HFSCREEN and LTHOMAS

 HFSCREEN = real    (truncate the long range Fock potential)

In combination with PBE potentials, attributing a value to HFSCREEN will switch from the PBE0 functional (in case LHFCALC=.TRUE.) to the closely related HSE03 or HSE06 functional [65,66,67].

The HSE03 and HSE06 functional replaces the slowly decaying long-ranged part of the Fock exchange, by the corresponding density functional counterpart. The resulting expression for the exchange-correlation energy is given by:

$$E_{\mathrm{xc}}^{\mathrm{HSE}}= \frac{1}{4}~E_{\mathrm{x}}^{... ...rm{x}}^{\mathrm{PBE,LR}}(\mu)~+~E_{\mathrm{c}}^{\mathrm{PBE}}.$$ (25)

As can be seen above, the separation of the electron-electron interaction into a short- and long-ranged part, labeled SR and LR respectively, is realized only in the exchange interactions. Electronic correlation is represented by the corresponding part of the PBE density functional.

The decomposition of the Coulomb kernel is obtained using the following construction ($\mu\equiv$ HFSCREEN):

$$\frac{1}{r}=S_{\mu}(r)+L_{\mu}(r)=\frac{{\rm erfc}(\mu r)}{r}+\frac{{\rm erf} (\mu r)}{r}$$ (26)

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

Note: It has been shown [65] that the optimum $\mu$, controlling the range separation is approximatively $0.2-0.3$ Å$^{-1}$. To conform with the HSE06 functional you need to select ( HFSCREEN=0.2) [65,66,67].

Using the decomposed Coulomb kernel and Eq. (6.13), one straightforwardly obtains:

$$E^{\rm SR}_{\rm x}(\mu)= -\frac{e^2}{2}\sum_{{\bf k}n,{\bf q... ...}({\bf r}') \phi_{{\bf k}n}({\bf r}')\phi_{{\bf q}m}({\bf r}).$$ (27)

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

$$V^{\rm SR}_{\bf k}\left( {\bf G},{\bf G}'\right) = \langle {\bf k}+{\bf G} \vert V^{\rm SR}_x [\mu] \vert {\bf k}+{\bf G}'\rangle$$

$$= -\frac{4\pi e^2}{\Omega} \sum_{m{\bf q}}f_{{\bf q}m}\sum_{{\bf G}... ...t^2} \times \left( 1-e^{-\vert{\bf k}-{\bf q}+{\bf G}''\vert^2 /4\mu^2} \right)$$ (28)

Clearly, the only difference to the reciprocal space representation of the complete (undecomposed) Fock exchange potential, given by Eq. (6.17), is the second factor in the summand in Eq. (6.22), representing the complementary error function in reciprocal space.

The short-ranged PBE exchange energy and potential, and their long-ranged counterparts, are arrived at using the same decomposition [Eq. (6.20)], in accordance with Heyd et al. [65] It is easily seen from Eq. (6.20) that the long-range term becomes zero for $\mu=0$, and the short-range contribution then equals the full Coulomb operator, whereas for $\mu \to \infty$ it is the other way around. Consequently, the two limiting cases of the HSE03/HSE06 functional [see Eq. (6.19)] are a true PBE0 functional for $\mu=0$, and a pure PBE calculation for $\mu \to \infty$.

LTHOMAS

If the flag LTHOMAS is set, a similar decomposition of the exchange functional into a long range and a short range part is used. This time it is more convenient to write the decomposition in reciprocal space:

$$\frac{4 \pi e^2}{\vert{\bf G}\vert^2}=S_{\mu}(\vert{\bf G}\v... ...t^2} -\frac{4 \pi e^2}{\vert{\bf G}\vert^2 +k_{TF}^2} \right),$$ (29)

where $q_{TF}$ is the Thomas Fermi screening length. Here, HFSCREEN is used to specify this parameter $q_{TF}$. VASP calculates this density dependent parameter and writes it to the OUTCAR file in the line:

  Thomas-Fermi vector in A             =   2.00000

Note however that the parameter depends on the electrons counted as valence electrons: For the determination of the value written to the OUTCAR file, VASP simply counts all electrons in the POTCAR file as valence electrons, whereas literature suggests that semi-core states and $d$-states should not be included in the determination of the Thomas Fermi screening length ( HFSCREEN can be manually set to any value). Details can be found in literature [69,70,71]. An important detail concerns that implementation of the density functional part in the screened exchange case. Literature suggests that a global enhancement factor $z$ (see Equ. (3.15) in Ref. [71]) should be used), whereas VASP implements a local density dependent enhancement factor $z= k_{TF}/\bar k$, where $\bar k$ is the Fermi wave vector corresponding to the local density (and not the average density as suggested in Ref. [71]). This is in the spirit of the local density approximation.

Note: A comprehensive study of the performance of the HSE03/HSE06 functional as compared to the PBE and PBE0 functionals can be found in Ref. [68].