Hybrid functionals: formalism: Difference between revisions

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The hybrid functionals can be categorized into two types: unscreened and range-separated (i.e., screened), as described in more details below.
In hybrid functionals the exchange energy is a mixture of semilocal (SL) and nonlocal Hartree-Fock (HF) types. They can be categorized into different families according to the type of semilocal approximation (LDA, GGA, or MGGA) or the treatment of the short- and long-range parts of the exchange. A rather general formula that encompasses the different families of hybrid functionals is given by


Note that the hybrid functionals are implemented within the generalized KS scheme{{cite|seidl:prb:96}}. Thus, the total energy is minimized with respect to the orbitals (instead of the electron density), which means that the HF exchange leads to a nonlocal operator as in the Hartree-Fock-Roothaan theory.
:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{SR}})E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}}</math>


=== Unscreened hybrid functionals ===
where '''<math>a_{\mathrm{SR}}</math> and <math>a_{\mathrm{LR}}</math> are the mixing parameters (fraction of HF exchange) at short- and long-range''', respectively, and '''<math>\mu</math> is the screening parameter''' that determines the separation between short range (SR) and long range (LR). The SR and LR components of the full-range <math>E_{\mathrm{x}}^{\mathrm{SL}}</math> and <math>E_{\mathrm{x}}^{\mathrm{HF}}</math> exchange energies are constructed such that <math>E_{\mathrm{x}}^{\mathrm{HF}}=E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu)+E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu)</math> and <math>E_{\mathrm{x}}^{\mathrm{SL}}=E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu)+E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu)</math> at all values of <math>\mu</math>.
 
In hybrid exchange-correlation functionals, the exchange component consists of a mixing of {{TAG|GGA}} (or meta-GGA) and Hartree-Fock exchange:
:<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}}


The HF exchange energy (full-range, SR, or LR) is given by
<span id="ExFockSR">
:<math>
:<math>
E_{\mathrm{xc}}^{\mathrm{PBE0}}=\frac{1}{4} E_{\mathrm{x}}^{\mathrm{HF}} +
E_{\mathrm{x,(SR/LR)}}^{\rm HF}(\mu)=
\frac{3}{4} E_{\mathrm{x}}^{\mathrm{PBE}} + E_{\mathrm{c}}^{\mathrm{PBE}},
-\frac{1}{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}'
v(\mu,|\mathbf{r}-\mathbf{r}'|)
\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>
</math>
</span>
</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:
<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 nonlocal 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 nonlocal 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  
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
Bloch states of the system, and <math>\{f_{n\mathbf{k}}\}</math> the corresponding
set of (possibly fractional) occupational numbers.
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">nonlocal Hartree-Fock potential is given by
The sums over <math>{\bf k}</math> and <math>{\bf q}</math> run over all k-points chosen to sample the Brillouin zone, whereas the sums over <math>m</math> and <math>n</math> run over all bands at these k-points.  
 
The corresponding nonlocal HF potential is given by
<span id="VxFock">
:<math>
:<math>
V_{\mathrm{x}}^{\mathrm{HF}}\left(\mathbf{r},\mathbf{r}'\right)=
V_{\mathrm{x,(SR/LR)}}^{\mathrm{HF}}\left(\mu,\mathbf{r},\mathbf{r}'\right)=
-\frac{e^2}{2}\sum_{m\mathbf{q}}f_{m\mathbf{q}}
-\sum_{m\mathbf{q}}f_{m\mathbf{q}}v(\mu,|\mathbf{r}-\mathbf{r}'|)\psi_{m\mathbf{q}}^{*}(\mathbf{r}')\psi_{m\mathbf{q}}(\mathbf{r})
\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>
</math>
</span>
</span>
The non-multiplicative form of the HF potential is such that hybrid functionals are implemented within the generalized KS scheme{{cite|seidl:prb:96}}. Thus, the total energy is minimized with respect to the orbitals (instead of the electron density), which means that the HF exchange leads to a nonlocal operator as in the Hartree-Fock-Roothaan theory.


where <math>u_{m\mathbf{q}}(\mathbf{r})</math> is the cell periodic part of the Bloch state,
Using the decomposition of the Bloch states <math>\psi_{m\mathbf{q}}</math> in plane waves,
<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>
:<math>
Line 73: Line 38:
</math>
</math>


the Hartree-Fock exchange potential may be written as  
the HF exchange potential can be written as  


