Sum rules for generalized electron-pair moments of Hartree–Fock orbitals

For many-electron systems, the generalized electron-pair density function g(q;a,b) represents the probability density function for the magnitude |ar_i+br_j| of two-electron vector ar_i+br_j to be q, where a and b are real-valued parameters. For Hartree–Fock wave functions, decomposition of g(q;a,b) into spin–orbital-pair components {g^ij}(q;a,b) shows that the second moments {_(a,b)}^ij, associated with {g^ij}(q;a,b), satisfy several rigorous sum rules which connect one- and two-electron properties of spin–orbitals i and j. The same is also true in momentum space. As an illustrative application, the orbital kinetic energies of the Rn atom are separated into the relative motion and center-of-mass motion contributions of electrons in two relevant orbitals