[gpaw-users] Questions on EquationofStates in pw mode

[Ole Holm Nielsen]

jun yan <junyan at stanford.edu> wrote:
>      When one performs the EOS in pw mode, the number of plane waves are forced to be the same with different volumes of cells (let's say the lattice parameters are scaled from 0.95 to 1.05) by using the command PW(ecut, cell=cell_at_scale_1.0). Using this way, the EOS fitting curve looks perfect. It however means that, with increasing cell volume, the effective plane wave cutoff energy is decreasing since the number of plane waves is fixed. In my calculations, the decreasing of plane wave cutoff energy results in increasing of the total energy. The consequence is thus the lattice parameters shift to smaller values using fixed number of plane waves compared to using fixed plane wave cutoff energy. My question is, which one is physically meaningful : fixing the number of plane waves or the effective plane wave energy cutoff ? I will think its the latter. However, the EOS based on using the same plane wave cutoff energy looks much worse than fixing the number of plane waves.!
  I!
>   appreciate if anyone has any insights on this. Thanks !

This is probably a FAQ. Conventional wisdom in the field of plane wave 
DFT calculations states that for varying unit cell volumes you *must* 
keep the plane wave *kinetic energy* cutoff constant in order to ensure 
a constant, unbiased real-space resolution of your plane wave basis set. 
  Probably every textbook dealing with the plane wave method will agree 
with this.

If you (wrongly) vary the volume and keep the *number* of plane waves 
constant (i.e., the basis set is unchanged), you'll have a finer real 
space resolution at smaller volumes, and hence lower total energies at 
those volumes than if you (correctly) kept the cutoff energy constant. 
The calculated equation of state (EOS), lattice constant, and bulk 
modulus will consequently be quite wrong.  The exception is if you have 
an unnecessarily high number of plane waves so that the calculation is 
equally well converged with respect to the basis set at all volumes.

I wrote a paper with two colleagues on this particular aspect: P. Gomes 
Dacosta et al., J. Phys. C: Solid State Phys. vol. 19 (1986), 3163-3172. 
We discussed the jumps in E_total as the basis set changes, and show how 
the pressure (stress theorem) can be used to improve the numbers, and 
even a correction formula Eq.(5) for E_total due to non-fully converged 
basis sets.  Hopefully this old paper may clarify the issue that you are 
asking about.

For the EOS it also makes a difference *which* mathematical formula you 
choose.  I don't know the current status in this field (high pressure 
physics), but many years ago we discussed whether the Birch EOS was more 
physically correct than the Murnaghan EOS for determining the bulk 
modulus (probably yes).  There may be newer EOS formulas which perform 
even better.  Please take a look at Wikipedia:
http://en.wikipedia.org/wiki/Birch%E2%80%93Murnaghan_equation_of_state
and
http://www.sklogwiki.org/SklogWiki/index.php/Rose-Vinet_%28Universal%29_equation_of_state

Best regards,
Ole

-- 
Ole Holm Nielsen
Department of Physics, Technical University of Denmark