The inhomogeneous electron gas was chosen as the basis of this chapter because it allows a controlled environment to examine the properties of exchange-correlation functionals, that is also free from the pseudopotential approximation. The work on the one-dimensional cosine-wave electron gas in Sec. 6.2 demonstrated the closely similarities between the WDA and the VMC method, which is highly promising. It is therefore hoped that the success of this study will stimulate further quantum Monte Carlo work so that the WDA can be accurately tested in a broad variety of density environments. These comparisons can continue without the use of consistent pseudopotentials, so long as the same, or very similar, reference densities are used.
The quasi-D system studied in Sec. 6.3 demonstrated the divergent trend of the LDA and GGA total energies and energy densities with respect to the WDA, which results from the inadequacy of a local/semi-local description of the exchange-correlation hole. This demonstrates the possibility of successfully applying the WDA to systems that exhibit low dimensional character such as semiconductor quantum well structures which have important device applications .
Finally, in the case of isotropic confinement examined in Sec. 6.4, the WDA yielded similar results as the LDA and GGA for the system containing a large number of electrons. However when there are few electrons in the system, the LDA and GGA underestimate total energies quite substantially in comparison to the WDA. This was discussed in the context of self-interaction effects, but is ultimately a direct consequence of the non-locality of the exchange-correlation hole, which was demonstrated by the WDA. An interesting application of the WDA would be to the determination of molecular reaction barriers and the calculation of corresponding exchange-correlation holes due to the predominance of self-interaction effects in these systems, especially since conventional functionals yield large errors for most reactions.