Hybrid functionals: formalism - Revision history

Ftran at 14:20, 25 February 2025

2025-02-25T14:20:37Z

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	=== Unscreened hybrid functionals ===		=== Unscreened hybrid functionals ===

	In hybrid exchange-correlation functionals, the exchange component consists of a mixing of ~~[[:Category:GGA~~\|GGA]] (or meta-GGA) and Hartree-Fock exchange:		In hybrid exchange-correlation functionals, the exchange component consists of a mixing of {{TAG\|GGA}} (or meta-GGA) and Hartree-Fock exchange:
	:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF}} + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA}} + E_{\mathrm{c}}^{\mathrm{GGA}},</math>		:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF}} + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA}} + E_{\mathrm{c}}^{\mathrm{GGA}},</math>
	where <math>\alpha</math> is the mixing parameter ({{TAG\|AEXX}}) that is typically in the range 0.1-0.5. Two examples of hybrid functionals, PBE0 and B3LYP, are given below.		where <math>\alpha</math> is the mixing parameter ({{TAG\|AEXX}}) that is typically in the range 0.1-0.5. Two examples of hybrid functionals, PBE0 and B3LYP, are given below.
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	==== Error function screening with short-range Hartree-Fock exchange ====		==== Error function screening with short-range Hartree-Fock exchange ====

	More popular in solid-state physics, are the screened hybrid functionals, where only the short-range (SR) exchange is mixed, while the long-range (LR) exchange is still fully ~~[[:Category:GGA~~\|GGA]]:		More popular in solid-state physics, are the screened hybrid functionals, where only the short-range (SR) exchange is mixed, while the long-range (LR) exchange is still fully {{TAG\|GGA}}:
	:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF,SR}}(\mu) + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA,SR}}(\mu) + E_{\mathrm{x}}^{\mathrm{GGA,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{GGA}},</math>		:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF,SR}}(\mu) + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA,SR}}(\mu) + E_{\mathrm{x}}^{\mathrm{GGA,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{GGA}},</math>
	where <math>\mu</math> is the screening parameter ({{TAG\|HFSCREEN}}) that determines the range separation. The most popular range-separated functional, HSE, is given below.		where <math>\mu</math> is the screening parameter ({{TAG\|HFSCREEN}}) that determines the range separation. The most popular range-separated functional, HSE, is given below.

Huebsch: /* References */

2024-06-20T05:46:52Z

References

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	<references/>		<references/>

	~~----~~
	[[Category:Exchange-correlation functionals]][[Category:Hybrid_functionals]][[Category:Theory]]		[[Category:Exchange-correlation functionals]][[Category:Hybrid_functionals]][[Category:Theory]]

Ftran at 12:25, 19 October 2023

2023-10-19T12:25:30Z

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	</span>		</span>

	where <math>r =\|{\bf r}-{\bf r}'\|</math>, and <math>\mu</math> (={{TAG\|HFSCREEN}}) is the parameter that defines the range separation, and is related to a characteristic distance, <math>2/\mu</math>, at which the short-range interactions become negligible.		where <math>r =\|{\bf r}-{\bf r}'\|</math>, and <math>\mu</math> (set by {{TAG\|HFSCREEN}}) is the parameter that defines the range separation, and is related to a characteristic distance, <math>2/\mu</math>, at which the short-range interactions become negligible.

	Note: It has been shown that the optimum <math>\mu</math>, controlling the range separation is approximately 0.2-0.3 Å<sup>-1</sup>.{{cite\|heyd:jcp:03}}{{cite\|heyd:jcp:04}}{{cite\|heyd:jcp:06}}{{cite\|krukau:jcp:06}}		Note: It has been shown that the optimum <math>\mu</math>, controlling the range separation is approximately 0.2-0.3 Å<sup>-1</sup>.{{cite\|heyd:jcp:03}}{{cite\|heyd:jcp:04}}{{cite\|heyd:jcp:06}}{{cite\|krukau:jcp:06}}
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	Screened hybrid functionals with Hartree-Fock exchange at long range are more popular in molecular chemistry, where a proper decay of the exchange-correlation potential at long range far from the nuclei may be important. These functionals are less useful for solid-state physics, in particular for bulk solids. Examples belonging to this class of functionals are (available in VASP):		Screened hybrid functionals with Hartree-Fock exchange at long range are more popular in molecular chemistry, where a proper decay of the exchange-correlation potential at long range far from the nuclei may be important. These functionals are less useful for solid-state physics, in particular for bulk solids. Examples belonging to this class of functionals are (available in VASP):

