Figure 1: Electron densities of atomic shells from n=1 to n=5 for Pb4+ (solid line), Pb22+ (dotted line), Pb54+ (short dash), Pb72+ (long dash).	Figure 2: Classic analog of atomic shells. Negatively charged spheres around a positively charged ball.

It is very well known that electrons in atoms are grouped in shells. What is not so well known is that these shells are very much independent of each other.

Figure 1 shows the result of self-consistent field calculations of electron densities for Pb⁴⁺, Pb²²⁺, Pb⁵⁴⁺ and Pb⁷²⁺ ions which have 5, 4, 3, and 2 shells correspondingly. One can see that removing the contribution of the upper-most shell from the self-consistent field has only small effect on the next shell and practically no effect on other shells. In the end all ions have almost the same electron densities for each shell. This is a surprising fact given that the electron energies are, in contrast, very different.

To understand this phenomenon one can use a classic analog of electron shells presented in Figure 2. A charged sphere creates no electric field inside itself and cannot affect electrons moving in its interior region. The classic analog of an electron energy is the energy to move an electron from the interior region to infinity. One has to cross all charged spheres to do this. Therefore, the energy does depend on the shells. However, electron movement in the interior region does not depend on this energy. In other words, an electron feels no effect from outer shells unless it crosses them.

Now, how does this help in calculations? Suppose we need to calculate the electron structure of an atom with several valence electrons. The independence of shells means that we can start the calculations from a positive ion with all valence electrons removed. The electron densities and the effective field will still be the same. This leads to the simplest form of perturbation theory with the number of terms in every order being several times smaller than in the case of any other initial approximation.

Calculations for Ge, Sn, Pb, Ba, Ra and their ions show that applying this scheme leads to a significant improvement of the accuracy of calculations compared to methods available before. Areas of research which are to benefit from this development include searches for variation of fundamental constants, study of parity and time invariance violation in atoms and many other important problems.

Vladimir Dzuba