ODDONLYGW and EVENONLYGW: 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:

$$\chi_{\mathbf{q}}^0 (\mathbf{G}, \mathbf{G}', \omega) = \fr... ...thbf{k}+\mathbf{q}}-\epsilon_{n\mathbf{k}} - \omega - i \eta }$$ (84)

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):

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

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. Note that the GW loop, involves a sum over $\bf k$, and a second one over $\bf q$ (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. 6.69.9 the index ${\bf q}$ can be restricted to the values

$${\vec q} = {\vec b}_1 \frac{n_1 C_1}{N_1} + {\vec b}_2 \frac... ...+ {\vec b}_3 \frac{n_3 C_3}{N_3}, \hspace{3mm}(n_i=0,..,N_i-1)$$ (85)

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.