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Hybrid functionals, which mix the Hartree-Fock and Kohn-Sham theories, can be more accurate than semilocal methods like GGA, in particular for nonmetallic systems. They are particularly suited for band gap calculation for instance. Hybrid functionals are available in VASP.
'''Hybrid functionals''' go beyond the semilocal approximations by mixing the Hartree-Fock (HF) and semilocal (SL) exchange{{cite|becke:jcp:93}}:
 
== Theoretical background ==
 
In hybrid functionals the exchange part consists of a linear combination of Hartree-Fock (HF) and semilocal (e.g., GGA) exchange:
:<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}}
E_{\mathrm{xc}}^{\mathrm{hybrid}}=a E_{\mathrm{x}}^{\mathrm{HF}} + (1-a)E_{\mathrm{x}}^{\mathrm{SL}} + E_{\mathrm{c}}^{\mathrm{SL}}
</math>
</math>
where <math>a_{x}</math> determines the relative amount of HF and semilocal exchange. There are essentially two types of hybrid functionals: (a) the ones where the HF exchange is applied at full interelectronic range (unscreened hybrids) and (b) the others where the HF exchange is applied either at short or at long interelectronic range (called screened or range-separated hybrids).
where <math>a</math> is the mixing parameter that determines the relative weights of HF and semilocal exchange.


=== Unscreened hybrid functionals ===
Depending on the type of systems or the property under consideration they can be more accurate than semilocal (GGA, meta-GGA) functionals. For instance, hybrid functionals are usually more suited for calculating the electronic and magnetic properties of nonmetallic systems. They are particularly recommended for bandgap calculations.{{cite|heyd:jcp:05}}{{cite|chen2018nonempirical}} Polarons{{cite|franchini:nrm:21}} or defect states{{cite|oba:prb:08}} are among properties that can also be better described by hybrid functionals.
Note that hybrid functionals are also often good at treating [[:Category:Strongly correlated electrons|strongly correlated electrons]].{{cite|liu2019assessing}}


In hybrid exchange-correlation functionals, the exchange component consists of a mixing of [[:Category:GGA|GGA]] (or meta-GGA) and Hartree-Fock exchange:
* {{NB|mind|
:<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>
*Evaluating the HF exchange is computationally very demanding, leading to '''a computational time that is one or several orders of magnitude larger than with semilocal functionals'''. Therefore, a proper assessment of the available computational resources and parallelization strategies becomes increasingly important.
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.
*Some properties cannot be computed using hybrid functionals because the corresponding implementations are not yet available. A non-exhaustive list includes [[Electron-phonon_interactions_theory|electron-phonon interactions]], [[Phonons from density-functional-perturbation theory|phonons from density-functional-perturbation theory]], and [[Calculating_the_chemical_shieldings|NMR shielding]]. However, note that hybrid functionals can be used if the phonons are calculated using the [[Phonons from finite differences|finite differences method]].}}


*[[list_of_hybrid_functionals#PBE0|PBE0]]:{{cite|perdew:jcp:1996}}
== Overview ==


:<math>
The hybrid functionals can be divided into families according to the type of semilocal approximation that is used (LDA, GGA, or MGGA) and the interelectronic range at which the HF exchange is applied: at full range (unscreened hybrids) or either at short or long range (called screened or range-separated hybrids). From the practical point of view, the short-range hybrid functionals like HSE06{{cite|krukau:jcp:06}} are preferable for periodic solids, since leading to faster convergence with respect to the number of k-points (or size of the unit cell).
E_{\mathrm{xc}}^{\mathrm{PBE0}}=\frac{1}{4} E_{\mathrm{x}}^{\mathrm{HF}} +
\frac{3}{4} E_{\mathrm{x}}^{\mathrm{PBE}} + E_{\mathrm{c}}^{\mathrm{PBE}},
</math>
</span>
:where <math>E_{x}^{\rm PBE}</math> and <math>E_{c}^{\rm PBE}</math> denote the exchange and correlation parts of the PBE density functional, respectively.