:<math>
:<math>
V_{\mathrm{x}}^{\mathrm{HF}}\left(\mathbf{r},\mathbf{r}'\right)=
V_{\mathrm{x,(SR/LR)}}^{\mathrm{HF}}\left(\mu,\mathbf{r},\mathbf{r}'\right)=
\sum_{\mathbf{k}}\sum_{\mathbf{G}\mathbf{G}'}
\sum_{\mathbf{k}}\sum_{\mathbf{G}\mathbf{G}'}
e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}
e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}
V_{\mathbf{k}}^{\mathrm{HF}}\left( \mathbf{G},\mathbf{G}'\right)
V_{\mathbf{k}}\left(\mu, \mathbf{G},\mathbf{G}'\right)
e^{-i(\mathbf{k}+\mathbf{G}')\cdot\mathbf{r}'}
e^{-i(\mathbf{k}+\mathbf{G}')\cdot\mathbf{r}'}
</math>
</math>
where
where
:<math>
V_{\mathbf{k}}\left(\mu, \mathbf{G},\mathbf{G}'\right)=
\langle \mathbf{k}+\mathbf{G} | V_{\mathrm{x,(SR/LR)}}^{\mathrm{HF}} | \mathbf{k}+\mathbf{G}'\rangle
</math>
=== Types of potentials ===
In most hybrid functionals proposed in the literature, the interelectronic Coulomb potential <math>v(\mu,|\mathbf{r}-\mathbf{r}'|)</math> has one of the following forms (<math>r=|\mathbf{r}-\mathbf{r}'|</math>):
* Full range (bare Colomb potential):
:<math>
v^{\mathrm{Coul}}(r)=\frac{1}{r}
</math>
<span id="VxFockRecip">
<span id="VxFockRecip">
:<math>
:<math>
V_\mathbf{k}^{\mathrm{HF}}\left( \mathbf{G},\mathbf{G}'\right)=
V_{\mathbf{k}}^{\mathrm{Coul}}\left( \mathbf{G},\mathbf{G}'\right)=
\langle \mathbf{k}+\mathbf{G} | V_{\mathrm{x}}^{\mathrm{HF}} | \mathbf{k}+\mathbf{G}'\rangle =
-\frac{4\pi}{\Omega} \sum_{m\mathbf{q}}f_{m\mathbf{q}}\sum_{\mathbf{G}''}
-\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>
* Short-range with error function:
:<math>
v_{\mathrm{SR}}^{\mathrm{erf}}(\mu,r)=\frac{\mathrm{erfc}(\mu r)}{r}
</math>
<span id="VxFockSRRecip">
:<math>
\begin{align}
V_{\mathbf{k},\mathrm{SR}}^{\mathrm{erf}}\left(\mu, \mathbf{G},\mathbf{G}'\right)=
-\frac{4\pi}{\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}'')}
\frac{C^*_{m\mathbf{q}}(\mathbf{G}'-\mathbf{G}'') C_{m\mathbf{q}}(\mathbf{G}-\mathbf{G}'')}
{|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2}
{|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2}
\left( 1-e^{-|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2 /(4\mu^2)} \right)
\end{align}
</math>
</span>
* Short-range with exponential function:
:<math>
v_{\mathrm{SR}}^{\mathrm{exp}}(\mu,r)=\frac{e^{-\mu r}}{r}
</math>
<span id="VxFockRecip">
:<math>
V_{\mathbf{k},\mathrm{SR}}^{\mathrm{exp}}\left(\mu, \mathbf{G},\mathbf{G}'\right)=
-\frac{4\pi}{\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 + \mu^2}
</math>
</math>
</span>
</span>
is the representation of the Hartree-Fock potential in reciprocal space.
The corresponding long-range potentials are given by <math>v_{\mathrm{LR}}^{\mathrm{erf/exp}}=v^{\mathrm{Coul}}-v_{\mathrm{SR}}^{\mathrm{erf/exp}}</math>.
 
In VASP, these expressions are implemented within the [[Projector-augmented-wave_formalism|PAW formalism]].{{cite|paier:jcp:05}}
In VASP, these expressions are implemented within the [[Projector-augmented-wave_formalism|PAW formalism]].{{cite|paier:jcp:05}}


=== Range-separated hybrid functionals ===
The families of hybrid functionals implemented in VASP are listed below along with examples, whose corresponding {{TAG|INCAR}} files can be found at the page [[list_of_hybrid_functionals|list of hybrid functionals]].
{{NB|important|The screening <math>\mu</math> ({{TAG|HFSCREEN}} tag) can be used only when the semilocal functional is PBE, PBEsol, or LDA ({{TAG|GGA}}{{=}}PE, PS, or CA, respectively). The other {{TAG|GGA}} and {{TAG|METAGGA}} functionals have no screened version available in VASP.}}


In range-separated hybrid functionals, the exchange is separated into short- and long-range components. This separation is determined by the screening parameter {{TAG|HFSCREEN}}.
== Families of hybrid functionals ==
{{NB|mind|{{TAG|HFSCREEN}} can be used only when {{TAG|GGA}}{{=}}PE, PS or CA. The other {{TAG|GGA}} and {{TAG|METAGGA}} functionals have no screened version available in VASP.}}


==== Error function screening with short-range Hartree-Fock exchange ====
=== HF exchange at full range ===


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 {{TAG|GGA}}:
There is no range separation, i.e. the same fraction of HF exchange is applied at full range, <math>a_{\mathrm{SR}}=a_{\mathrm{LR}}=a</math> ({{TAG|AEXX}} tag):
:<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:
:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=a E_{\mathrm{x}}^{\mathrm{HF}} + (1-a)E_{\mathrm{x}}^{\mathrm{SL}} + E_{\mathrm{c}}^{\mathrm{SL}}</math>
: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 (see the discussion on the [[Coulomb singularity]]) is replaced by the corresponding part of the PBE density functional counterpart. The resulting expression for the exchange-correlation energy is given by:
{{NB|mind|
*These functionals are set with {{TAG|LHFCALC}}{{=}}.TRUE. By default {{TAG|AEXX}}{{=}}0.25, but can be set to another value.
*The semilocal part can be of the LDA, GGA or MGGA type.
}}
 
These are the original and most simple forms of hybrid functionals. Two examples, PBE0 and B3LYP, are given below.
 
