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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}}. Polarons{{cite|franchini:nrm:21}} or defect states{{cite|oba:prb:08}} are among properties that 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}}
Note that hybrid functionals are also 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'''.


== Overview ==
== 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{{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 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.
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.


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.
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.
 
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.
 
VASP offers a convenient way to generate the [band structure with hybrid functionals|Bandstructure_calculation_using_hybrid_functionals]
 
== Technical points ==


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.
* {{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.}}


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).
* 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]].
To make the computations feasible requires special treatment of the [[Coulomb singularity]].


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).
* 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|angyan:jpa:2006}} However, this option should be use with care for metals.
 
* 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 ==
=== Tutorials ===
=== Tutorials ===
Line 43: Line 47:
*The B3LYP functional applied to solid state systems and its failure for metals{{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}} 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}}
*Various benchmark studies using hybrid functionals: HSEsol hybrid functional,{{cite|schimka:jcp:11}} Analysis of the HSE parameter space,{{cite|moussa:jcp:12}}  
parameter space


== References ==
== References ==

Revision as of 19:53, 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 also often good at treating strongly correlated electrons.[6]

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 with respect to the orbitals (instead of the electron density) as in the HF theory.

VASP offers a convenient way to generate the [band structure with hybrid functionals|Bandstructure_calculation_using_hybrid_functionals]

Technical points

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.
  • 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.[10] However, this option should be use with care for metals.
  • 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 parameter space,[22]

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. ↑ a b 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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