In this paper we investigate Coulomb correlation effects in the He atom by studying the structure of the static and dynamic Coulomb hole charge distributions as determined by the analytical 39-parameter correlated wave function of Kinoshita. The static Coulomb hole, which is defined in terms of the radial electron-electron distribution function, shows that as a result of Coulomb repulsion there is a reduction in probability of electron approach within a distance of one atomic unit and an increase in this probability for greater separations. The dynamic Coulomb hole defined directly in terms of the pair-correlation density describes the probability of finding one of the electrons given the position of the other. We thus demonstrate how the two electrons are correlated as a function of the nonuniform density of the atom. We also investigate the correlation potential Wc(r) of the work formalism, which is the work done to move an electron in the force field of the dynamic Coulomb hole charge. The structure of Wc(r) is similar to the exchange potential Wx(r) (which is the work done to move an electron in the force field of the Fermi hole), in that it is attractive, monotonic, and has zero slope at the nucleus. However, as a result of the structure of the Coulomb hole for asymptotic electron positions, and the fact that its total charge is zero, the potential Wc(r) vanishes rapidly in the classically forbidden region.
Thus, the asymptotic structure of the exchange-correlation potential Wxc(r) of the work formalism is that of Wx(r) which is (-1/r). We also detemine via the Kinoshita wave function the correlation potential μc(r) of Kohn-Sham theory, which differs from Wc(r) in that it also incorporates the effects of the correlation contribution to the kinetic energy. Consequently, it is less attractive than Wc(r), but also has zero slope at the nucleus. However, as is known, the potential μc(r) is nonmonotonic, since it goes positive within the atom, then becomes negative in the classically forbidden region, finally vanishing asymptotically as a negative function. Since the exchange potentials of the work formalism and Kohn-Sham theory are the same for this atom, and because Wc(r) is strictly representative of Coulomb correlations, we attribute the nonmonotonicity and positiveness of the Kohn-Sham potential μc(r) to the correlation kinetic energy. This conclusion is consistent with the result that the difference between the correlation energies determined within the work formalism from the dynamic Coulomb hole and Kohn-Sham theory is equal to the correlation contribution to the kinetic energy.
Slamet, Marlina and Viraht Sahni. "Coulomb Holes and Correlation Potentials in the Helium Atom." Physical Review A 51.4 (1995): 2815-2825.