MetaGGA

Meta-GGA, or meta-generalized gradient approximation, is a class of exchange-correlation functionals used in density functional theory. Meta-GGAs extend GGAs by incorporating the kinetic energy density, τ(r), in addition to the electron density and its gradient. This extra ingredient allows a more detailed description of electron localization and bonding, delivering higher accuracy for many systems while remaining less expensive than hybrids that mix exact exchange.

Meta-GGAs are designed to satisfy exact constraints and known norms and are often nonempirical or semi-empirical. Notable examples include TPSS and revTPSS, as well as the strongly constrained and appropriately normed (SCAN) functional and its efficient variant r2SCAN. Other families include the Minnesota meta-GGAs such as M06-L, which are more empirical.

The class generally provides improved thermochemistry, noncovalent interactions, and barrier heights relative to GGAs. They are widely used in molecular and solid-state calculations. However, meta-GGAs are more computationally demanding than GGAs and require careful numerical integration grids. They can exhibit residual self-interaction errors and, by themselves, do not capture dispersion forces; dispersion corrections or nonlocal terms are often added (e.g., D3/D4 or VV10).

In summary, meta-GGAs occupy an intermediate tier in DFT, offering improved accuracy over GGAs with modest cost increases, and they remain popular for a broad range of applications in chemistry and materials science.