Coulomb singularity: Difference between revisions
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In the unscreened HF exchange, the bare Coulomb operator | In the unscreened HF exchange, the bare Coulomb operator | ||
:<math> | :<math> | ||
\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} | V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} | ||
</math> | </math> | ||
is singular in the reciprocal space | is singular in the reciprocal space | ||
Revision as of 09:05, 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
- [math]\displaystyle{ V(q)=\frac{4\pi}{q^2} }[/math]
at [math]\displaystyle{ \mathbf{q}=\mathbf{k}'-\mathbf{k}+\mathbf{G} }[/math] in reciprocal space
- [math]\displaystyle{ V(G)=\frac{4\pi e^2}{G^2} }[/math]
diverges for small G vectors. To alleviate this issue and improve the convergence of the exact exchange integral with respect to supercell size (or 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 within the limit of large supercell sizes.