1. More electron atoms

2. 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

3. Sphere model Number of states:
Considering the two different spin-quantum-numbers: 2n² states n 1 2 3 4
Name of the sphere K L M N

4. Charge-distribution
Charge-distribution of a complete sphere is sphere-symmetric
=> Summation over the squares of the sphere-plane-functions

5. Radialdistribution

6. Hundt´s rule
Full sphere and sub-spheres don´t contribute to the total angular momentum
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)

8. 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

9. Volumes and iononizing energies

10. Volumes and iononizing energies

11. Theoretical models
Model of independent Electrons
Hartree-method

12. Model of independent electrons
We look at one electron in a effectic sphere-symmetric potential due to the nucleus and the other electrons
The wavefunction has the same angular-part, but a different spatial-part because we have no coulomb potential

13. Model of independent electrons
Effective potential
Need iteration methods to get better wave-function, if we don´t know it
Screening due to the
charge-distribution
of the other electrons
Attraction of the
charge of the nucleus

14. 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…

15. 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

16. The Hartree-method Total wavefunction:
BUT: wavefunction need to be antisymmetric=>

17. 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…

18. Couling schemes
L-S-coupling (Russel-Saunders)
j-j-coupling

19. 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:

20. 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

21. Coupling-schemes L-S- and j-j-coupling are both borderline cases
The spectra of the most atoms is a mixture of both cases