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ODDONLYGW and EVENONLYGW and NKRED: reducing the $ k$-grid for the response functions

ODDONLYGW= .TRUE. | .FALSE. $ \qquad$ EVENONLYGW= .TRUE. | .FALSE.

ODDONLYGW allows to avoid the inclusion of the $ \Gamma $-point in the evaluation of response functions. The independent particle polarizability $ \chi_{{\mathbf{q}}}^0 ({\mathbf{G}}, {\mathbf{G}}', \omega)$ is given by:

$\displaystyle \chi_{{\mathbf{q}}}^0 ({\mathbf{G}}, {\mathbf{G}}', \omega) = \fr...
...ilon_{n'{\mathbf{k}}+{\mathbf{q}}}-\epsilon_{n{\mathbf{k}}} - \omega - i \eta }$ (6.80)

If the $ \Gamma $ point is included in the summation over $ {\mathbf{k}}$, convergence is very slow for some materials (e.g. GaAs).

To deal with this problem the flag ODDONLYGW has been included. In the automatic mode, the $ k$-grid is given by (see Sec. 5.5.3):

$\displaystyle {\vec k} = {\vec b}_1 \frac{n_1}{N_1} + {\vec b}_2 \frac{n_2}{N_2...
...ac{n_3}{N_3} ,\qquad n_1=0...,N_1-1 \quad n_2=0...,N_2-1 \quad n_3=0...,N_3-1.
$

If the three integers $ n_i$ sum to an odd value, the $ k$-point is included in the previous summation in the GW routine (ODDONLYGW=.TRUE.). Note that other routines (linear optical properties) presently do not recognize this flag. EVENONLYGW=.TRUE. is only of limited use and restricts the summation to $ k$-points with $ n_1+n_2+n_3$ being even ($ \Gamma $-point and from there on ever second k-point included).

Accelerations are also possible by evaluating the response function itself at a restricted number of $ \bf q$-points (see also Sec. 6.73.1). Note that the GW loop, involves a sum over $ \bf k$, and a second one over the momentum transfer vector $ \bf q$ (the index in the response function). To some extend both can be varied independently. The former one by using ODDONLYGW, and the latter one using the Hartree-Fock related flags NKRED, NKREDX, NKREDY, NKREDZ and EVENONLY, ODDONLY. As explained in Sec. 6.71.9 the index $ {\bf q}$ can be restricted to the values

$\displaystyle {\vec q} = {\vec b}_1 \frac{n_1 C_1}{N_1} + {\vec b}_2 \frac{n_2 C_2}{N_2} + {\vec b}_3 \frac{n_3 C_3}{N_3}, \hspace{3mm}(n_i=0,..,N_i-1)$ (6.81)

The integer grid reduction factors 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.


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