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 three main 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 the exchange-correlation potential

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

Meta-GGA

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

leading to a non-multiplicative exchange-correlation potential:

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

the functionals of the generalized-gradient approximation (GGA) and

 that, in addition to the electron density ${\displaystyle n}$ and the gradient ${\displaystyle \nabla n}$, depend also on 

• the kinetic-energy density ${\displaystyle \tau}$, and/or
• the Laplacian of the electron density ${\displaystyle \nabla^{2}n}$.

Thus, the exchange-correlation energy can be written as

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

which leads to the exchange-correlation potential having the form

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

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. Note that as in most other codes, meta-GGAs are implemented in VASP (see METAGGA) within the generalized KS scheme^[4].

How to

A meta-GGA functional can be used by specifying

• the METAGGA tag, or
• XC tag

in the INCAR file.

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