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 [13] and parameterised by
Perdew and Zunger [14]. Although a gross approximation, LDA
has been found to give good results in a wide range of solid state
systems [15]. 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
[16] and in representing weak inter-molecular bonds
[17]. For this reason the GGA due to Perdew and Wang
[18] 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 [18] 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 [19]. 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.