Hybrid functionals: formalism - Revision history

Ftran at 14:21, 17 March 2026

2026-03-17T14:21:06Z

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	The corresponding long-range potentials are given by <math>v_{\mathrm{LR}}^{\mathrm{erf/exp}}=v^{\mathrm{bare}}-v_{\mathrm{SR}}^{\mathrm{erf/exp}}</math>.		The corresponding long-range potentials are given by <math>v_{\mathrm{LR}}^{\mathrm{erf/exp}}=v^{\mathrm{bare}}-v_{\mathrm{SR}}^{\mathrm{erf/exp}}</math>.

	In VASP, these expressions are implemented within the [[Projector-augmented-wave_formalism\|PAW formalism]].{{cite\|paier:jcp:05}}		In VASP, these expressions are implemented within the [[Projector-augmented-wave_formalism\|PAW formalism]].{{cite\|paier:jcp:05}}{{cite\|angyan:jpa:2006}}

	The families of hybrid functionals implemented in VASP are listed below along with examples, whose corresponding {{TAG\|INCAR}} files can be found at the page [[list_of_hybrid_functionals\|list of hybrid functionals]].		The families of hybrid functionals implemented in VASP are listed below along with examples, whose corresponding {{TAG\|INCAR}} files can be found at the page [[list_of_hybrid_functionals\|list of hybrid functionals]].

Ftran at 15:40, 4 March 2026

2026-03-04T15:40:15Z

← Older revision		Revision as of 15:40, 4 March 2026
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	The non-multiplicative form of the HF potential is such that hybrid functionals are implemented within the generalized KS scheme{{cite\|seidl:prb:96}}. Thus, the total energy is minimized with respect to the orbitals (instead of the electron density), which means that the HF exchange leads to a nonlocal operator as in the Hartree-Fock-Roothaan theory.		The non-multiplicative form of the HF potential is such that hybrid functionals are implemented within the generalized KS scheme{{cite\|seidl:prb:96}}. Thus, the total energy is minimized with respect to the orbitals (instead of the electron density), which means that the HF exchange leads to a nonlocal operator as in the Hartree-Fock-Roothaan theory.

	=== Expressions of Hartree-Fock ~~energy and~~ potential for the plane-wave basis set ===		=== Expressions of the Hartree-Fock potential for the plane-wave basis set ===

	Using the decomposition of the Bloch states <math>\psi_{m\mathbf{q}}</math> in plane waves,		Using the decomposition of the Bloch states <math>\psi_{m\mathbf{q}}</math> in plane waves,

Ftran at 22:18, 27 February 2026

2026-02-27T22:18:42Z

← Older revision		Revision as of 22:18, 27 February 2026
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	For most hybrid functionals proposed in the literature, the interelectronic Coulomb potential <math>v(\mu,\|\mathbf{r}-\mathbf{r}'\|)</math> is one of these types (<math>r=\|\mathbf{r}-\mathbf{r}'\|</math>):		For most hybrid functionals proposed in the literature, the interelectronic Coulomb potential <math>v(\mu,\|\mathbf{r}-\mathbf{r}'\|)</math> is one of these types (<math>r=\|\mathbf{r}-\mathbf{r}'\|</math>):
	* Full range (bare ~~Colomb~~ potential):		* Full range (bare Coulomb potential):
	:<math>		:<math>
	v^{\mathrm{bare}}(r)=\frac{1}{r}		v^{\mathrm{bare}}(r)=\frac{1}{r}

Ftran: /* HF exchange at short range and long range (error-function screening) with different mixings */

2026-02-27T22:16:50Z

HF exchange at short range and long range (error-function screening) with different mixings

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	:<math>		:<math>
	E_{\mathrm{xc}}^{\mathrm{RS-DDH}}=~~a_{~~\~~mathrm~~{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + ~~(1-a_{~~\~~mathrm~~{SR}})E_{\mathrm{x,SR}}^{\mathrm{PBE}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}}		E_{\mathrm{xc}}^{\mathrm{RS-DDH}}=\frac{1}{4} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + \frac{3}{4}E_{\mathrm{x,SR}}^{\mathrm{PBE}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}}
	</math>		</math>
	:where <math>a_{\mathrm{SR}}=1/4</math> and <math>a_{\mathrm{LR}}=\varepsilon^{-1}</math> is chosen as the inverse of the dielectric constant <math>\varepsilon^{-1}</math>.		:where <math>a_{\mathrm{SR}}=1/4</math> and <math>a_{\mathrm{LR}}=\varepsilon^{-1}</math> is chosen as the inverse of the dielectric constant <math>\varepsilon^{-1}</math>.

