On-top hole

The LSD and PW91-GGA system-averaged holes agree at zero interelec-tronic separation, where both are nearly exact. In Sect. 2.3, we discuss how the near-universality of this on-top hole density provides the missing link between real atoms and molecules and the uniform electron gas. Except very close to the nucleus, the local on-top hole density is also accurately represented by LSD, even in the classically-forbidden tail region of the electron density [18]. 

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]. 

The solid curve was obtained from Yasuhara s expression for the on-top hole for a uniform electron gas [55]. The circles indicate the LSD values of /< (0)>- Their proximity to the uniform-gas curve indicates that the approximation... 

The importance of the accuracy of the on-top hole in LSD cannot be over-stressed. This accuracy is not just in the system-average, but is also there on a point-wise basis. In Fig. 7, we plot the ratio of the full-coupling strength... 

Fig. 6. Universal curve for the system-averaged on-top hole density in spin-unpolarized systems. The solid curve is for the uniform gas. The circles indicate values calculated within LSD, while the crosses indicate essentially exact results, and the plus signs indicate less accurate Cl results. The high-density (r, - 0) and low-density (r, ->oo) limits behave respectively like the weak-coupling U - 0) and strong-coup-ling (X- oo) limits...

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. 

Acknowledgements. This work has been supported by NSF grant DMR95-21353 and by the Deutsche Forschungsgemeinschaft. We thank Andreas Savin for supplying us with accurate values of the local on-top hole density in the He atom. 

Burke K, Perdew JP, Ernzerhof M, Why semilocal functionals work Accuracy of the on-top hole density, in preparation for J Chem Phys... 

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