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