Category:Hybrid functionals
Hybrid functionals go beyond the semilocal approximations by mixing the Hartree-Fock (HF) and Kohn-Sham methods[1]. 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. Polarons and defect states 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.
Theoretical background
In hybrid functionals the exchange part consists of a linear combination of HF and semilocal (e.g., GGA) exchange:
- [math]\displaystyle{ 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]\displaystyle{ 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[2] 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[3], 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
- Tutorial for hybrid calculations.
Lectures
- Lecture on hybrid functionals.
How to
- List of available hybrid functionals and how to specify them in the INCAR file.
- Downsampling of the Hartree-Fock operator to reduce the computational cost.
- band-structure calculation using hybrid functionals.
Further reading
- A comprehensive study of the performance of the HSE03/HSE06 functional compared to the PBE and PBE0 functionals[4].
- The B3LYP functional applied to solid state systems[5].
- Applications of hybrid functionals to selected materials: Ceria,[6] lead chalcogenides,[7] CO adsorption on metals,[8][9] defects in ZnO,[10] excitonic properties,[11] SrTiO and BaTiO.[12]
References
- ↑ A. D. Becke, J. Chem. Phys. 98, 5648 (1993).
- ↑ A. V. Krukau , O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006).
- ↑ A. Seidl, A. Görling, P. Vogl, J.A. Majewski, and M. Levy, Phys. Rev. B 53, 3764 (1996).
- ↑ J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Ángyán, J. Chem. Phys. 124, 154709 (2006).
- ↑ J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).
- ↑ J. L. F. Da Silva, M. V. Ganduglia-Pirovano, J. Sauer, V. Bayer, and G. Kresse, Phys. Rev. B 75, 045121 (2007).
- ↑ Hummer, A. Grüneis, and G. Kresse, Phys. Rev. B 75, 195211 (2007).
- ↑ A. Stroppa, K. Termentzidis, J. Paier, G. Kresse, and J. Hafner, Phys. Rev. B 76, 195440 (2007).
- ↑ A. Stroppa and G. Kresse, New Journal of Physics 10, 063020 (2008).
- ↑ F. Oba, A. Togo, I. Tanaka, J. Paier, and G. Kresse, Phys. Rev. B 77, 245202 (2008).
- ↑ J. Paier, M. Marsman, and G. Kresse, Phys. Rev. B 78, 121201(R) (2008).
- ↑ R. Wahl, D. Vogtenhuber, and G. Kresse, Phys. Rev. B 78, 104116 (2008).
Pages in category "Hybrid functionals"
The following 43 pages are in this category, out of 43 total.