More electron atoms.

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Presentation transcript:

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-quantum-numbers: 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 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)

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 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

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

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