Exchange-Correlation Functionals

KS Theory allows the kinetic energy to be computed to a chemical accuracy, so `all' that remains is an accurate form for the exchange-correlation energy functional, $E_{xc}[\rho]$. The exact form is obviously unknown, and with the accuracy of DFT determined mainly by the functional used, it is no surprise that finding new functionals is the focus of much modern research.

The simplest form for $E_{xc}[\rho]$ is the Dirac exchange term, forming Hartree-Fock-Slater (HFS) theory [76]. Slater also pointed out that E_x^D30 systematically underestimated the exchange energy by about 10% and proposed multiplying the Dirac coefficient by 1.1, resulting in the semi-empirical $X_{\alpha}$ theory. While the HFS total energies are not as accurate as HF theory, for thermochemistry, HFS theory is a big improvement over HF theory [100]. This is due to a convenient cancellation of errors arising from HFS systematically underestimating the total energy.

The natural extension of HFS theory is to add the VWN functional for the correlation energy, thus using the uniform electron gas to model exchange and correlation effects. The resulting theory is termed the Local Spin Density approximation (LSDA). The LSDA is an improvement over HFS theory, yet VWN makes no account for the correction of the kinetic energy, $T_{c}[\rho]$, where
$$\begin{align}T_{c}[\rho] = T[\rho]-T_{s}[\rho] \end{align}$$
which can reach the magnitude of the correlation energy itself [77]. As VWN usually overestimates E_c by a factor of two, inclusion of T_c could have a dramatic effect on the accuracy of the VWN functional.