*[[list_of_hybrid_functionals#B3LYP|B3LYP]]{{cite|stephens:jpc:1994}}, well known and popular amongst quantum chemists:
The different families of hybrid functionals available in VASP are described in details at [[Hybrid_functionals: formalism|formalism of the HF method and hybrids]] along with examples and links to the corresponding {{TAG|INCAR}} files.


<span id="B3LYP">
Details about the implementation of the unscreened hybrid functionals can be found in the work of Paier et al.,{{cite|paier:jcp:05}} while details specific to the screened hybrid functionals can be found in Refs. {{cite|paier:jcp:06}}{{cite|angyan:jpa:2006}} Refs. {{cite|cui2018doubly}}{{cite|liu2019assessing}} report the development of dielectric-dependent hybrid functionals, which provide very accurate band gaps and are also available in VASP.
:<math>
\begin{align}
E_{\mathrm{x}}^{\mathrm{B3LYP}} &=0.8 E_{\mathrm{x}}^{\mathrm{LDA}}+
0.2 E_{\mathrm{x}}^{\mathrm{HF}} + 0.72 \Delta E_{\mathrm{x}}^{\mathrm{B88}}, \\
E_{\mathrm{c}}^{\mathrm{B3LYP}} &=0.19 E_{\mathrm{c}}^{\mathrm{VWN3}}+
0.81 E_{\mathrm{c}}^{\mathrm{LYP}},
\end{align}
</math>
</span>
:where <math>E_{x}^{\rm B3LYP}</math> and <math>E_{c}^{\rm B3LYP}</math> are the B3LYP exchange and correlation energy contributions, respectively. <math>E_{x}^{\rm B3LYP}</math> consists of 80% of LDA exchange plus 20% of non-local Hartree-Fock exchange, and 72% of the gradient corrections of the Becke88 exchange functional. <math>E_{c}^{\rm B3LYP}</math> consists of 81% of LYP correlation energy, which contains a local and a semilocal (gradient dependent) part, and 19% of the (local) Vosko-Wilk-Nusair correlation functional III, which is fitted to the correlation energy in the random phase approximation RPA of the homogeneous electron gas.


The non-local Hartree-Fock exchange energy, <math>E_{x}</math>, can be written as
Note that as in most other codes, hybrid functionals are implemented in VASP within the generalized KS scheme,{{cite|seidl:prb:96}} which means that the total energy is minimized, as in the HF theory, with respect to the orbitals instead of the electron density.


<span id="ExFock">
VASP offers a convenient way to generate the [[Band-structure calculation using hybrid functionals|band structure with hybrid functionals]].
:<math>
E_{\mathrm{x}}^{\mathrm{HF}}= -\frac{e^2}{2}\sum_{n\mathbf{k},m\mathbf{q}}
f_{n\mathbf{k}} f_{m\mathbf{q}} \times
\int\int d^3\mathbf{r} d^3\mathbf{r}'
\frac{\psi_{n\mathbf{k}}^{*}(\mathbf{r})\psi_{m\mathbf{q}}^{*}(\mathbf{r}')
\psi_{n\mathbf{k}}(\mathbf{r}')\psi_{m\mathbf{q}}(\mathbf{r})}
{\vert \mathbf{r}-\mathbf{r}' \vert}
</math>
</span>


with <math>\{\psi_{n\mathbf{k}}(\mathbf{r})\}</math> being the set of one-electron
== Technical points ==
Bloch states of the system, and <math>\{f_{n\mathbf{k}}\}</math> the corresponding
set of (possibly fractional) occupational numbers.
The sums over <math>{\bf k}</math> and <math>{\bf q}</math> run over all <math>{\bf k}</math> points chosen to sample the Brillouin zone (BZ), whereas the sums over <math>m</math> and <math>n</math> run over all bands at these <math>{\bf k}</math> points. The corresponding <span id="VxFock">non-local Hartree-Fock potential is given by


