Category:Exchange-correlation functionals: Difference between revisions

Theoretical Background

In the KS formulation of DFT^[1][2], the total energy is given by

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

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 ${\displaystyle \rho=\sum_{i}\left\vert\psi_{i}\right\vert^{2}}$ that are used to evaluate ${\displaystyle E_{\rm tot}}$ are obtained by solving self-consistently the KS equations

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

The only terms in ${\displaystyle E_{\rm tot}}$ and in the KS equations that are not known exactly are the exchange-correlation energy functional ${\displaystyle E_{\rm xc}}$ and potential ${\displaystyle v_{\rm xc}=\delta E_{\rm xc}/\delta\rho}$. 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 can be found in the subcategories listed below.

How to

• GGA and LDA: GGA.
• Meta-GGA: METAGGA.
• Hybrid functionals: Specific hybrid functionals.
• L(S)DA (on-site interactions): LDAUTYPE.
• van der Waals:
  • Main tag for van der Waals algorithm: IVDW
  • DFT-D2 method.
  • DFT-D3 method.
  • DDsC dispersion correction.
  • Many-body dispersion energy.
  • Tkatchenko-Scheffler method.
  • Tkatchenko-Scheffler method with iterative Hirshfeld partitioning.
  • Self-consistent screening in Tkatchenko-Scheffler method.
  • VdW-DF functional of Langreth and Lundqvist et al.