Coulomb singularity: Difference between revisions

Revision as of 12:11, 10 May 2022

The bare Coulomb operator

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{1}{\vert\mathbf{r}-\mathbf{r}'\vert} }

in the unscreened HF exchange has a representation in the reciprocal space that is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q)=\frac{4\pi}{q^2} }

It has a singularity at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0} , and to alleviate this issue and to 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. These methods are described below.

Probe-charge Ewald method

Auxiliary function methods

Truncation methods

In this method the bare Coulomb operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(\vert\mathbf{r}-\mathbf{r}'\vert)} is truncated by multiplying it by the step function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)} , and in the reciprocal this leads to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q)=\frac{4\pi}{q^{2}}\left(1-\cos(q R_{\text{c}})\right) }

whose value at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle q=0} is finite and is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q=0)=2\pi R_{\text{c}}^{2}} . The screened Coulomb operators

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{e^{-\lambda\left\vert\mathbf{r}-\mathbf{r}'\right\vert}}{\left\vert\mathbf{r}-\mathbf{r}'\right\vert} }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(\vert\mathbf{r}-\mathbf{r}'\vert)=\frac{\text{erfc}\left({-\lambda\left\vert\mathbf{r}-\mathbf{r}'\right\vert}\right)}{\left\vert\mathbf{r}-\mathbf{r}'\right\vert} }

have representations in the reciprocal space that are given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q)=\frac{4\pi}{q^{2}+\lambda^{2}} }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q)=\frac{4\pi}{q^{2}}\left(1-e^{-q^{2}/\left(4\lambda^2\right)}\right) }

respectively. Thus, the screened potentials have no singularity at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle q=0} . Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of k-points to multiply these screened operators by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)} , which in the reciprocal space gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q)=\frac{4\pi}{q^{2}+\lambda^{2}} \left( 1-e^{-\lambda R_{\text{c}}}\left(\frac{\lambda}{q} \sin\left(qR_{\text{c}}\right) + \cos\left(qR_{\text{c}}\right)\right)\right) }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q)=\frac{4\pi}{q^{2}} \left( 1-\cos(qR_{\text{c}})\text{erfc}\left(\lambda R_{\text{c}}\right) - e^{-q^{2}/\left(4\lambda^2\right)} \Re\left({\text{erf}\left(\lambda R_{\text{c}} + \text{i}\frac{q}{2\lambda}\right)}\right)\right) }

respectively, with the following values at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle q=0} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q=0)=\frac{4\pi}{\lambda^{2}}\left(1-e^{-\lambda R_{\text{c}}}\left(\lambda R_{\text{c}} + 1\right)\right) }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle V(q=0)=2\pi\left(R_{\text{c}}^{2}\text{erfc}(\lambda R_{\text{c}}) - \frac{R_{\text{c}}e^{-\lambda^{2}R_{\text{c}}^{2}}}{\sqrt{\pi}\lambda} + \frac{\text{erf}(\lambda R_{\text{c}})}{2\lambda^{2}}\right) }

[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 9: / Line 9: @@
 It has a singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math>, and to alleviate this issue and to 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. These methods are described below.
+=== Probe-charge Ewald method ===
+=== Auxiliary function methods ===
 === Truncation methods ===
@@ Line 59: / Line 63: @@
 \frac{\text{erf}(\lambda R_{\text{c}})}{2\lambda^{2}}\right)
 </math>
-=== Auxiliary function methods ===