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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}}{{cite|liu2019assessing}}. Polarons{{cite|franchini:nrm:21}} or defect states{{cite|oba:prb:08}} are among properties that can also be better described by hybrid 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'''.
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 ==
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).
 
In hybrid functionals the exchange part consists of a linear combination of HF and semilocal (e.g., GGA) exchange:
:<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.
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.

Revision as of 12:14, 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][4]. Polarons[5] or defect states[6] are among properties that can also be better described by hybrid 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.

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[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).

Note that as in most other codes, hybrid functionals are implemented in VASP within the generalized KS scheme[8], which means that the total energy is minimized with respect to the orbitals (instead of the electron density) as in the Hartree-Fock theory.

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

Read more about formalism of the HF method and hybrids.

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

Mind: Hybrid functionals are often good at treating strongly correlated electrons.

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[10].
  • Applications of hybrid functionals to selected materials: Ceria,[11] lead chalcogenides,[12] CO adsorption on metals,[13][14] defects in ZnO,[6] excitonic properties,[15] SrTiO and BaTiO.[16]

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. ↑ 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).
  5. ↑ C. Franchini, M. Reticcioli, M. Setvin, and U. Diebold, Polarons in Materials, Nat. Rev. Mat. 6, 560 (2021).
  6. ↑ a b F. Oba, A. Togo, I. Tanaka, J. Paier, and G. Kresse, Phys. Rev. B 77, 245202 (2008).
  7. ↑ A. V. Krukau , O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006).
  8. ↑ A. Seidl, A. Görling, P. Vogl, J.A. Majewski, and M. Levy, Phys. Rev. B 53, 3764 (1996).
  9. ↑ J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Ángyán, J. Chem. Phys. 124, 154709 (2006).
  10. ↑ J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).
  11. ↑ J. L. F. Da Silva, M. V. Ganduglia-Pirovano, J. Sauer, V. Bayer, and G. Kresse, Phys. Rev. B 75, 045121 (2007).
  12. ↑ Hummer, A. Grüneis, and G. Kresse, Phys. Rev. B 75, 195211 (2007).
  13. ↑ A. Stroppa, K. Termentzidis, J. Paier, G. Kresse, and J. Hafner, Phys. Rev. B 76, 195440 (2007).
  14. ↑ A. Stroppa and G. Kresse, New Journal of Physics 10, 063020 (2008).
  15. ↑ J. Paier, M. Marsman, and G. Kresse, Phys. Rev. B 78, 121201(R) (2008).
  16. ↑ R. Wahl, D. Vogtenhuber, and G. Kresse, Phys. Rev. B 78, 104116 (2008).
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