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

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 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

Attraction of thecharge of the nucleus

Screening due to thecharge-distributionof the other electrons

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