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Hybrid functionals, which mix the Hartree-Fock and Kohn-Sham theories, can be more accurate than semilocal methods like GGA, in particular for nonmetallic systems. They are particularly suited for band gap calculation for instance. Hybrid functionals are available in VASP.
'''Hybrid functionals''' mix the Hartree-Fock (HF) and Kohn-Sham theories{{cite|becke:jcp:93}} and can be more accurate than semilocal methods, e.g., {{TAG|GGA}}, in particular for nonmetallic systems. They are for instance suited for band-gap calculations. Several hybrid functionals are available in VASP.


== Theoretical background ==
== Theoretical background ==


In hybrid functionals the exchange part consists of a linear combination of Hartree-Fock (HF) and semilocal (e.g., GGA) exchange:
In hybrid functionals the exchange part consists of a linear combination of HF and semilocal (e.g., GGA) exchange:
:<math>
:<math>
E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF}} + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA}} + E_{\mathrm{c}}^{\mathrm{GGA}}
E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF}} + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA}} + E_{\mathrm{c}}^{\mathrm{GGA}}
</math>
</math>
where <math>a_{x}</math> determines the relative amount of HF and semilocal exchange. There are essentially two types of hybrid functionals: (a) the ones where the HF exchange is applied at full interelectronic range (unscreened hybrids) and (b) the others where the HF exchange is applied either at short or at long interelectronic range (called screened or range-separated hybrids).
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 either at short or at 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).


=== Unscreened hybrid functionals ===
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.


In hybrid exchange-correlation functionals, the exchange component consists of a mixing of [[:Category:GGA|GGA]] (or meta-GGA) and Hartree-Fock exchange:
It is important to mention that hybrid functionals are computationally more expensive than semilocal methods.
:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF}} + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA}} + E_{\mathrm{c}}^{\mathrm{GGA}},</math>
where <math>\alpha</math> is the mixing parameter ({{TAG|AEXX}}) that is typically in the range 0.1-0.5. Two examples of hybrid functionals, PBE0 and B3LYP, are given below.


*[[list_of_hybrid_functionals#PBE0|PBE0]]:{{cite|perdew:jcp:1996}}
Read more about [[Hybrid_functionals: formalism|formalism of the HF method and hybrids]].


:<math>
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).
E_{\mathrm{xc}}^{\mathrm{PBE0}}=\frac{1}{4} E_{\mathrm{x}}^{\mathrm{HF}} +
To make the computations feasible requires special treatment of the [[Coulomb singularity]].
\frac{3}{4} E_{\mathrm{x}}^{\mathrm{PBE}} + E_{\mathrm{c}}^{\mathrm{PBE}},
</math>
</span>
:where <math>E_{x}^{\rm PBE}</math> and <math>E_{c}^{\rm PBE}</math> denote the exchange and correlation parts of the PBE density functional, respectively.


*[[list_of_hybrid_functionals#B3LYP|B3LYP]]{{cite|stephens:jpc:1994}}, well known and popular amongst quantum chemists:
== How to ==
 
<span id="B3LYP">
:<math>
\begin{align}
E_{\mathrm{x}}^{\mathrm{B3LYP}} &=0.8 E_{\mathrm{x}}^{\mathrm{LDA}}+
0.2 E_{\mathrm{x}}^{\mathrm{HF}} + 0.72 \Delta E_{\mathrm{x}}^{\mathrm{B88}}, \\
E_{\mathrm{c}}^{\mathrm{B3LYP}} &=0.19 E_{\mathrm{c}}^{\mathrm{VWN3}}+
0.81 E_{\mathrm{c}}^{\mathrm{LYP}},
\end{align}
</math>
</span>
:where <math>E_{x}^{\rm B3LYP}</math> and <math>E_{c}^{\rm B3LYP}</math> are the B3LYP exchange and correlation energy contributions, respectively. <math>E_{x}^{\rm B3LYP}</math> consists of 80% of LDA exchange plus 20% of non-local Hartree-Fock exchange, and 72% of the gradient corrections of the Becke88 exchange functional. <math>E_{c}^{\rm B3LYP}</math> consists of 81% of LYP correlation energy, which contains a local and a semilocal (gradient dependent) part, and 19% of the (local) Vosko-Wilk-Nusair correlation functional III, which is fitted to the correlation energy in the random phase approximation RPA of the homogeneous electron gas.
 
