Coulomb singularity

In the unscreened HF exchange, the bare Coulomb operator

$V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert}$

is singular in the reciprocal space at $q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0$:

$V(q)=\frac{4\pi}{q^2}$

To alleviate this issue and improve the convergence of the exact exchange with respect to the supercell size (or the k-point mesh density) different methods have been proposed: the auxiliary function methods, probe-charge Ewald (HFALPHA), and Coulomb truncation methods (HFRCUT). These mostly involve modifying the Coulomb Kernel in a way that yields the same result as the unmodified kernel in the limit of large supercell sizes.