Exchange-Correlation Energy Densities

To be consistent with the VMC study the WDA energy density $e_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r})$ is calculated as,

$$e_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r}) = n({\bf r}) \, \varepsil... ...x{{WDA}}}({\bf r},{\bf r'})}{\vert\,{\bf r}-{\bf r'} \,\vert} \, d{\bf r'} \, ,$$ (6.8)

similarly, the LDA energy density is determined as,

$$e_{\hbox{{XC}}}^{\hbox{{LDA}}}({\bf r}) = n({\bf r}) \, \varepsilon_{\hbox{{XC}}}^{\hbox{{hom}}}({\bf r}) \, .$$ (6.9)

GGA energy densities are not presented since they cannot be rigorously defined - it is possible to add a function to $e_{\hbox{{XC}}}^{\hbox{{GGA}}}({\bf r}) = n({\bf r}) \, f_{\hbox{{XC}}}[n({\bf r}),\vert \nabla n({\bf r}) \vert]$, which although leaves $E_{\hbox{{XC}}}^{\hbox{{GGA}}}[n({\bf r})]$ unchanged, modifies $e_{\hbox{{XC}}}^{\hbox{{GGA}}}({\bf r})$ point-wise. It should be noted that attempts have been made to correct this deficiency of the GGA [164], however, determining differences with respect to just the LDA, $\Delta e_{\hbox{{XC}}}({\bf r}) = e_{\hbox{{XC}}}^{\hbox{{LDA}}}({\bf r}) - e_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r})$, is sufficient for the purposes of comparison with the VMC work.

Figure 6.3: The panels display the density together with the Laplacian of the density $\nabla^2 n({\bf r})$, and show the exchange-correlation energy difference $\Delta e_{\hbox {{XC}}}(y) = e_{\hbox {{XC}}}^{\hbox {{LDA}}}(y)$ - $e_{\hbox {{XC}}}^{\hbox {{WDA}}}(y)$ for the cosine-wave electron gas systems with $q = 1.12 \, k_{\hbox {{F}}}^{\hbox {{0}}}$, $1.56 \, k_{\hbox {{F}}}^{\hbox {{0}}}$ and $2.18 \, k_{\hbox {{F}}}^{\hbox {{0}}}$.

Fig. 6.3 shows plots of $\Delta e_{\hbox{{XC}}}$, $n({\bf r})$ and the corresponding Laplacian $\nabla ^2 n({\bf r})$, along the direction of inhomogeneity, $y$, for all three systems. It is clear that the resemblance between $\Delta e_{\hbox{{XC}}}(y)$ and $\nabla ^2 n({\bf r})$ is again observed - in extremely close agreement with the VMC results shown in Fig. 6.1. Despite the differences in the densities used in the present study, the magnitude of the deviations are also in very good agreement with the VMC results.

Fig. 6.4 displays $\Delta e_{\hbox{{XC}}}(y)$ calculated for the $q=1.12 \, k_{\hbox {{F}}}^0$ system using all twelve pair-correlation models described in Chapter $5$, labelled WDA$1$ to WDA$12$.

Figure 6.4: The energy density difference $\Delta e_{\hbox {{XC}}}(y) = e_{\hbox {{XC}}}^{\hbox {{LDA}}}$(y) - $e_{\hbox {{XC}}}^{\hbox {{WDA}}}(y)$ calculated for the $q=1.12 \, k_{\hbox {{F}}}^0$ system using the twelve model pair-correlation functions defined in Table 5.1.

Except for the less physical models given by WDA$7$ and WDA$10$, all the functions exhibit the characteristic Laplacian-type deviations, but with varying amplitudes. A general trend is observed for the size of the minima that occur around the peaks in the density, in that they become larger (more negative) for the three models within each group. Other general trends are exhibited that are in complete accordance with the self-consistent properties calculated in Chapter $5$, for example the smallest and largest variations in $\Delta e_{\hbox{{XC}}}(y)$ within a particular group, occur for the functions in groups $2$ and $3$ respectively.