:<math>
* The unscreened Coulomb potential used to evaluate the exchange integral in HF has an integrable Coulomb singularity that leads to slow convergence with respect to supercell size (or equivalently '''k''' point sampling). To make the computations feasible requires special treatment of the [[Coulomb singularity]].
V_{\mathrm{x}}\left(\mathbf{r},\mathbf{r}'\right)=
-\frac{e^2}{2}\sum_{m\mathbf{q}}f_{m\mathbf{q}}
\frac{\psi_{m\mathbf{q}}^{*}(\mathbf{r}')\psi_{m\mathbf{q}}(\mathbf{r})}
{\vert \mathbf{r}-\mathbf{r}' \vert}
= -\frac{e^2}{2}\sum_{m\mathbf{q}}f_{m\mathbf{q}}
e^{-i\mathbf{q}\cdot\mathbf{r}'}
\frac{u_{m\mathbf{q}}^{*}(\mathbf{r}')u_{m\mathbf{q}}(\mathbf{r})}
{\vert \mathbf{r}-\mathbf{r}' \vert}
e^{i\mathbf{q}\cdot\mathbf{r}},
</math>
</span>


where <math>u_{m\mathbf{q}}(\mathbf{r})</math> is the cell periodic part of the Bloch state,
* The [[Downsampling of the Hartree-Fock operator|Downsampling of the HF operator]] allows for the use of a coarser '''k''' point sampling for the HF operator, and therefore faster calculations.{{cite|paier:jcp:06}} However, this option should be use with care for metals.
<math>\psi_{n\mathbf{q}}(\mathbf{r})</math>, at <math>{\bf k}</math> point, <math>{\bf q}</math>, with band index ''m''.
Using the decomposition of the Bloch states, <math>\psi_{m\mathbf{q}}</math>, in plane waves,


:<math>
* The Adaptively Compressed Exchange Operator,{{cite|linlin:jctc:2016}} that allows for a more efficient evaluation of the Fock operator, is used if the Davidson algorithm ({{TAG|ALGO}} = Normal, the default) is selected (see {{TAG|LFOCKACE}} for more details).
\psi_{m\mathbf{q}}(\mathbf{r})=
\frac{1}{\sqrt{\Omega}}
\sum_\mathbf{G}C_{m\mathbf{q}}(\mathbf{G})e^{i(\mathbf{q}+\mathbf{G}) \cdot \mathbf{r}}
</math>


the Hartree-Fock exchange potential may be written as
== Additional resources ==
=== Tutorials ===
*Tutorial for {{Tutorial|hybrids:part1|hybrid calculations}}.


:<math>
=== Lectures ===
V_{\mathrm{x}}\left(\mathbf{r},\mathbf{r}'\right)=
*Lecture on {{Video|hybrid:henrique:2022|hybrid functionals}}.
\sum_{\mathbf{k}}\sum_{\mathbf{G}\mathbf{G}'}
e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}
V_{\mathbf{k}}\left( \mathbf{G},\mathbf{G}'\right)
e^{-i(\mathbf{k}+\mathbf{G}')\cdot\mathbf{r}'}
</math>
where
<span id="VxFockRecip">
:<math>
V_\mathbf{k}\left( \mathbf{G},\mathbf{G}'\right)=
\langle \mathbf{k}+\mathbf{G} | V_{\mathrm{x}} | \mathbf{k}+\mathbf{G}'\rangle =
-\frac{4\pi e^2}{\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}'')}
{|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2}
</math>
</span>
is the representation of the Hartree-Fock potential in reciprocal space.
In VASP, these expressions are implemented within the [[PAW_method|PAW formalism]].{{cite|paier:jcp:05}}
 
=== Range-separated hybrid functionals ===
 
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]]:
:<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.
 
