A Variable Cell calculation for Bulk Silicon.

In our bulk Silicon calculation we had a rigid supercell shape and the starting atomic positions of the silicon atoms were those known to be correct. What effect will a variable cell calculation have?

In the cell file

FIX_ALL_CELL : FALSE

In the .param file our keywords are;

finite_basis_corr : 2

which ensures that an automatic finite basis set correction is made

The complete output is in the link. We see that the first three steps are indeed singlepoint energy calculations

Calculating finite basis set correction with  3 cut-off energies.
Calculating total energy with cut-off of  270.000eV.
------------------------------------------------------------------------ <-- SCF
SCF loop      Energy                           Energy gain       Timer   <-- SCF
                                               per atom          (sec)   <-- SCF
------------------------------------------------------------------------ <-- SCF
Initial   0.00000000E+000                                          4.05  <-- SCF
      1  -2.07999179E+002                    1.03999590E+002      15.06  <-- SCF
      2  -2.16644967E+002                    4.32289377E+000      25.37  <-- SCF
      3  -2.16547612E+002                   -4.86773393E-002      35.44  <-- SCF
      4  -2.16472419E+002                   -3.75966977E-002      45.52  <-- SCF
      5  -2.16473346E+002                    4.63609274E-004      51.23  <-- SCF
      6  -2.16473342E+002                   -2.09197016E-006      55.58  <-- SCF
      7  -2.16473343E+002                    7.21451053E-007      59.95  <-- SCF
      8  -2.16473343E+002                    1.95459391E-008      64.28  <-- SCF
------------------------------------------------------------------------ <-- SCF
Final energy =  -216.4733431455     eV (not corrected for finite basis set)

Calculating total energy with cut-off of  275.000eV.
------------------------------------------------------------------------ <-- SCF
SCF loop      Energy                           Energy gain       Timer   <-- SCF
                                               per atom          (sec)   <-- SCF
------------------------------------------------------------------------ <-- SCF
Initial  -2.16473343E+002                                         65.79  <-- SCF
      1  -2.16474422E+002                    5.39545221E-004      72.46  <-- SCF
      2  -2.16474423E+002                    2.77186149E-007      76.73  <-- SCF
      3  -2.16474423E+002                    2.10512829E-007      81.13  <-- SCF
------------------------------------------------------------------------ <-- SCF
Final energy =  -216.4744232114     eV (not corrected for finite basis set)

Calculating total energy with cut-off of  280.000eV.
------------------------------------------------------------------------ <-- SCF
SCF loop      Energy                           Energy gain       Timer   <-- SCF
                                               per atom          (sec)   <-- SCF
------------------------------------------------------------------------ <-- SCF
Initial  -2.16474423E+002                                         82.48  <-- SCF
      1  -2.16475528E+002                    5.52276233E-004      89.09  <-- SCF
      2  -2.16475528E+002                    3.59413754E-007      93.47  <-- SCF
      3  -2.16475529E+002                    2.76001709E-007      97.85  <-- SCF
------------------------------------------------------------------------ <-- SCF
Final energy =  -216.4755290347     eV (not corrected for finite basis set)

 For future reference: finite basis dEtot/dlog(Ecut) =      -0.061086eV
 Total energy corrected for finite basis set =    -216.475512 eV

With the finite basis set correction in place the calculation can proceed as normal. The calculation completes and we have a silicon bond length of 2.328 Å which is 0.17% out and an improvement of about 1.7% on our fixed cell calculation.

We get a bonus. The calculation of the finite basis set correction provides information that can be used to estimate the bulk modulus of silicon based upon the energy changes as the cell size changes.

BFGS: Final bulk modulus = 9.99180951E+001 GPa

The known bulk modulus of silicon is 98.8 GPa at room temperature (Kittel page 59 [5]) and our zero temperature estimation is only 1 % out. A more rigorous way to calculate this quantity would be to plot energy versus cell volume for a variety of cell volumes (perhaps individually done as fixed cell calculations) and to fit a curve to this plot. The bulk modulus B can then be calculated from

$$B=V*\frac{d^{2}E}{dV^{2}}$$ (1)

In summary the user should become more inclined to a finite basis set correction whenever there is uncertainty about whether the final expected configuration of the supercell might be very different from the initial configuration.