fermi operator with application to the electronic structure analysis of metallic systems, Phys. Rev. B, 79(11), 2009, 115133.CrossRefGoogle Scholar
  13. [13]
    Lin, L., Lu, J., Ying, L., et al, Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems, Comm. Math. Sci., 2009, submitted.Google Scholar
  14. [14]
    Ozaki, T., Continued fraction representation of the Fermi-Dirac function for large-scale electronic structure calculations, Phys. Rev. B, 75(3), 2007, 035123.CrossRefGoogle Scholar
  15. [15]
    Ying, L., Biros, G. and Zorin, D., A kernel-independent adaptive fast multipole algorithm in two and three dimensions, J. Comput. Phys., 196(2), 2004, 591–626.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of Mathematics and ICESUniversity of Texas at Austin, 1 University Station/C1200AustinUSA
  3. 3.Department of Mathematics and PACMPrinceton UniversityPrincetonUSA

Personalised recommendations