*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:
 
:<math>
E_{\mathrm{xc}}^{\mathrm{HSE}}= \frac{1}{4}E_{\mathrm{x}}^{\mathrm{SR,HF}}(\mu)
+ \frac{3}{4} E_{\mathrm{x}}^{\mathrm{PBE,SR}}(\mu)
+ E_{\mathrm{x}}^{\mathrm{PBE,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}}.
</math>


The decomposition of the Coulomb kernel is obtained using the following construction:
=== How to ===
*[[Hybrid_functionals: formalism|Formalism of the HF method and hybrids]]. The various families are described.
*[[List_of_hybrid_functionals|List of available hybrid functionals]] and how to specify them in the {{FILE|INCAR}} file.
*[[Downsampling_of_the_Hartree-Fock_operator|Downsampling of the Hartree-Fock operator]] to reduce the computational cost.
*[[Band-structure calculation using hybrid functionals|band-structure calculation using hybrid functionals]].


<span id="SRLR">
=== Further reading ===
:<math>
In addition to the works already cited above, the selected publications listed below describe methodological developments or computational studies performed using the hybrid functionals implemented in VASP.
\frac{1}{r}=S_{\mu}(r)+L_{\mu}(r)=\frac{\mathrm{erfc}(\mu r)}{r}+\frac{\mathrm{erf}(\mu r)}{r},
</math>
</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.
*The B3LYP functional applied to solid-state systems and its failure for metals{{cite|paier:jcp:07}}.
 
*Applications of hybrid functionals to selected materials: Ceria,{{cite|juarez:prb:07}} lead chalcogenides,{{cite|hummer:prb:07}} CO adsorption on metals,{{cite|stroppa:prb:07}}{{cite|stroppa:njp:08}} excitonic properties,{{cite|paier:prb:08}} SrTiO and BaTiO, {{cite|wahl:prb:08}} perovskites,{{cite|franchini:jpcm:14}} and transition-metal oxides.{{cite|gopidi:prb:26}}
Note: It has been shown that the optimum <math>\mu</math>, controlling the range separation is approximately 0.2-0.3 &Aring;<sup>-1</sup>.{{cite|heyd:jcp:03}}{{cite|heyd:jcp:04}}{{cite|heyd:jcp:06}}{{cite|krukau:jcp:06}}
*HSEsol hybrid functional.{{cite|schimka:jcp:11}}
To select [[List_of_hybrid_functionals#HSE06|the HSE06 functional]] you need to select ({{TAG|HFSCREEN}}=0.2).
*Analysis of the HSE parameter space.{{cite|moussa:jcp:12}}
 
*Automated workflow for non-empirical Wannier-localized optimal tuning of range-separated hybrid functionals.{{cite|gant:cpc:26}}
Using the decomposed Coulomb kernel one may straightforwardly rewrite the non-local [[#ExFock|Hartree-Fock exhange energy]]:
 
<span id="ExFockSR">
:<math>
E^{\rm SR,HF}_{\mathrm{x}}(\mu)=
-\frac{e^2}{2}\sum_{n\mathbf{k},m\mathbf{q}}
f_{n\mathbf{k}} f_{m\mathbf{q}}
\int \int d^3\mathbf{r} d^3\mathbf{r}'
\frac{\mathrm{erfc}(\mu|\mathbf{r}-\mathbf{r}'|)}{|\mathbf{r}-\mathbf{r}'|}
\times \psi_{n\mathbf{k}}^{*}(\mathbf{r})\psi_{m\mathbf{q}}^{*}(\mathbf{r}')
\psi_{n\mathbf{k}}(\mathbf{r}')\psi_{m\mathbf{q}}(\mathbf{r}).
</math>
</span>
 
The representation of the corresponding short-range Hartree-Fock potential in reciprocal space is given by
 
<span id="VxFockSRRecip">
:<math>
\begin{align}
V^{\mathrm{SR}}_\mathbf{k}\left( \mathbf{G},\mathbf{G}'\right)&=
\langle \mathbf{k}+\mathbf{G} | V^{\rm SR}_x [\mu] | \mathbf{k}+\mathbf{G}'\rangle \\
&=-\frac{4\pi e^2}{\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}'')}
{|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2}
\times \left( 1-e^{-|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2 /4\mu^2} \right).
\end{align}
</math>
</span>
 
The only difference to the reciprocal space representation of the complete [[#VxFockRecip|Hartree-Fock exchange potential]] is the second factor in the summand above, representing the complementary error function in reciprocal space.
 
