Summary

This chapter demonstrated that the form of the analytic function used to model the coupling-constant averaged pair-correlation function in the WDA is crucially important. WDA properties are indeed sensitive to the type of model employed. This was observed by the wide range of lattice constants and bulk moduli calculated for Si using twelve example models. The extremely weak nature of the bonding reported by Charlesworth in Ref. [150] for the WDA using the same model functions, was not observed in this study. In great contrast to Charlesworth, some of the models yielded lattice constants shorter than experiment and most were closer to experiment than the LDA. Hopefully the new results presented here, which are substantiated by the comparison of XC holes with the variational Monte Carlo method, will stimulate further research into model pair-correlation functions for use in the WDA.

Although the specific choice of model functions provided informative insights into general trends in the structural properties and demonstrated close links with the corresponding XC holes, they all possess certain deficiencies. One particular fault is the violation of the non-negativity constraint. The models $f_7(u)$ and $f_{10}(u)$ violated this condition most severely of the twelve models considered, and also produced the greatest errors in the structural properties and holes.

In order to tackle such problems a new prescription was proposed that satisfies an extra exact constraint known as the Kimball cusp condition. In addition, the new prescription attempted to resolve the fact that the on-top pair-correlation function should be in the range $[0,0.5]$, by including two extra parameters, namely $m$ and $\kappa$. Although this objective was not achieved precisely, a definite improvement over the original formulation was attained when examining the homogeneous electron gas. The inclusion of the additional parameters does not invalidate the non-empirical nature of the functional, since the parameters were invoked in order to satisfy exact constraints. A model function that satisfies the on-top condition for the homogeneous electron gas, for a broad range of densities, must first be devised before applied to real inhomogeneous systems. Hopefully this approach to testing the models will prove universally applicable, i.e. for the majority of real physical systems.

Another direction that could be pursued in the development of model pair-correlation functions is to devise models for exchange and correlation separately. Incorporating coupling-constant $(\lambda)$ dependence may also be useful due to the greater amount of near-exact data that could be utilised - QMC simulations are again a good source of this type of data since they must calculate the pair-correlation function at several values of $\lambda$ in order to determine ($\lambda$-averaged) density functional exchange-correlation quantities.