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 [q(2)]((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.