The Kohn-Sham DFT approach to the solution of the many-body Schrödinger equation has not required any approximations thus far. However, the exchange-correlation energy, in Equation , is defined as the difference between the true functional and the remaining terms. As the true form of F is unknown, we must use an approximation for .
A number of possible approximations may be made. The simplest, known as the Local Density Approximation (LDA), defines as
where is the exchange-correlation energy per unit volume of a homogeneous electron gas of density . The values of were calculated by Ceperly and Alder using Quantum Monte Carlo techniques  and parameterised by Perdew and Zunger . Although a gross approximation, LDA has been found to give good results in a wide range of solid state systems . Generalised Gradient Approximations (GGAs) add a term in the gradient of the electron density to the parameterisation of . Although GGAs do not offer a consistent improvement over LDA in all types of system, they have been shown to improve on the LDA for calculations of molecular structures  and in representing weak inter-molecular bonds . For this reason the GGA due to Perdew and Wang  has been used in this thesis. In cases where the external potential is spin dependent, an approximation must be made to which depends on both the total electronic density and the polarisation , where and are the densities of spin up and spin down electrons respectively. We have used the spin dependent GGA also due to Perdew and Wang  for the spin-dependent calculations presented in this thesis.
The reasons for the success of these approximations are not well understood, although this may be partially attributed to the fact that both obey the sum rule for the exchange-correlation hole in the electron density . Certainly, LDA and GGAs give rise to a systematic overestimation of the electronic binding energy. However, differences in energies may be accurately computed and it is these which are important for the estimation of physical and chemical properties.