https://beta.vasp.at/wiki/index.php?action=history&feed=atom&title=Coulomb_singularity
Coulomb singularity - Revision history
2026-04-23T01:14:13Z
Revision history for this page on the wiki
MediaWiki 1.43.8
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=21754&oldid=prev
Ftran at 13:26, 18 October 2023
2023-10-18T13:26:57Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:26, 18 October 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l111">Line 111:</td>
<td colspan="2" class="diff-lineno">Line 111:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|FOCKCORR}},</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|FOCKCORR}},</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|Hybrid functionals: formalism}},</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|Hybrid functionals: formalism}},</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{TAG|</del>Downsampling of the Hartree-Fock operator<del style="font-weight: bold; text-decoration: none;">}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[</ins>Downsampling of the Hartree-Fock operator<ins style="font-weight: bold; text-decoration: none;">]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td></tr>
</table>
Ftran
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=21724&oldid=prev
Ftran at 13:04, 18 October 2023
2023-10-18T13:04:35Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:04, 18 October 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8">Line 8:</td>
<td colspan="2" class="diff-lineno">Line 8:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It has an (integrable) singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math> that leads to a very slow convergence of the results with respect to the cell size or number of '''k''' points. In order to alleviate this issue different methods have been proposed: the auxiliary function {{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation{{cite|spenceralavi:prb:08}} methods (selected with {{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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It has an (integrable) singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math> that leads to a very slow convergence of the results with respect to the cell size or number of '''k''' points. In order to alleviate this issue different methods have been proposed: the auxiliary function {{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation{{cite|spenceralavi:prb:08}} methods (selected with {{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.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>These methods can also be applied to the [[Hybrid functionals: formalism#Thomas-Fermi exponential screening with short-range Hartree-Fock exchange|Thomas-Fermi]] and [[Hybrid functionals: formalism#Range-separated hybrid functionals|error function]]<del style="font-weight: bold; text-decoration: none;">, </del>screened Coulomb operators given by</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>These methods can also be applied to the [[Hybrid functionals: formalism#Thomas-Fermi exponential screening with short-range Hartree-Fock exchange|Thomas-Fermi]] and [[Hybrid functionals: formalism#Range-separated hybrid functionals|error function]] screened Coulomb operators given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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}</div></td></tr>
</table>
Ftran
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=21723&oldid=prev
Ftran at 13:04, 18 October 2023
2023-10-18T13:04:23Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:04, 18 October 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8">Line 8:</td>
<td colspan="2" class="diff-lineno">Line 8:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It has an (integrable) singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math> that leads to a very slow convergence of the results with respect to the cell size or number of '''k''' points. In order to alleviate this issue different methods have been proposed: the auxiliary function {{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation{{cite|spenceralavi:prb:08}} methods (selected with {{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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It has an (integrable) singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math> that leads to a very slow convergence of the results with respect to the cell size or number of '''k''' points. In order to alleviate this issue different methods have been proposed: the auxiliary function {{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation{{cite|spenceralavi:prb:08}} methods (selected with {{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.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>These methods can also be applied to the [[Hybrid functionals: formalism#Thomas-Fermi exponential screening with short-range Hartree-Fock exchange|Thomas-Fermi]]<del style="font-weight: bold; text-decoration: none;">, </del>and [[Hybrid functionals: formalism#Range-separated hybrid functionals|error function]], screened Coulomb operators given by</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>These methods can also be applied to the [[Hybrid functionals: formalism#Thomas-Fermi exponential screening with short-range Hartree-Fock exchange|Thomas-Fermi]] and [[Hybrid functionals: formalism#Range-separated hybrid functionals|error function]], screened Coulomb operators given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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}</div></td></tr>
</table>
Ftran
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=21722&oldid=prev
Ftran at 13:03, 18 October 2023
2023-10-18T13:03:39Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:03, 18 October 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8">Line 8:</td>