*[[list_of_hybrid_functionals#PBE0|PBE0]]:{{cite|perdew:jcp:1996}}  


:<math>
:<math>
E_{\mathrm{xc}}^{\mathrm{HSE}}= \frac{1}{4}E_{\mathrm{x}}^{\mathrm{HF,SR}}(\mu)
E_{\mathrm{xc}}^{\mathrm{PBE0}}=\frac{1}{4} E_{\mathrm{x}}^{\mathrm{HF}} +
+ \frac{3}{4} E_{\mathrm{x}}^{\mathrm{PBE,SR}}(\mu)
\frac{3}{4} E_{\mathrm{x}}^{\mathrm{PBE}} + E_{\mathrm{c}}^{\mathrm{PBE}}
+ E_{\mathrm{x}}^{\mathrm{PBE,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}}.
</math>
</math>
</span>
:It is based on the PBE GGA functional{{cite|perdew:prl:1996}} and <math>a=1/4</math>.


The decomposition of the Coulomb kernel is obtained using the following construction:
*[[list_of_hybrid_functionals#B3LYP|B3LYP]]{{cite|stephens:jpc:1994}}, well known and popular amongst quantum chemists:  


<span id="SRLR">
<span id="B3LYP">
:<math>
:<math>
\frac{1}{r}=S_{\mu}(r)+L_{\mu}(r)=\frac{\mathrm{erfc}(\mu r)}{r}+\frac{\mathrm{erf}(\mu r)}{r},
\begin{align}
E_{\mathrm{x}}^{\mathrm{B3LYP}} &=0.8 E_{\mathrm{x}}^{\mathrm{LDA}}+
0.2 E_{\mathrm{x}}^{\mathrm{HF}} + 0.72 (E_{\mathrm{x}}^{\mathrm{B88}}-E_{\mathrm{x}}^{\mathrm{LDA}}) +
0.19 E_{\mathrm{c}}^{\mathrm{VWN3}}+ 0.81 E_{\mathrm{c}}^{\mathrm{LYP}}
\end{align}
</math>
</math>
</span>
</span>
:The exchange part consists of 80% of LDA exchange plus 20% of HF exchange, and 72% of the gradient corrections of the B88 GGA functional.{{cite|becke:pra:1988}} The correlation consists of 81% of LYP{{cite|lee:prb:1988}} correlation energy, which contains a LDA and a GGA part, and 19% of the LDA Vosko-Wilk-Nusair correlation functional III,{{cite|vosko1980}} which was fitted to the correlation energy in the random phase approximation of the homogeneous electron gas.


where <math>r =|{\bf r}-{\bf r}'|</math>, and <math>\mu</math> (set by {{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.
=== HF exchange at short range (error-function screening) ===


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}}
The HF exchange is used only at short-range (the long-range part is fully semilocal, <math>a_{\mathrm{LR}}=0</math>) and the screening is done with the error function:
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 nonlocal [[#ExFock|Hartree-Fock exhange energy]]:
:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{SR}})E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}}</math>
 
The mixing <math>a_{\mathrm{SR}}</math> and screening <math>\mu</math> are controlled by the {{TAG|AEXX}} and {{TAG|HFSCREEN}} tags, respectively.
 
The most popular range-separated functional, HSE, is given below.
*[[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}}:
 
:<math>
E_{\mathrm{xc}}^{\mathrm{HSE}}= \frac{1}{4}E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu)
+ \frac{3}{4} E_{\mathrm{x,SR}}^{\mathrm{PBE}}(\mu)
+ E_{\mathrm{x,LR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}}
</math>
 
:They are based on the PBE GGA functional{{cite|perdew:prl:1996}} and <math>a_{\mathrm{SR}}=1/4</math>. 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}} [[List_of_hybrid_functionals#HSE03|HSE03]] and [[List_of_hybrid_functionals#HSE06|HSE06]] correspond to {{TAG|HFSCREEN}}=0.3 and 0.2, respectively. Note that the two limit cases of HSE are [[List_of_hybrid_functionals#PBE0|PBE0]] at <math>\mu=0</math> and PBE at <math>\mu\rightarrow\infty</math>.
 
=== HF exchange at short range and long range (error-function screening) with different mixings ===
 
The fraction of HF exchange is fixed to <math>a_{\mathrm{SR}}=1</math> at short range and is given by <math>a_{\mathrm{LR}}</math> at long-range:


<span id="ExFockSR">
:<math>
:<math>
E^{\rm HF,SR}_{\mathrm{x}}(\mu)=
E_{\mathrm{xc}}^{\mathrm{hybrid}}=E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}}
-\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>
</math>
</span>


The representation of the corresponding short-range Hartree-Fock potential in reciprocal space is given by
These functionals are selected with {{TAG|LMODELHF}}=.TRUE. The mixing <math>a_{\mathrm{LR}}</math> and screening <math>\mu</math> are controlled by the {{TAG|AEXX}} and {{TAG|HFSCREEN}} tags, respectively.
{{NB|mind|<math>a_{\mathrm{SR}}</math> is fixed to 1 and can not be changed.}}
 
This functional form has been used in the context of dielectric-dependent hybrids. An example based on PBE is given below.


