More electron atoms
Structure
• Due to the Pauli-principle only two
electrons can be in the ground state
• Further electrons need to be in higher states
• Pauli-principle must still be fulfilled
• In the ground state of the atom the total
energy of the electrons must be minimal
Sphere model
• Number of states:
• Considering the two different spin-quantumnumbers: 2n² states
n
1
2
3
4
Name of the sphere
K
L
M
N
Charge-distribution
• Charge-distribution of a complete sphere is
sphere-symmetric
=> Summation over the squares of the
sphere-plane-functions
Radialdistribution
Hundt´s rule
1. Full sphere and sub-spheres don´t contribute to
the total angular momentum
2. In the ground state the total spin has the
maximum value allowed by the pauli-principle
Sometimes it´s energetic more convinient to start
another sphere bevor completing the previous
sphere (lower l means higher probability to be
near the nucleus => lower energy)
Volumes and iononizing energies
• Volumes increase from the top to the bottom
and right to left in the Periodic-system
• Iononizing energies decrease from the top to
the bottom and from right to left in the
Periodic-system
Volumes and iononizing energies
Volumes and iononizing energies
Theoretical models
• Model of independent Electrons
• Hartree-method
Model of independent electrons
• We look at one electron in a effectic spheresymmetric potential due to the nucleus and
the other electrons
• The wavefunction has the same angularpart, but a different spatial-part because we
have no coulomb potential
Model of independent electrons
• Effective potential
Attraction of the
charge of the nucleus
Screening due to the
charge-distribution
of the other electrons
• Need iteration methods to get better wave-function, if we
don´t know it
The Hartree-method
• Start with a sphere-symmetric-potential considering
the screening of the other electrons
• For example:
Parameter a and b need to be adjusted…
The Hartree-method
• With the potential and the Schrödinger-equation for electron i
• We do this for all electrons
• Derive the new potential:
• Derive new
• Compare the difference between the old and the new values
for E and , if it´s larger than given difference borders, start
again with the new wavefunctions
The Hartree-method
• Total wavefunction:
• BUT: wavefunction need to be antisymmetric=>
The Hartree-method
• The handicap is that we still neglect the
interaction between the electrons
• A solution is the Hartree-Fock-method, but
this is too ugly for this presentation…
Couling schemes
• L-S-coupling (Russel-Saunders)
• j-j-coupling
L-S-coupling
• The interaction of magnetic momentum and the
spinmomentum of one electron is smaller than the
interaction between the spinmomenta si and
magnetic momenta li of all electrons
• Then the li and the si couple to:
• Total angular momentum:
j-j-coupling
• The interaction of magnetic-momentum and the
spin-momentum of one electron is bigger than the
interaction between the spin-momenta si and
magnetic-momenta li of all electrons
• =>total angular-momentum
• Only at atom with high Z
Coupling-schemes
• L-S- and j-j-coupling are both borderline
cases
• The spectra of the most atoms is a mixture
of both cases