Coulomb singularity: Difference between revisions

Revision as of 09:06, 10 May 2022

In the unscreened HF exchange, the bare Coulomb operator

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

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

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

[1]

[2]

[3]

@@ Line 3: / Line 3: @@
 V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert}
 </math>
-is singular in the reciprocal space
+is singular in the reciprocal space at <math>\mathbf{q}=\mathbf{k}'-\mathbf{k}+\mathbf{G}=0</math>
 :<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>
 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.