Coulomb singularity

In the unscreened HF exchange, the bare Coulomb operator

$V(\vert \mathbf {r} -\mathbf {r} '\vert )={\frac {1}{\vert \mathbf {r} -\mathbf {r} '\vert }}$

is singular in the reciprocal space at $q=\vert \mathbf {k} '-\mathbf {k} +\mathbf {G} \vert =0$:

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

To alleviate this issue and to improve the convergence of the exact exchange with respect to the supercell size (or the 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 in the limit of large supercell sizes. These methods are described below.

Truncation methods

The potential $V(\vert \mathbf {r} -\mathbf {r} '\vert )$ is truncated by multiplying it by the step function $\theta (R_{\text{c}}-\left\vert \mathbf {r} -\mathbf {r} '\right\vert )$, which removes the singularity in the reciprocal space:

${\frac {4\pi }{\left\vert \mathbf {q} \right\vert ^{2}}}\left(1-\cos(\left\vert \mathbf {q} \right\vert R_{\text{c}})\right)$