Rational approximation to the Fermi-Dirac function with applications in density functional theory
Roger B. Sidje, Yousef Saad
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.
Continued fraction
Diagonal of matrix inverse
Electronic structure calculations
Fermi-Dirac
Rational Chebyshev
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Density functional theory Engineering & Materials Science
Rational Approximation Mathematics
Matrix Function Mathematics
Paul Adrien Maurice Dirac Mathematics
Hamiltonian Matrix Mathematics
Partial fractions Mathematics
Chebyshev Approximation Mathematics
Sidje, R. B., & Saad, Y. (2011). Rational approximation to the Fermi-Dirac function with applications in density functional theory. Numerical Algorithms, 56(3), 455-479. https://doi.org/10.1007/s11075-010-9397-6
Rational approximation to the Fermi-Dirac function with applications in density functional theory. / Sidje, Roger B.; Saad, Yousef.
In: Numerical Algorithms, Vol. 56, No. 3, 01.03.2011, p. 455-479.
Sidje, RB & Saad, Y 2011, 'Rational approximation to the Fermi-Dirac function with applications in density functional theory', Numerical Algorithms, vol. 56, no. 3, pp. 455-479. https://doi.org/10.1007/s11075-010-9397-6
Sidje RB, Saad Y. Rational approximation to the Fermi-Dirac function with applications in density functional theory. Numerical Algorithms. 2011 Mar 1;56(3):455-479. https://doi.org/10.1007/s11075-010-9397-6
Sidje, Roger B. ; Saad, Yousef. / Rational approximation to the Fermi-Dirac function with applications in density functional theory. In: Numerical Algorithms. 2011 ; Vol. 56, No. 3. pp. 455-479.
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title = "Rational approximation to the Fermi-Dirac function with applications in density functional theory",
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.",
keywords = "Continued fraction, Density functional theory, Diagonal of matrix inverse, Electronic structure calculations, Fermi-Dirac, Rational Chebyshev",
author = "Sidje, {Roger B.} and Yousef Saad",
journal = "Numerical Algorithms",
T1 - Rational approximation to the Fermi-Dirac function with applications in density functional theory
AU - Sidje, Roger B.
AU - Saad, Yousef
N2 - 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.
AB - 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.
KW - Continued fraction
KW - Diagonal of matrix inverse
KW - Electronic structure calculations
KW - Fermi-Dirac
KW - Rational Chebyshev
JO - Numerical Algorithms
JF - Numerical Algorithms