Electron–electron coalescence and counterbalance densities for atoms in Hartree–Fock theory

Abstract

The electron–electron coalescence I(0) and counterbalance E(0) densities are probability densities of finding any two electrons, respectively, at the same position and at the reflection points in the three-dimensional space. For a single Slater determinant wave function, these electron-pair properties are shown to be exactly expressible in terms of the spin-traced one-electron density function ρ(r) and its orbital components ρi(r): I(0)=(1/4){〈ρ〉−ΔI} and E(0)=2{〈ρ〉−ΔE}, where 〈ρ〉 is the average electron density, and ΔI and ΔE are linear combinations of overlaps between two orbital densities, that depend on the electronic configuration and the LS multiplet state of the atom under consideration. For the atoms He through Ne in their experimental ground state, the explicit forms of ΔI and ΔE are derived, and the electron–electron coalescence and counterbalance densities obtained from the numerical Hartree–Fock calculations are discussed.