Coulomb singularity: Difference between revisions

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=== Truncation methods ===
=== Truncation methods ===


The potential <math>V(\vert\mathbf{r}-\mathbf{r}'\vert)</math> is truncated by multiplying it by the step function <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, which removes the singularity in the reciprocal space:
The potential <math>V(\vert\mathbf{r}-\mathbf{r}'\vert)</math> is truncated by multiplying it by the step function <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, and in the reciprocal this leads to
:<math>
:<math>
\frac{4\pi}{\left\vert\mathbf{q}\right\vert^{2}}\left(1-\cos(\left\vert\mathbf{q}\right\vert R_{\text{c}})\right)
\frac{4\pi}{\left\vert\mathbf{q}\right\vert^{2}}\left(1-\cos(\left\vert\mathbf{q}\right\vert R_{\text{c}})\right)
</math>
which has no singularity at <math>q=0</math>, but the value
:<math>
V(0)=2\pi R_{\text{c}}^{2}
</math>
</math>



Revision as of 09:37, 10 May 2022

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{ \frac{4\pi}{\left\vert\mathbf{q}\right\vert^{2}}\left(1-\cos(\left\vert\mathbf{q}\right\vert 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]


Auxiliary function methods