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NKRED, NKREDX, NKREDY, NKREDZ and EVENONLY, ODDONLY

NKRED= [integer]

NKREDX= [integer] $ \qquad$ NKREDY= [integer] $ \qquad$ NKREDZ= [integer]

EVENONLY= [logical] $ \qquad$ ODDONLY= [logical]

NKRED, or alternatively, NKREDX, NKREDY, and NKREDZ are the grid reduction factors that may be used to evaluate the Hartree-Fock kernel (see Eq. 6.57) at a subgrid of $ q$-points. Under certain circumstances this is possible without much loss of accuracy (see Ref. [99]). Whether the errors remain small, depends on the range of the exchange interactions in the compound of choice. This can be understood along the following lines:

Consider the description of a certain bulk system, using a supercell made up of $ N$ primitive cells, in such a way that, $ \{{\bf A}_i'\}$, the lattice vectors of the supercell are given by $ {\bf A}_i'=n_i{\bf A}_i$ ($ i=1,2,3$), where $ \{{\bf A}_i\}$ are the lattice vectors of the primitive cell. Let $ R_{\rm max}=2/\mu$ be the distance for which

$\displaystyle \frac{{\rm erfc}(\mu\vert{\bf r}-{\bf r}'\vert)}{\vert{\bf r}-{\b...
...ox 0, \hspace{3mm}{\rm for}\hspace{2mm}{\vert{\bf r}-{\bf r}'\vert}>R_{\rm max}$ (6.68)

When the nearest neighbour distance between the periodically repeated images of the supercell $ R_{\rm NN}>2R_{\rm max}$ (i.e. $ R_{\rm NN}>4/\mu$), the short-ranged Fock potential, $ V^{\rm SR}_x [\mu]$, can be represented exactly, sampling the BZ at the $ \Gamma $-point only, i.e.,

$\displaystyle V_x[\mu]\left({\bf r},{\bf r}'\right)= -\frac{e^2}{2}\sum_m f_{{\...
...\frac{{\rm erfc}(\mu\vert{\bf r}-{\bf r}'\vert)} {\vert {\bf r}-{\bf r}' \vert}$ (6.69)

This is equivalent to a representation of the bulk system using the primitive cell and a $ n_1\times n_2\times n_3$ sampling of the BZ,

$\displaystyle V_x[\mu]\left({\bf r},{\bf r}'\right)= -\frac{e^2}{2}\sum_{{\bf q...
...\frac{{\rm erfc}(\mu\vert{\bf r}-{\bf r}'\vert)} {\vert {\bf r}-{\bf r}' \vert}$ (6.70)

where the set of $ \bf q$ vectors is given by

$\displaystyle \{{\bf q}\}=\{i{\bf G}_1+j{\bf G}_2+k{\bf G}_3\},$ (6.71)

for $ i=1,..,n_1$, $ j=1,..,n_2$, and $ k=1,..,n_3$, with $ {\bf G}_{1,2,3}$ being the reciprocal lattice vectors of the supercell.

In light of the above it is clear that the number of $ q$-points needed to represent the short-ranged Fock potential decreases with decreasing $ R_{\rm max}$ (i.e., with increasing $ \mu$). Furthermore, one should realize that the maximal range of the exchange interactions is not only limited by the $ {\rm erfc}(\mu\vert{\bf r}-{\bf r}'\vert)/\vert{\bf r}-{\bf r}'\vert$ kernel, but depends on the extend of the spatial overlap of the orbitals as well [this can easily be shown for the Fock exchange energy when one adopts a Wannier representation of the orbitals in Eqs. (6.57) or (6.65)]; $ R_{\rm max}$, as defined in Eq. (6.68), 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-ranged 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 $ {\bf q}$ in Eq. (6.66) to a subset, $ \{{\bf q_k}\}$, of the full ( $ N_{1}\times N_{2}\times N_{3}$) $ k$-point set, $ \{{\bf k}\}$, for which the following holds

$\displaystyle {\bf q_k} = {\bf b}_1 \frac{n_1 C_1}{N_1} + {\bf b}_2 \frac{n_2 C_2}{N_2} + {\bf b}_3 \frac{n_3 C_3}{N_3}, \hspace{3mm}(n_i=0,..,N_i-1)$ (6.72)

where $ {\bf b}_{1,2,3}$ are the reciprocal lattice vectors of the primitive cell, and $ C_i$ is the integer grid reduction factor along reciprocal lattice direction $ {\bf b}_i$. This leads to a reduction in the computational workload to:

$\displaystyle \frac{1}{C_1 C_2 C_3}$ (6.73)

The integer grid reduction factor are either set separately through $ C_1$=NKREDX, $ C_2$=NKREDY, and $ C_3$=NKREDZ, or simultaneously through $ C_1=C_2=C_3$=NKRED. The flag EVENONLY chooses a subset of $ k-$points with $ C_1=C_2=C_3=1$, and $ n_1+n_2+n_2$ even. It reduces the computational work load for HF type calculations by a factor two, but is only sensible for high symmetry cases (such as sc, fcc or bcc cells).

Note: From occurrence of the range-separation parameter $ \mu$ in the equation above, one should not obtain the impression that the grid reduction can only be useful in conbination with the HSE03/HSE06 functional (see Sec. 6.71.8). It can be applied to the PBE0 and pure Hartree-Fock cases as well, although from the above, it might be clear that the range seperated HSE functional will allow for a larger reduction of the grid than the conventional hybrid functionals (see Ref. [99]).


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