Low Average Density

In the low density case the results are very different from those just described. Fig. 6.16 shows the density distribution in a plane taken through the centre of the cell, calculated for the most strongly confined density, $v_0^{\hbox{{max}}} = 2016 \varepsilon_{\hbox{ F}}^0$, for this $2$ electron system. Presented in Table 6.3 are the total exchange-correlation energy differences $\Delta E_{{\hbox{XC}}}^{{\hbox{LDA}}}/N$ and $\Delta E_{{\hbox{XC}}}^{{\hbox{GGA}}}/N$, relative to the WDA, for several densities that span a large range of confinement strengths. The results contrast those obtained in the high density case on two counts. Firstly, the deviations are about one magnitude greater in comparison, for both functionals, although the GGA is in better agreement. Secondly, and possibly more striking, is the fact that the energy differences are hardly affected by the change in density inhomogeneity as $v_0$ increases. The LDA deviations vary between $10 - 12 \%$, whereas for the GGA they remain almost constant at $7 \%$, throughout the range of confinements considered.

Figure 6.16: Isotropically confined electron density distribution for the system with low average density ($r_s = 4.3 a_0$), plotted in a plane taken through the centre of the confining potential with $v_0 = v_0^{\hbox {{max}}}$.

Table 6.3: The total exchange-correlation energy $E_{{\hbox {XC}}}^{{\hbox {WDA}}}/N$, and the difference relative to the LDA and the GGA, for various confinement strengths in the $r_s = 4.3 a_0$ system.

$v_0 / \varepsilon_{\hbox{ F}}^{\hbox{ 0}}$	$E_{{\hbox {XC}}}^{{\hbox {WDA}}}/N$	$\Delta E_{{\hbox{XC}}}^{{\hbox{LDA}}}/N$	$\Delta E_{{\hbox{XC}}}^{{\hbox{GGA}}}/N$
2.0	-0.3321	+0.0336 (+10 %)	+0.0230 (+7 %)
20	-0.4042	+0.0452 (+11 %)	+0.0305 (+8 %)
101	-0.6155	+0.0740 (+11 %)	+0.0455 (+7 %)
303	-0.7789	+0.0935 (+12 %)	+0.0534 (+7 %)
2016	-0.8831	+0.8530 (+12 %)	+0.0581 (+7 %)

These results can be explained by considering the self-interaction effect which is more prominent than in the high average density, simply because of the much smaller number of electrons. The WDA contains a more accurate account of self-interaction effects than the LDA and GGA, as described in Sec. 4.2.4, and if the WDA is considered to be close to the exact result, then Table 6.3 shows that the GGA provides an improvement over the LDA for self-interaction errors. Also, other than being coincidental, the distinct lack of variation in the total energy differences as the amount of density localisation increases, indicates that the self-interaction error is overwhelming the error caused by the inhomogeneity in the density.

Once more, the source of the discrepancies between the different functionals can be rationalised in terms of their respective descriptions of the exchange-correlation hole. When an electron moves out from the main density distribution in the centre of the cell, into the tail of the density, the LDA hole, as always, stays centred on the electron, whereas the WDA hole will stay localised at the density peak in the centre. A clear demonstration of this effect is given in Fig. 6.17 which shows the WDA hole for an electron at three positions moving from the centre to one of the corners of the unit cell.

Figure 6.17: WDA exchange-correlation holes in the $N = 2$ electron system with $v_0 = v_0^{\hbox {{max}}}$, calculated for an electron located at three points, moving along a diagonal direction from the centre of the density distribution, to a corner of the unit cell. The position of the electron in the $x-y$ plane is marked in each case.

From the experience gained in the previous chapters, it is apparent that the electron will become separated from its hole as soon as it moves away from the central density distribution, and when the electron is situated in the corner of the cell, as in Fig. 6.17(c), its hole is located over $3 \hbox{\AA}$ away (since $l = 4.71 \hbox{\AA}$ for this system). Also, despite the extremely low value of the local density in this region, $r_s \sim 75 a_0$ at this point, the WDA hole can still be observed on the same scale as when it is situated at the density peak in Fig. 6.17(a), where the local density has a value $r_s \sim 0.7 a_0$. The LDA will therefore give an underestimated (less negative) value for the energy-density $\varepsilon_{\hbox{{XC}}}({\bf r})$ compared to the WDA, since its hole will be shallower than the WDA as a result of sampling only the local density $n({\bf r})$. The importance of retaining the non-local density dependence $n({\bf r'})$ in the formulation of a model for the XC hole cannot be overstated in these circumstances.

Figure 6.18: The difference between the LDA and WDA energy densities, $\Delta\varepsilon_{\hbox{{XC}}}({\bf r}) = \varepsilon_{\hbox{{XC}}}^{\hbox{{LDA}}}({\bf r}) - \varepsilon_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r})$ (in Hartrees), calculated in a plane through the centre of the cell, for the $N = 2$ electron system with $v_0 = v_0^{\hbox {{max}}}$. $\Delta \varepsilon _{\hbox {{XC}}}$ is plotted throughout the entire plane in (a), whereas just half of the plane is displayed in (b), in order to expose the behaviour near the centre of the density.

As a note, it may appear that the hole in Fig. 6.17(c) does not satisfy the sum rule when compared with the holes presented in Figs. 6.17(a) and (b). The reason is because the electron is situated at the corner of the unit cell, so from periodic boundary conditions there are hole contributions emanating from the neighbouring unit cells that are not present when the electron is located near the centre of the cell.

Fig. 6.18 shows the energy-density difference $\Delta \varepsilon _{\hbox {{XC}}}({\bf r}) = \varepsilon _{\hbox {{XC}}}^{\hbox {{LDA}}}({\bf r}) - \varepsilon _{\hbox {{XC}}}^{\hbox {{WDA}}}({\bf r})$ calculated in a plane going through the centre of the cell. The fact that $\Delta\varepsilon_{\hbox{{XC}}}({\bf r})$ is positive at all points in the plane demonstrates that $\varepsilon_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r})$ is indeed more negative than $\varepsilon_{\hbox{{XC}}}^{\hbox{{LDA}}}({\bf r})$, for the reasons just given concerning the XC holes. The positive total energy differences given in Table 6.3 are therefore explained. It is presumed that the GGA is behaving in a similar way as the LDA, although the GGA description of the hole appears to be marginally better, judging from the closer agreement with the WDA total energies in Table 6.3.