<span id="VxFockSRRecip">
*[[List_of_hybrid_functionals#DD-RSH-CAM (dielectric-dependent hybrid)|DD-RSH-CAM]]:{{cite|chen2018nonempirical}}{{cite|cui2018doubly}}
:<math>
:<math>
\begin{align}
E_{\mathrm{xc}}^{\mathrm{DD-RSH-CAM}}=E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}}
V^{\mathrm{HF,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>
</math>
</span>
:where <math>a_{\mathrm{LR}}=\varepsilon^{-1}</math> is chosen as the inverse of the dielectric constant <math>\varepsilon^{-1}</math>.


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.
=== HF exchange at long range (error-function screening) ===


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>.
The HF exchange is used only at long-range (the short-range part is fully semilocal, <math>a_{\mathrm{SR}}=0</math>) and the screening is done with the error function:


<span id="RSHX"></span>
:<math>E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}}</math>


==== Error function screening with long-range Hartree-Fock exchange ====
These functionals are selected with {{TAG|LRHFCALC}}=.TRUE. The mixing <math>a_{\mathrm{LR}}</math> and screening <math>\mu</math> are controlled by the {{TAG|AEXX}} and {{TAG|HFSCREEN}} tags, respectively.
{{NB|mind|{{TAG|LRHFCALC}}{{=}}.TRUE. automatically sets {{TAG|AEXX}}{{=}}1. However, {{TAG|AEXX}} can be set to another value.}}
{{NB|important|When {{TAG|AEXX}}{{=}}1 (the default for {{TAG|LRHFCALC}}{{=}}.TRUE.), the correlation <math>E_{\mathrm{c}}^{\mathrm{SL}}</math> is not included. However, it can be included by setting {{TAG|ALDAC}}{{=}}1.0 and {{TAG|AGGAC}}{{=}}1.0.}}


Screened hybrid functionals with Hartree-Fock exchange at long range are more popular in molecular chemistry, where a proper decay of the exchange-correlation potential at long range far from the nuclei may be important. These functionals are less useful for solid-state physics, in particular for bulk solids. Examples belonging to this class of functionals are (available in VASP):
Long-range hybrid functionals are more popular in molecular chemistry, where a proper decay of the exchange-correlation potential at long range far from the nuclei may be important, and thus less useful for bulk solids. Examples belonging to this class of functionals are:


*[[List_of_hybrid_functionals#RSHXLDA|RSHXLDA]] and [[List_of_hybrid_functionals#RSHXPBE|RSHXPBE]]:{{cite|iikura:jcp:2001}}{{cite|gerber:cpl:2005}}{{cite|gerber:jcp:2007}}
*[[List_of_hybrid_functionals#RSHXLDA|RSHXLDA]] and [[List_of_hybrid_functionals#RSHXPBE|RSHXPBE]]:{{cite|iikura:jcp:2001}}{{cite|gerber:cpl:2005}}{{cite|gerber:jcp:2007}}
:In the RSHXLDA and RSHXPBE functionals the exchange operator is decomposed into short-range LDA/PBE and long-range Hartree-Fock:
:<math>
:<math>
E_{\mathrm{xc}}^{\mathrm{RSHXLDA}}= E_{\mathrm{x}}^{\mathrm{LDA,SR}}(\mu)
E_{\mathrm{xc}}^{\mathrm{RSHXLDA}} = E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + E_{\mathrm{x,SR}}^{\mathrm{LDA}}(\mu) + E_{\mathrm{c}}^{\mathrm{LDA}}
+ E_{\mathrm{x}}^{\mathrm{HF,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{LDA}},
</math>
</math>
:<math>
:<math>
E_{\mathrm{xc}}^{\mathrm{RSHXPBE}}= E_{\mathrm{x}}^{\mathrm{PBE,SR}}(\mu)
E_{\mathrm{xc}}^{\mathrm{RSHXPBE}} = E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + E_{\mathrm{x,SR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}}
+ E_{\mathrm{x}}^{\mathrm{HF,LR}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}},
</math>
</math>
:where <math>\mu</math> (set by {{TAG|HFSCREEN}}) is the parameter that defines the range separation. The use of the long-range Hartree-Fock exchange is activated with the {{TAG|LRHFCALC}} tag. This functional can only be used when the short-range density functional part is LDA or PBE. When LDA is chosen, a value of <math>\mu=0.75</math> &Aring;<sup>-1</sup> is recommended for solids.{{cite|gerber:jcp:2007}}
:When LDA is chosen, a value of <math>\mu=0.75</math> &Aring;<sup>-1</sup> is recommended for solids.{{cite|gerber:jcp:2007}}
 
=== HF exchange at short range (exponential screening) ===


==== Thomas-Fermi exponential screening with short-range Hartree-Fock exchange ====
The HF exchange is used only at short-range (the long-range part is fully semilocal, <math>a_{\mathrm{LR}}=0</math>) and the screening is done with the exponential function:
In the case of Thomas-Fermi screening (activated with the {{TAG|LTHOMAS}} tag), the Coulomb kernel is again decomposed in a short-range and a long-range part with the exponential function.{{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>E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{SR}})E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}}</math>
 
