Average electron momenta in many-electron atoms

In many-electron atoms, the average electron momentum 〈p〉 represents the mean momentum of a single electron when all the electron motions are averaged. If any two electrons are considered simultaneously, however, the average momentum 〈p〉 splits into two different momenta, low momentum 〈p<〉 and high momentum 〈p>〉. For the 102 atoms He through Lr in their ground states, the momenta 〈p<〉 and 〈p>〉 are systematically examined at the Hartree–Fock limit level. It is also shown that the sum 〈p>〉+〈p<〉 and the difference 〈p>〉−〈p<〉 of the two momenta constitute upper and lower bounds to the electron-pair relative momentum 〈p12〉=〈|p1−p2|〉 and to the electron-pair center-of-mass momentum 〈P〉=〈|p1+p2|/2〉. The tightness of the bounds is discussed for the 102 atoms.