The non-local Hartree-Fock exchange energy, <math>E_{x}</math>, can be written as
 
<span id="ExFock">
:<math>
E_{\mathrm{x}}^{\mathrm{HF}}= -\frac{e^2}{2}\sum_{n\mathbf{k},m\mathbf{q}}
f_{n\mathbf{k}} f_{m\mathbf{q}} \times
\int\int d^3\mathbf{r} d^3\mathbf{r}'
\frac{\psi_{n\mathbf{k}}^{*}(\mathbf{r})\psi_{m\mathbf{q}}^{*}(\mathbf{r}')
\psi_{n\mathbf{k}}(\mathbf{r}')\psi_{m\mathbf{q}}(\mathbf{r})}
{\vert \mathbf{r}-\mathbf{r}' \vert}
</math>
</span>
 
with <math>\{\psi_{n\mathbf{k}}(\mathbf{r})\}</math> being the set of one-electron
Bloch states of the system, and <math>\{f_{n\mathbf{k}}\}</math> the corresponding
set of (possibly fractional) occupational numbers.
The sums over <math>{\bf k}</math> and <math>{\bf q}</math> run over all <math>{\bf k}</math> points chosen to sample the Brillouin zone (BZ), whereas the sums over <math>m</math> and <math>n</math> run over all bands at these <math>{\bf k}</math> points. The corresponding <span id="VxFock">non-local Hartree-Fock potential is given by
 
:<math>
V_{\mathrm{x}}\left(\mathbf{r},\mathbf{r}'\right)=
-\frac{e^2}{2}\sum_{m\mathbf{q}}f_{m\mathbf{q}}
\frac{\psi_{m\mathbf{q}}^{*}(\mathbf{r}')\psi_{m\mathbf{q}}(\mathbf{r})}
{\vert \mathbf{r}-\mathbf{r}' \vert}
= -\frac{e^2}{2}\sum_{m\mathbf{q}}f_{m\mathbf{q}}
e^{-i\mathbf{q}\cdot\mathbf{r}'}
\frac{u_{m\mathbf{q}}^{*}(\mathbf{r}')u_{m\mathbf{q}}(\mathbf{r})}
{\vert \mathbf{r}-\mathbf{r}' \vert}
e^{i\mathbf{q}\cdot\mathbf{r}},
</math>
</span>
 
where <math>u_{m\mathbf{q}}(\mathbf{r})</math> is the cell periodic part of the Bloch state,
<math>\psi_{n\mathbf{q}}(\mathbf{r})</math>, at <math>{\bf k}</math> point, <math>{\bf q}</math>, with band index ''m''.
Using the decomposition of the Bloch states, <math>\psi_{m\mathbf{q}}</math>, in plane waves,
 
:<math>
\psi_{m\mathbf{q}}(\mathbf{r})=
\frac{1}{\sqrt{\Omega}}
\sum_\mathbf{G}C_{m\mathbf{q}}(\mathbf{G})e^{i(\mathbf{q}+\mathbf{G}) \cdot \mathbf{r}}
</math>
 
the Hartree-Fock exchange potential may be written as
 
:<math>
V_{\mathrm{x}}\left(\mathbf{r},\mathbf{r}'\right)=
\sum_{\mathbf{k}}\sum_{\mathbf{G}\mathbf{G}'}
e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}
V_{\mathbf{k}}\left( \mathbf{G},\mathbf{G}'\right)
e^{-i(\mathbf{k}+\mathbf{G}')\cdot\mathbf{r}'}
</math>
where
<span id="VxFockRecip">
:<math>
V_\mathbf{k}\left( \mathbf{G},\mathbf{G}'\right)=
\langle \mathbf{k}+\mathbf{G} | V_{\mathrm{x}} | \mathbf{k}+\mathbf{G}'\rangle =
-\frac{4\pi e^2}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
\frac{C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')}
{|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2}
</math>
</span>
is the representation of the Hartree-Fock potential in reciprocal space.
In VASP, these expressions are implemented within the [[PAW_method|PAW formalism]].{{cite|paier:jcp:05}}
 
=== Range-separated hybrid functionals ===
 
More popular in solid-state physics, are the screened hybrid functionals, where only the short-range (SR) exchange is mixed, while the long-range (LR) exchange is still fully [[:Category:GGA|GGA]]:
:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=\alpha E_{\mathrm{x}}^{\mathrm{HF,SR}}(\mu) + (1-\alpha)E_{\mathrm{x}}^{\mathrm{GGA,SR}}(\mu) + E_{\mathrm{x}}^{\mathrm{GGA,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{GGA}},</math>
where <math>\mu</math> is the screening parameter ({{TAG|HFSCREEN}}) that determines the range separation. The most popular range-separated functional, HSE, is given below.
 
