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Many electron systems
The atomic shell model revisited
=> electron-electron interaction
At first Hww is neglected, for each electron we get a hydrogen problem
Due to Pauli principle, each term can only be occupied with one
electron => This yeilds the Periodic table, works regularly only for the
first 18 electrons. After the 3p the 4s and not the 3d shell is filled.
This is due to Hww
Atomic shell model
Estimate the effect of Wss
For small distances r->0, the electron sees the unshielded nucleas
for large distances the nucleus and (Z-1) electrons form
an almost spherical charge distribution(core)
V (r )  
4 0 r
Ze 2 1
V (r )  
4 0 r
Effective potential showing
Screening of the nuclear charge by
the electrons
Alkali atom : consist of a full noble gas configuration with an additiona
valance electron = “Leucht” electron
(nl , l )  quantum defect
 EG
me  mEG
me  mEG
with mEG = mass of the noble gas core
Level-scheme of Li
Model of an Alkali-atom
valence electron
Table: Quantum defect (n, l )
note:for large l the quantum defect disappears
The Helium atom = simplest many electron atom= 2 electrons
For time being let us neglect V, or set V=0
Write eigen function as a product of hydrogen functions
The total wave functin for Femions (particles with s=
special case
This holds for the symmetric spin function
electrons) must be antisymmetric
Space wave function of two particles:
Probability density for both cases
Spin wave function of the two electronic system
Total spin wave function is
Total spin wave function is antisymmetric
Total wave function is product wave function of space and spin part and always antisymmetric
Wave Function of the ground state of helium with S=0
note: changing the direction of the spin costs ≈ 40eV and this is
happening without a spin dependance of the Hamilton operator
Effect of the electron-electron interaction
a. for singulet s=0
ground state
due to the second electron
note: The good agreement between calculated and experimental value
b. for the first excited state of the singulet respectively triplet system:
The exchange energy is ~ 0.4ev, Spin Triplet state is lower
Energy levels of the excited-terms of the helium, shown is the
effect of the direct integral J and the exchange integral K
Doubly excited states in helium
larger than -24.6 eV the one electron ionization of helium
Helium atom
• Para helium S=0
• Ortho helium S=1
The allowed electric dipole transitions
are indicated