Category:Semilocal functionals

The local and semilocal exchange-correlation functionals depend locally on quantities like the electron density ${\displaystyle n}$ or the kinetic-energy density ${\displaystyle \tau}$. Most of them can be classified into one of the three subcategories listed below, depending on the variables on which ${\displaystyle E_{\mathrm{xc}}}$ depends.

Local density approximation (LDA)

The LDA functionals are purely local in the sense that they depend solely on ${\displaystyle n}$:

${\displaystyle E_{\mathrm{xc}}^{\mathrm{LDA}}=\int\epsilon_{\mathrm{xc}}^{\mathrm{LDA}}(n)d^{3}r }$

with a corresponding exchange-correlation potential calculated as

${\displaystyle v_{\mathrm{xc}}^{\mathrm{LDA}} = \frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{LDA}}}{\partial n} }$

The most common LDA functionals, e.g,. GGA=CA ^[1][2][3], provide the (nearly) exact exchange-correlation energy for the homogeneous electron gas.

Generalized-gradient approximation (GGA)

In the GGA, there is an additional dependency on the gradient of ${\displaystyle n}$:

${\displaystyle E_{\mathrm{xc}}^{\mathrm{GGA}}=\int\epsilon_{\mathrm{xc}}^{\mathrm{GGA}}(n,\nabla n)d^{3}r }$

leading to an additional term in the potential compared to LDA:

${\displaystyle v_{\mathrm{xc}}^{\mathrm{GGA}} = \frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{GGA}}}{\partial n} - \nabla\cdot\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{GGA}}}{\partial\nabla n} }$

The GGA that has been the most commonly used in solid-state physics is PBE.^[4]

Meta-GGA

Compared to the GGAs, the meta-GGA functionals depend additionally on the kinetic-energy density ${\displaystyle \tau}$ and/or the Laplacian of the electron density ${\displaystyle \nabla^{2}n}$:

${\displaystyle E_{\mathrm{xc}}^{\mathrm{MGGA}}=\int\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}(n,\nabla n,\nabla^{2}n,\tau)d^{3}r }$

leading to

${\displaystyle \hat{v}_{\mathrm{xc}}\psi_{i} = \frac{\delta E_{\mathrm{xc}}^{\mathrm{MGGA}}}{\delta\psi_{i}^{*}} = \left(\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}}{\partial n} - \nabla\cdot\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}}{\partial\nabla n} + \nabla^2\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}}{\partial\nabla^2 n} \right)\psi_{i} - \frac{1}{2}\nabla\cdot\left(\frac{\partial\epsilon_{\mathrm{xc}}^{\mathrm{MGGA}}}{\partial \tau} \nabla\psi_{i}\right) }$

where the last term is of non-multiplicative nature and arises due to the dependency of the functional on ${\displaystyle \tau}$.^[5][6] With such a non-multiplicative potential the method lies outside the traditional Kohn-Sham scheme,^[7] but rather belongs to the generalized Kohn-Sham scheme.^[8]

Although meta-GGAs are slightly more expensive than GGAs, they are still fast to evaluate and appropriate for very large systems. Furthermore, meta-GGAs can be more accurate than GGAs and more broadly applicable.

How to

• LDA and GGA: GGA
• Meta-GGA: METAGGA

How to do a band-structure calculation using meta-GGA functionals.