Category:Exchange-correlation functionals: Difference between revisions

In the Kohn-Sham (KS) formulation of density-functional theory (DFT)^[1][2], the total energy is given by

${\displaystyle E_{\rm tot}^{\rm KS-DFT} = -\frac{1}{2}\sum_{i}\int\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r})d^{3}r - \sum_{A}\int\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n({\bf r})d^{3}r + \frac{1}{2}\int\int\frac{n({\bf r})n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert}d^{3}rd^{3}r' + E_{\rm xc} + \frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert} }$

where the terms on the right-hand side represent the non-interacting kinetic energy of the electrons, the electrons-nuclei attraction energy, the classical Coulomb electron-electron repulsive energy, the exchange-correlation energy, and the nuclei-nuclei repulsion energy, respectively. The KS orbitals ${\displaystyle \psi_{i}}$ and the electronic density ${\displaystyle n=\sum_{i}\left\vert\psi_{i}\right\vert^{2}}$ that are used to evaluate ${\displaystyle E_{\rm tot}^{\rm KS-DFT}}$ are obtained by solving self-consistently the KS equations

${\displaystyle \left(-\frac{1}{2}\nabla^{2} -\sum_{A}\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert} + \int\frac{n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert}d^{3}r' + v_{\rm xc}({\bf r})\right)\psi_{i}({\bf r}) = \epsilon_{i}\psi_{i}({\bf r}). }$

The only terms in ${\displaystyle E_{\rm tot}^{\rm KS-DFT}}$ and in the KS equations that are not known exactly are the exchange-correlation energy functional ${\displaystyle E_{\rm xc}}$ and potential ${\displaystyle v_{\rm xc}=\delta E_{\rm xc}/\delta n}$. Therefore, the accuracy of the calculated properties depends strongly on the approximations used for ${\displaystyle E_{\rm xc}}$ and ${\displaystyle v_{\rm xc}}$.

Several hundreds of approximations for the exchange and correlation have been proposed^[3]. They can be classified into families: the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid. There is also the possibility to include van der Waals corrections or an on-site Coulomb repulsion using DFT+U on top of another functional. More details on the different types of approximations available in VASP and how to use them can be found in the pages and subcategories listed below.

How to

• Semilocal functionals:
  • LDA and GGA: GGA
  • Meta-GGA: METAGGA
• Hybrids: LHFCALC, AEXX, HFSCREEN and list of hybrid functionals
• DFT+U: LDAU and LDAUTYPE
• Atom-pairwise and many-body methods for van der Waals interactions (selected with the IVDW tag):
  • Methods from Grimme et al.:
    • DFT-D2^[4]
    • DFT-D3^[5][6]
    • DFT-D4^[7] (available as of VASP.6.2 as external package)
  • Methods from Tkatchenko, Scheffler et al.:
    • Tkatchenko-Scheffler method^[8]
    • Tkatchenko-Scheffler method with iterative Hirshfeld partitioning^[9][10]
    • Self-consistent screening in Tkatchenko-Scheffler method^[11]
    • Many-body dispersion energy^[11][12]
    • Many-body_dispersion_energy_with_fractionally_ionic_model_for_polarizability^[13][14]
  • dDsC dispersion correction^[15][16]
  • DFT-ulg^[17]
• Nonlocal vdW-DF functionals for van der Waals interactions: LUSE_VDW and IVDW_NL

References