FOCKCORR - Revision history

Ftran at 08:26, 20 October 2023

2023-10-20T08:26:11Z

← Older revision		Revision as of 08:26, 20 October 2023
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	{{sc\|HFRCUT\|Examples\|Examples that use this tag}}		{{sc\|HFRCUT\|Examples\|Examples that use this tag}}
	----		----
	~~[[The_VASP_Manual\|Contents]]~~

	[[Category:INCAR tag]][[Category:Hybrid_functionals]]		[[Category:INCAR tag]][[Category:Hybrid_functionals]]

Miranda.henrique at 14:17, 11 May 2022

2022-05-11T14:17:00Z

← Older revision		Revision as of 14:17, 11 May 2022
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	:<math>		:<math>
	\langle \mathbf{k}+\mathbf{G}' \| V^\text{HF}_\text{x} \| \mathbf{k}+\mathbf{G} \rangle =		\langle \mathbf{k}+\mathbf{G}' \| V^\text{HF}_\text{x} \| \mathbf{k}+\mathbf{G} \rangle =
	-~~\frac{4\pi e^2}{\Omega}~~ \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}		- \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
	C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') \Phi(\mathbf{k}-\mathbf{q}+\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')		C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') \Phi(\mathbf{k}-\mathbf{q}+\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')
	</math>		</math>
Line 36:		Line 36:
	\langle \mathbf{k}+\mathbf{G}' \|\hat{V}^\text{HF}_{\text{x}} \| \psi_{\mathbf{k}n} \rangle		\langle \mathbf{k}+\mathbf{G}' \|\hat{V}^\text{HF}_{\text{x}} \| \psi_{\mathbf{k}n} \rangle
	&=		&=
	-~~\frac{4\pi e^2}{\Omega}~~ \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}		- \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
	C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') \Phi(\mathbf{k}-\mathbf{q}+\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')		C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') \Phi(\mathbf{k}-\mathbf{q}+\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')
	C_{n\mathbf{k}}(\mathbf{G})\\		C_{n\mathbf{k}}(\mathbf{G})\\
	&=		&=
	-~~\frac{4\pi e^2}{\Omega}~~ \chi\sum_{m\mathbf{q}}f_{m\mathbf{q}}		- \chi\sum_{m\mathbf{q}}f_{m\mathbf{q}}
	C_{m\mathbf{q}}(\mathbf{G})		C_{m\mathbf{q}}(\mathbf{G})
	\end{aligned}		\end{aligned}

Miranda.henrique at 14:00, 11 May 2022

2022-05-11T14:00:20Z

← Older revision		Revision as of 14:00, 11 May 2022
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	:<math>		:<math>
	\langle \mathbf{k}+\mathbf{G} \| V^\text{HF}_\text{x} \| \mathbf{k}+\mathbf{G}' \rangle =		\langle \mathbf{k}+\mathbf{G}' \| V^\text{HF}_\text{x} \| \mathbf{k}+\mathbf{G} \rangle =
	-\frac{4\pi e^2}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}		-\frac{4\pi e^2}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
	C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') \Phi(\mathbf{k}-\mathbf{q}+\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')		C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') \Phi(\mathbf{k}-\mathbf{q}+\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')
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	:<math>		:<math>
	\begin{aligned}		\begin{aligned}
	\langle \mathbf{k}+\mathbf{G} \|\hat{V}^\text{HF}_{\text{x}} \| \psi_{\mathbf{k}n} \rangle		\langle \mathbf{k}+\mathbf{G}' \|\hat{V}^\text{HF}_{\text{x}} \| \psi_{\mathbf{k}n} \rangle
	&=		&=
	-\frac{4\pi e^2}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}		-\frac{4\pi e^2}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}

Vaspmaster at 15:30, 5 May 2022

2022-05-05T15:30:54Z

← Older revision		Revision as of 15:30, 5 May 2022
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	Note that in the case {{TAG\|FOCKCORR}}=2 the corrections are only applied to orbitals in the <math>\mathbf{q}</math> regular grid used to describe the exact exchange potential so this method cannot be used to compute band structures where this potential is applied to orbitals <math>n\mathbf{k}</math> not in the <math>m\mathbf{q}</math> set.		Note that in the case {{TAG\|FOCKCORR}}=2 the corrections are only applied to orbitals in the <math>\mathbf{q}</math> regular grid used to describe the exact exchange potential so this method cannot be used to compute band structures where this potential is applied to orbitals <math>n\mathbf{k}</math> not in the <math>m\mathbf{q}</math> set.
	{{NB\|warning\| {{TAG\|FOCKCORR}} ~~set to~~ 2 should not be used when computing the band structure along a path with the [[Si_HSE_bandstructure\|0-weight scheme]] or {{TAG\|KPOINTS_OPT}} }}		{{NB\|warning\| {{TAG\|FOCKCORR}}{{=}}2 should '''not''' be used when computing the band structure along a path with the [[Si_HSE_bandstructure\|0-weight scheme]] or {{TAG\|KPOINTS_OPT}} }}

