Category:Exchange-correlation functionals: Difference between revisions

Theoretical Background

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

[math]\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} }[/math]

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. The orbitals [math]\displaystyle{ \psi_{i} }[/math] and the electron density [math]\displaystyle{ \rho=\sum_{i}\left\vert\psi_{i}\right\vert^{2} }[/math] that are used to evaluate [math]\displaystyle{ E_{\rm tot} }[/math] are obtained by solving self-consistently the KS equations

[math]\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}) }[/math]

The only terms in [math]\displaystyle{ E_{\rm tot} }[/math] and in the KS equations that are not known exactly are the exchange-correlation energy functional [math]\displaystyle{ E_{\rm xc} }[/math] and potential [math]\displaystyle{ v_{\rm xc}=\delta E_{\rm xc}/\delta\rho }[/math]. Therefore, the accuracy of the calculated properties depends mainly on the approximations used for [math]\displaystyle{ E_{\rm xc} }[/math] and [math]\displaystyle{ v_{\rm xc}=\delta E_{\rm xc}/\delta\rho }[/math]. 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 approxmations can eb found at the following pages.

• Hybrid functionals: Hartree-Fock and HF/DFT hybrid functionals.
• L(S)DA (on-site interactions): LDAUTYPE.
• van der Waals:
  • 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.

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.