Coulomb singularity

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Revision as of 09:05, 10 May 2022 by Ftran (talk | contribs)

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.