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:

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