Coulomb singularity

In the unscreened HF exchange, the bare Coulomb operator

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

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.