Coulomb singularity
In the unscreened HF exchange, the bare Coulomb operator
- [math]\displaystyle{ V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} }[/math]
is singular in the reciprocal space at [math]\displaystyle{ q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0 }[/math]:
- [math]\displaystyle{ V(q)=\frac{4\pi}{q^2} }[/math]
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[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 in the limit of large supercell sizes. These methods are described below.
Truncation methods
The potential [math]\displaystyle{ V(\vert\mathbf{r}-\mathbf{r}'\vert) }[/math] is truncated by multiplying it by the step function [math]\displaystyle{ \theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert) }[/math], and in the reciprocal this leads to
- [math]\displaystyle{ V(q)=\frac{4\pi}{q^{2}}\left(1-\cos(q R_{\text{c}})\right) }[/math]
which has no singularity at [math]\displaystyle{ q=0 }[/math], but the value
- [math]\displaystyle{ V(0)=2\pi R_{\text{c}}^{2} }[/math]
The screened potentials
- [math]\displaystyle{ V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{e^{-\lambda\left\vert\mathbf{r}-\mathbf{r}'\right\vert}}{\left\vert\mathbf{r}-\mathbf{r}'\right\vert} }[/math]
- [math]\displaystyle{ V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{\text{erfc}\left({-\lambda\left\vert\mathbf{r}-\mathbf{r}'\right\vert}\right)}{\left\vert\mathbf{r}-\mathbf{r}'\right\vert} }[/math]
whose representation in the reciprocal space are given by
- [math]\displaystyle{ \frac{4\pi}{q^{2}+\lambda^{2}} }[/math]
- [math]\displaystyle{ \frac{4\pi}{q^{2}} \left(1-e^{-q^{2}/\left(4\lambda^2\right)}\right) }[/math]