The exponential screening, also called Thomas-Fermi (TF) screening,{{cite|bylander:prb:90}}{{cite|seidl:prb:96}}{{cite|picozzi:prb:00}} is activated by setting {{TAG|LTHOMAS}}=.TRUE.. The mixing <math>a_{\mathrm{SR}}</math> and screening <math>\mu=k_{\rm TF}</math> are controlled by the {{TAG|AEXX}} and {{TAG|HFSCREEN}} tags, respectively.
{{NB|mind|{{TAG|LTHOMAS}}{{=}}.TRUE. automatically sets {{TAG|AEXX}}{{=}}1. However, {{TAG|AEXX}} can be set to another value.}}
{{NB|important|
*When {{TAG|AEXX}}{{=}}1 (the default for {{TAG|LTHOMAS}}{{=}}.TRUE.), the correlation <math>E_{\mathrm{c}}^{\mathrm{SL}}</math> is not included. However, it can be included by setting {{TAG|ALDAC}}{{=}}1.0 and {{TAG|AGGAC}}{{=}}1.0.
*This functional should be used only with LDA ({{TAG|GGA}}{{=}}CA).}}
 
The sX-LDA functional, which uses <math>a_{\mathrm{SR}}=1</math>, is probably the first hybrid using an exponential screening:
 
*[[List_of_hybrid_functionals#sX-LDA|sX-LDA]]:{{cite|bylander:prb:90}}
:<math>
:<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}+
E_{\mathrm{xc}}^{\mathrm{sX-LDA}} = E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + E_{\mathrm{x,LR}}^{\mathrm{LDA}}(\mu) + E_{\mathrm{c}}^{\mathrm{LDA}}
\left( \frac{4 \pi e^2}{|\mathbf{G}|^2} -\frac{4 \pi e^2}{|\mathbf{G}|^2 +k_{\mathrm{TF}}^2} \right),
</math>
</math>
</span>


where <math>k_{\rm TF}</math> (set by {{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:
For typical semiconductors, a Thomas-Fermi screening length <math>\mu=k_{\rm TF}</math> 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 <math>k_{\rm TF}</math> from the number of valence electrons (read from the {{FILE|POTCAR}} file) and the volume (leading to an average density <math>\bar{n}</math>) and writes the corresponding value of <math>k_{\rm TF}=\sqrt{4k_{\rm F}/\pi}</math>, where <math>k_{\rm F}=(3\pi^2\bar{n})^{1/3}</math> to the {{FILE|OUTCAR}} file (note that this value is only printed for information; it is not used during the calculation):
   Thomas-Fermi vector in A            =  2.00000
   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.  
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 not recommended.


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.
Another important detail concerns the implementation of the local LDA part in VASP. Literature [see Eqs. (3.10), (3.14), and (3.15) in Ref. {{cite|seidl:prb:96}}] suggests to use in the enhancement factor <math>F(z)</math> a position-independent variable <math>z=k_{\rm TF}/\bar{k}_{\rm F}</math> where the Fermi wave vector <math>\bar{k}_{\rm F}=(3\pi^2 \bar{n})^{1/3}</math> is calculated using the average density <math>\bar{n}</math> in the unit cell.
However, implemented in VASP is a position-dependent variable <math>z({\bf r})=k_{\rm TF}/k_{\rm F}({\bf r})</math>, where <math>k_{\rm F}({\bf r})=(3\pi^2 n({\bf r}))^{1/3}</math> is the Fermi wave vector calculated with the local density <math>n({\bf r})</math>.


== Related tags and articles ==
== Related tags and articles ==

Latest revision as of 20:59, 21 October 2025

In hybrid functionals the exchange energy is a mixture of semilocal (SL) and nonlocal Hartree-Fock (HF) types. They can be categorized into different families according to the type of semilocal approximation (LDA, GGA, or MGGA) or the treatment of the short- and long-range parts of the exchange. A rather general formula that encompasses the different families of hybrid functionals is given by

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{SR}})E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}} }[/math]

where [math]\displaystyle{ a_{\mathrm{SR}} }[/math] and [math]\displaystyle{ a_{\mathrm{LR}} }[/math] are the mixing parameters (fraction of HF exchange) at short- and long-range, respectively, and [math]\displaystyle{ \mu }[/math] is the screening parameter that determines the separation between short range (SR) and long range (LR). The SR and LR components of the full-range [math]\displaystyle{ E_{\mathrm{x}}^{\mathrm{SL}} }[/math] and [math]\displaystyle{ E_{\mathrm{x}}^{\mathrm{HF}} }[/math] exchange energies are constructed such that [math]\displaystyle{ E_{\mathrm{x}}^{\mathrm{HF}}=E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu)+E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) }[/math] and [math]\displaystyle{ E_{\mathrm{x}}^{\mathrm{SL}}=E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu)+E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) }[/math] at all values of [math]\displaystyle{ \mu }[/math].

The HF exchange energy (full-range, SR, or LR) is given by

[math]\displaystyle{ E_{\mathrm{x,(SR/LR)}}^{\rm HF}(\mu)= -\frac{1}{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}' v(\mu,|\mathbf{r}-\mathbf{r}'|) \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]

with [math]\displaystyle{ \{\psi_{n\mathbf{k}}(\mathbf{r})\} }[/math] being the set of one-electron Bloch states of the system, and [math]\displaystyle{ \{f_{n\mathbf{k}}\} }[/math] the corresponding set of (possibly fractional) occupational numbers. The sums over [math]\displaystyle{ {\bf k} }[/math] and [math]\displaystyle{ {\bf q} }[/math] run over all k-points chosen to sample the Brillouin zone, whereas the sums over [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] run over all bands at these k-points. The corresponding nonlocal HF potential is given by

[math]\displaystyle{ V_{\mathrm{x,(SR/LR)}}^{\mathrm{HF}}\left(\mu,\mathbf{r},\mathbf{r}'\right)= -\sum_{m\mathbf{q}}f_{m\mathbf{q}}v(\mu,|\mathbf{r}-\mathbf{r}'|)\psi_{m\mathbf{q}}^{*}(\mathbf{r}')\psi_{m\mathbf{q}}(\mathbf{r}) }[/math]

The non-multiplicative form of the HF potential is such that hybrid functionals are implemented within the generalized KS scheme[1]. Thus, the total energy is minimized with respect to the orbitals (instead of the electron density), which means that the HF exchange leads to a nonlocal operator as in the Hartree-Fock-Roothaan theory.

