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Task 3: Structure optimization

Determining lattice constant​

Determine the lattice constant of Silicon by computing the total energy as a function of the lattice constant. You can try a lattice range from 9.6 to 10.8 Borh.

The lattice constant that minimizes the total energy is your optimal lattice constant.

notebooks/silicon-scf.ipynb
x, y = np.loadtxt('../src/silicon/etot-vs-alat.dat', delimiter=' ', unpack=True)
plt.plot(x, y, "o-", markersize=5, label='Etot vs alat')
plt.xlabel('alat (Bohr)')
plt.ylabel('Etot (Ry)')
plt.legend(frameon=False)
plt.show()
etot-vs-alat

Further structure optimization in Quantum ESPRESSO​

What we have just done ourselves, can be done by Quantum ESPRESSO using its structure optimization modes. There are two types of structural optimization calculations in Quantum espresso: (1) relax: where only the atomic positions are allowed to vary, and (2) vc-relax: which allows to vary both the atomic positions and lattice constants.

For these calculations, one must add this to the input file:

&ions
/

&cell
/
src/silicon/si_relax.in
&control
    calculation = 'vc-relax'
    prefix = 'silicon'
    outdir = './tmp/'
    pseudo_dir = './pseudos/'
    etot_conv_thr = 1e-5
    forc_conv_thr = 1e-4
/

&system
    ibrav=2, celldm(1) =14,
    nat=2, ntyp=1,
    ecutwfc=30
/

&electrons
    conv_thr=1e-8
/

&ions
/

&cell
    cell_dofree='ibrav'
/

ATOMIC_SPECIES
  Si  28.0855  Si.pz-vbc.UPF

ATOMIC_POSITIONS (alat)
  Si 0.00 0.00 0.00 0 0 0
  Si 0.25 0.25 0.25 0 0 0

K_POINTS (automatic)
  6 6 6 1 1 1

Perform the plane wave calculation:

pw.x -inp si_relax.in > si_relax.out

This produces following output (see the si_relax.out for more details, look for "Final enthalpy"):

     Final enthalpy =     -15.8536258868 Ry
Begin final coordinates
     new unit-cell volume =    265.89380 a.u.^3 (    39.40140 Ang^3 )
     density =      2.36728 g/cm^3

CELL_PARAMETERS (alat= 14.00000000)
  -0.364556379   0.000000000   0.364556379
   0.000000000   0.364556379   0.364556379
  -0.364556379   0.364556379   0.000000000

ATOMIC_POSITIONS (alat)
Si            0.0000000000        0.0000000000        0.0000000000    0   0   0
Si            0.1822781896        0.1822781896        0.1822781896    0   0   0
End final coordinates

Lattice constant = 0.364556379 * 14 / 0.5 = 10.2076 Bohr.