Practical Density Functional TheoryPractical Density Functional Theory I. Abdolhosseini Sarsari1,...
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Practical Density Functional Theory I. Abdolhosseini Sarsari 1 , 1 Department of Physics Isfahan University of Technology April 29, 2015 Collaborators at IUT : M. Alaei and S. J. Hashemifar Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology ) Practical DFT April 29, 2015 1 / 10
Transcript of Practical Density Functional TheoryPractical Density Functional Theory I. Abdolhosseini Sarsari1,...
Practical Density Functional Theory
I. Abdolhosseini Sarsari1,
1Department of PhysicsIsfahan University of Technology
April 29, 2015
Collaborators at IUT : M. Alaei and S. J. Hashemifar
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 1 / 10
Table of contents
1 Reducing the number of k-points
2 Irreducible Brillouin zone integrationTetrahedron methodSmearing method parametersRight smearing parameters
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 2 / 10
Reducing the number of k-points
Properties like the electron density, total energy, etc. can be evaluated by integrationover k inside the BZ.
fi =1
Ω
∫BZ
d~kfi(~k)
fi =1
NkΣ~kfi(
~k)
fi = Σkw~kfi(~k)
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 3 / 10
Reducing the number of k-points
Example:
fi = Σkw~kfi(~k)
fi =1
4fi(~k4,4) +
1
4fi(~k3,3) +
1
2fi(~k4,3)
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 4 / 10
Irreducible Brillouin zone integration
fi = Σ~kw~kfi(~k)θ(εi(~k) − εF )
In a Semiconductor: density of states vanish smoothly before the gap.In a Metal: Brillouin zone can be divided into regions that are occupied andunoccupied by electrons.
Functions that are integrated change discontinuously from nonzero values to zeroat the Fermi surface.Problem: converging.
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 5 / 10
Irreducible Brillouin zone integration
fi = Σ~kw~kfi(~k)θ(εi(~k) − εF )
In a Semiconductor: density of states vanish smoothly before the gap.In a Metal: Brillouin zone can be divided into regions that are occupied andunoccupied by electrons.Functions that are integrated change discontinuously from nonzero values to zeroat the Fermi surface.
Problem: converging.
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 5 / 10
Irreducible Brillouin zone integration
fi = Σ~kw~kfi(~k)θ(εi(~k) − εF )
In a Semiconductor: density of states vanish smoothly before the gap.In a Metal: Brillouin zone can be divided into regions that are occupied andunoccupied by electrons.Functions that are integrated change discontinuously from nonzero values to zeroat the Fermi surface.Problem: converging.
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 5 / 10
Irreducible Brillouin zone integration
fi = Σ~kw~kfi(~k)θ(εi(~k) − εF )
In a Semiconductor: density of states vanish smoothly before the gap.In a Metal: Brillouin zone can be divided into regions that are occupied andunoccupied by electrons.Functions that are integrated change discontinuously from nonzero values to zeroat the Fermi surface.Problem: converging.
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 5 / 10
Irreducible Brillouin zone integration
fi = Σ~kw~kfi(~k)θ(εi(~k) − εF )
In a Semiconductor: density of states vanish smoothly before the gap.In a Metal: Brillouin zone can be divided into regions that are occupied andunoccupied by electrons.Functions that are integrated change discontinuously from nonzero values to zeroat the Fermi surface.Problem: converging.
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 5 / 10
Irreducible Brillouin zone integration
Aim: Improving convergence with respect to Brillouin zone sampling in metals
No special efforts: very large numbers of k points are needed to getwell-converged results.