Ftran at 20:58, 10 February 2026

2026-02-10T20:58:51Z

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	In hybrid functionals ~~the exchange energy~~ is a mixture of semilocal (SL) and nonlocal Hartree-Fock (HF) types. They can be categorized into different families according to the type of semilocal approximation (LDA, GGA, or MGGA) or the treatment of the short- and long-range parts of the exchange. A rather general formula that encompasses the different families of hybrid functionals is given by		The exchange energy in hybrid functionals is a mixture of semilocal (SL) and nonlocal Hartree-Fock (HF) types. They can be categorized into different families according to the type of semilocal approximation (LDA, GGA, or MGGA) or the treatment of the short- and long-range parts of the exchange. A rather general formula that encompasses the different families of hybrid functionals is given by

	:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{SR}})E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}}</math>		:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{SR}})E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}}</math>
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	</math>		</math>
	</span>		</span>
	[[File:Screened_operator.png\|340px\|thumb\|right\|Short- and long-range potentials using the error or exponential screening compared to the bare potential.]]		[[File:Screened_operator.png\|340px\|thumb\|right\|Short- and long-range potentials using the error or exponential screening compared to the bare potential. <math>\mu=1</math> is used.]]
	* Short-range with exponential function:		* Short-range with exponential function:
	:<math>		:<math>

Ftran at 11:29, 9 February 2026

2026-02-09T11:29:48Z

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	where		where
	*~~'''~~<math>a_{\mathrm{SR}}</math> and <math>a_{\mathrm{LR}}</math> are the mixing parameters (fraction of HF exchange) at short and long range''', respectively.		*<math>a_{\mathrm{SR}}</math> and <math>a_{\mathrm{LR}}</math> are the '''mixing parameters (fraction of HF exchange) at short and long range''', respectively.
	*~~'''~~<math>\mu</math> is the screening parameter''' that determines the separation between short range (SR) and long range (LR).		*<math>\mu</math> is the '''screening parameter''' that determines the separation between short range (SR) and long range (LR).

	The SR and LR components of the full-range <math>E_{\mathrm{x}}^{\mathrm{SL}}</math> and <math>E_{\mathrm{x}}^{\mathrm{HF}}</math> exchange energies are constructed such that at all values of <math>\mu</math>		The SR and LR components of the full-range <math>E_{\mathrm{x}}^{\mathrm{SL}}</math> and <math>E_{\mathrm{x}}^{\mathrm{HF}}</math> exchange energies are constructed such that at all values of <math>\mu</math>

Ftran at 10:49, 9 February 2026

2026-02-09T10:49:45Z

← Older revision		Revision as of 10:49, 9 February 2026
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	* Full range (bare Colomb potential):		* Full range (bare Colomb potential):
	:<math>		:<math>
	v^{\mathrm{~~Coul~~}}(r)=\frac{1}{r}		v^{\mathrm{bare}}(r)=\frac{1}{r}
	</math>		</math>
	<span id="VxFockRecip">		<span id="VxFockRecip">
	:<math>		:<math>
	V_{\mathbf{k}}^{\mathrm{~~Coul~~}}\left( \mathbf{G},\mathbf{G}'\right)=		V_{\mathbf{k}}^{\mathrm{bare}}\left( \mathbf{G},\mathbf{G}'\right)=
	-\frac{4\pi}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}		-\frac{4\pi}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
	\frac{C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')}		\frac{C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')}
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	</math>		</math>
	</span>		</span>
	The corresponding long-range potentials are given by <math>v_{\mathrm{LR}}^{\mathrm{erf/exp}}=v^{\mathrm{~~Coul~~}}-v_{\mathrm{SR}}^{\mathrm{erf/exp}}</math>.		The corresponding long-range potentials are given by <math>v_{\mathrm{LR}}^{\mathrm{erf/exp}}=v^{\mathrm{bare}}-v_{\mathrm{SR}}^{\mathrm{erf/exp}}</math>.

