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More electron atoms

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.

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More electron atoms

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  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

  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)

  7. 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

  8. Volumes and iononizing energies

  9. Volumes and iononizing energies

  10. Theoretical models • Model of independent Electrons • Hartree-method

  11. 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

  12. 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

  13. 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…

  14. 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

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

  16. 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…

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

  18. 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:

  19. 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

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

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