Coulomb singularity

In the unscreened HF exchange, the bare Coulomb operator

[math]\displaystyle{ V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} }[/math]

is singular in the reciprocal space at [math]\displaystyle{ \mathbf{q}=\mathbf{k}'-\mathbf{k}+\mathbf{G}=0 }[/math]:

[math]\displaystyle{ V(q)=\frac{4\pi}{q^2} }[/math]

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.