Using the decomposition of the Bloch states [math]\displaystyle{ \psi_{m\mathbf{q}} }[/math] in plane waves,

[math]\displaystyle{ \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 HF exchange potential can be written as

[math]\displaystyle{ V_{\mathrm{x,(SR/LR)}}^{\mathrm{HF}}\left(\mu,\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(\mu, \mathbf{G},\mathbf{G}'\right) e^{-i(\mathbf{k}+\mathbf{G}')\cdot\mathbf{r}'} }[/math]

where

[math]\displaystyle{ V_{\mathbf{k}}\left(\mu, \mathbf{G},\mathbf{G}'\right)= \langle \mathbf{k}+\mathbf{G} | V_{\mathrm{x,(SR/LR)}}^{\mathrm{HF}} | \mathbf{k}+\mathbf{G}'\rangle }[/math]

Types of potentials

In most hybrid functionals proposed in the literature, the interelectronic Coulomb potential [math]\displaystyle{ v(\mu,|\mathbf{r}-\mathbf{r}'|) }[/math] has one of the following forms ([math]\displaystyle{ r=|\mathbf{r}-\mathbf{r}'| }[/math]):

  • Full range (bare Colomb potential):
[math]\displaystyle{ v^{\mathrm{Coul}}(r)=\frac{1}{r} }[/math]

[math]\displaystyle{ V_{\mathbf{k}}^{\mathrm{Coul}}\left( \mathbf{G},\mathbf{G}'\right)= -\frac{4\pi}{\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]

  • Short-range with error function:
[math]\displaystyle{ v_{\mathrm{SR}}^{\mathrm{erf}}(\mu,r)=\frac{\mathrm{erfc}(\mu r)}{r} }[/math]

[math]\displaystyle{ \begin{align} V_{\mathbf{k},\mathrm{SR}}^{\mathrm{erf}}\left(\mu, \mathbf{G},\mathbf{G}'\right)= -\frac{4\pi}{\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} \left( 1-e^{-|\mathbf{k}-\mathbf{q}+\mathbf{G}''|^2 /(4\mu^2)} \right) \end{align} }[/math]

  • Short-range with exponential function:
[math]\displaystyle{ v_{\mathrm{SR}}^{\mathrm{exp}}(\mu,r)=\frac{e^{-\mu r}}{r} }[/math]

[math]\displaystyle{ V_{\mathbf{k},\mathrm{SR}}^{\mathrm{exp}}\left(\mu, \mathbf{G},\mathbf{G}'\right)= -\frac{4\pi}{\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 + \mu^2} }[/math]

The corresponding long-range potentials are given by [math]\displaystyle{ v_{\mathrm{LR}}^{\mathrm{erf/exp}}=v^{\mathrm{Coul}}-v_{\mathrm{SR}}^{\mathrm{erf/exp}} }[/math].

In VASP, these expressions are implemented within the PAW formalism.[2]

The families of hybrid functionals implemented in VASP are listed below along with examples, whose corresponding INCAR files can be found at the page list of hybrid functionals.

Important: The screening [math]\displaystyle{ \mu }[/math] (HFSCREEN tag) can be used only when the semilocal functional is PBE, PBEsol, or LDA (GGA=PE, PS, or CA, respectively). The other GGA and METAGGA functionals have no screened version available in VASP.

Families of hybrid functionals

HF exchange at full range

There is no range separation, i.e. the same fraction of HF exchange is applied at full range, [math]\displaystyle{ a_{\mathrm{SR}}=a_{\mathrm{LR}}=a }[/math] (AEXX tag):

[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]
Mind:
  • These functionals are set with LHFCALC=.TRUE. By default AEXX=0.25, but can be set to another value.
  • The semilocal part can be of the LDA, GGA or MGGA type.

These are the original and most simple forms of hybrid functionals. Two examples, PBE0 and B3LYP, are given below.

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{PBE0}}=\frac{1}{4} E_{\mathrm{x}}^{\mathrm{HF}} + \frac{3}{4} E_{\mathrm{x}}^{\mathrm{PBE}} + E_{\mathrm{c}}^{\mathrm{PBE}} }[/math]

It is based on the PBE GGA functional[4] and [math]\displaystyle{ a=1/4 }[/math].
  • B3LYP[5], well known and popular amongst quantum chemists:

[math]\displaystyle{ \begin{align} E_{\mathrm{x}}^{\mathrm{B3LYP}} &=0.8 E_{\mathrm{x}}^{\mathrm{LDA}}+ 0.2 E_{\mathrm{x}}^{\mathrm{HF}} + 0.72 (E_{\mathrm{x}}^{\mathrm{B88}}-E_{\mathrm{x}}^{\mathrm{LDA}}) + 0.19 E_{\mathrm{c}}^{\mathrm{VWN3}}+ 0.81 E_{\mathrm{c}}^{\mathrm{LYP}} \end{align} }[/math]

The exchange part consists of 80% of LDA exchange plus 20% of HF exchange, and 72% of the gradient corrections of the B88 GGA functional.[6] The correlation consists of 81% of LYP[7] correlation energy, which contains a LDA and a GGA part, and 19% of the LDA Vosko-Wilk-Nusair correlation functional III,[8] which was fitted to the correlation energy in the random phase approximation of the homogeneous electron gas.