The short-range PBE exchange energy and potential, and their long-range counterparts, are arrived at using the same [[#SRLR|decomposition]], in accordance with Heyd ''et al''.{{cite|heyd:jcp:03}} It is easily seen that the long-range term in the [[#SRLR|decomposed Coulomb kernel]] becomes zero for <math>\mu=0</math>, and the short-range contribution then equals the full Coulomb operator, whereas for <math>\mu\rightarrow\infty</math> it is the other way around. Consequently, the two limiting cases of the HSE functional are a true [[List_of_hybrid_functionals#PBE0|PBE0]] functional for <math>\mu=0</math>, and a pure PBE calculation for <math>\mu\rightarrow\infty</math>.
 
<span id="Thomas_Fermi"></span>
 
==== Thomas-Fermi screening ====
In the case of [[LTHOMAS|Thomas-Fermi screening]], the Coulomb kernel is again decomposed in a short-range and a long-range part.{{cite|bylander:prb:90}}{{cite|seidl:prb:96}}{{cite|picozzi:prb:00}} This decomposition can be conveniently written in reciprocal space:
 
<span id="ThomasFermi">
:<math>
\frac{4 \pi e^2}{|\mathbf{G}|^2}=S_{\mu}(|\mathbf{G}|)+L_{\mu}(|\mathbf{G}|)=\frac{4 \pi e^2}{|\mathbf{G}|^2 +k_{\mathrm{TF}}^2}+
\left( \frac{4 \pi e^2}{|\mathbf{G}|^2} -\frac{4 \pi e^2}{|\mathbf{G}|^2 +k_{\mathrm{TF}}^2} \right),
</math>
</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 &Aring;<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
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.
 
Another important detail concerns the implementation of the density-functional part in the screened exchange case. Literature suggests that a global enhancement factor <math>z</math> (see Eq. 3.15){{cite|seidl:prb:96}} should be used, whereas VASP implements a local-density-dependent enhancement factor <math>z=k_{\rm TF}/k</math> , where <math>k</math> is the Fermi wave vector corresponding to the local density (and not the average density as suggested Seidl ''et al''.{{cite|seidl:prb:96}}. The VASP implementation is in the spirit of the ''local'' density approximation.
 
== How to ==
 
[[List_of_hybrid_functionals|List of available hybrid functionals]] and how to specify them in {{FILE|INCAR}}.
 
[[Downsampling_of_the_Hartree-Fock_operator|Downsampling of the Hartree-Fock operator]].
 
== Further reading ==
*A comprehensive study of the performance of the HSE03/HSE06 functional compared to the PBE and PBE0 functionals.{{cite|paier:jcp:06}}
*The B3LYP functional applied to solid state systems.{{cite|paier:jcp:07}}
*Applications of hybrid functionals to selected materials: Ceria,{{cite|juarez:prb:07}} lead chalcogenides,{{cite|hummer:prb:07}} CO adsorption on metals,{{cite|stroppa:prb:07}}{{cite|stroppa:njp:08}} defects in ZnO,{{cite|oba:prb:08}} excitonic properties,{{cite|paier:prb:08}} SrTiO and BaTiO.{{cite|wahl:prb:08}}
 
 
{{sc|Hartree-Fock and HF/DFT hybrid functionals|Examples|Examples that use this tag}}


== References ==
== References ==
<references/>
<references/>
----


[[Category:VASP|Hybrids]][[Category:Exchange-correlation functionals]]
[[Category:VASP|Hybrids]][[Category:Exchange-correlation functionals]]

Latest revision as of 14:48, 11 June 2026

Hybrid functionals go beyond the semilocal approximations by mixing the Hartree-Fock (HF) and semilocal (SL) exchange[1]:

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{hybrid}}=a E_{\mathrm{x}}^{\mathrm{HF}} + (1-a)E_{\mathrm{x}}^{\mathrm{SL}} + E_{\mathrm{c}}^{\mathrm{SL}} }[/math]

where [math]\displaystyle{ a }[/math] is the mixing parameter that determines the relative weights of HF and semilocal exchange.

