View source for Downsampling of the Hartree-Fock operator

<span id="downsampling"></span>
=== Downsampling ===
Consider the description of a certain bulk system, using a supercell made up
of <math>N</math> primitive cells, in such a way that, {'''A'''<sub> i</sub>' }, the lattice vectors
of the supercell are given by '''A'''<sub> i</sub>' =''n''<sub> i</sub>'''A'''<sub> i</sub> (i=1,2,3), 
where {'''A'''<sub> i</sub>} are the lattice vectors of the primitive cell.
Let <math>R_{\rm max}=2/\mu</math> be the distance for which

<span id="rmax">
:<math>
\frac{\mathrm{erfc}\left(\mu|\mathbf{r}-\mathbf{r}'|\right)}{|\mathbf{r}-\mathbf{r}'|} \approx 0 
\quad \mathrm{for}\quad {|\mathbf{r}-\mathbf{r}'|}>R_\mathrm{max}.
</math>
</span>

When the nearest neighbour distance between the periodically repeated images
of the supercell <math>R_{NN}>2R_{\rm max}</math> (i.e. <math>R_{NN}>4/\mu</math>), the 
short-range Hartree-Fock potential can be represented exactly, sampling the BZ at the <math>\Gamma</math>-point only, i.e.,

:<math>
V^{\mathrm{SR}}_x\left(\mathbf{r},\mathbf{r}'\right)=
-\frac{e^2}{2}\sum_m f_{m{\Gamma}}
u_{m{\Gamma}}^{*}(\mathbf{r}')u_{m{\Gamma}}(\mathbf{r})
\frac{\mathrm{ erfc}(\mu|\mathbf{r}-\mathbf{r}'|)}
{\vert \mathbf{r}-\mathbf{r}' \vert}.
</math>

This is equivalent to a representation of the bulk system using the 
primitive cell and a <math>n_{1}\times n_{2}\times n_{3}</math> sampling of the BZ, 

<span id="VxFockSR">
:<math>
V^{\mathrm{SR}}_x\left(\mathbf{r},\mathbf{r}'\right)=
-\frac{e^2}{2}\sum_{m'\mathbf{q}}f_{m'\mathbf{q}}
e^{-i\mathbf{q}\cdot\mathbf{r}'}u_{m'\mathbf{q}}^{*}(\mathbf{r}')
u_{m'\mathbf{q}}(\mathbf{r})e^{i\mathbf{q}\cdot\mathbf{r}}
\times\frac{\mathrm{ erfc}(\mu|\mathbf{r}-\mathbf{r}'|)}
{\vert \mathbf{r}-\mathbf{r}' \vert}
</math>
</span>

where the set of <math>{\mathbf{q}}</math> vectors is given by

:<math>
\{\mathbf{q}\}=\{i\mathbf{G}_1+j\mathbf{G}_2+k\mathbf{G}_3\}, 
</math>

for ''i'' =1,..,''n''<sub> 1</sub>, ''j'' =1,..,n<sub> 2</sub>, and ''k'' =1,..,n<sub> 3</sub>, with '''G'''<sub>1,2,3</sub> being the reciprocal lattice vectors of the supercell.

In light of the above it is clear that the number of <math>{\mathbf{q}}</math>-points needed to represent
the short-range Hartree-Fock potential decreases with decreasing <math>R_{\rm max}</math>
(i.e., with increasing <math>\mu</math>).
Furthermore, one should realize that the maximal range of the exchange interactions
is not only limited by the <math>{\rm erfc}(\mu|{\bf r}-{\bf r}'|)/|{\bf r}-{\bf r}'|</math>
kernel, but depends on the extent of the spatial overlap of the orbitals as well 
(this can easily be shown for the [[#ExFock|Hartree-Fock exchange energy]] when one adopts a Wannier 
representation of the orbitals);
<math>R_{\rm max}=2/\mu</math>; (as defined [[#rmax|above]]), therefore, provides an upper limit
for the range of the exchange interactions, consistent with maximal spatial
overlap of the orbitals.

