The
Helium atom
1. General facts
• consists of a nucleus with atomic number Z=2
and two electrons
1. General facts
• the state of the electrons is given by the
wavefunction
• the potential energy:
• operator of the kinetic energy:
(center of mass system, μ  me)
1. General facts
• Schrödinger equation (1.4) :
• because of the last term from the potential
energy (1.2), the potential isn´t any more
spherically symmetric (it depends on the
angle between and )
→ the Schrödinger equation can´t be solved
analytically
2. Approximate models
a) Independent electron approach
because of the repulsion of the two e→
→
3.term in (1.2) can be neglected
→ now we can use the product function:
Product approach
→ the
Schrödinger equation (1.4) is separated
into two equations, one for each e- :
(identical to Schrödinger-equation for H-atom)
Product approach
• total energy for the Helium atom
(Z=2 and both electrons in state n=1):
• experimental value:
• difference of 40%
Subtitute model
b) Substituted potential method
• one electron moves in a potential due to the
coulomb-potential of the nucleus and the
second electron
• the first electron is assumed as a spherical
symmetric cloud around the nucleus and
screens the coulomb-potential of the nucleus
Subtitute model
energy of the electron
in full coulomb-potetial
of the nucleus
energy of the electron in
the screened coulombpotential of the nucleus
and the first electron
experimental value: S = 0.65  Zeff = 1.35
while Z = 2
Variation method
c) Variation method
•
repulsion term
•
(  is our trial function)
Variation method
• guessed wavefunction for one electron:
• total energy of the electron:
Variation method
• with two electrons:
 repulsion energy:
 so we get a total energy of:
our energy, due to the guessed
function, is equal E0 or higher
Variation method
• to determine Zeff we have to minimise the
difference in the energies
Variation method
• for Helium (Z=2) we get:
• experimental value:
3. Symmetry of the wavefunction
• due to the exchange symmetry of the two
electrons, the spatial wavefunction must be
symmetric or antisymmetric
The electron spin
S=1
Tripplet
S=0
Singlet
Termscheme
The Truth
Helium has the shape of an
pineapple and taste like chicken