Coulomb singularity: Difference between revisions
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V(0)=2\pi R_{\text{c}}^{2} | V(0)=2\pi R_{\text{c}}^{2} | ||
</math> | </math> | ||
The screened | The screened Coulomb operators | ||
:<math> | :<math> | ||
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} | 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} | ||
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V(q)=\frac{4\pi}{q^{2}}\left(1-e^{-q^{2}/\left(4\lambda^2\right)}\right) | V(q)=\frac{4\pi}{q^{2}}\left(1-e^{-q^{2}/\left(4\lambda^2\right)}\right) | ||
</math> | </math> | ||
respectively. Thus, the screened potentials have no singularity at <math>q=0</math>. | respectively. Thus, the screened potentials have no singularity at <math>q=0</math>. Nevertheless, it is still beneficial for accelerating convergence with respect to the number of k-points to multiply these screened operators by <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>. | ||
=== Auxiliary function methods === | === Auxiliary function methods === | ||
Revision as of 11:47, 10 May 2022
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[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]
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 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].