HF exchange at short range (error-function screening)

The HF exchange is used only at short-range (the long-range part is fully semilocal, [math]\displaystyle{ a_{\mathrm{LR}}=0 }[/math]) and the screening is done with the error function:

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{SR}})E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}} }[/math]

The mixing [math]\displaystyle{ a_{\mathrm{SR}} }[/math] and screening [math]\displaystyle{ \mu }[/math] are controlled by the AEXX and HFSCREEN tags, respectively.

The most popular range-separated functional, HSE, is given below.

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{HSE}}= \frac{1}{4}E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + \frac{3}{4} E_{\mathrm{x,SR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{x,LR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}} }[/math]
They are based on the PBE GGA functional[4] and [math]\displaystyle{ a_{\mathrm{SR}}=1/4 }[/math]. It has been shown that the optimum [math]\displaystyle{ \mu }[/math], controlling the range separation is approximately 0.2-0.3 Å-1.[9][10][11][12] HSE03 and HSE06 correspond to HFSCREEN=0.3 and 0.2, respectively. Note that the two limit cases of HSE are PBE0 at [math]\displaystyle{ \mu=0 }[/math] and PBE at [math]\displaystyle{ \mu\rightarrow\infty }[/math].

HF exchange at short range and long range (error-function screening) with different mixings

The fraction of HF exchange is fixed to [math]\displaystyle{ a_{\mathrm{SR}}=1 }[/math] at short range and is given by [math]\displaystyle{ a_{\mathrm{LR}} }[/math] at long-range:

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{hybrid}}=E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}} }[/math]

These functionals are selected with LMODELHF=.TRUE. The mixing [math]\displaystyle{ a_{\mathrm{LR}} }[/math] and screening [math]\displaystyle{ \mu }[/math] are controlled by the AEXX and HFSCREEN tags, respectively.

Mind: [math]\displaystyle{ a_{\mathrm{SR}} }[/math] is fixed to 1 and can not be changed.

This functional form has been used in the context of dielectric-dependent hybrids. An example based on PBE is given below.

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{DD-RSH-CAM}}=E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}} }[/math]
where [math]\displaystyle{ a_{\mathrm{LR}}=\varepsilon^{-1} }[/math] is chosen as the inverse of the dielectric constant [math]\displaystyle{ \varepsilon^{-1} }[/math].

HF exchange at long range (error-function screening)

The HF exchange is used only at long-range (the short-range part is fully semilocal, [math]\displaystyle{ a_{\mathrm{SR}}=0 }[/math]) and the screening is done with the error function:

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{LR}} E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + (1-a_{\mathrm{LR}})E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}} }[/math]

These functionals are selected with LRHFCALC=.TRUE. The mixing [math]\displaystyle{ a_{\mathrm{LR}} }[/math] and screening [math]\displaystyle{ \mu }[/math] are controlled by the AEXX and HFSCREEN tags, respectively.

Mind: LRHFCALC=.TRUE. automatically sets AEXX=1. However, AEXX can be set to another value.
Important: When AEXX=1 (the default for LRHFCALC=.TRUE.), the correlation [math]\displaystyle{ E_{\mathrm{c}}^{\mathrm{SL}} }[/math] is not included. However, it can be included by setting ALDAC=1.0 and AGGAC=1.0.

Long-range hybrid functionals are more popular in molecular chemistry, where a proper decay of the exchange-correlation potential at long range far from the nuclei may be important, and thus less useful for bulk solids. Examples belonging to this class of functionals are:

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{RSHXLDA}} = E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + E_{\mathrm{x,SR}}^{\mathrm{LDA}}(\mu) + E_{\mathrm{c}}^{\mathrm{LDA}} }[/math]
[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{RSHXPBE}} = E_{\mathrm{x,LR}}^{\mathrm{HF}}(\mu) + E_{\mathrm{x,SR}}^{\mathrm{PBE}}(\mu) + E_{\mathrm{c}}^{\mathrm{PBE}} }[/math]
When LDA is chosen, a value of [math]\displaystyle{ \mu=0.75 }[/math] Å-1 is recommended for solids.[17]

HF exchange at short range (exponential screening)

The HF exchange is used only at short-range (the long-range part is fully semilocal, [math]\displaystyle{ a_{\mathrm{LR}}=0 }[/math]) and the screening is done with the exponential function:

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{hybrid}}=a_{\mathrm{SR}} E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + (1-a_{\mathrm{SR}})E_{\mathrm{x,SR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{x,LR}}^{\mathrm{SL}}(\mu) + E_{\mathrm{c}}^{\mathrm{SL}} }[/math]

The exponential screening, also called Thomas-Fermi (TF) screening,[18][1][19] is activated by setting LTHOMAS=.TRUE.. The mixing [math]\displaystyle{ a_{\mathrm{SR}} }[/math] and screening [math]\displaystyle{ \mu=k_{\rm TF} }[/math] are controlled by the AEXX and HFSCREEN tags, respectively.

