Asymptotic behaviour of the electron-pair radial coalescence density of atoms

The electron-pair radial coalescence density D2(r, r) is the diagonal element function of the electron-pair radial density D2(r1, r2) and represents the probability density of two electrons being on the same sphere with a radius r. Theoretical analysis of atomic coalescence densities for the asymptotic short-range (r ≪ 1) and long-range (r ≫ 1) regions shows that the Hartree–Fock (HF) radial coalescence density D2(r, r) has rigorous equality relations with the familiar single-electron radial density D(r) in these particular regions. Numerical examinations confirm the theoretical results for the 53 atoms He through Xe. The effect of electron correlations on these HF equalities is discussed theoretically and numerically.