Results for the Homogeneous Electron Gas

As a first brief test, the pair-correlation function $g_{\hbox {{XC}}}({\bf r},{\bf r'})$ obtained from the new model is examined in the case of the homogeneous electron gas, for densities ranging from high $(r_s=0.1)$, to low $(r_s=10.0)$ values.

Two versions of the new model are considered which employ the values $m=2$, and $m=4$, and are labelled $G^{\hbox{{Cusp}}}_1[{\bf r},{\bf r'};\tilde{n}]$ and $G^{\hbox{{Cusp}}}_2[{\bf r},{\bf r'};\tilde{n}]$ respectively. As mentioned in Sec. 5.5.1, another objective of the new model, apart from incorporating the cusp condition, is to tackle the problem of the on-top pair-correlation function violating the exact range of values $[0,0.5]$. Therefore in each of the two cases considered, the value of $\kappa$ was set by attempting to keep the on-top values within this specified range. The optimal values, $\kappa = 0.75$ and $\kappa = 1.08$, were obtained for $G^{\hbox{{Cusp}}}_1[{\bf r},{\bf r'};\tilde{n}]$ and $G^{\hbox{{Cusp}}}_2[{\bf r},{\bf r'};\tilde{n}]$ respectively. Table 5.5 compares the on-top value of the homogeneous electron gas pair-correlation function $g_{\hbox{{XC}}}(r=0)$, calculated for the range of densities $0.1 \le r_s \le 10.0$, with the original Gaussian model $G^{{a}}[{\bf r},{\bf r'};\tilde{n}]$, and the two new versions.

Table 5.5: The on-top value of the pair-correlation function, $g_{\hbox{{XC}}}({\bf r},{\bf r})$, for the homogeneous electron gas at various $r_s$ values. Results are presented for the original Gaussian pair-correlation function that violates the cusp condition $G^{{a}}[{\bf r},{\bf r'};\tilde{n}]$, and two versions of the new model that satisfy the cusp condition, $G^{\hbox{{Cusp}}}_1[{\bf r},{\bf r'};\tilde{n}]$ with $m=2, \kappa = 0.75$, and $G^{\hbox{{Cusp}}}_2[{\bf r},{\bf r'};\tilde{n}]$ with $m=4, \kappa = 1.08$.

It is observed from these results that although the new models still yield $g_{\hbox{{XC}}}(r=0)$ greater than $0.5$ in the high density case, and lower then $0$ in the low density regime, they are an improvement over $G^{{a}}[{\bf r},{\bf r'};\tilde{n}]$. For $r_s = 0.1$, the on-top values are $0.564$, $0.538$ and $0.516$ for $G^{{a}}[{\bf r},{\bf r'};\tilde{n}]$, $G^{\hbox{{cusp}}}_1[{\bf r},{\bf r'};\tilde{n}]$, and $G^{\hbox{{cusp}}}_2[{\bf r},{\bf r'};\tilde{n}]$ respectively, whereas for $r_s = 10.0$, the values are $-0.118$, $-0.026$ and $-0.015$ respectively. The function $G^{\hbox{{cusp}}}_2[{\bf r},{\bf r'};\tilde{n}]$ is therefore the most successful in this respect.

The effect on the overall pair-correlation function is observed in Fig. 5.11 which shows $g_{\hbox {{XC}}}(r)$ calculated using $G^{{a}}[{\bf r},{\bf r'};\tilde{n}]$ and $G^{\hbox{{Cusp}}}_1[{\bf r},{\bf r'};\tilde{n}]$, for several $r_s$ values.

Figure 5.11: The pair-correlation function $g_{\hbox{{XC}}}({\bf r},{\bf r'})$ calculated for the homogeneous electron gas at various $r_s$ values, determined using the new model $G_1^{\hbox{{Cusp}}}[{\bf r},{\bf r'};\tilde{n}]$, that incorporates the Kimball cusp condition (solid lines), and the original Gaussian model $G^{{a}}[{\bf r},{\bf r'};\tilde{n}]$ (dotted lines).

It is clear that while both models demonstrate different behaviour near the cusp, as expected, they are very similar as $r$ increases, as intended. However, an unfortunate feature of the new models is that in the low density regime, they yield a negative gradient at $r = 0$, which is a consequence of satisfying the cusp condition (5.10) when $g_{\hbox {{XC}}}(r)$ goes negative. An example of this is shown for the $r_s = 10.0$ case. This unphysical characteristic can only be eliminated by satisfying the non-negativity constraint for all $r_s$.