**The central field approximation.
**

The treatment is based on an approximate method where the central part of the electron-electron interaction is included from the beginning. The Hamilton function of the system is written

The corresponding Hamilton operator is

H_{1} is uncomplicated, leading to a system of uncoupled differential
equations which may be solved using a wavefunction that is a product of
one-electron wavefunctions in the usual manner. At the moment, we also
restrict the discussion to atoms which are not too heavy, which allows
us to treat H_{3} as a perturbation. Remaining is H_{2},
which we know leads to coupled differential equations. However, H_{2}
may be separated into two parts, one of which is centrally symmetric. In
a first approximation, this part is maintained along with H_{1}
in the Schrödinger equation. This is the important __central field
approximation__, which may be stated as follows:

In the central field approximation, each electron moves independently of the other electrons in the atom in a spherically -symmetric average field created by the nucleus and the other electrons.

This approximation provides a much better description of the atom than
if only H_{1} is considered. The Hamiltonian may for each electron
be written

where the potential energy operator V(r_{i}) is spherically
symmetric. The Hamiltonian reduces to a sum of one-electron operators,
each giving rise to a one-electron eigenfunction
,
which is also referred to as an *orbital*. For each one-electron operator
H_{i }an eigenvalue equation can be formulated as

The wavefunction
for the Hamilton operator H_{0} may then be written in terms of
a product of one-electron functions, i.e.

while the energy eigenvalue E_{0} of H_{0 }is obtained
as a sum of one-electron energies, i.e.

Since the potential is spherically symmetric, each wavefunction
will as in the one-electron case be an eigenfunction to the operators **l**_{i}^{2},
**s**_{i}^{2}, l_{iz} and s_{iz}. The
state for each electron is therefore defined by the four quantum numbers
n, l, m_{l}, m_{s} or n, l, j, m dependent upon which representation
is preferred.

The energy for each electron is determined solely by n and l, which
means that the wavefunctions are degenerate with respect to m_{l }and
m_{s}. According to the Pauli exclusion principle we may therefore
have a maximum of 2(2l+1) electrons with the same n and l.

The 2(2l+1) electrons with the same n and l are said to form a closed shell, often called subshell. All closed shells form symmetric charge distributions.

Last change: 19 February 2001