## Numerical treatment of many electron atoms

### A Computer Exercise in Atomic Physics

#### Originally by Eva Lindroth, Stockholm. Adapted by L. Kocbach, Bergen Oktober - November 1995 version

For pure two-body systems, like the hydrogen atom, it is possible to solve the Schroedinger equation analytically. The other elements in the periodic table are, however, many-body systems where the motion of every electron is coupled to the motion of all the other electrons. To study such a system we have to rely on some approximation scheme.

In this exercise you will meet one widely used approximation method called the Hartree-Fock method. It is based on the rather natural approximation that each electron moves in the average potential from the nucleus and the other electrons. This assumption leads to the independent-particle model, which essentially reduce the many-electron problem to the problem of solving a number of coupled single-particle equations.

The single-particle equations are solved in an iterative way which will be described below. Hartree made the first calculation based on these ideas already 1928, but calculations of this type are of course best suited for computers. Today there are several computer codes available for anyone who are interested in atomic properties. You will work with one of the first such codes, written by Herman and Skillman in 1961.

The Hartree-Fock approximation is a fast and reliable method for a wide range of atomic systems, but it is just a first approximation. Nowadays there are several calculation schemes developed which can produce much more accurate results. For very few electron systems, as helium, the ``many-body problem'' can be solved more or less exactly ( at present the non-relativistic ground state energy of helium is known with fifteen significant figures).

General many-electron systems cannot be treated with such a precision, but a large part of the electron correlation, i.e. effects beyond the independent particle model, can be accounted for with methods such as configuration interaction (CI) or perturbation theory. Here you will make a small CI calculation on helium and beryllium to get a feeling for the limitations of the independent particle model.

In fact, in the present version of the exercises, we will work only with the Hartree method.

### Question 1.

What is the difference between Hartree Method and the Hartree Fock method. If you do not know the answer, the next section might help you.