Pole-Based approximation of the Fermi-Dirac function

Abstract

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.

Keywords

Contour integral / Fermi-Dirac function / Rational approximation

References

[1] Baroni S., Giannozzi P.. Towards very large-scale electronic-structure calculations. Europhys. Lett., 1992, 176: 547-552

[2] Ceriotti, M., Kühne, T. D. and Parrinello, M., A hybrid approach to Fermi operator expansion, 2008, preprint. arXiv:0809.2232v1

[3] Ceriotti M., Kühne T. D., Parrinello M.. An effcient and accurate decomposition of the Fermi operator. J. Chem. Phys., 2008, 129(2): 024707

[4] Goedecker S., Colombo L.. Efficient linear scaling algorithm for tight-binding molecular dynamics. Phys. Rev. Lett., 1994, 73(1): 122-125

[5] Goedecker S., Teter M.. Tight-binding electronic-structure calculations and tight-binding molecular dynamics with localized orbitals. Phys. Rev. B, 1995, 51(15): 9455-9464

[6] Goedecker S.. Linear scaling electronic structure methods. Rev. Mod. Phys., 1999, 71(4): 1085-1123

[7] Greengard L., Rokhlin V.. A fast algorithm for particle simulations. J. Comput. Phys., 1987, 73(2): 325-348

[8] Hale N., Higham N. J., Trefethen L. N.. Computing A ^α, log(A), and related matrix functions by contour integrals. SIAM J. Numer. Anal., 2008, 46(5): 2505-2523

[9] Krajewski F. R., Parrinello M.. Stochastic linear scaling for metals and nonmetals. Phys. Rev. B, 2005, 71(23): 233105

[10] Liang W. Z., Baer R., Saravanan C.. Fast methods for resumming matrix polynomials and Chebyshev matrix polynomials. J. Comput. Phys., 2004, 194(2): 575-587

[11] Liang W. Z., Saravanan C., Shao Y. H.. Improved Fermi operator expansion methods for fast electronic structure calculations. J. Chem. Phys., 2003, 119(8): 4117-4125

[12] Lin L., Lu J., Car R.. Multipole representation of the Fermi operator with application to the electronic structure analysis of metallic systems. Phys. Rev. B, 2009, 79(11): 115133

[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.

[14] Ozaki T.. Continued fraction representation of the Fermi-Dirac function for large-scale electronic structure calculations. Phys. Rev. B, 2007, 75(3): 035123

[15] Ying L., Biros G., Zorin D.. A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys., 2004, 196(2): 591-626