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Rational approximation to the Fermi–Dirac function with applications in density functional theory

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Abstract

We are interested in computing the Fermi–Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from density functional theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We describe Lanczos and sparse direct methods to address these systems. Each approach has advantages and disadvantages that are illustrated with experiments.

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Authors and Affiliations

  1. Department of Mathematics, The University of Alabama, Tuscaloosa, AL, 35487, USA

    Roger B. Sidje

  2. Computer Science & Engineering, University of Minnesota, Twin Cities, Minneapolis, MN, 55455, USA

    Yousef Saad

Authors
  1. Roger B. Sidje
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  2. Yousef Saad
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Correspondence to Roger B. Sidje.

Additional information

Work supported by NSF under grant 0325218, by DOE under grant DE-FG02-03ER25585, and by the Minnesota Supercomputing Institute.

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Sidje, R.B., Saad, Y. Rational approximation to the Fermi–Dirac function with applications in density functional theory. Numer Algor 56, 455–479 (2011). https://doi.org/10.1007/s11075-010-9397-6

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  • DOI: https://doi.org/10.1007/s11075-010-9397-6

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