# Solution LAB1 diamond lattice parameter

### Lattice parameter of diamond

Given the convergence study of the total energy as a function of the k-point grid (as done in previous exercises), here we set the k-grid to a converged value, say

``` nk=8
```

and perform a series of total energy calculations at different lattice parameters, for several values of the kinetic energy cutoff.

Here we consider the following values:

``` ecutwfc_list="20 30 40 50 60 80 100 120 140 160"
for cutwfc in \$ecutwfc_list
do
# run scf calculations for multiple alat values
done
```

For each cutoff, Etot vs alat is fit (eg using a python script provided) and the minimum is extracted.

For instance, once data are extracted and cast in the form:

``` \$> cat results_50Ry.dat
# calculations using ecutwfc=50 Ry
# alat [bohr]    etot [Ry]
6.5388      -22.75492438
6.6062      -22.75814079
6.6736      -22.75965543
6.7410      -22.75832711
6.8084      -22.75526586
6.8758      -22.75029450
6.9432      -22.74340707
```

The script

``` ./LabCQM/tools/analyze_lattice.py
```

can be used to extract the position of the minimum of the curve as well as to plot the behaviour of etot vs alat, e.g.:

``` ./analyze_lattice.py  results_50Ry.dat
```

The scripts developed in previous exercises can be used to run multiple calculations and to extract the data.

### Results as a function of the cutoff

At first, it is important to inspect the behaviour of the total energy as a function of the lattice parameter.

Some plots for `ecutwfc = 20, 30, 40 Ry` and `ecutwfc=50, 60, 80 Ry` are reported below:

In the following you can find the fitted lattice parameter as a function of the kinetic energy cutoff used in the calculations.

``` # Ecut     alat
20      6.8354274
30      6.686406
40      6.727857
50      6.6744762
60      6.6694212
80      6.664164
100      6.6647706
120      6.6639618
140      6.6639618
160      6.6639618
```

A decent convergence can be already obtained using values of the kinetic energy cutoff as low as 60-80 Ry.