*HSE:
:In the range-separated [[List_of_hybrid_functionals#HSE03|HSE03]]{{cite|heyd:jcp:03}}{{cite|heyd:jcp:04}}{{cite|heyd:jcp:06}} and [[List_of_hybrid_functionals#HSE06|HSE06]]{{cite|krukau:jcp:06}} hybrid functionals the slowly decaying long-range part of the Hartree-Fock exchange interaction is replaced by the corresponding part of the PBE density functional counterpart. The resulting expression for the exchange-correlation energy is given by:
 
:<math>
E_{\mathrm{xc}}^{\mathrm{HSE}}= \frac{1}{4}E_{\mathrm{x}}^{\mathrm{SR,HF}}(\mu)
+ \frac{3}{4} E_{\mathrm{x}}^{\mathrm{PBE,SR}}(\mu)
+ E_{\mathrm{x}}^{\mathrm{PBE,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}}.
</math>
 
The decomposition of the Coulomb kernel is obtained using the following construction:
 
<span id="SRLR">
:<math>
\frac{1}{r}=S_{\mu}(r)+L_{\mu}(r)=\frac{\mathrm{erfc}(\mu r)}{r}+\frac{\mathrm{erf}(\mu r)}{r},
</math>
</span>
 
where <math>r =|{\bf r}-{\bf r}'|</math>, and <math>\mu</math> (={{TAG|HFSCREEN}}) is the parameter that defines the range separation, and is related to a characteristic distance, <math>2/\mu</math>, at which the short-range interactions become negligible.
 
Note: It has been shown that the optimum <math>\mu</math>, controlling the range separation is approximately 0.2-0.3 &Aring;<sup>-1</sup>.{{cite|heyd:jcp:03}}{{cite|heyd:jcp:04}}{{cite|heyd:jcp:06}}{{cite|krukau:jcp:06}}
To select [[List_of_hybrid_functionals#HSE06|the HSE06 functional]] you need to select ({{TAG|HFSCREEN}}=0.2).
 
Using the decomposed Coulomb kernel one may straightforwardly rewrite the non-local [[#ExFock|Hartree-Fock exhange energy]]:
 
<span id="ExFockSR">
:<math>
E^{\rm SR,HF}_{\mathrm{x}}(\mu)=
-\frac{e^2}{2}\sum_{n\mathbf{k},m\mathbf{q}}
f_{n\mathbf{k}} f_{m\mathbf{q}}
\int \int d^3\mathbf{r} d^3\mathbf{r}'
\frac{\mathrm{erfc}(\mu|\mathbf{r}-\mathbf{r}'|)}{|\mathbf{r}-\mathbf{r}'|}
\times \psi_{n\mathbf{k}}^{*}(\mathbf{r})\psi_{m\mathbf{q}}^{*}(\mathbf{r}')
\psi_{n\mathbf{k}}(\mathbf{r}')\psi_{m\mathbf{q}}(\mathbf{r}).
</math>
</span>
 
The representation of the corresponding short-range Hartree-Fock potential in reciprocal space is given by
 
<span id="VxFockSRRecip">
:<math>
\begin{align}
V^{\mathrm{SR}}_\mathbf{k}\left( \mathbf{G},\mathbf{G}'\right)&=
\langle \mathbf{k}+\mathbf{G} | V^{\rm SR}_x [\mu] | \mathbf{k}+\mathbf{G}'\rangle \\
&=-\frac{4\pi e^2}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
\frac{C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')}
{|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2}
\times \left( 1-e^{-|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2 /4\mu^2} \right).
\end{align}
</math>
</span>
 
The only difference to the reciprocal space representation of the complete [[#VxFockRecip|Hartree-Fock exchange potential]] is the second factor in the summand above, representing the complementary error function in reciprocal space.
 
The short-range PBE exchange energy and potential, and their long-range counterparts, are arrived at using the same [[#SRLR|decomposition]], in accordance with Heyd ''et al''.{{cite|heyd:jcp:03}} It is easily seen that the long-range term in the [[#SRLR|decomposed Coulomb kernel]] becomes zero for <math>\mu=0</math>, and the short-range contribution then equals the full Coulomb operator, whereas for <math>\mu\rightarrow\infty</math> it is the other way around. Consequently, the two limiting cases of the HSE functional are a true [[List_of_hybrid_functionals#PBE0|PBE0]] functional for <math>\mu=0</math>, and a pure PBE calculation for <math>\mu\rightarrow\infty</math>.
 