Depending on the type of systems or the property under consideration they can be more accurate than semilocal (GGA, meta-GGA) functionals. For instance, hybrid functionals are usually more suited for calculating the electronic and magnetic properties of nonmetallic systems. They are particularly recommended for bandgap calculations.[2][3] Polarons[4] or defect states[5] are among properties that can also be better described by hybrid functionals. Note that hybrid functionals are also often good at treating strongly correlated electrons.[6]

Mind:
  • Evaluating the HF exchange is computationally very demanding, leading to a computational time that is one or several orders of magnitude larger than with semilocal functionals. Therefore, a proper assessment of the available computational resources and parallelization strategies becomes increasingly important.
  • Some properties cannot be computed using hybrid functionals because the corresponding implementations are not yet available. A non-exhaustive list includes electron-phonon interactions, phonons from density-functional-perturbation theory, and NMR shielding. However, note that hybrid functionals can be used if the phonons are calculated using the finite differences method.

Overview

The hybrid functionals can be divided into families according to the type of semilocal approximation that is used (LDA, GGA, or MGGA) and the interelectronic range at which the HF exchange is applied: at full range (unscreened hybrids) or either at short or long range (called screened or range-separated hybrids). From the practical point of view, the short-range hybrid functionals like HSE06[7] are preferable for periodic solids, since leading to faster convergence with respect to the number of k-points (or size of the unit cell).

The different families of hybrid functionals available in VASP are described in details at formalism of the HF method and hybrids along with examples and links to the corresponding INCAR files.

Details about the implementation of the unscreened hybrid functionals can be found in the work of Paier et al.,[8] while details specific to the screened hybrid functionals can be found in Refs. [9][10] Refs. [11][6] report the development of dielectric-dependent hybrid functionals, which provide very accurate band gaps and are also available in VASP.

Note that as in most other codes, hybrid functionals are implemented in VASP within the generalized KS scheme,[12] which means that the total energy is minimized, as in the HF theory, with respect to the orbitals instead of the electron density.

VASP offers a convenient way to generate the band structure with hybrid functionals.

Technical points

  • The unscreened Coulomb potential used to evaluate the exchange integral in HF has an integrable Coulomb singularity that leads to slow convergence with respect to supercell size (or equivalently k point sampling). To make the computations feasible requires special treatment of the Coulomb singularity.
  • The Downsampling of the HF operator allows for the use of a coarser k point sampling for the HF operator, and therefore faster calculations.[9] However, this option should be use with care for metals.
  • The Adaptively Compressed Exchange Operator,[13] that allows for a more efficient evaluation of the Fock operator, is used if the Davidson algorithm (ALGO = Normal, the default) is selected (see LFOCKACE for more details).

Additional resources

Tutorials

Lectures

How to

Further reading

In addition to the works already cited above, the selected publications listed below describe methodological developments or computational studies performed using the hybrid functionals implemented in VASP.

  • The B3LYP functional applied to solid-state systems and its failure for metals[14].
  • Applications of hybrid functionals to selected materials: Ceria,[15] lead chalcogenides,[16] CO adsorption on metals,[17][18] excitonic properties,[19] SrTiO and BaTiO, [20] perovskites,[21] and transition-metal oxides.[22]
  • HSEsol hybrid functional.[23]
  • Analysis of the HSE parameter space.[24]
  • Automated workflow for non-empirical Wannier-localized optimal tuning of range-separated hybrid functionals.[25]