Useful algorithms to improve the slow convergence:Tetrahedron methodSmearing method= Electronic tempreture
Trick : replacing step function with a smoother function : partial occupation at the Fermilevel
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 6 / 10
Irreducible Brillouin zone integration
Aim: Improving convergence with respect to Brillouin zone sampling in metals
No special efforts: very large numbers of k points are needed to getwell-converged results.Useful algorithms to improve the slow convergence:
Tetrahedron methodSmearing method= Electronic tempreture
Trick : replacing step function with a smoother function : partial occupation at the Fermilevel
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 6 / 10
Irreducible Brillouin zone integration
Aim: Improving convergence with respect to Brillouin zone sampling in metals
No special efforts: very large numbers of k points are needed to getwell-converged results.Useful algorithms to improve the slow convergence:
Tetrahedron methodSmearing method= Electronic tempreture
Trick : replacing step function with a smoother function : partial occupation at the Fermilevel
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 6 / 10
Irreducible Brillouin zone integration
Aim: Improving convergence with respect to Brillouin zone sampling in metals
No special efforts: very large numbers of k points are needed to getwell-converged results.Useful algorithms to improve the slow convergence:
Tetrahedron methodSmearing method= Electronic tempreture
Trick : replacing step function with a smoother function : partial occupation at the Fermilevel
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 6 / 10
Irreducible Brillouin zone integration
Aim: Improving convergence with respect to Brillouin zone sampling in metals
No special efforts: very large numbers of k points are needed to getwell-converged results.Useful algorithms to improve the slow convergence:
Tetrahedron methodSmearing method= Electronic tempreture
Trick : replacing step function with a smoother function : partial occupation at the Fermilevel
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 6 / 10
Irreducible Brillouin zone integration Tetrahedron method
Tetrahedron method
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 7 / 10
Irreducible Brillouin zone integration Smearing method parameters
The idea of these methods is to force the function being integrated to be continuous bysmearing out the discontinuity.An example of a smearing function: Fermi-Dirac function.
f(k − k0σ
) = [exp(k − k0σ
) + 1]−1
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 8 / 10
Irreducible Brillouin zone integration Smearing method parameters
Smearing method parameters
Fermi-Dirac smearingReduced occupancies below εF are not compensated new occupancies above εF .
Gaussian smearingSmearing parameter, σ has no physical interpretation.Entropy and the free energy cannot be written in terms of f.
Method of Methfessel-PaxtonYields negative occupation numbers!
Marzari-Vanderbilt : cold smearing
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 9 / 10
Irreducible Brillouin zone integration Smearing method parameters
Smearing method parameters
Fermi-Dirac smearingReduced occupancies below εF are not compensated new occupancies above εF .
Gaussian smearingSmearing parameter, σ has no physical interpretation.Entropy and the free energy cannot be written in terms of f.
Method of Methfessel-PaxtonYields negative occupation numbers!
Marzari-Vanderbilt : cold smearing
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 9 / 10
Irreducible Brillouin zone integration Smearing method parameters
Smearing method parameters
Fermi-Dirac smearingReduced occupancies below εF are not compensated new occupancies above εF .
Gaussian smearingSmearing parameter, σ has no physical interpretation.Entropy and the free energy cannot be written in terms of f.
Method of Methfessel-PaxtonYields negative occupation numbers!
Marzari-Vanderbilt : cold smearing
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 9 / 10
Irreducible Brillouin zone integration Smearing method parameters
Smearing method parameters
Fermi-Dirac smearingReduced occupancies below εF are not compensated new occupancies above εF .
Gaussian smearingSmearing parameter, σ has no physical interpretation.Entropy and the free energy cannot be written in terms of f.
Method of Methfessel-PaxtonYields negative occupation numbers!
Marzari-Vanderbilt : cold smearing
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 9 / 10
Irreducible Brillouin zone integration Smearing method parameters
Smearing method parameters
Fermi-Dirac smearingReduced occupancies below εF are not compensated new occupancies above εF .
Gaussian smearingSmearing parameter, σ has no physical interpretation.Entropy and the free energy cannot be written in terms of f.
Method of Methfessel-PaxtonYields negative occupation numbers!
Marzari-Vanderbilt : cold smearing
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 9 / 10
Irreducible Brillouin zone integration Right smearing parameters
Figure: Force acting on an iron atom in a 2-atom unit cell, plotted as a function of smearing and fordifferent Monkhorst-Pack samplings of the Brillouin Zone (different colored curves). The 2-atomsimple cubic cell breaks symmetry with a displacement along the (111) direction by 5 percent ofthe initial 1NN atomic distance. Here we use a PAW pseudo potential (pslibrary 0.2.1) with a PBEparametrization for the XC functional and Marzari-Vanderbilt smearing [Marzari Lectures].
Isfahan University of Technology (IUT) ( Department of Physics Isfahan University of Technology )Practical DFT April 29, 2015 10 / 10