<span id="Thomas_Fermi"></span>
 
==== Thomas-Fermi screening ====
In the case of [[LTHOMAS|Thomas-Fermi screening]], the Coulomb kernel is again decomposed in a short-range and a long-range part.{{cite|bylander:prb:90}}{{cite|seidl:prb:96}}{{cite|picozzi:prb:00}} This decomposition can be conveniently written in reciprocal space:
 
<span id="ThomasFermi">
:<math>
\frac{4 \pi e^2}{|\mathbf{G}|^2}=S_{\mu}(|\mathbf{G}|)+L_{\mu}(|\mathbf{G}|)=\frac{4 \pi e^2}{|\mathbf{G}|^2 +k_{\mathrm{TF}}^2}+
\left( \frac{4 \pi e^2}{|\mathbf{G}|^2} -\frac{4 \pi e^2}{|\mathbf{G}|^2 +k_{\mathrm{TF}}^2} \right),
</math>
</span>
 
where <math>k_{\rm TF}</math> (={{TAG|HFSCREEN}}) is the Thomas-Fermi screening length. For typical semiconductors, a Thomas-Fermi screening length of about 1.8 &Aring;<sup>-1</sup> yields reasonable band gaps. In principle, however, the Thomas-Fermi screening length depends on the valence-electron density; VASP determines this parameter from the number of valence electrons (read from the {{FILE|POTCAR}} file) and the volume and writes the corresponding value to the {{FILE|OUTCAR}} file:
  Thomas-Fermi vector in A            =  2.00000
Since VASP counts the semi-core states and ''d''-states as valence electrons, although these states do not contribute to the screening, the values reported by VASP are often incorrect.


Another important detail concerns the implementation of the density-functional part in the screened exchange case. Literature suggests that a global enhancement factor <math>z</math> (see Eq. 3.15){{cite|seidl:prb:96}} should be used, whereas VASP implements a local-density-dependent enhancement factor <math>z=k_{\rm TF}/k</math> , where <math>k</math> is the Fermi wave vector corresponding to the local density (and not the average density as suggested Seidl ''et al''.{{cite|seidl:prb:96}}. The VASP implementation is in the spirit of the ''local'' density approximation.
[[List_of_hybrid_functionals|List of available hybrid functionals]] and how to specify them in the {{FILE|INCAR}} file.


== How to ==
[[Downsampling_of_the_Hartree-Fock_operator|Downsampling of the Hartree-Fock operator]] to reduce the computational cost.


[[List_of_hybrid_functionals|List of available hybrid functionals]] and how to specify them in {{FILE|INCAR}}.
How to do a [[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 ==
== 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{{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}} defects in ZnO,{{cite|oba:prb:08}} excitonic properties,{{cite|paier:prb:08}} SrTiO and BaTiO.{{cite|wahl:prb:08}}
{{sc|Hartree-Fock and HF/DFT hybrid functionals|Examples|Examples that use this tag}}


== References ==
== References ==

Latest revision as of 12:30, 19 October 2023

Hybrid functionals mix the Hartree-Fock (HF) and Kohn-Sham theories[1] and can be more accurate than semilocal methods, e.g., GGA, in particular for nonmetallic systems. They are for instance suited for band-gap calculations. Several hybrid functionals are available in VASP.

Theoretical background

In hybrid functionals the exchange part consists of a linear combination of HF and semilocal (e.g., GGA) exchange:

where 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 at 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).

Note that as in most other codes, hybrid functionals are implemented in VASP within the generalized KS scheme[2], 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.

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.

How to do a 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[3].
  • The B3LYP functional applied to solid state systems[4].
  • Applications of hybrid functionals to selected materials: Ceria,[5] lead chalcogenides,[6] CO adsorption on metals,[7][8] defects in ZnO,[9] excitonic properties,[10] SrTiO and BaTiO.[11]

References

  1. ↑ A. D. Becke, J. Chem. Phys. 98, 5648 (1993).
  2. ↑ A. Seidl, A. Görling, P. Vogl, J.A. Majewski, and M. Levy, Phys. Rev. B 53, 3764 (1996).
  3. ↑ J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Ángyán, J. Chem. Phys. 124, 154709 (2006).
  4. ↑ J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).
  5. ↑ J. L. F. Da Silva, M. V. Ganduglia-Pirovano, J. Sauer, V. Bayer, and G. Kresse, Phys. Rev. B 75, 045121 (2007).
  6. ↑ Hummer, A. Grüneis, and G. Kresse, Phys. Rev. B 75, 195211 (2007).
  7. ↑ A. Stroppa, K. Termentzidis, J. Paier, G. Kresse, and J. Hafner, Phys. Rev. B 76, 195440 (2007).
  8. ↑ A. Stroppa and G. Kresse, New Journal of Physics 10, 063020 (2008).
  9. ↑ F. Oba, A. Togo, I. Tanaka, J. Paier, and G. Kresse, Phys. Rev. B 77, 245202 (2008).
  10. ↑ J. Paier, M. Marsman, and G. Kresse, Phys. Rev. B 78, 121201(R) (2008).
  11. ↑ R. Wahl, D. Vogtenhuber, and G. Kresse, Phys. Rev. B 78, 104116 (2008).
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