Coulomb singularity
In the HF exchange, the Coulomb operator [math]\displaystyle{ 1/\vert\mathbf{r}-\mathbf{r}'\vert }[/math] in reciprocal space
- [math]\displaystyle{ V(G)=\frac{4\pi e^2}{G^2} }[/math]
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[1], probe-charge Ewald [2] (HFALPHA), and Coulomb truncation methods[3] (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.