Coulomb singularity: Difference between revisions
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\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} | \frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} | ||
</math> | </math> | ||
is singular | is singular in the reciprocal space | ||
:<math> | |||
V(q)=\frac{4\pi}{q^2} | |||
</math> | |||
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 09:03, 10 May 2022
In the unscreened HF exchange, the bare Coulomb operator
is singular in 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.