When NKRED should not be applied

In metallic systems, NKRED must be used with great care, and results might be wrong, if NKRED is applied. Problematic cases include electron or hole doped semiconductors or insulators. If two electrons are added to a bulk TiO$_2$ cell containing 72 atoms, and calculations are performed using $2\times 2 \times 2$ k-points, the following results are obtained for the one-electron energies and occupancies with and without NKRED=2 (AEXX=0.2 ; HFSCREEN = 0.2):

 k-point   1: 0.0000    0.0000    0.0000
             DOPED NKRED = 2           DOPED NKRED = 1              UNDOPED CASE
  band No.  band energies occupation   band energies occupation   band energies occupation
valence bands
    262       2.4107      2.00000        2.4339      2.00000        2.4082      2.00000
    263       2.4107      2.00000        2.4339      2.00000        2.4082      2.00000
    264       2.8522      2.00000        2.8597      2.00000        2.8566      2.00000
conduction bands						 
    265       5.4046      2.00000        5.8240      1.87262        5.8126      0.00000
    266       5.4908      2.00000        5.8695      1.62151        5.8424      0.00000
    267       5.4894      2.00000        5.8695      1.62192        5.8424      0.00000

 k-point   2: 0.5000    0.0000    0.0000
             DOPED NKRED = 2           DOPED NKRED = 1              UNDOPED CASE
  band No.  band energies occupation   band energies occupation  band energies occupation
valence bands
    262       2.0015      2.00000        2.0144      2.00000       2.0160      2.00000       
    263       2.5961      2.00000        2.6072      2.00000       2.6046      2.00000
    264       2.5961      2.00000        2.6072      2.00000       2.6045      2.00000
conduction bands					                              
    265       6.1904      0.00000        6.1335      0.00435       6.0300      0.00000
    266       6.1904      0.00000        6.1335      0.00435       6.0300      0.00000
    267       6.1907      0.00000        6.1340      0.00426       6.0305      0.00000

 k-point   3 :  0.5000    0.5000    0.0000
             DOPED NKRED = 2           DOPED NKRED = 1              UNDOPED CASE
  band No.  band energies occupation   band energies occupation  band energies occupation
valence bands
    262       2.4237      2.00000        2.4433      2.00000       2.4287      2.00000
    263       2.4238      2.00000        2.4432      2.00000       2.4287      2.00000
    264       2.4239      2.00000        2.4433      2.00000       2.4287      2.00000
conduction bands						                      
    265       5.8966      0.42674        5.9100      1.24121       5.8817      0.00000
    266       5.8780      0.54128        5.9100      1.24143       5.8817      0.00000
    267       5.8826      0.50661        5.9100      1.24261       5.8817      0.00000

Without NKRED, the one electron energies are pretty similar to the one electron energies in the undoped system (last two columns), whereas using NKRED a strong reduction of the ``gap'' between the valence and conduction band is observed, in particular, close to the conduction band minimum (in this case the $\Gamma$ point). This result is an artefact of the approximation used for NKRED=2. The non-local exchange operator cancels the self-interaction present in the Hartree-potential. For NKRED=2 and $2\times 2 \times 2$ k-points, the non-local exchange operator at each k-point is evaluated using the one-electron orbitals at this k-point only, e.g.:

$$V_{\bf k}\left( {\bf G},{\bf G}'\right)= \langle {\bf k}+{\bf G} ... ...}}({\bf G}'-{\bf G}'') C_{m{\bf k}}({\bf G}-{\bf G}'')} {\vert{\bf G}''\vert^2}$$ (6.76)

The sum over $\bf q$, which is present in Equ. (6.63), is replaced by the single k-point ${\bf k}$. This reduces the self-interaction for states that have originally an occupancy larger one, concomitantly pulling those states to lower energies. Initially empty states (occupancy smaller one) are pushed up slightly. Since this is clearly an artefact, NKRED must be used with uttermost care for large supercells with coarse k-point sampling. Please always check whether occupancies are similar at all k-points, if this is not the case, the calculations should be double checked without NKRED.

Since Hartree-Fock type calculations using $2\times 2 \times 2$ k-points without NKRED, are roughly 64 times more expensive than those using the $\Gamma$ point only, it might seem impossible to do anything but $\Gamma$ point only calculations. However, VASP allows to generate special k-points using generating lattices (see Sec. 5.5.3). Particularly usefull for Hartree-Fock type calculations, are the following k-point sets

k-point set generating a bcc like lattice in the BZ ->  2 k-points in BZ
0
direct
 0.5 0.5 0.5
 -.5 -.5 0.5
 0.5 -.5 -.5
 0 0 0

This KPOINTS file generates two 2 k-points, one at the $\Gamma$-point and one along the space diagonal at the BZ boundary ($R$-point).

The second KPOINTS file generates 4 k-points, one at the $\Gamma$-point and three at the $S$-points (the latter ones might be symmetry equivalent for cubic cells).

k-point set generating an fcc lattice ->  4 k-points in BZ
0
direct
 0.5 0.5 0.0
 0.0 0.5 0.5
 0.5 0.0 0.5
 0 0 0

Using such grids, sensible and fairly rapidly converging results are obtained e.g. for electron and hole doped materials, even if the conduction or valence band is partially occupied or depleted. For instance for TiO$_2$ the following energies are obained:

Gamma only     TOTEN  =      -837.759900 eV
2 k-points     TOTEN  =      -838.039157 eV
4 k-points     TOTEN  =      -838.129712 eV
2x2x2          TOTEN  =      -838.104787 eV
2x2x2 NKRED=2  TOTEN  =      -838.418681 eV

Note that results using NKRED not improved compared to $\Gamma$ only calculations.