It is thus well conceivable that the situation arises where the short-range Hartree-Fock 
potential may be represented on a considerably coarser mesh of points in the BZ than 
the other contributions to the Hamiltonian.
To take advantage of this situation one may, for instance, restrict the sum over 
<math>{\bf q}</math> in the [[#VxFockSR|short range exchange potential]] to a subset, {<math>{\bf q_{k}}</math>}, of the full <math>N_{1}\times N_{2}\times N_{3}</math> <math>{\bf k}</math>-point set, {<math>{\bf k}</math>}, for which the following holds

:<math>
\mathbf{q_k} = \mathbf{b}_1 \frac{n_1 C_1}{N_1} + \mathbf{b}_2 \frac{n_2 C_2}{N_2}
+ \mathbf{b}_3 \frac{n_3 C_3}{N_3},\quad(n_i=0,..,N_i-1)
</math>

where '''b'''<sub>1,2,3</sub> are the reciprocal lattice vectors of the primitive cell,
and <math>C_{i}</math> is the integer grid reduction factor along reciprocal lattice direction
'''b'''<sub> i</sub>. This leads to a reduction in the computational workload by a factor:

:<math>
\frac{1}{C_1 C_2 C_3}
</math>

'''Note''': From the above, one should not get the impression that the grid reduction can only be used (or is useful) only in conjunction with range-separated functionals (e.g., HSE03/HSE06). It can be applied, for instance, in the PBE0 and pure HF cases as well, although from the above it might be clear that the range-separated functionals, in general, will allow for a larger reduction of the grid.{{cite|paier:jcp:06}}

<span id="downsamplingCaveat"></span>
==== Caveat: when one should not use downsampling ====
In metallic systems, downsampling the exact exchange potential ({{TAG|NKRED}}, {{TAG|NKREDX}}, {{TAG|NKREDY}}, and/or {{TAG|NKREDZ}} &ne; 1) must be used with great care, and results might be wrong, if downsampling is applied. Problematic cases include electron or hole doped semiconductors or insulators. If two electrons are added to a bulk TiO<sub>2</sub> cell containing 72 atoms, and calculations are performed using <math>2\times2\times2</math> <math>{\bf k}</math>-points, the following results are obtained for the one-electron energies and occupancies with and without {{TAG|NKRED}}=2 ({{TAG|LHFCALC}}=.TRUE. ; {{TAG|AEXX}}=0.2 ; {{TAG|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 {{TAG|NKRED}}, the one electron energies are pretty similar to the one electron energies in the undoped system (last two columns), whereas using {{TAG|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  point). This result is an artifact of the approximation used for {{TAG|NKRED}}=2. The non-local exchange operator cancels the self-interaction present in the Hartree-potential. For {{TAG|NKRED}}=2 and <math>2\times2\times2</math> <math>{\bf k}</math>-points, the non-local exchange operator at each <math>{\bf k}</math>-point is evaluated using the one-electron orbitals at this <math>{\bf k}</math>-point only, e.g.:

:<math>
V_{\mathbf{k}}\left( \mathbf{G},\mathbf{G}'\right)=
\langle \mathbf{k}+\mathbf{G} | V_x | \mathbf{k}+\mathbf{G}'\rangle =
-\frac{4\pi e^2}{\Omega} f_{m\mathbf{k}}\sum_{\mathbf{G}''}
\frac{C^*_{m\mathbf{k}}(\mathbf{G}'-\mathbf{G}'') C_{m\mathbf{k}}(\mathbf{G}-\mathbf{G}'')}
{|\mathbf{G}''|^2}
</math>

The sum over <math>{\bf q}</math> in the [[#VxFockRecip|Hartree-Fock exchange potential]] reduces to a single <math>{\bf k}</math>-point. This reduces the self-interaction for states that originally have an occupancy larger than one, concomitantly pulling those states to lower energies. Initially, empty states (occupancy smaller one) are pushed up slightly. Since this is an artifact, {{TAG|NKRED}} must be used with utmost care for large supercells with coarse <math>{\bf k}</math>-point sampling. Please always check whether occupancies are similar at all <math>{\bf k}</math> points if this
is not the case. The calculations should be double-checked without downsampling.

Since HF type calculations using <math>2\times2\times2</math> <math>{\bf k}</math> points without {{TAG|NKRED}}, are roughly 64 times more expensive than those using the <math>\Gamma</math>-point only, it might seem impossible to do anything but <math>\Gamma</math>-point only calculations. However, VASP allows to generate special <math>{\bf k}</math> points using generating lattices. 

The following <math>{\bf k}</math>-point sets are particularly useful for HF-type calculations:

 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 {{FILE|KPOINTS}} file generates two 2 <math>{\bf k}</math> points, one at the <math>\Gamma</math>-point and one along the space diagonal at the BZ boundary (''R'' point).

The following {{FILE|KPOINTS}} file generates 4 <math>{\bf k}</math> points, one at the <math>\Gamma</math> 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, the following energies are obtained for TiO<sub>2</sub>:

 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 {{TAG|NKRED}}=2  TOTEN  =      -838.418681 eV

Note that results using {{TAG|NKRED}} not improved compared to <math>\Gamma</math>-only calculations.