<td colspan="2" class="diff-lineno">Line 8:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It has an (integrable) singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math> that leads to a very slow convergence of the results with respect to the cell size or number of '''k''' points. In order to alleviate this issue different methods have been proposed: the auxiliary function {{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation{{cite|spenceralavi:prb:08}} methods (selected with {{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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It has an (integrable) singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math> that leads to a very slow convergence of the results with respect to the cell size or number of '''k''' points. In order to alleviate this issue different methods have been proposed: the auxiliary function {{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation{{cite|spenceralavi:prb:08}} methods (selected with {{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.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>These methods can also be applied to the Thomas-Fermi and error function screened Coulomb operators given by</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>These methods can also be applied to the <ins style="font-weight: bold; text-decoration: none;">[[Hybrid functionals: formalism#</ins>Thomas-Fermi <ins style="font-weight: bold; text-decoration: none;">exponential screening with short-range Hartree-Fock exchange|Thomas-Fermi]], </ins>and <ins style="font-weight: bold; text-decoration: none;">[[Hybrid functionals: formalism#Range-separated hybrid functionals|</ins>error function<ins style="font-weight: bold; text-decoration: none;">]], </ins>screened Coulomb operators given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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}</div></td></tr>
</table>
Ftran
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=20461&oldid=prev
Miranda.henrique: /* Spherical truncation */
2023-06-15T12:10:32Z
<p><span class="autocomment">Spherical truncation</span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:10, 15 June 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l75">Line 75:</td>
<td colspan="2" class="diff-lineno">Line 75:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>V(q)=\frac{4\pi}{q^{2}}\left(1-\cos(q R_{\text{c}})\right)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>V(q)=\frac{4\pi}{q^{2}}\left(1-\cos(q R_{\text{c}})\right)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>whose value at <math>q=0</math> is finite and is given by <math>V(q=0)=2\pi R_{\text{c}}^{2}</math>, where the truncation radius <math>R_{\text{c}}</math> is chosen as <math>R_{\text{c}}=\left(3/\left(4\pi\right)N_{\mathbf{k}}\Omega\right)^{1/3}</math> with <math>N_{\mathbf{k}}</math> being the number of <math>k</math>-points in the full Brillouin zone.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>whose value at <math>q=0</math> is finite and is given by <math>V(q=0)=2\pi R_{\text{c}}^{2}</math>, where the truncation radius <math>R_{\text{c}}</math> <ins style="font-weight: bold; text-decoration: none;">([[HFRCUT]]) </ins>is <ins style="font-weight: bold; text-decoration: none;">by default </ins>chosen as <math>R_{\text{c}}=\left(3/\left(4\pi\right)N_{\mathbf{k}}\Omega\right)^{1/3}</math> with <math>N_{\mathbf{k}}</math> being the number of <math>k</math>-points in the full Brillouin zone.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The screened potentials have no singularity at <math>q=0</math>. Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of '''k''' points to multiply these screened operators by <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, which in the reciprocal space gives</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The screened potentials have no singularity at <math>q=0</math>. Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of '''k''' points to multiply these screened operators by <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, which in the reciprocal space gives</div></td></tr>
</table>
Miranda.henrique
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=18310&oldid=prev
Miranda.henrique at 20:34, 19 July 2022
2022-07-19T20:34:34Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:34, 19 July 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td>
<td colspan="2" class="diff-lineno">Line 7:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>V(q)=\frac{4\pi}{q^2}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>V(q)=\frac{4\pi}{q^2}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>It has an (integrable) singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math> that leads to a very slow convergence of the results with respect to the cell size or number of k<del style="font-weight: bold; text-decoration: none;">-</del>points. In order to alleviate this issue different methods have been proposed: the auxiliary function {{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation{{cite|spenceralavi:prb:08}} methods (selected with {{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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>It has an (integrable) singularity at <math>q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0</math> that leads to a very slow convergence of the results with respect to the cell size or number of <ins style="font-weight: bold; text-decoration: none;">'''</ins>k<ins style="font-weight: bold; text-decoration: none;">''' </ins>points. In order to alleviate this issue different methods have been proposed: the auxiliary function {{cite|gygi:prb:86}}, probe-charge Ewald {{cite|massidda:prb:93}} ({{TAG|HFALPHA}}), and Coulomb truncation{{cite|spenceralavi:prb:08}} methods (selected with {{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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>These methods can also be applied to the Thomas-Fermi and error function screened Coulomb operators given by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>These methods can also be applied to the Thomas-Fermi and error function screened Coulomb operators given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l77">Line 77:</td>
<td colspan="2" class="diff-lineno">Line 77:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>whose value at <math>q=0</math> is finite and is given by <math>V(q=0)=2\pi R_{\text{c}}^{2}</math>, where the truncation radius <math>R_{\text{c}}</math> is chosen as <math>R_{\text{c}}=\left(3/\left(4\pi\right)N_{\mathbf{k}}\Omega\right)^{1/3}</math> with <math>N_{\mathbf{k}}</math> being the number of <math>k</math>-points in the full Brillouin zone.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>whose value at <math>q=0</math> is finite and is given by <math>V(q=0)=2\pi R_{\text{c}}^{2}</math>, where the truncation radius <math>R_{\text{c}}</math> is chosen as <math>R_{\text{c}}=\left(3/\left(4\pi\right)N_{\mathbf{k}}\Omega\right)^{1/3}</math> with <math>N_{\mathbf{k}}</math> being the number of <math>k</math>-points in the full Brillouin zone.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The screened potentials have no singularity at <math>q=0</math>. Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of k<del style="font-weight: bold; text-decoration: none;">-</del>points to multiply these screened operators by <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, which in the reciprocal space gives</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The screened potentials have no singularity at <math>q=0</math>. Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of <ins style="font-weight: bold; text-decoration: none;">'''</ins>k<ins style="font-weight: bold; text-decoration: none;">''' </ins>points to multiply these screened operators by <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, which in the reciprocal space gives</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>V(q)=\frac{4\pi}{q^{2}+\lambda^{2}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>V(q)=\frac{4\pi}{q^{2}+\lambda^{2}}</div></td></tr>
