Density Functional Theory - Approximations (exchange-correlation Functionals)

Approximations (exchange-correlation Functionals)

The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas. However, approximations exist which permit the calculation of certain physical quantities quite accurately. In physics the most widely used approximation is the local-density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:

The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron spin:

Highly accurate formulae for the exchange-correlation energy density have been constructed from quantum Monte Carlo simulations of jellium.

Generalized gradient approximations (GGA) are still local but also take into account the gradient of the density at the same coordinate:

Using the latter (GGA) very good results for molecular geometries and ground-state energies have been achieved.

Potentially more accurate than the GGA functionals are the meta-GGA functionals, a natural development after the GGA (generalized gradient approximation). Meta-GGA DFT functional in its original form includes the second derivative of the electron density (the Laplacian). Meta-GGA includes only the density and its first derivative in the exchange-correlation potential.

The functionals of this type are: B98, TPSS and M06. These functionals include a further term in the expansion, depending on the density, the gradient of the density and the Laplacian (second derivative) of the density.

Difficulties in expressing the exchange part of the energy can be relieved by including a component of the exact exchange energy calculated from Hartree–Fock theory. Functionals of this type are known as hybrid functionals.