Coulomb singularity

The bare Coulomb operator

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

in the unscreened HF exchange has a representation in the reciprocal space that is given by

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

It has a singularity at $q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0$, and 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 bare Coulomb operator $V(\vert\mathbf{r}-\mathbf{r}'\vert)$ is truncated by multiplying it by the step function $\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)$, and in the reciprocal this leads to

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

whose value at $q=0$ is finite and is given by

$V(0)=2\pi R_{\text{c}}^{2}$

The screened Coulomb operators

$V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{e^{-\lambda\left\vert\mathbf{r}-\mathbf{r}'\right\vert}}{\left\vert\mathbf{r}-\mathbf{r}'\right\vert}$

and

$V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{\text{erfc}\left({-\lambda\left\vert\mathbf{r}-\mathbf{r}'\right\vert}\right)}{\left\vert\mathbf{r}-\mathbf{r}'\right\vert}$

have representations in the reciprocal space that are given by

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

and

$V(q)=\frac{4\pi}{q^{2}}\left(1-e^{-q^{2}/\left(4\lambda^2\right)}\right)$

respectively. Thus, the screened potentials have no singularity at $q=0$. Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of k-points to multiply these screened operators by $\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)$, which in the reciprocal space gives

$\frac{4\pi}{q^{2}+\lambda^{2}} \left( 1-e^{-\lambda R_{\text{c}}}\left(\frac{\lambda}{q} \sin\left(qR_{\text{c}}\right) + \cos\left(qR_{\text{c}}\right)\right)\right)$

and

$\frac{4\pi}{\left\vert\mathbf{q}\right\vert^{2}} \left( 1-\cos(\left\vert\mathbf{q}\right\vert R_{\text{c}})\text{erfc}\left(\lambda R_{\text{c}}\right) - e^{-\left\vert\mathbf{q}\right\vert^{2}/\left(4\lambda^2\right)} \Re\left({\text{erf}\left(\lambda R_{\text{c}} + \text{i}\frac{\left\vert\mathbf{q}\right\vert}{2\lambda}\right)}\right)\right)$

respectively.