Coulomb singularity

In the HF exchange, the Coulomb operator $s$ 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.