Introduction

The reason for developing functionals is so that the properties of matter can be predicted more accurately, hence the determination of real atomic, molecular and solid state quantities represents an important and necessary test of any new functional. However, to obtain a true assessment of an exchange-correlation functional that is implemented within a plane-wave formalism, it is essential to use consistent pseudopotentials in the calculations - as demonstrated in Sec. 3.3.6. The generation of WDA pseudopotentials may be complicated by the need to include shell-partitioning techniques [85,147], whereby the core intra-shell exchange-correlation interactions are accounted for by the WDA, while the inter-shell regions are described with the LDA. However as yet, no attempt has been made to do this. While the self-consistent WDA calculations performed in the previous chapter demonstrated general trends, they are by no means conclusive results. So an extensive study of solid state systems will provide only limited insight into the WDA whilst inconsistent pseudopotentials are used. Although the development of WDA pseudopotentials is of utmost importance to the WDA method as a whole, they are not the focus of this thesis and so will not be discussed further.

This chapter focuses on analysing the WDA in various inhomogeneous electron gas systems. The reason for choosing such model systems is because the electron gas does not contain ions, so the pseudopotential issue is automatically resolved. Also, the density can be completely controlled, so exchange-correlation interactions can be explored in a range of pre-determined density regimes. There exist a plethora of electron gas systems that can be examined quite easily, however the particular examples chosen here were stimulated by a recent quantum Monte Carlo study performed by Nekovee  et al. [161], which applied a strong one-dimensional cosinusoidal potential to a three-dimensional electron gas. These systems will be re-examined using the WDA, along with other extensions including the quasi-$2$D limit of the electron gas, and strong isotropic confinement in three dimensions.