Category:Exchange-correlation functionals: Difference between revisions

In the KS formulation of 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 orbitals ${\displaystyle \psi_{i}}$ and the electron 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 mainly 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 several types, like the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid. Functionals that include van der Waals corrections have also been proposed. 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 and list of hybrid functionals
• DFT+U: LDAU and LDAUTYPE
• Atom-pairwise methods for van der Waals interactions (selected with the IVDW tag):
  • Methods from Grimme et al.:
    • DFT-D2^[4]
    • DFT-D3
    • DFT-D4 (available as of VASP.6.2 as external package)
  • Methods from Tkatchenko, Scheffler et al.:
    • Tkatchenko-Scheffler method
    • Tkatchenko-Scheffler method with iterative Hirshfeld partitioning
    • Self-consistent screening in Tkatchenko-Scheffler method
    • Many-body dispersion energy
    • Many-body_dispersion_energy_with_fractionally_ionic_model_for_polarizability
  • dDsC dispersion correction
  • DFT-ulg
• Nonlocal vdW-DF functionals for van der Waals interactions: LUSE_VDW

References