Hybrid functionals: formalism

Unscreened hybrid functionals

In hybrid exchange-correlation functionals, the exchange component consists of a mixing of GGA (or meta-GGA) and Hartree-Fock exchange:

[math]\displaystyle{ 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]\displaystyle{ \alpha }[/math] is the mixing parameter (AEXX) that is typically in the range 0.1-0.5. Two examples of hybrid functionals, PBE0 and B3LYP, are given below.

• PBE0:

[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]

where [math]\displaystyle{ E_{x}^{\rm PBE} }[/math] and [math]\displaystyle{ E_{c}^{\rm PBE} }[/math] denote the exchange and correlation parts of the PBE density functional, respectively.

• B3LYP, 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 \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]

where [math]\displaystyle{ E_{x}^{\rm B3LYP} }[/math] and [math]\displaystyle{ E_{c}^{\rm B3LYP} }[/math] are the B3LYP exchange and correlation energy contributions, respectively. [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ E_{x} }[/math], can be written as

[math]\displaystyle{ 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]

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 [math]\displaystyle{ {\bf k} }[/math] points chosen to sample the Brillouin zone (BZ), whereas the sums over [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] run over all bands at these [math]\displaystyle{ {\bf k} }[/math] points. The corresponding non-local Hartree-Fock potential is given by

[math]\displaystyle{ 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]

where [math]\displaystyle{ u_{m\mathbf{q}}(\mathbf{r}) }[/math] is the cell periodic part of the Bloch state, [math]\displaystyle{ \psi_{n\mathbf{q}}(\mathbf{r}) }[/math], at [math]\displaystyle{ {\bf k} }[/math] point, [math]\displaystyle{ {\bf q} }[/math], with band index m. 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 Hartree-Fock exchange potential may be written as

[math]\displaystyle{ 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

[math]\displaystyle{ 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]

is the representation of the Hartree-Fock potential in reciprocal space. In VASP, these expressions are implemented within the PAW formalism.

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 GGA:

[math]\displaystyle{ 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]\displaystyle{ \mu }[/math] is the screening parameter (HFSCREEN) that determines the range separation. The most popular range-separated functional, HSE, is given below.

• HSE:

In the range-separated HSE03 and HSE06 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]\displaystyle{ 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:

[math]\displaystyle{ \frac{1}{r}=S_{\mu}(r)+L_{\mu}(r)=\frac{\mathrm{erfc}(\mu r)}{r}+\frac{\mathrm{erf}(\mu r)}{r}, }[/math]

where [math]\displaystyle{ r =|{\bf r}-{\bf r}'| }[/math], and [math]\displaystyle{ \mu }[/math] (=HFSCREEN) is the parameter that defines the range separation, and is related to a characteristic distance, [math]\displaystyle{ 2/\mu }[/math], at which the short-range interactions become negligible.

Note: It has been shown that the optimum [math]\displaystyle{ \mu }[/math], controlling the range separation is approximately 0.2-0.3 Å^-1. To select the HSE06 functional you need to select (HFSCREEN=0.2).

Using the decomposed Coulomb kernel one may straightforwardly rewrite the non-local Hartree-Fock exhange energy:

[math]\displaystyle{ 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]

The representation of the corresponding short-range Hartree-Fock potential in reciprocal space is given by

[math]\displaystyle{ \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]

The only difference to the reciprocal space representation of the complete 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 decomposition, in accordance with Heyd et al. It is easily seen that the long-range term in the decomposed Coulomb kernel becomes zero for [math]\displaystyle{ \mu=0 }[/math], and the short-range contribution then equals the full Coulomb operator, whereas for [math]\displaystyle{ \mu\rightarrow\infty }[/math] it is the other way around. Consequently, the two limiting cases of the HSE functional are a true PBE0 functional for [math]\displaystyle{ \mu=0 }[/math], and a pure PBE calculation for [math]\displaystyle{ \mu\rightarrow\infty }[/math].

Thomas-Fermi screening

In the case of Thomas-Fermi screening, the Coulomb kernel is again decomposed in a short-range and a long-range part. This decomposition can be conveniently written in reciprocal space:

[math]\displaystyle{ \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]

where [math]\displaystyle{ k_{\rm TF} }[/math] (=HFSCREEN) is the Thomas-Fermi screening length. For typical semiconductors, a Thomas-Fermi screening length 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 this parameter from the number of valence electrons (read from the POTCAR file) and the volume and writes the corresponding value to the 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]\displaystyle{ z }[/math] (see Eq. 3.15) should be used, whereas VASP implements a local-density-dependent enhancement factor [math]\displaystyle{ z=k_{\rm TF}/k }[/math] , where [math]\displaystyle{ k }[/math] is the Fermi wave vector corresponding to the local density (and not the average density as suggested Seidl et al.. The VASP implementation is in the spirit of the local density approximation.

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