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Hybrid functionals, which mix the Hartree-Fock (HF) and Kohn-Sham theories{{cite|becke:jcp:93}}, can be more accurate than semilocal methods like GGA, in particular for nonmetallic systems. They are suited for band gap calculations 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}}:
:<math>
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>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.{{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}}
 
* {{NB|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_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]].}}
 
== 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{{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).
 
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.


== Theoretical background ==
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.


In hybrid functionals the exchange part consists of a linear combination of HF and semilocal (e.g., GGA) exchange:
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.
:<math>
 
E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF}} + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA}} + E_{\mathrm{c}}^{\mathrm{GGA}}
VASP offers a convenient way to generate the [[Band-structure calculation using hybrid functionals|band structure with hybrid functionals]].
</math>
 
where <math>\alpha</math> determines the relative amount of HF and semilocal exchange. The hybrid functionals can be divided into families according to the interelectronic range at which the HF exchange is applied: at full range (unscreened hybrids) or at either short or long range (called screened or range-separated hybrids). From the practical point of view the short-range hybrid functionals like HSE are preferable for periodic solids, since leading to faster convergence with respect to the number of k-points (or size of the unit cell).
== 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 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.
 
* 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).


More detail about the formalism of the HF method and hybrids can be found [[Hybrid_functionals: formalism|here]].
== Additional resources ==
=== Tutorials ===
*Tutorial for {{Tutorial|hybrids:part1|hybrid calculations}}.


== How to ==
=== Lectures ===
*Lecture on {{Video|hybrid:henrique:2022|hybrid functionals}}.


[[List_of_hybrid_functionals|List of available hybrid functionals]] and how to specify them in {{FILE|INCAR}}.
=== 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]].


[[Downsampling_of_the_Hartree-Fock_operator|Downsampling of the Hartree-Fock operator]].
=== 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.


== Further reading ==
*The B3LYP functional applied to solid-state systems and its failure for metals{{cite|paier:jcp:07}}.
*A comprehensive study of the performance of the HSE03/HSE06 functional compared to the PBE and PBE0 functionals.{{cite|paier:jcp:06}}
*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}}
*The B3LYP functional applied to solid state systems.{{cite|paier:jcp:07}}
*HSEsol hybrid functional.{{cite|schimka:jcp:11}}
*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}}
*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}}


== 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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