	In VASP, these expressions are implemented within the [[Projector-augmented-wave_formalism\|PAW formalism]].{{cite\|paier:jcp:05}}		In VASP, these expressions are implemented within the [[Projector-augmented-wave_formalism\|PAW formalism]].{{cite\|paier:jcp:05}}

Ftran at 10:47, 9 February 2026

2026-02-09T10:47:42Z

← Older revision		Revision as of 10:47, 9 February 2026
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	</span>		</span>
	The non-multiplicative form of the HF potential is such that hybrid functionals are implemented within the generalized KS scheme{{cite\|seidl:prb:96}}. Thus, the total energy is minimized with respect to the orbitals (instead of the electron density), which means that the HF exchange leads to a nonlocal operator as in the Hartree-Fock-Roothaan theory.		The non-multiplicative form of the HF potential is such that hybrid functionals are implemented within the generalized KS scheme{{cite\|seidl:prb:96}}. Thus, the total energy is minimized with respect to the orbitals (instead of the electron density), which means that the HF exchange leads to a nonlocal operator as in the Hartree-Fock-Roothaan theory.

			=== Expressions of Hartree-Fock energy and potential for the plane-wave basis set ===

	Using the decomposition of the Bloch states <math>\psi_{m\mathbf{q}}</math> in plane waves,		Using the decomposition of the Bloch states <math>\psi_{m\mathbf{q}}</math> in plane waves,
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	</math>		</math>

	=== Types of potentials ===		== Types of potentials ==

	For most hybrid functionals proposed in the literature, the interelectronic Coulomb potential <math>v(\mu,\|\mathbf{r}-\mathbf{r}'\|)</math> is one of these types (<math>r=\|\mathbf{r}-\mathbf{r}'\|</math>):		For most hybrid functionals proposed in the literature, the interelectronic Coulomb potential <math>v(\mu,\|\mathbf{r}-\mathbf{r}'\|)</math> is one of these types (<math>r=\|\mathbf{r}-\mathbf{r}'\|</math>):

Ftran at 10:38, 9 February 2026

2026-02-09T10:38:52Z

← Older revision		Revision as of 10:38, 9 February 2026
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	</math>		</math>
	</span>		</span>
			[[File:Screened_operator.png\|340px\|thumb\|right\|Short- and long-range potentials using the error or exponential screening compared to the bare potential.]]
	* Short-range with exponential function:		* Short-range with exponential function:
	:<math>		:<math>

Ftran at 22:07, 6 February 2026

2026-02-06T22:07:22Z

← Older revision		Revision as of 22:07, 6 February 2026
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	*'''<math>\mu</math> is the screening parameter''' that determines the separation between short range (SR) and long range (LR).		*'''<math>\mu</math> is the screening parameter''' that determines the separation between short range (SR) and long range (LR).

	The SR and LR components of the full-range <math>E_{\mathrm{x}}^{\mathrm{SL}}</math> and <math>E_{\mathrm{x}}^{\mathrm{HF}}</math> exchange energies are constructed such that <math>E_{\mathrm{x}}^{\mathrm{HF}}=E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu)+E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu)</math> ~~and~~ <math>E_{\mathrm{x}}^{\mathrm{SL}}=E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu)+E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu)</math> ~~at all values of <math>\mu</math>.~~		The SR and LR components of the full-range <math>E_{\mathrm{x}}^{\mathrm{SL}}</math> and <math>E_{\mathrm{x}}^{\mathrm{HF}}</math> exchange energies are constructed such that at all values of <math>\mu</math>
			*<math>E_{\mathrm{x}}^{\mathrm{HF}}=E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu)+E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu)</math>
			*<math>E_{\mathrm{x}}^{\mathrm{SL}}=E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu)+E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu)</math>

	The HF exchange energy (full-range, SR, or LR) is given by		The HF exchange energy (full-range, SR, or LR) is given by