Multi electron atomsMulti electron atoms

•Atoms with Z>1 contain >1 electron. This changes the atomic structure considerably because in addition to the electron-nucleus interaction, there is the repulsive electron-electron interaction.

•Calculations show that allowed electron energies are no longer solely determined by the single quantum number, n.

•Several distinct electron states (orbits) exist, all with the same n, forming a `shell' of states.

•In general, these states have different energies.

•The number of different orbital states in a shell of a given n is n2.

•Atoms with Z>1 contain >1 electron. This changes the atomic structure considerably because in addition to the electron-nucleus interaction, there is the repulsive electron-electron interaction.

•Calculations show that allowed electron energies are no longer solely determined by the single quantum number, n.

•Several distinct electron states (orbits) exist, all with the same n, forming a `shell' of states.

•In general, these states have different energies.

•The number of different orbital states in a shell of a given n is n2.

Multi electron atomsMulti electron atoms

•The electrons interact not only with the nucleus but also among themselves. It is difficult to get the wave function.

•Electron configuration: How do the electrons fill the shells and subshells? How to get its ground state?

•In the ground states of atoms, electrons occupy the lowest energy states available consistent with the exclusion principle.

•The electrons interact not only with the nucleus but also among themselves. It is difficult to get the wave function.

•Electron configuration: How do the electrons fill the shells and subshells? How to get its ground state?

•In the ground states of atoms, electrons occupy the lowest energy states available consistent with the exclusion principle.

The Pauli exclusion principleThe Pauli exclusion principle

The Pauli exclusion principle states that only one electron can be in a given state, which is labeled by four quantum numbers (n,l,ml,ms):

o n, Principal quantum number, the energy levelo l, Orbital quantum number, orbital angular momentum,o ml (Orbital) magnetic quantum number,

o ms Spin (magnetic) quantum number,

The Pauli exclusion principle states that only one electron can be in a given state, which is labeled by four quantum numbers (n,l,ml,ms):

o n, Principal quantum number, the energy levelo l, Orbital quantum number, orbital angular momentum,o ml (Orbital) magnetic quantum number,

o ms Spin (magnetic) quantum number,

Hund’s ruleHund’s rule

Greatest stability results if the atomic orbitals (AO) in a degenerate set are half-filled with electrons before any of them are filled

Greatest stability results if the atomic orbitals (AO) in a degenerate set are half-filled with electrons before any of them are filled

Aufbau principleAufbau principle

a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy orbitals are filled before electrons are placed in higher-energy orbitals.

a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy orbitals are filled before electrons are placed in higher-energy orbitals.

Orbitals are filled in the order of increasing n+l;

Where two orbitals have the same value of n+l, they are filled in order of increasing n.

This gives the following order for filling the orbitals:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, and 7p

Orbitals are filled in the order of increasing n+l;

Where two orbitals have the same value of n+l, they are filled in order of increasing n.

This gives the following order for filling the orbitals:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, and 7p

• Each shell is identified by a letter according to its n value: The innermost shell (n = 1) is called the K-shell, the next innermost shell (n = 2) is called the L-shell, etc. •No two electrons can occupy precisely the same state.

• Each shell is identified by a letter according to its n value: The innermost shell (n = 1) is called the K-shell, the next innermost shell (n = 2) is called the L-shell, etc. •No two electrons can occupy precisely the same state.

nn ShellShell Maximum number of electrons 2n2

11 KK 22

22 LL 88

33 MM 1818

44

…………..NN 3232

s + ss + s

s - ss - s

pz - pzpz - pz

pz + pzpz + pz

px(y) - px(y)px(y) - px(y)

px(y) + px(y)px(y) + px(y)

Molecular OrbitalsMolecular Orbitals MO → LCAOMO → LCAO

s orbital

pz orbital

px orbital py orbital

φA Combination with φB No combination with φB

s  s, pz, dz² px, py, dx²-y², dxy, dyz, dxz

pz s, pz, dz² px, py, dx²-y², dxy, dyz, dxz

px #  px, dxz s, py, dx²-y², dz², dxy, dyz

dxz # px, dxz s, py, dx²-y², dz², dxy, dyz

dx²-y² dx²-y² s, px, py, pz, dz², dxy, dyz, dxz

dz² s, pz, dz² px, py, dx²-y², dxy, dyz, dxz

Energy diagramEnergy diagram

s

s

s

s

px py pxpy

pzpz

u*

u*

u*

g

g*

g

g

u

Dissociation energy eV:

gsgsgPususUpg[2px]g[2py]u[2px]u[2py]

gs

gs

gP

us

us

Up

g[2px]

g[2py]

u[2px]

u[2py]

SäkulargleichungSäkulargleichung

•Eine Verschmelzung unterschiedlicher Strukturen bezeichnet man als Hybridisierung. Es entsteht ein Hybrid.

•In der Chemie verschmelzen verschiedene Elektronenorbitale (s, p, d) zu neuen Hybridorbitalen.

•Hybridisierung tritt nur bei Atomen auf, die kovalente Bindungen zu anderen Atomen aufweisen.

•Hybridisierung erzeugt energetisch stabile zumeist dreidimensionale geometrische Strukturen. •Nur Valenzorbitale (äußerste Schale, auch wenn im Grundzustand nicht besetzt) eines Atoms können an der Hybridisierung teilnehmen.

•Hybridorbitale erzeugen stets -Bindungen. -Bindungen - "Doppelbindungen" - entstehen aus nicht hybridisierten p-Orbitalen

•Eine Verschmelzung unterschiedlicher Strukturen bezeichnet man als Hybridisierung. Es entsteht ein Hybrid.

•In der Chemie verschmelzen verschiedene Elektronenorbitale (s, p, d) zu neuen Hybridorbitalen.

•Hybridisierung tritt nur bei Atomen auf, die kovalente Bindungen zu anderen Atomen aufweisen.

•Hybridisierung erzeugt energetisch stabile zumeist dreidimensionale geometrische Strukturen. •Nur Valenzorbitale (äußerste Schale, auch wenn im Grundzustand nicht besetzt) eines Atoms können an der Hybridisierung teilnehmen.

•Hybridorbitale erzeugen stets -Bindungen. -Bindungen - "Doppelbindungen" - entstehen aus nicht hybridisierten p-Orbitalen

HybridisierungHybridisierung

HybridisierungHybridisierung

s – Orbital + p – Orbital sp-Hybridorbitale

s – Orbital + 2p – Orbitale sp2 -Hybridorbitale

s – Orbital + 3p – Orbitale sp3 Hybridorbitale