Band-structure calculation using hybrid functionals

Band-structure calculations for hybrid functionals require multiple steps. Below we give a step-by-step introduction and an example. Additionally, we provide some advice reduce computational and human effort.

Step-by-step instructions

For hybrid functionals, the Hamiltonian cannot be expressed in terms of the electronic charge density alone. Instead, the Kohn-Sham orbitals on a regular k mesh are required for any calculation within the formalism of hybrid functionals. The regular k mesh must be supplied in the KPOINTS file. Consequently, restarting a hybrid calculation requires the WAVECAR file of the previous self-consistent-field (SCF) run. This is in contrast to density-functional theory (DFT) where the electronic charge density written to the CHGCAR file suffices to restart a DFT calculation.

Step 1: Run an SCF calculation to obtain a converged WAVECAR file.

Band-structure calculations generally compute the Kohn-Sham orbitals and eigenenergies along a path in reciprocal space which usually connects high-symmetry points in the first Brillouin zone. Some external tools help to identify the high-symmetry points and k points along a high-symmetry path for materials of any symmetry.

Step 2: Determine the high-symmetry points along which VASP should compute the band structure.

There are two options to simultaneously supply a regular k mesh and k points along a high-symmetry path to VASP.

1. Read an explicit list of k points with zero-weighted k points.

Here, the explicit list of the irreducible k points of the regular k mesh can be copied from the IBZKPT file of a previous run to the KPOINTS file. These irreducible k points must be weighted by their multiplicity according to the symmetry of the system. Additionally, the k points along a high-symmetry path must be added to the KPOINTS file with the value of all weights set to zero.

2. Read an additional KPOINTS_OPT file that can specify the high-symmetry path in line mode.

Generally, the KPOINTS file and the KPOINTS_OPT file accept the same format. But again, the regular k mesh needs to be supplied in the KPOINTS file and the high-symmetry path in the KPOINTS_OPT file. We therefore recommend using the Γ-centered mesh or Monkhorst-Pack mesh and providing the high-symmetry path in line mode, respectively.

The KPOINTS_OPT method is more convenient because it allows using the automatic generation modes for the k points. The computational cost and memory requirement can vary for the two methods due to the scaling with the number of k points.

Step 3: Supply a regular k mesh and k points along a high-symmetry path either using the explicit list including zero-weighted k points or using a KPOINTS_OPT file and restart the hybrid calculation from the converged WAVECAR file.

Recommendations and advice

The KPOINTS_OPT file can also be provided when starting an SCF calculation from scratch. In that case, VASP computes the band energies for the k points of the KPOINTS_OPT file after SCF is reached. While this seems to make step 1 obsolete, mind that it affects how VASP can treat the Coulomb-convergence during the SCF calculation. That is, FOCKCORR=2 cannot be used when computing the band structure with either option of supplying the k points.

The method using an explicit list including zero-weighted k points should not be used from scratch for performance reasons. Constructing the Fock exchange dominates the computational cost and memory requirement and it scales with the number of k points. Thus, each SCF step will get unnecessarily expensive when including zero-weighted k points.

Tip: For both methods, restart from a converged hybrid SCF calculation to seize all available options for optimal performance.

As mentioned, the computational cost and memory requirement can vary for the two methods due to the scaling with the number of k points. It is still possible to achieve very fine sampling along the k path with both methods: For the KPOINTS_OPT method, set an appropriate batch size, i.e., the KPOINTS_OPT_NKBATCH tag. For the explicit list including zero-weighted k points, VASP may exceed the available memory if the number of zero-weighted k points is large. In that case, split the hybrid band-structure calculation into multiple calculations. For each calculation, add part of the zero-weighted k points.

Tip: Make fine sampling computationally feasible using the KPOINTS_OPT_NKBATCH tag or splitting the calculation with part of the zero-weighted k points.

To understand how the two methods work in practice, try using them with a DFT calculation as if it were a hybrid calculation.

Finally, let us stress a major difference in hybrid band-structure calculations and DFT band-structure calculations. For density functionals, the electronic charge density suffices to define the Hamiltonian and no regular k mesh is required during DFT band-structure calculations. If no regular k mesh is provided, the electronic charge density must be fixed during the DFT band-structure calculation by setting ICHARG=11 in the INCAR file.

Warning: The electronic charge density must not be fixed for any hybrid calculation, i.e., never set ICHARG=11!

Example of k points for hybrid band-structure calculation

For instance, for cubic-diamond Si with the following POSCAR file

 cd Si
 5.5
  0.0    0.5     0.5 
  0.5    0.0     0.5 
  0.5    0.5     0.0 
   Si
   2
 Fractional
  -0.125 -0.125 -0.125
   0.125  0.125  0.125

we can generate a regular k mesh using the following KPOINTS file

 Regular k-points mesh
 0
 Monkhorst-Pack method
  3 3 3 
  0 0 0

The resulting IBZKPT file contains the following lines:

 Automatically generated mesh
      4
 Reciprocal lattice
   0.00000000000000    0.00000000000000    0.00000000000000             1
   0.33333333333334    0.00000000000000   -0.00000000000000             8
   0.33333333333334    0.33333333333334   -0.00000000000000             6
  -0.33333333333334    0.33333333333334    0.00000000000000            12

For the explicit k-points list, copy the regular k mesh from the IBZKPT file and add, e.g., 5 k points from Γ to X with zero weight:

 Explicit k-points list
      9
 Reciprocal lattice
   0.00000000000000    0.00000000000000    0.00000000000000             1
   0.33333333333334    0.00000000000000   -0.00000000000000             8
   0.33333333333334    0.33333333333334   -0.00000000000000             6
  -0.33333333333334    0.33333333333334    0.00000000000000            12
   0.00000000       0.00000000       0.00000000 0   
   0.12500000       0.00000000       0.12500000 0
   0.25000000       0.00000000       0.25000000 0
   0.37500000       0.00000000       0.37500000 0
   0.50000000       0.00000000       0.50000000 0

For the KPOINTS_OPT method, the same path from Γ to X can be specified by creating the following KPOINTS_OPT file

 k points for band structure
 5  ! intersections 
 line-mode
 Fractional
   0.0000000000     0.0000000000     0.0000000000 Γ
   0.5000000000     0.0000000000     0.5000000000 X 

And continue using the following KPOINTS file

 Regular k-points mesh
 0
 Monkhorst-Pack method
  3 3 3 
  0 0 0

Related tags and articles

KPOINTS, KPOINTS_OPT, Hybrid functionals