Electron–electron coalescence and counterbalance functions for atoms

Abstract

For many-electron atoms, spherically averaged electron–electron coalescence h0(R) and counterbalance d0(u) functions are studied which, respectively, represent the probability densities that any electron pairs with zero interelectronic distance are located at a radius R from the nucleus and that any electron pairs with zero center-of-mass radius have a relative distance u. For the exact and Hartree–Fock (HF) wave functions, cusp relations h0′(0)/h0(0)=−4Z and d0′(0)/d0(0)=−2Z are derived theoretically, where the prime denotes the first derivative and Z is nuclear charge. At the Hartree–Fock limit level, both functions h0(R) and d0(u) are found to be monotonically decreasing with a single maximum at R=0 or u=0 for all the 102 atoms He through Lr. The long-range asymptotic behavior of the coalescence and counterbalance functions is governed in general by the orbital energy of the highest occupied atomic orbital.