How to choose the best SIGMA value in VASP?

Question

I am trying to test how to choose the SIGMA value in VASP, so I did a little bit of test. Here is the input file:

INCAR

SYSTEM=FCC Si
#Start parameter for this run:
ISTART=0
ICHARG=2

#Electronic Relaxation 1
ENCUT=240

#Ionic relaxation

#SMEARING
ISMEAR=0
SIGMA=0.1

POSCAR

fcc Si:
3.9
0.5 0.5 0.0
0.0 0.5 0.5
0.5 0.0 0.5
1
cartesian
0 0 0

KPOINTS

K-POINTS SETUP
0
GAMMA
11 11 11
0 0 0

With different SIGMA values, here is the Total energy VS Sigma Value:

My question is:
It seems the total energy won't converge to a fixed limit as the sigma decreases, since the sigma value is used to generate orbital occupation numbers, I am guessing it doesn't mean the sigma value should be as small as possible. Is there any standard procedure to optimize the sigma value or is there a range of sigma values I could choose?

Answer 1

Following the suggestion by Ian, let's first translate SIGMA into electronic structure language: VASP's SIGMA parameter defines the smearing width of the Fermi-Dirac [1] occupation function f(E) in electron volts.

There are at least two different reasons for choosing a finite smearing:

1. You want to simulate a finite electronic temperature.
2. You are dealing with a metal (or material with a very small Kohn-Sham gap), and you would like to speed up convergence with respect to the density of the k-point grid.

From your post, it seems neither is the case - you don't mention electronic temperature, and you are simulating silicon, i.e. a semiconductor (while semi-local density functionals will underestimate its band gap, the resulting gap is not close to 0, see e.g. here).

Further points:

• Electronic smearing is not a convergence parameter. Physically speaking, at T=0K, electronic smearing is 0.
• If the energy gap of your material is much larger than the chosen smearing value, its effect on the electronic occupation (and thus on the total energy) is negligible. This is why VASP gets away with using a default smearing width of 0.2eV (which would correspond to a temperature of roughly 2300K) - for semiconductors and insulators it should not have an effect, while for metals it helps with convergence as a function of the k-point density.
• Your plot does not show the magnitude of the effect you are seeing. I would expect it to be negligible - if not, one may need to investigate (or a VASP user may comment).

[1] By default, VASP uses Methfessel-Paxton smearing, but this detail is not required for the discussion here.