Charge density reconstitution from approximate exchange-correlation holes

The electronic charge density and the expectation of any monoelectronic operator can be reconstructed by a density-weighted integral of the exchange-correlation hole. This property can be used to test and improve approximate holes that are not spherically averaged. As an example, we test the Becke–Roussel exhange-hole model in its nonspherical formulation on small atoms, finding that it accurately reproduces the density in the core region, while missing some of the features of the valence and core-valence regions. The same test on the local-density approximation shows poor results, as expected.