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, , for this electron system. Presented in Table 6.3 are the total exchange-correlation energy differences and , 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 increases. The LDA deviations vary between , whereas for the GGA they remain almost constant at , throughout the range of confinements considered.
|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.
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 calculated in a plane going through the centre of the cell. The fact that is positive at all points in the plane demonstrates that is indeed more negative than , 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.