Electron-electron radial and angular holes in the Hartree-Fock theory of atoms

The two-electron radial D-2(r(1), r(2)) and angular A(2)(Omega(1), Omega(2)) density functions are the probability densities that one electron is located at a radius r(1) and another at r(2) and that one electron is located along a direction Omega(1) = (theta(1), phi(1)) and another along Omega(2) = (theta(2), phi(2)), respectively, when any two electrons are considered simultaneously. Within the Hartree-Fock framework, these densities are the sums of contributions D-2(ij) (r(1), r(2)) and A(2)(ij)(Omega(1), Omega(2)) from a pair of spin orbitals i and j. Theoretical analyses of the contributions D-2(ij)(r(1), r(2)) and A(2)(ij)(Omega(1), Omega(2)) for atoms show that there exist an 'electron-electron radial hole' D-2(ij) (r, r) = 0 and an 'electron-electron angular hole' A(2)(ij)(Omega, +/-Omega) = 0 for a pair of spin orbitals with particular conditions. The radial and angular holes add new holes to the electron-electron coalescence ( or Fermi) hole for two spin orbitals with the same spin and the electron-electron counterbalance hole for two spin orbitals with the same spin and the same spatial inversion symmetry.