## 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.