# 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
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 = 0.5 # 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 -1.761837494527 NaN 5.06e-02 4.7 2 -1.762190485497 -3.53e-04 9.80e-03 1.0 3 -1.762239269615 -4.88e-05 6.90e-04 3.3 4 -1.762241610022 -2.34e-06 1.11e-04 3.3 5 -1.762241620378 -1.04e-08 1.06e-05 2.3

`scfres.occupation[1]`

9-element Array{Float64,1}: 1.999999999907795 1.999997586291818 0.004016898611702055 3.000523120348313e-15 1.1115675460661067e-18 1.1115011746909567e-18 7.978132870801505e-19 7.977513813948098e-19 3.1114397656090493e-22

`scfres.energies`

Energy breakdown: Kinetic 0.7180651 AtomicLocal 0.3145392 AtomicNonlocal 0.3265751 Ewald -2.1544222 PspCorrection -0.1026056 Hartree 0.0055003 Xc -0.8610489 Entropy -0.0088446 total -1.762241620378

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

`plot_dos(scfres)`