Quantum theory successfully predicts the shell structure of atoms and can explain the observed features of the periodic table (see end of chapter 5). The shell structure and the ionization energies of atoms can be probed by a technique known as photoelectron spectroscopy.
The technique works very much like the photoelectric effect in metals except that we use incident EM radiation (usually X-rays or UV rays) to knock electrons out of individuals atoms, thereby leading to measures of the ionization energies. In this method, the ionization energy IE plays the same role as the work function of a metal.
Suppose a photon of energy strikes an atom or cation in order to probe one of the ionization energies . The electron will be ejected from the atom or ion and have a residual kinetic energy . Thus, by energy conservation
The essential approximations made in these relations are, firstly, the approximations inherent in the HF theory. But beyond this, it is also assumed that the energies do not change much as the electrons are sequentially stripped out of the atom. Obviously, these energies should change, as the screening and repulsion effects of the other electrons is reduced, and the electrons are drawn closer to the positively charged nucleus, but the effects are small. This type of approximation is known as the frozen core approximation.