Charge Density Differences

Molecular studies have revealed that the LDA and GGA give a reasonable description of the exchange-correlation potential in regions close to a molecule, but break down with increasing distance [37]. This is evident since the true potential decays asymptotically with an inverse distance behaviour, whereas the LDA and GGA potentials behave exponentially at large distance from a finite charge distribution. As a consequence, conventional functionals such as PW$91$ lead to over-diffuse electron densities. This also has an effect on the occupied-unoccupied eigenvalue differences in atoms and molecules [128], which are greatly underestimated.

The charge density difference of Si between HCTH and PW$91$ has been computed to investigate whether HCTH provides any improvement regarding the diffuse nature of the density. Fig. 3.5(a) shows the crystal lattice of Si, and the colour contours in Figs. 3.5(b) and (c) display the positive difference between the densities calculated as, PW91 - HCTH and HCTH - PW91 respectively. The density differences are superimposed onto the Si crystal lattice in order to illustrate the regions where charge has been transferred. It is clear from these plots that the HCTH density is greatest near the atomic sites and bonds, whereas the PW$91$ density is higher in the interstitial regions of the lattice. The superposition of Figs. 3.5(b) and (c) given in 3.5(d) clearly shows the transfer of density.

Figure 3.5: The crystal lattice of Si is displayed in (a) while the positive contributions to the density differences are shown by the contours, for PW$91$ minus HCTH (b) and HCTH minus PW$91$ (c). The superposition of (b) and (c) is shown in (d) to illustrate the net flow of charge from the interstitials to the bonding regions in going from PW91 to HCTH.

If the over-diffuse nature of the density in the molecular environment extends into the solid state, then the above results indicate that HCTH yields a more accurate description of the exchange-correlation potential than PW$91$. This would also explain why HCTH band gaps are generally more accurate than those obtained with PW$91$.