Category:Exchange-correlation functionals

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

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

where

[math]\displaystyle{ T_{\rm s}[\{\psi_{i}\}]=-\frac{1}{2}\sum_{i=1}^{N}\int \psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r})d^{3}r }[/math]

is the non-interacting kinetic energy of the electrons,

[math]\displaystyle{ U_{\rm H}[\rho] = \int v_{\rm ext}({\bf r})\rho({\bf r})d^{3}r }[/math]

is the Classical Coulomb Hartree term,

[math]\displaystyle{ V_{\rm en}[\rho] = \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', }[/math]

is the electrons-nuclei attraction energy and

[math]\displaystyle{ V_{\rm nn} = \frac{1}{2}\sum_{A\ne B} \frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert} }[/math]

is the nuclei-nuclei repulsion energy.

Theoretical Background

• 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

• Hybrid functionals: Specific hybrid functionals.
• Meta GGA's: METAGGA.
• 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.