</table>
Miranda.henrique
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=18308&oldid=prev
Miranda.henrique: Miranda.henrique moved page Construction:Coulomb singularity to Coulomb singularity
2022-07-19T20:27:24Z
<p>Miranda.henrique moved page <a href="/wiki/Construction:Coulomb_singularity" class="mw-redirect" title="Construction:Coulomb singularity">Construction:Coulomb singularity</a> to <a href="/wiki/Coulomb_singularity" title="Coulomb singularity">Coulomb singularity</a></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<tr class="diff-title" lang="en">
<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:27, 19 July 2022</td>
</tr><tr><td colspan="2" class="diff-notice" lang="en"><div class="mw-diff-empty">(No difference)</div>
</td></tr></table>
Miranda.henrique
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=18207&oldid=prev
Miranda.henrique: /* Related tags and articles */
2022-07-19T13:37:16Z
<p><span class="autocomment">Related tags and articles</span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:37, 19 July 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l110">Line 110:</td>
<td colspan="2" class="diff-lineno">Line 110:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|HFRCUT}},</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|HFRCUT}},</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|FOCKCORR}},</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|FOCKCORR}},</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|<del style="font-weight: bold; text-decoration: none;">Hybrid_functionals</del>: formalism}},</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|<ins style="font-weight: bold; text-decoration: none;">Hybrid functionals</ins>: formalism}},</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|<del style="font-weight: bold; text-decoration: none;">Downsampling_of_the_Hartree</del>-<del style="font-weight: bold; text-decoration: none;">Fock_operator</del>}}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|<ins style="font-weight: bold; text-decoration: none;">Downsampling of the Hartree</ins>-<ins style="font-weight: bold; text-decoration: none;">Fock operator</ins>}}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td></tr>
</table>
Miranda.henrique
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=17751&oldid=prev
Ftran at 16:13, 11 May 2022
2022-05-11T16:13:32Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:13, 11 May 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l109">Line 109:</td>
<td colspan="2" class="diff-lineno">Line 109:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Related tags and articles ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Related tags and articles ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|HFRCUT}},</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|HFRCUT}},</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{TAG|FOCKCORR}},</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|Hybrid_functionals: formalism}},</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|Hybrid_functionals: formalism}},</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|Downsampling_of_the_Hartree-Fock_operator}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TAG|Downsampling_of_the_Hartree-Fock_operator}}</div></td></tr>
</table>
Ftran
https://beta.vasp.at/wiki/index.php?title=Coulomb_singularity&diff=17750&oldid=prev
Ftran at 16:01, 11 May 2022
2022-05-11T16:01:29Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:01, 11 May 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32">Line 32:</td>
<td colspan="2" class="diff-lineno">Line 32:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Probe-charge Ewald ===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Probe-charge Ewald ===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A similar approach to the auxiliary function method described above is the probe-charge Ewald method {{cite|massidda:prb:93}}. In this case, the auxiliary function <math>F(q)</math> is chosen to have the form of the Coulomb kernel times a Gaussian function <math>e^{-\alpha q^2}</math> with <del style="font-weight: bold; text-decoration: none;">an </del>width <math>\alpha</math> ({{TAG|HFALPHA}}) comparable to the Brillouin zone diameter.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A similar approach to the auxiliary function method described above is the probe-charge Ewald method {{cite|massidda:prb:93}}. In this case, the auxiliary function <math>F(q)</math> is chosen to have the form of the Coulomb kernel times a Gaussian function <math>e^{-\alpha q^2}</math> with <ins style="font-weight: bold; text-decoration: none;">a </ins>width <math>\alpha</math> ({{TAG|HFALPHA}}) comparable to the Brillouin zone diameter.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This function is used to regularize the Coulomb integral that is evaluated in the regular '''k''' point grid with the divergent part being evaluated by analytical integration of the Coulomb kernel (see eq. 29 in ref. {{cite|massidda:prb:93}}).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This function is used to regularize the Coulomb integral that is evaluated in the regular '''k''' point grid with the divergent part being evaluated by analytical integration of the Coulomb kernel (see eq. 29 in ref. {{cite|massidda:prb:93}}).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The value of the integral of the bare Coulomb potential is (see eq. 31 in ref. {{cite|massidda:prb:93}})</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The value of the integral of the bare Coulomb potential is (see eq. 31 in ref. {{cite|massidda:prb:93}})</div></td></tr>
</table>
Ftran