Resultant exchange-correlation hole

Figure 2-2. Fermi, Coulomb and the resulting total exchange-correlation holes for H2 at three different intemuclear distances the position of the probe electron is marked with an arrow (adapted from Baerends and Gritsenko, J. Phys. Chem. A, 101, 5390 (1997), with permission by the American Chemical Society).

While the LSD exchange-correlation hole is accurate for small interelec-tronic separations (Sect. 2.3), it is less satisfactory at large separations, as discussed in Sect. 2.5. For example, consider the hole for an electron which has wandered out into the classically-forbidden tail region around an atom (or molecule). The exact hole remains localized around the nucleus, and in Sect. 2.5 we give explicit results for its limiting form as the electron moves far away [19]. The LSD hole, however, becomes more and more diffuse as the density at the electron s position gets smaller, and so is quite incorrect. The weighted density approximation (WDA) and the self-interaction correction (SIC) both yield more accurate (but not exact) descriptions of this phenomenon. 

We found that our method results in electron densities nx(r) which deviate from the reference density by less than 1%. This is shown in figure 1(a) where the density is plotted as a function of A along a line parallel to the direction in which the external potential varies (we call this the ( direction). While the density is fixed, all other physical quantities vary smoothly and monotically with A. As an example we consider the spherically-averaged exchange-correlation hole... 

We now turn to our results for nxc and exc. The spherically averaged exchange-correlation hole, nxc(r,s), obtained from our adiabatic calculations is shown in figure 2(a) together with the LDA approximation (28) to this quantity... 

Sterne and Inkson have recently proposed a similar construction of an extreme tight-binding model for E, where the exchange-correlation hole is directly identified with the Lorentz sphere. This results(apart from a different expression uged for p and the... 

This approximation for xc[n] has proved amazingly successful, even when applied to systems that are quite different from the electron liquid that forms the reference system for the LDA. A partial explanation for this success of the LDA is systematic error cancellation typically, the LDA underestimates Ac but overestimates Ax, resulting in unexpectedly good values of Axe. This error cancellation is not accidental, but systematic, and is caused by the fact that for any density the LDA exchange-correlation hole satisfies the correct sum rule on the exchange-correlation hole Hxc(r, r ), /d r x ( > rO = -1> which is only possible if integrated errors in cancel with those of... 

Spin density is found in the molecular plane because of spin polarization, which is an effect arising from exchange correlation. The Fermi hole that surrounds the unpaired electron allows other electrons of the same spin to localize above and below the molecular plane slightly more than can electrons of opposite spin. Thus, if the unpaired electron is a, we would expect there to be a slight excess of density in the molecular plane as a result, the hyperfine splitting should be negative (see Section 9.1.3), and this is indeed the situation observed experimentally. An ROHF wave function, because it requires the spatial distribution of both spins in the doubly occupied orbitals to be identical, cannot represent this physically realistic situation. 

---