Calculating the bulk modulus of Silicon

Many times when doing an analysis of a (novel) material, you have to validate your model against values from real experiments. One of those values is the bulk modulus of a material which we are going to calculate for bulk silicon.

If you are looking at a crystal with a well known structure, the simulation study gets particularly easy since you can specify the atomic coordinates in terms of an irreducible cell (note the SCALED keyword in the &COORD section):

silicon.inp

&GLOBAL
  PROJECT silicon
  RUN_TYPE ENERGY
  PRINT_LEVEL MEDIUM
&END GLOBAL
&FORCE_EVAL
  METHOD Quickstep
  STRESS_TENSOR ANALYTICAL
  &DFT
    BASIS_SET_FILE_NAME  BASIS_SET
    POTENTIAL_FILE_NAME  POTENTIAL
    &POISSON
      PERIODIC XYZ
    &END POISSON
    &SCF
      SCF_GUESS ATOMIC
      EPS_SCF 1.0E-6
      MAX_SCF 500
    &END SCF
    &XC
      &XC_FUNCTIONAL PBE
      &END XC_FUNCTIONAL
    &END XC
  &END DFT
  &SUBSYS
    &KIND Si
      ELEMENT   Si
      BASIS_SET DZVP-GTH-PBE
      POTENTIAL GTH-PBE
    &END KIND
    &CELL
      ABC 5.430697500 5.430697500 5.430697500
      PERIODIC XYZ
    &END CELL
    &COORD
      SCALED
      Si    0    0    0
      Si    0    2/4  2/4
      Si    2/4  2/4  0
      Si    2/4  0    2/4
      Si    3/4  1/4  3/4
      Si    1/4  1/4  1/4
      Si    1/4  3/4  3/4
      Si    3/4  3/4  1/4
    &END COORD
  &END SUBSYS
&END FORCE_EVAL

• By scaling the lattice constant (for example between $0.97 \cdot a$ and $1.1 \cdot a$) you can now run the simulation for different volumes and get a volume-energy curve. You may want to use and adapt the script from the previous exercise
• Fit this curve to the Birch–Murnaghan equation of state to recover the bulk modulus $B_0$