Mind: LTHOMAS=.TRUE. automatically sets AEXX=1. However, AEXX can be set to another value.
Important:
  • When AEXX=1 (the default for LTHOMAS=.TRUE.), the correlation [math]\displaystyle{ E_{\mathrm{c}}^{\mathrm{SL}} }[/math] is not included. However, it can be included by setting ALDAC=1.0 and AGGAC=1.0.
  • This functional should be used only with LDA (GGA=CA).

The sX-LDA functional, which uses [math]\displaystyle{ a_{\mathrm{SR}}=1 }[/math], is probably the first hybrid using an exponential screening:

[math]\displaystyle{ E_{\mathrm{xc}}^{\mathrm{sX-LDA}} = E_{\mathrm{x,SR}}^{\mathrm{HF}}(\mu) + E_{\mathrm{x,LR}}^{\mathrm{LDA}}(\mu) + E_{\mathrm{c}}^{\mathrm{LDA}} }[/math]

For typical semiconductors, a Thomas-Fermi screening length [math]\displaystyle{ \mu=k_{\rm TF} }[/math] of about 1.8 Å-1 yields reasonable band gaps. In principle, however, the Thomas-Fermi screening length depends on the valence-electron density. VASP determines [math]\displaystyle{ k_{\rm TF} }[/math] from the number of valence electrons (read from the POTCAR file) and the volume (leading to an average density [math]\displaystyle{ \bar{n} }[/math]) and writes the corresponding value of [math]\displaystyle{ k_{\rm TF}=\sqrt{4k_{\rm F}/\pi} }[/math], where [math]\displaystyle{ k_{\rm F}=(3\pi^2\bar{n})^{1/3} }[/math] to the OUTCAR file (note that this value is only printed for information; it is not used during the calculation): 

 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 not recommended.

Another important detail concerns the implementation of the local LDA part in VASP. Literature [see Eqs. (3.10), (3.14), and (3.15) in Ref. [1]] suggests to use in the enhancement factor [math]\displaystyle{ F(z) }[/math] a position-independent variable [math]\displaystyle{ z=k_{\rm TF}/\bar{k}_{\rm F} }[/math] where the Fermi wave vector [math]\displaystyle{ \bar{k}_{\rm F}=(3\pi^2 \bar{n})^{1/3} }[/math] is calculated using the average density [math]\displaystyle{ \bar{n} }[/math] in the unit cell. However, implemented in VASP is a position-dependent variable [math]\displaystyle{ z({\bf r})=k_{\rm TF}/k_{\rm F}({\bf r}) }[/math], where [math]\displaystyle{ k_{\rm F}({\bf r})=(3\pi^2 n({\bf r}))^{1/3} }[/math] is the Fermi wave vector calculated with the local density [math]\displaystyle{ n({\bf r}) }[/math].

Related tags and articles

AEXX, ALDAX, ALDAC, AGGAX, AGGAC, AMGGAX, AMGGAC, LHFCALC, LMODELHF, LTHOMAS, LRHFCALC, HFSCREEN, List of hybrid functionals, Downsampling of the Hartree-Fock operator, Coulomb singularity

References

  1. ↑ a b c A. Seidl, A. Görling, P. Vogl, J.A. Majewski, and M. Levy, Phys. Rev. B 53, 3764 (1996).
  2. ↑ J. Paier, R. Hirschl, M. Marsman, and G. Kresse, J. Chem. Phys. 122, 234102 (2005).
  3. ↑ J. P. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982 (1996).
  4. ↑ a b J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett., 77, 3865 (1996).
  5. ↑ P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, J. Phys. Chem. 98, 11623 (1994).
  6. ↑ A. D. Becke, Density-functional exchange-energy approximation with correct asymptotic behavior, Phys. Rev. A 38, 3098 (1988).
  7. ↑ C. Lee, W. Yang, and R. G. Parr, Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density, Phys. Rev. B 37, 785 (1988).
  8. ↑ S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980).
  9. ↑ a b J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).
  10. ↑ a b J. Heyd and G. E. Scuseria, J. Chem. Phys. 121, 1187 (2004).
  11. ↑ a b J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 124, 219906 (2006).
  12. ↑ a b A. V. Krukau , O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006).
  13. ↑ 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).
  14. ↑ 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).
  15. ↑ H. Iikura, T. Tsuneda, T. Yanai, and K. Hirao, A long-range correction scheme for generalized-gradient-approximation exchange functionals, J. Chem. Phys. 115, 3540 (2001).
  16. ↑ I. C. Gerber and J. G. Ángyán, Hybrid functional with separated range, Chem. Phys. Lett. 415, 100 (2005).
  17. ↑ a b I. C. Gerber, J. G. Ángyán, M. Marsman, and G. Kresse, Range separated hybrid density functional with long-range Hartree-Fock exchange applied to solids, J. Chem. Phys. 127, 054101 (2007).
  18. ↑ a b D. M. Bylander and L. Kleinman, Phys. Rev. B 41, 7868 (1990).
  19. ↑ S. Picozzi, A. Continenza, R. Asahi, W. Mannstadt, A.J. Freeman, W. Wolf, E. Wimmer, and C.B. Geller, Phys. Rev. B 61, 4677 (2000).
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