	*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}		*[[List_of_hybrid_functionals#RSHXLDA\|RSHXLDA]] and [[List_of_hybrid_functionals#RSHXPBE\|RSHXPBE]]:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}
	:In the RSHXLDA and RSHXPBE functionals the exchange operator is decomposed into short-range LDA/PBE and long-range Hartree-Fock:		:In the RSHXLDA and RSHXPBE functionals the exchange operator is decomposed into short-range LDA/PBE and long-range Hartree-Fock:
	:<math>		:<math>
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	+ E_{\mathrm{x}}^{\mathrm{HF,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}},		+ E_{\mathrm{x}}^{\mathrm{HF,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}},
	</math>		</math>
	:where <math>\mu</math> (={{TAG\|HFSCREEN}}) is the parameter that defines the range separation. The use of the long-range Hartree-Fock exchange is activated with the {{TAG\|LRHFCALC}} tag. This functional can only be used when the short-range density functional part is LDA or PBE. When LDA is chosen, a value of <math>\mu=0.75</math> Å<sup>-1</sup> is recommended for solids.{{cite\|gerber:jcp:2007}}		:where <math>\mu</math> (set by {{TAG\|HFSCREEN}}) is the parameter that defines the range separation. The use of the long-range Hartree-Fock exchange is activated with the {{TAG\|LRHFCALC}} tag. This functional can only be used when the short-range density functional part is LDA or PBE. When LDA is chosen, a value of <math>\mu=0.75</math> Å<sup>-1</sup> is recommended for solids.{{cite\|gerber:jcp:2007}}

	==== Thomas-Fermi exponential screening with short-range Hartree-Fock exchange ====		==== Thomas-Fermi exponential screening with short-range Hartree-Fock exchange ====
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	</span>		</span>

	where <math>k_{\rm TF}</math> (={{TAG\|HFSCREEN}}) is the Thomas-Fermi screening length. For typical semiconductors, a Thomas-Fermi screening length of about 1.8 Å<sup>-1</sup> yields reasonable band gaps. In principle, however, the Thomas-Fermi screening length depends on the valence-electron density; VASP determines this parameter from the number of valence electrons (read from the {{FILE\|POTCAR}} file) and the volume and writes the corresponding value to the {{FILE\|OUTCAR}} file:		where <math>k_{\rm TF}</math> (set by {{TAG\|HFSCREEN}}) is the Thomas-Fermi screening length. For typical semiconductors, a Thomas-Fermi screening length of about 1.8 Å<sup>-1</sup> yields reasonable band gaps. In principle, however, the Thomas-Fermi screening length depends on the valence-electron density; VASP determines this parameter from the number of valence electrons (read from the {{FILE\|POTCAR}} file) and the volume and writes the corresponding value to the {{FILE\|OUTCAR}} file:
	Thomas-Fermi vector in A = 2.00000		Thomas-Fermi vector in A = 2.00000
	Since VASP counts the semi-core states and ''d''-states as valence electrons, although these states do not contribute to the screening, the values reported by VASP are often incorrect.		Since VASP counts the semi-core states and ''d''-states as valence electrons, although these states do not contribute to the screening, the values reported by VASP are often incorrect.

Ftran at 12:19, 19 October 2023

2023-10-19T12:19:11Z

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	*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}		*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}
	:In RSHXLDA and RSHXPBE functionals the exchange operator is decomposed into short-range LDA/PBE and long-range Hartree-Fock:		:In the RSHXLDA and RSHXPBE functionals the exchange operator is decomposed into short-range LDA/PBE and long-range Hartree-Fock:
	:<math>		:<math>
	E_{\mathrm{xc}}^{\mathrm{RSHXLDA}}= E_{\mathrm{x}}^{\mathrm{LDA,SR}}(\mu)		E_{\mathrm{xc}}^{\mathrm{RSHXLDA}}= E_{\mathrm{x}}^{\mathrm{LDA,SR}}(\mu)
	+ E_{\mathrm{x}}^{\mathrm{HF,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{LDA}}.		+ E_{\mathrm{x}}^{\mathrm{HF,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{LDA}},
	</math>		</math>
			:<math>
	~~:and similarly for~~ RSHXPBE, :where <math>\mu</math> (={{TAG\|HFSCREEN}}) is the parameter that defines the range separation. The use of the long-range Hartree-Fock exchange is activated with the {{TAG\|LRHFCALC}} tag. This functional can only be used when the short-range density functional part is LDA or PBE. When LDA is chosen, a value of <math>\mu=0.75</math> Å<sup>-1</sup> is recommended for solids.{{cite\|gerber:jcp:2007}}		E_{\mathrm{xc}}^{\mathrm{RSHXPBE}}= E_{\mathrm{x}}^{\mathrm{PBE,SR}}(\mu)
			+ E_{\mathrm{x}}^{\mathrm{HF,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}},
			</math>
			:where <math>\mu</math> (={{TAG\|HFSCREEN}}) is the parameter that defines the range separation. The use of the long-range Hartree-Fock exchange is activated with the {{TAG\|LRHFCALC}} tag. This functional can only be used when the short-range density functional part is LDA or PBE. When LDA is chosen, a value of <math>\mu=0.75</math> Å<sup>-1</sup> is recommended for solids.{{cite\|gerber:jcp:2007}}

	==== Thomas-Fermi exponential screening with short-range Hartree-Fock exchange ====		==== Thomas-Fermi exponential screening with short-range Hartree-Fock exchange ====

