Coulomb singularity: Difference between revisions

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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>
1/\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>  
:<math>V(G)=\frac{4\pi e^2}{G^2}</math>  
diverges for small G vectors.
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}}).
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.
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.

Revision as of 08:59, 10 May 2022

In the HF exchange, the Coulomb operator

is singular ikn the reciprocal space at  in reciprocal space 

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.