Meta-Generalised Gradient Approximation

An active line of research into functionals that go beyond the GGA at the moment is the meta-GGA (MGGA) form [65,66]. MGGAs include additional semi-local information beyond the first-order density gradient contained in the GGA, such as higher order density gradients, or more popular is the inclusion of the kinetic energy density $\tau({\bf r})$ which involves derivatives of the occupied Kohn-Sham orbitals,

$$\tau({\bf r}) = \frac{1}{2} \sum_{i}^{occ} \vert \nabla \psi_i({\bf r}) \vert^2 \, .$$ (2.46)

The integrated $\tau({\bf r})$ is equivalent to the usual non-interacting kinetic energy $T_s[n({\bf r})]$, given by relation (1.53), i.e. $T_s[n({\bf r})] = \int \tau({\bf r}) \, d{\bf r}$. The MGGA may be written with the general form,

$$E_{\hbox{{XC}}}^{\hbox{{MGGA}}}[n({\bf r})] = \int f[n({\bf r}),\... ...({\bf r}), \tau({\bf r}),\mu({\bf r}), \ldots \gamma({\bf r})] \, d{\bf r} \, ,$$ (2.47)

where $\mu({\bf r}), \ldots \gamma({\bf r})$ are other possible semi-local quantities (i.e. defined locally at ${\bf r}$) that could be used in the construction of MGGAs.

There are several MGGA forms now in existence  [67,68,69,70,71,72,73,74,75,76] and some improvement has been obtained over the GGA in a limited number of tests [77]. However a few cautionary words should be said about the MGGA. At present, MGGA calculations for solids are performed inconsistently because they resort to using GGA orbitals and densities to evaluate $E_{\hbox{{XC}}}^{\hbox{{MGGA}}}[n({\bf r})]$, since the orbital dependence does not permit an easy evaluation of a multiplicative exchange-correlation potential $v_{\hbox {{XC}}}({\bf r})$. Therefore properties are only calculated at experimental structures. To achieve self-consistency using a multiplicative potential, computationally expensive methods such as the optimised effective potential (OEP) [78,79] must be invoked, however this has yet to be implemented - indeed it may eventually prove too costly for practical computations. Another point to highlight is that all MGGA forms are constructed using experimental molecular data to define the form. This will have the effect of introducing an element of bias into the character of the functional. This issue is investigated in Chapter $3$ with regard to GGA functionals.