Electron-electron counterbalance hole in Hartree–Fock theory

Abstract

The electronic extracule density E(R) and its spherical average d(R) are probability densities of finding the center-of-mass vector (rj+rk)/2 and its magnitude |rj+rk|/2 of any two electrons j and k to be R and R, respectively. Theoretical analysis of spin–orbital-pair contributions Ejk(R) and djk(R) to the extracule densities shows that within the Hartree–Fock framework, there exists an “electron-electron counterbalance hole” Ejk(0)=djk(0)=0 between two electrons in spin–orbitals j and k with the same spin and the same spatial inversion symmetry, which implies that the center-of-mass vector (or distance) of these electrons cannot be zero or these electrons cannot be exactly at the opposite positions in space with respect to the inversion center. The same is also true in momentum space. The electron-electron counterbalance hole may be considered as a complement to the familiar electron-electron coalescence or Fermi hole for two electrons with the same spin. We also point out that the electron-electron counterbalance and coalescence densities are rigorously proportional for a pair of spin–orbitals with different spins.