On-Top Exchange-Correlation Hole

To see this in more detail, consider the ratio of the system-averaged on-top exchange-correlation hole to the system-average of the density itself, , as defined in Eq. (20). This ratio satisfies the inequalities... 

Fig. 7. Local on-top exchange-correlation hole at full coupling strength divided by density as a function of r in the He atom. The nuclear cusp produces greater LSD error for r-0...

As mentioned above, LSD yields a reasonable description of the exchange-correlation hole, because it satisfies several exact conditions. However, since the correlation hole satisfies a zero sum rule, the scale of the hole must be set by its value at some value of . The local approximation is most accurate at points near the electron. In fact, while not exact at m = 0, LSD is highly accurate there. Thus the on-top hole provides the missing link between the uniform electron gas and real atoms and molecules [18]. 

By (1.95), the LSD on-top exchange hole rix (r, r) is exact, at least when the Kohn-Sham wavefunction is a single Slater determinant. The LSD on-top correlation hole n (r,r) is nonexact [63] (except in the high-density, low-density, fully spin-polarized, or slowly-varying limit), but it is often quite realistic [64]. By (1.85), its cusp is then also realistic. 

Successful density functional approximations such as the PW91 GGA or the self-interaction correction (SIC) [57] to LSD recover [19] LSD values for the on-top hole density and cusp. The weighted density approximation (WDA) [41,42], which recovers the LSD exchange hole density but not the LSD correlation hole density [19] in the limit u -> 0, needs improvement in this respect. 

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