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!

5 thoughts on “Band structure calculations in QE using hybrid functionals

  1. Hi Christoph,

    thanks for the tutorial and references; it was very insightful. I was wondering how to plot the results. We essentially have to harvest the orbital eigenvalues and match with the kpts file from open_grid.x right?

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  2. Dear zww4855, could you let me know what exactly you want to plot? The bands? The easiest way would be to let wannier90 do the work by using bands_plot = .true. in the win file and then plot the prefix_band.dat

    In my example I get quite nice bands, at least close to the Gamma point; if you want to improve the bands you probably have to get better MLWFs!

    HTH!
    Chris

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  3. Hi Christoph,

    Thanks for sharing your experience of hybrid functional band structure calculation!

    I have a problem with the script of open_grid.x , everytime the script will show the segmentation fault when I tried to unfold the reduced grids obtained by paw pseudopotentials. However, it works fine with the results obtained by the NC pseudo, do you know why? Or can the open_grid.x grid only deal with the results from NC pseudo?

    Thanks!
    Ruishen

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