Coulomb singularity

The bare Coulomb operator

[math]\displaystyle{ V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} }[/math]

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

[math]\displaystyle{ V(q)=\frac{4\pi}{q^2} }[/math]

It has a singularity at [math]\displaystyle{ q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0 }[/math], 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 [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]

whose value at [math]\displaystyle{ q=0 }[/math] is finite and is given by

[math]\displaystyle{ V(0)=2\pi R_{\text{c}}^{2} }[/math]

The screened Coulomb operators

[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]

and

[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]

have representations in the reciprocal space that are given by

[math]\displaystyle{ V(q)=\frac{4\pi}{q^{2}+\lambda^{2}} }[/math]

and

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

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

[math]\displaystyle{ \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) }[/math]

and

[math]\displaystyle{ \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) }[/math]

respectively.