Hybrid band structure calculations (QE 7.0 update)

Dear all,

I have received several emails and comments reporting issues with the pw.x + open_grid.x + wannier90 approach for band structure calculations with hybrid XC functionals. I decided to quickly test the latest version of QE and found that all is well – at least for my simple test case.

The attached run file will let you calculate the bands of bulk MgO with hybrid functionals (I tested HSE, B3LYP and PBE0 but that choice was entirely arbitrary and does not reflect any meaningful selection). I also tested that both, norm-conserving (ONCV) and ultrasoft/PAW pseudopotentials work at this stage and found no problem.

Below you will see the bands obtained for the ONCV case – note that this is just a test of the code’s capabilities and not converged (wiggles in the bands indicate that the wannierization failed to localize the states sufficiently which might require adjustment of the wannierization window)

And we can also quickly check if the hybrid functionals do what we expect them to do – fix the “band-gap problem”. Indeed, we find that the band-gap is increased from the PZ case (~4.1 eV) to a maximum value of 7.2 eV (using PBE0) – with literature values usually falling between 5.7 and 7.7 this is not a bad match at all 🙂

As always, use these with caution and make sure your calculations are converged! A run script that tests above-mentioned cases as well as an example of the output for HSE using ONCV pseudopotentials are available for download below.

Have fun!

Band structure calculations in QE using hybrid functionals

In recent years the interest in hybrid functionals (that is the incorporation of parts of Hartree-Fock exchange in calculations based on common approximations of exchange-correlations such as LDA, GGA and so on) has steadily increased owing to its improvement over most common functionals, especially when it comes to band-gap calculation of extended solids – see for example this work for a comparison. Quantum Espresso now offers a variety of hybrid functionals (for a complete list see the header of funct.f90) but currently the code can only use hybrid functionals for self-consistent calculations. The reason being that:
The problem is quite fundamental, because in order to get the Fock operator at a certain k-point you need the wavefunctions on a grid that is commensurate with it, this can only be done self-consistently. (L. Paulatto)
But there is a quite elegant solution via maximally localized Wannier functions. If you worked with wannier functions before you know they required the “full grid” (conveniently generated using the tool kmesh.pl) but a recently introduced tool allows to “unfold” the SCF calculation of the reduced grid to the full grid using the executable open_grid.x. An example is provided with recent versions of QE as well. In the following I will use this approach to improve the band-gap of MgO. Using PBE pseudopotentials the band-gap is underestimated at about 4.3 eV but using the HSE hybrid the band-gap can be improved to about 7 eV, which is within acceptable range of the experimental value of ~7.7 eV. The workflow is fairly straightforward:
  1. run a (converged) SCF calculation with input_dft=’HSE’ and a number of empty bands. You have to ensure convergence with respect to the usual parameters (k-points, cutoff, …) AND the mesh for the Fock operator (nqx)
  2.  unfold the reduced grid onto the full grid using open_grid.x
  3.  wannierize the obtained wave-functions using wannier90.x and plot the band-structure along a desired high symmetry path
The wannierization is by far the most tricky part in this particular example but by projecting on O:p and Mg:s one can accurately describe  the valence band and an additional single conduction band. For the resulting band-structure see the figure below.
Figure 1. Comparison of the band-structure of MgO on PBE level (purple) and the corresponding wannierzation (grey) as well as the final HSE hybrid calculation with vastly improved band-gap of 7.4 eV (blue)
So this is actually quite straight-forward. please note that in above calculation the Fock operator was calculated on a very coarse 1x1x1 grid. The calculation on such a coarse grid actually seems to over-estimate the band-gap and a converged energy can be obtained on a 6x6x6 nqx grid with ~6.7 eV. The entire calculation can be run using the attached script – I hope you find it helpful!