Coulomb singularity

In the HF exchange, the Coulomb operator

$1/\vert\mathbf{r}-\mathbf{r}'\vert$

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

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

diverges for small G vectors. To alleviate this issue and improve the convergence of the exact exchange integral with respect to supercell size (or 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 within the limit of large supercell sizes.