References

  1. ↑ A. D. Becke, J. Chem. Phys. 98, 5648 (1993).
  2. ↑ J. Heyd, J. E. Peralta, G. E. Scuseria, and R. L. Martin, Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhof screened hybrid functional, J. Chem. Phys. 123, 174101 (2005).
  3. ↑ W. Chen, G. Miceli, G.M. Rignanese, and A. Pasquarello, Nonempirical dielectric-dependent hybrid functional with range separation for semiconductors and insulators, Phys. Rev. Mater. 2, 073803 (2018).
  4. ↑ C. Franchini, M. Reticcioli, M. Setvin, and U. Diebold, Polarons in Materials, Nat. Rev. Mat. 6, 560 (2021).
  5. ↑ F. Oba, A. Togo, I. Tanaka, J. Paier, and G. Kresse, Phys. Rev. B 77, 245202 (2008).
  6. ↑ a b P. Liu, C. Franchini, M. Marsman, and G. Kresse, Assessing model-dielectric-dependent hybrid functionals on the antiferromagnetic transition-metal monoxides MnO, FeO, CoO, and NiO, J. Phys.: Condens. Matter 32, 015502 (2020).
  7. ↑ A. V. Krukau , O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006).
  8. ↑ J. Paier, R. Hirschl, M. Marsman, and G. Kresse, J. Chem. Phys. 122, 234102 (2005).
  9. ↑ a b J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Ángyán, J. Chem. Phys. 124, 154709 (2006).
  10. ↑ J. G. Ángyán, I. Gerber, and M. Marsman, Spherical harmonic expansion of short-range screened Coulomb interactions, J. Phys. A: Math. Gen. 39, 8613 (2006).
  11. ↑ Z.H. Cui, Y.C. Wang, M.Y. Zhang, X. Xu, and H. Jiang, Doubly Screened Hybrid Functional: An Accurate First-Principles Approach for Both Narrow- and Wide-Gap Semiconductors J. Phys. Chem. Lett., 9, 2338-2345 (2018).
  12. ↑ A. Seidl, A. Görling, P. Vogl, J.A. Majewski, and M. Levy, Phys. Rev. B 53, 3764 (1996).
  13. ↑ L. Lin, J. Chem. Theory Comput. 12, 2242-2249 (2016).
  14. ↑ J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).
  15. ↑ J. L. F. Da Silva, M. V. Ganduglia-Pirovano, J. Sauer, V. Bayer, and G. Kresse, Phys. Rev. B 75, 045121 (2007).
  16. ↑ Hummer, A. Grüneis, and G. Kresse, Phys. Rev. B 75, 195211 (2007).
  17. ↑ A. Stroppa, K. Termentzidis, J. Paier, G. Kresse, and J. Hafner, Phys. Rev. B 76, 195440 (2007).
  18. ↑ A. Stroppa and G. Kresse, New Journal of Physics 10, 063020 (2008).
  19. ↑ J. Paier, M. Marsman, and G. Kresse, Phys. Rev. B 78, 121201(R) (2008).
  20. ↑ R. Wahl, D. Vogtenhuber, and G. Kresse, Phys. Rev. B 78, 104116 (2008).
  21. ↑ C. Franchini, Hybrid functionals applied to perovskites, J. Phys.: Condens. Matter 26, 253202 (2014).
  22. ↑ H. R. Gopidi, R. Zhang, Y. Wang, A. Patra, J. Sun, A. Ruzsinszky, J. P. Perdew, and P. Canepa, Reducing self-interaction error in transition-metal oxides with different exact-exchange fractions for energy and density, Phys. Rev, B 113, 165115 (2026).
  23. ↑ L. Schimka, J. Harl, and G. Kresse, J. Chem. Phys. 134, 024116 (2011).
  24. ↑ J. E. Moussa, P. A. Schultz, and J. R. Chelikowsky, Analysis of the Heyd-Scuseria-Ernzerhof density functional parameter space, J. Chem. Phys. 136, 204117 (2012).
  25. ↑ S. E. Gant, F. Ricci, G. Ohad, A. Ramasubramaniam, L. Kronik, and J. B. Neaton, Automated workflow for non-empirical Wannier-localized optimal tuning of range-separated hybrid functionals, Comput. Phys. Commun. 320, 109995 (2026).
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