Coulomb singularity: Difference between revisions

Revision as of 08:59, 10 May 2022

In the HF exchange, the Coulomb operator

{\displaystyle 1/\vert\mathbf{r}-\mathbf{r}'\vert }

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

V(G)=\frac{4\pi e^2}{G^2}

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.

[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).

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[2]

[3]

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-In the HF exchange, the Coulomb operator <math>1/\vert\mathbf{r}-\mathbf{r}'\vert</math> in reciprocal space
+In the HF exchange, the Coulomb operator
+:<math>
+/\vert\mathbf{r}-\mathbf{r}'\vert
+</math>
+ is singular ikn the reciprocal space at <math>\mathbf{q}=\mathbf{k}'-\mathbf{k}+\mathbf{G}</math> in reciprocal space
 :<math>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{{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.