The Scalar Fields: $\alpha ({\bf r})$ and $\beta ({\bf r})$

Before proceeding with the WDA potential, the scalar fields $\alpha ({\bf r})$ and $\beta ({\bf r})$ that define the WDA hole are first described. As stated previously, these parameters are obtained by demanding that the WDA explicitly satisfies the sum rule and the energy density in the limit of constant density, $n({\bf r})=n$, i.e.

$$n \int \, G^{\hbox{{WDA}}}[u,\tilde{n}({\bf r })] \, d{\bf r} = -1 \, ,$$ (4.38)

$$n \int \frac{G^{\hbox{{WDA}}}[u,\tilde{n}({\bf r})]}{u}\,d{\bf r} = \varepsilon_{\hbox{{XC}}}^{\hbox{ {LDA}}}(n) \, .$$ (4.39)

For a homogeneous electron gas, the weighted density is identical to the actual density, i.e. $\tilde{n}({\bf r}) = n = \hbox{constant}$, therefore relations (4.33) and (4.37) simplify to give,

$$-1 = \tilde{n} \, \alpha({\bf r}) \beta^{3}({\bf r}) F_{1}(0)$$ (4.40)

$$\varepsilon_{\hbox{{XC}}}^{\hbox{ {LDA}}}(\tilde{n}) = \frac{1}{2} \tilde{n} \, \alpha({\bf r}) \beta^{2}({\bf r}) F_{2}(0) \, ,$$ (4.41)

where the constants, $F_{1}(0)$ and $F_{2}(0)$ are obtained by setting $q = 0$ in relations (4.32) and (4.36),

$$F_{1}(0) = 4 \pi \int_{0}^{\infty} u^2 \, f(u) \, du$$ (4.42)

$$F_{2}(0) = 4 \pi \int_{0}^{\infty} u \, f(u) \, du \, .$$ (4.43)

Rearranging (4.40) and (4.41) gives rise to the relations that define $\alpha ({\bf r})$ and $\beta ({\bf r})$:

$$\alpha({\bf r}) = \frac{-1}{\tilde{n} \beta({\bf r})^3 \, F_{1}(0)} \,,$$ (4.44)

$$\beta({\bf r}) = \frac{- F_{2}(0)}{2 \, \varepsilon_{\hbox{{XC}}}^{\hbox{{LDA}}}(\tilde{n}) \, F_{1}(0) } \, .$$ (4.45)