Ftran at 12:15, 19 October 2023

2023-10-19T12:15:56Z

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	*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}		*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}
	:In ~~the range-separated hybrid~~ RSHXLDA and RSHXPBE functionals the exchange operator is decomposed into short-range LDA/PBE and long-range Hartree-Fock:		:In RSHXLDA and RSHXPBE functionals the exchange operator is decomposed into short-range LDA/PBE and long-range Hartree-Fock:
	:<math>		:<math>
	E_{\mathrm{xc}}^{\mathrm{RSHXLDA}}= E_{\mathrm{x}}^{\mathrm{LDA,SR}}(\mu)		E_{\mathrm{xc}}^{\mathrm{RSHXLDA}}= E_{\mathrm{x}}^{\mathrm{LDA,SR}}(\mu)

Ftran at 12:13, 19 October 2023

2023-10-19T12:13:12Z

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	</math>		</math>

	and similarly for RSHXPBE, :where <math>\mu</math> (={{TAG\|HFSCREEN}}) is the parameter that defines the range separation. The use of the long-range Hartree-Fock exchange is activated with the {{TAG\|LRHFCALC}} tag. This functional can only be used when the short-range density functional part is LDA or PBE. When LDA is chosen, a value of <math>\mu=0.75</math> Å<sup>-1</sup> is recommended for solids.{{cite\|gerber:jcp:2007}}		:and similarly for RSHXPBE, :where <math>\mu</math> (={{TAG\|HFSCREEN}}) is the parameter that defines the range separation. The use of the long-range Hartree-Fock exchange is activated with the {{TAG\|LRHFCALC}} tag. This functional can only be used when the short-range density functional part is LDA or PBE. When LDA is chosen, a value of <math>\mu=0.75</math> Å<sup>-1</sup> is recommended for solids.{{cite\|gerber:jcp:2007}}

	==== Thomas-Fermi exponential screening with short-range Hartree-Fock exchange ====		==== Thomas-Fermi exponential screening with short-range Hartree-Fock exchange ====

Ftran at 12:12, 19 October 2023

2023-10-19T12:12:44Z

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	==== Error function screening with long-range Hartree-Fock exchange ====		==== Error function screening with long-range Hartree-Fock exchange ====

	Screened hybrid functionals with Hartree-Fock exchange at long range are more popular in molecular chemistry, where a proper decay of the exchange-correlation potential at long range far from the nuclei may be important. These functionals are less useful for solid-state physics, in particular for bulk solids. ~~Neverthels~~		Screened hybrid functionals with Hartree-Fock exchange at long range are more popular in molecular chemistry, where a proper decay of the exchange-correlation potential at long range far from the nuclei may be important. These functionals are less useful for solid-state physics, in particular for bulk solids. Examples belonging to this class of functionals are (available in VASP):

	*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}		*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}

Ftran at 12:10, 19 October 2023

2023-10-19T12:10:43Z

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	==== Error function screening with long-range Hartree-Fock exchange ====		==== Error function screening with long-range Hartree-Fock exchange ====

			Screened hybrid functionals with Hartree-Fock exchange at long range are more popular in molecular chemistry, where a proper decay of the exchange-correlation potential at long range far from the nuclei may be important. These functionals are less useful for solid-state physics, in particular for bulk solids. Neverthels

	*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}		*RSHXLDA and RSHXPBE:{{cite\|iikura:jcp:2001}}{{cite\|gerber:cpl:2005}}{{cite\|gerber:jcp:2007}}

Ftran at 13:16, 18 October 2023

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	{{TAG\|HFSCREEN}},		{{TAG\|HFSCREEN}},
	[[List of hybrid functionals]],		[[List of hybrid functionals]],
			[[Downsampling of the Hartree-Fock operator]],
	[[Coulomb singularity]]		[[Coulomb singularity]]

Ftran at 09:23, 18 October 2023

2023-10-18T09:23:35Z

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	*HSE:		*HSE:
	:In the range-separated [[List_of_hybrid_functionals#HSE03\|HSE03]]{{cite\|heyd:jcp:03}}{{cite\|heyd:jcp:04}}{{cite\|heyd:jcp:06}} and [[List_of_hybrid_functionals#HSE06\|HSE06]]{{cite\|krukau:jcp:06}} hybrid functionals the slowly decaying long-range part of the Hartree-Fock exchange interaction is replaced by the corresponding part of the PBE density functional counterpart. The resulting expression for the exchange-correlation energy is given by:		:In the range-separated [[List_of_hybrid_functionals#HSE03\|HSE03]]{{cite\|heyd:jcp:03}}{{cite\|heyd:jcp:04}}{{cite\|heyd:jcp:06}} and [[List_of_hybrid_functionals#HSE06\|HSE06]]{{cite\|krukau:jcp:06}} hybrid functionals the slowly decaying long-range part of the Hartree-Fock exchange interaction (see the discussion on the [[Coulomb singularity]]) is replaced by the corresponding part of the PBE density functional counterpart. The resulting expression for the exchange-correlation energy is given by:

	:<math>		:<math>