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where <math>a</math> is the mixing parameter that determines the relative weights of HF and semilocal exchange.
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}}{{cite|liu2019assessing}}. Polarons{{cite|franchini:nrm:21}} and defect states can also be better described by 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 often good at treating [[:Category:Strongly correlated electrons|strongly correlated electrons]].{{cite|liu2019assessing}}


However, be aware that 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'''.
However, be aware that 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'''.


== Theoretical background ==
== Overview ==


In hybrid functionals the exchange part consists of a linear combination of HF and semilocal (e.g., GGA) exchange:
The hybrid functionals can be divided into families according to the type of semilocal approximation (LDA, GGA, or MGGA) or 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).
:<math>
E_{\mathrm{xc}}^{\mathrm{hybrid}}=a E_{\mathrm{x}}^{\mathrm{HF}} + (1-a)E_{\mathrm{x}}^{\mathrm{GGA}} + E_{\mathrm{c}}^{\mathrm{GGA}}
</math>
where <math>a</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 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).


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 with respect to the orbitals (instead of the electron density) as in the Hartree-Fock theory.
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.


It is important to mention that hybrid functionals are computationally more expensive than semilocal methods.
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.


Read more about [[Hybrid_functionals: formalism|formalism of the HF method and hybrids]].
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 with respect to the orbitals (instead of the electron density) as in the HF theory.


The unscreened Coulomb potential used to evaluate the exchange integral in Hartree-Fock has an integrable singularity that leads to slow convergence with respect to supercell size (or equivalently '''k''' point sampling).
On a technical side, the unscreened Coulomb potential used to evaluate the exchange integral in HF has an integrable 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]].
To make the computations feasible requires special treatment of the [[Coulomb singularity]].
{{NB|mind|Hybrid functionals are often good at treating [[:Category:Strongly correlated electrons|strongly correlated electrons]].}}
 
Finally, we mention that the Adaptively Compressed Exchange Operator,{{cite|linlin:jctc:2016}}) that allows for more efficient evaluation of the Fock operator, is used if the Davidson algorithm ({{TAG|ALGO}} = Normal) is used (see {{TAG|LFOCKACE}} for more details).


== Additional resources ==
== Additional resources ==
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=== How to ===
=== How to ===
*[[Hybrid_functionals: formalism|Formalism of the HF method and hybrids]]
*[[List_of_hybrid_functionals|List of available hybrid functionals]] and how to specify them in the {{FILE|INCAR}} file.
*[[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.
*[[Downsampling_of_the_Hartree-Fock_operator|Downsampling of the Hartree-Fock operator]] to reduce the computational cost.
Line 41: Line 41:
=== Further reading ===
=== Further reading ===
*A comprehensive study of the performance of the HSE03/HSE06 functional compared to the PBE and PBE0 functionals{{cite|paier:jcp:06}}.
*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}}.
*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}} defects in ZnO,{{cite|oba:prb:08}} excitonic properties,{{cite|paier:prb:08}} SrTiO and BaTiO.{{cite|wahl:prb:08}}
*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}}
*Various benchmark studies using hybrid functionals: HSEsol hybrid functional,{{cite|schimka:jcp:11}} Analysis of the HSE,{{cite|moussa:jcp:12}}
parameter space


== References ==
== References ==

Latest revision as of 15:35, 10 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 often good at treating strongly correlated electrons.[6]

However, be aware that 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.

Overview

The hybrid functionals can be divided into families according to the type of semilocal approximation (LDA, GGA, or MGGA) or 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 with respect to the orbitals (instead of the electron density) as in the HF theory.

On a technical side, the unscreened Coulomb potential used to evaluate the exchange integral in HF has an integrable 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.

Finally, we mention that the Adaptively Compressed Exchange Operator,[13]) that allows for more efficient evaluation of the Fock operator, is used if the Davidson algorithm (ALGO = Normal) is used (see LFOCKACE for more details).


Additional resources

Tutorials

Lectures

How to

Further reading

  • A comprehensive study of the performance of the HSE03/HSE06 functional compared to the PBE and PBE0 functionals[9].
  • 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]
  • Various benchmark studies using hybrid functionals: HSEsol hybrid functional,[21] Analysis of the HSE,[22]

parameter space

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. ↑ L. Schimka, J. Harl, and G. Kresse, J. Chem. Phys. 134, 024116 (2011).
  22. ↑ 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).
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