homogeneous vs. randomized k/q-mesh in BZ integration

[NFH]

Dear EPW-team,

first of all congratulations to the refurbished, extended version of EPW. I was using EPW2.3
for quite some time as a benchmark for my own clutter of el-ph code, but the new version is
really a pleasure.

My question:

In the example for doped Diamond http://epw.org.uk/Documentation/B-dopedDiamond
in Fig. 3f) the phonon linewidths for the highest phonon mode is shown - randomized 50^3
Sobol k-points and a smearing of 100meV was used. I note \gamma:max~7meV around
the \Gamma point.
In the next Fig. 4, the same calculation is done with an homogeneous/unshifted 50^3 grid.
Here I note \gamma:max~11meV around the \Gamma point.

While I know that convergence with random meshes is much faster and a 50^3 homogenous
grid is far from convergence, I wonder why the discrepancy between both "integration method"
in the absolute value of the selfenergy is so large.

Is there a certain criterion (or empirical knowledge) on the convergence dependency
of Brillouin-zone integrations with randomized/homogenous meshes?

thanks for your effort!
Nicki

CAMd, DTU Lyngby

[sponce]

Dear Nicki,

Thank you for your interest in EPW.

There are many things that can affect this.

1) Figure 3 from the tutorial was done with an old version of EPW (v~2).
2) The height and smoothness of the phonon self energy depends on
- the k-point grid on which it is integrated
- the smearing for the i\delta in the denominator (here 0.1 eV)
- the temperature at which the calculation is done (this affects the Fermi-Dirac and Bose-Einstein distribution functions)
3) The most likely reasons is the following: there are three ways of computing the im. part of Eq. (4) of the new EPW paper:
- taking the imaginary part directly
- computing it with \delta (you can show that this is analytically the same, however it will be numerically slightly different)
- you can use the "delta delta" approximation.

In any case, the reference for the new version of EPW is Fig. 5 of the new EPW paper. Note that the calculations are done at 300K.

Hope it helps,

Samuel