Coulomb singularity: Difference between revisions

Revision as of 09:12, 10 May 2022

In the unscreened HF exchange, the bare Coulomb operator

{\displaystyle V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} }

is singular in the reciprocal space at $q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0$ :

{\displaystyle V(q)=\frac{4\pi}{q^2} }

To alleviate this issue and 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.

[gygi:prb:86-1] F. Gygi and A. Baldereschi, Phys. Rev. B 34, 4405(R) (1986).

[massidda:prb:93-2] S. Massidda, M. Posternak, and A. Baldereschi, Phys. Rev. B 48, 5058 (1993).

[spenceralavi:prb:08-3] J. Spencer and A. Alavi, Phys. Phys. Rev. B 77, 193110 (2008).

[1]

[2]

[3]

@@ Line 7: / Line 7: @@
 V(q)=\frac{4\pi}{q^2}
 </math>
+To alleviate this issue and 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{{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation methods{{cite|spenceralavi:prb:08}} ({{TAG|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.
-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{{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation methods{{cite|spenceralavi:prb:08}} ({{TAG|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.