The energies of one-electron atoms

The term one-electron atom is intended to cover the hydrogen atom as well as all ions which consist of one nucleus and one electron, such as for example He , Li+ or Be+. The total energy of such an atom may, as a good first approximation, be described as the sum of the kinetic energy of the nucleus KE , the kinetic energy of the electron KEg and the electrostatic potential energy PE(r)  

Here A/n, Xn, jn and Zn denote the mass and Cartesian coordinates of the nucleus, while me, Xe, Te and Ze denote the mass and Cartesian coordinates of the electron. The nuclear charge is -l-Ze and r the distance from the nucleus to the electron. This expression implies that the kinetic energy is zero when the two particles are at rest, and the potential energy is zero when the electron and nucleus are infinitely far apart. 

Equation (1.3) is not exact relativistic effects (such as the variation of mass with velocity or the electron spin ) have been neglected. We shall have to return to this point later. 

In order to separate the motion of the electron relative to the nucleus from the motion of the atom as a whole, we introduce the Cartesian coordinates of the atom s center of mass, X, Y andZ  

The two kinetic energy terms in equation (1.3) may then be rearranged to yield 

---