# Temperature and metallic systems

In this example we consider the modeling of a magnesium lattice as a simple example for a metallic system. For our treatment we will use the PBE exchange-correlation functional. First we import required packages and setup the lattice. Again notice that DFTK uses the convention that lattice vectors are specified column by column.

```
using DFTK
using Plots
using Unitful
using UnitfulAtomic
a = 3.01794 # bohr
b = 5.22722 # bohr
c = 9.77362 # bohr
lattice = [[-a -a 0]; [-b b 0]; [0 0 -c]]
Mg = ElementPsp(:Mg, psp=load_psp("hgh/pbe/Mg-q2"))
atoms = [Mg => [[2/3, 1/3, 1/4], [1/3, 2/3, 3/4]]];
```

Next we build the PBE model and discretize it. Since magnesium is a metal we apply a small smearing temperature to ease convergence using the Fermi-Dirac smearing scheme. Note that both the `Ecut`

is too small as well as the minimal $k$-point spacing `kspacing`

far too large to give a converged result. These have been selected to obtain a fast execution time. By default `PlaneWaveBasis`

chooses a `kspacing`

of `2π * 0.022`

inverse Bohrs, which is much more reasonable.

```
kspacing = 0.945 / u"angstrom" # Minimal spacing of k-points,
# in units of wavevectors (inverse Bohrs)
Ecut = 5 # kinetic energy cutoff in Hartree
temperature = 0.01 # Smearing temperature in Hartree
model = model_DFT(lattice, atoms, [:gga_x_pbe, :gga_c_pbe];
temperature=temperature,
smearing=DFTK.Smearing.FermiDirac())
kgrid = kgrid_size_from_minimal_spacing(lattice, kspacing)
basis = PlaneWaveBasis(model, Ecut, kgrid=kgrid);
```

Finally we run the SCF. Two magnesium atoms in our pseudopotential model result in four valence electrons being explicitly treated. Nevertheless this SCF will solve for eight bands by default in order to capture partial occupations beyond the Fermi level due to the employed smearing scheme.

`scfres = self_consistent_field(basis);`

n Free energy Eₙ-Eₙ₋₁ ρout-ρin Diag --- --------------- --------- -------- ---- 1 -1.761934041189 NaN 5.05e-02 5.0 2 -1.762196054229 -2.62e-04 9.85e-03 1.0 3 -1.762239801991 -4.37e-05 6.52e-04 3.3 4 -1.762241612243 -1.81e-06 1.10e-04 3.7 5 -1.762241620240 -8.00e-09 2.15e-05 2.3

`scfres.occupation[1]`

9-element Array{Float64,1}: 1.9999999999077955 1.9999975862989727 0.0040168978631028905 3.0005622057700963e-15 1.1115228237423443e-18 1.111445559767648e-18 7.978234283522822e-19 7.977548011104854e-19 3.310140169977531e-22

`scfres.energies`

Energy breakdown: Kinetic 0.7180667 AtomicLocal 0.3145438 AtomicNonlocal 0.3265690 Ewald -2.1544222 PspCorrection -0.1026056 Hartree 0.0055004 Xc -0.8610493 Entropy -0.0088446 total -1.762241620240

The fact that magnesium is a metal is confirmed by plotting the density of states around the Fermi level.

`plot_dos(scfres)`