	In previous versions of VASP,		In previous versions of VASP,

Vaspmaster at 15:13, 5 May 2022

2022-05-05T15:13:30Z

← Older revision		Revision as of 15:13, 5 May 2022
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	{{TAGDEF\|FOCKCORR\|~~[integer]~~\|~~1 or~~ 2 if {{TAG\|LMAXFOCKAE}}>0 }}		{{TAGDEF\|FOCKCORR\|1 {{!}} 2}}
			{{DEF\|FOCKCORR\|2\|if {{TAG\|LMAXFOCKAE}}>0\|1\|else}}

	Description: The tag {{TAG\|FOCKCORR}} determines how the Coulomb convergence corrections are applied.		Description: The tag {{TAG\|FOCKCORR}} determines how the Coulomb convergence corrections are applied.

Vaspmaster: Created page with "{{TAGDEF|FOCKCORR|[integer]|1 or 2 if {{TAG|LMAXFOCKAE}}>0 }} Description: The tag {{TAG|FOCKCORR}} determines how the Coulomb convergence corrections are applied. ---- The..."

2022-05-05T15:08:50Z

Created page with "{{TAGDEF|FOCKCORR|[integer]|1 or 2 if {{TAG|LMAXFOCKAE}}>0 }} Description: The tag {{TAG|FOCKCORR}} determines how the Coulomb convergence corrections are applied. ---- The..."

New page

{{TAGDEF|FOCKCORR|[integer]|1 or 2 if {{TAG|LMAXFOCKAE}}>0 }}

Description: The tag {{TAG|FOCKCORR}} determines how the Coulomb convergence corrections are applied.

----

The Coulomb potential 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.

These corrections are implemented in VASP either by changing the <math>\mathbf{G}=0</math> component of the Coulomb kernel when {{TAG|FOCKCORR}}=1
:<math>
\Phi(\mathbf{G}) =
\left\{
\begin{array}{lr}
\frac{4\pi e^2}{\Omega} \frac{1}{G^2} & \mathbf{G} \neq 0\\
\chi & \mathbf{G} = 0
\end{array}
\right.
</math>

with <math>\chi</math> being the value of the correction and depends on whether {{TAG|HFALPHA}} or {{TAG|HFRCUT}} are set,
or by including the original orbital scaled by the convergence correction when {{TAG|FOCKCORR}}=2

:<math>
\langle \mathbf{k}+\mathbf{G} | V^\text{HF}_\text{x} | \mathbf{k}+\mathbf{G}' \rangle =
-\frac{4\pi e^2}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') \Phi(\mathbf{k}-\mathbf{q}+\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')
</math>

:<math>
\begin{aligned}
\langle \mathbf{k}+\mathbf{G} |\hat{V}^\text{HF}_{\text{x}} | \psi_{\mathbf{k}n} \rangle
&=
-\frac{4\pi e^2}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') \Phi(\mathbf{k}-\mathbf{q}+\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')
C_{n\mathbf{k}}(\mathbf{G})\\
&=
-\frac{4\pi e^2}{\Omega} \chi\sum_{m\mathbf{q}}f_{m\mathbf{q}}
C_{m\mathbf{q}}(\mathbf{G})
\end{aligned}
</math>

For Hartree-Fock or hybrid functional calculations, either {{TAG|FOCKCORR}}=1 or {{TAG|FOCKCORR}}=2 can be used and should yield the same results when {{TAG|LMAXFOCKAE}}=-1 and there are no aliasing errors in the exact exchange (see {{TAG|PRECFOCK}} for more details).
For post-DFT methods such as ACDFT, GW, and BSE the {{TAG|FOCKCORR}}=2 should be used because the overlap densities are reconstructed in the plane-wave grid (see {{TAG|LMAXFOCKAE}} tag).

Note that in the case {{TAG|FOCKCORR}}=2 the corrections are only applied to orbitals in the <math>\mathbf{q}</math> regular grid used to describe the exact exchange potential so this method cannot be used to compute band structures where this potential is applied to orbitals <math>n\mathbf{k}</math> not in the <math>m\mathbf{q}</math> set.
{{NB|warning| {{TAG|FOCKCORR}} set to 2 should not be used when computing the band structure along a path with the [[Si_HSE_bandstructure|0-weight scheme]] or {{TAG|KPOINTS_OPT}} }}

In previous versions of VASP,
{{TAG|FOCKCORR}}=1 was used when {{TAG|ALGO}}=Normal; {{TAG|LFOCKACE}}=.FALSE. and
{{TAG|FOCKCORR}}=2 when {{TAG|ALGO}}=All or {{TAG|ALGO}}=Normal; {{TAG|LFOCKACE}}=.TRUE. .
{{NB|mind| Only available as of VASP 6.3.1.}}

== Related tags and articles ==
{{TAG|HFRCUT}},
{{TAG|HFALPHA}}

{{sc|HFRCUT|Examples|Examples that use this tag}}
----
[[The_VASP_Manual|Contents]]

[[Category:INCAR tag]][[Category:Hybrid_functionals]]