Slide 1/16
Where Are We Going…?
• Week 10: Orbitals and TermsRussell-Saunders coupling of orbital and spin angular momentaFree-ion terms for p2
• Week 11: Terms and ionization energiesFree-ion terms for d2 Ionization energies for 2p and 3d elements
• Week 12: Terms and levelsSpin-orbit couplingTotal angular momentum
• Week 13: Levels and ionization energiesj-j couplingIonization energies for 6p elements
Many Electron Atoms
• For any 2 e- atom or ion, the Schrödinger equation cannot be solved for every electron:
• Treatment leads to configurations
for example: He 1s2, C 1s2 2s2 2p2
rij
e2
HH-like = ri
Ze ½ mvi2 +
ii
• Inclusion of interelectron repulsion leads to terms
for example: p2 1D, 3F and 1S
characterized by S and L quantum numbers
energy given by Hund’s 1st and 2nd rules
(2S+1)(2L+1) degenerate
i≠j
Slide 3/16
Magnetism Due To Spin
• Electron(s) with spin angular momentum generate a magnetic field perpendicular to plane of loop
magnitude related to S
direction related to MS
Slide 4/16
Magnetism Due To Orbit
• Electron(s) with orbital angular momentum generate a magnetic field perpendicular to plane of loop
magnitude related to L
direction related to ML
Orbital Magnetism
• Electrons generate magnetism through their orbital motion• This is associated with an ability to rotate an orbital about an axis into
an identical and degenerate orbital.
x
y
z
x
y
z
rotation of a px orbital by 90° gives a py orbital and vice versa: generating magnetism about the z-direction
Slide 6/16
Orbital Magnetism
• To be able to do this: the orbitals involved must have the same energy there must not be an electron in the second orbital with the same
spin as that in the first orbital. If there is, the electron cannot orbit without breaking the Pauli principle.
free orbitals available for
electron to hop into:orbital magnetism
free orbital available for electron to hop
into:orbital magnetism
no free orbital available for electron to hop into:no orbital magnetism
rotation of a px orbital by 90° gives a py orbital and vice versa: generating magnetism about the z-direction
L = 1 L = 1 L = 0
Slide 7/16
Spin Orbit Coupling
• There is a magnetic interaction between the magnetism generated by the spin and orbital motions
results in different values for the total angular momentum, J
orbital magnetism spin magnetism
lowest energy highest energy
Russell – Saunders Coupling
• The magnetic interaction increases with the atomic number
for most of the periodic table, electrostatic >> magnetic
• Treat electrostatic to give terms characterized by L and S
l1 + l2 + … = L, s1 + s2 + … = S
rij
e2
i≠j
H = HH-like + λL.S+
• Then treat spin-orbit second to give levels:
L + S = J
J is the total angular momentum
configurations terms levels
Slide 9/16
Russell – Saunders Coupling
• For each L and S value:
J = L + S, L + S – 1, L + S – 2 …. L – S
Each level, MJ = J, J -1, J - 2, … -J (2J+1 values)
2S+1L
J
Slide 10/16
Hund’s 3rd Rule
• For less than half-filled shells, smallest J lies lowest
p2: ground term is 3P with S = 1 and L = 1
J = 2, 1 and 0
less than half-filled:
3P
3P0
3P1
3P2
Slide 11/16
Hund’s 3rd Rule
• For more than half-filled shells, highest J lies lowest
p4: ground term is 3P with S = 1 and L = 1
J = 2, 1 and 0
more than half-filled:
3P
3P2
3P1
3P0
Slide 12/16
Magnetism
• The magnetic moment is given by:
where g is the Landé splitting factor,
12effμ g[J(J 1)]
[S(S 1) L(L 1)]3g2 2J(J 1)
• p2: ground level is 3P0 with J = 0, S = 1, L = 1
μeff = 0 (p2 is diamagnetic, at least at low temperature)
• p4: ground level is 3P2 with J = 2, S = 1, L = 1
g = 3/2 and μeff = 3.68 B.M. (B.M. = “Bohr Magnetons”)
Slide 13/16
0
5
10
15
20
25
1 2 3 4 5 6pn
ion
iza
tion
en
erg
y (e
V)
2p
3p4p5p6p
Ionization Energies: (iii) Hund’s 3rd Rule
• For 6p, there is a decrease between p2 and p3
No half-filled shell effect!
p-block ionization energies: M M+
5
6
7
8
9
10
11
12
1 2 3 4 5 6pn
ion
iza
tion
en
erg
y (e
V) 6p
Slide 14/16
j-j Coupling
• For very heavy elements, magnetic coupling becomes large
• Then add individual j values together to give J
j1 + j2 + … = J
• Treat spin-orbit first to give spin-orbitals for each electron:
j = l + s each value is (2j+1) degenerate
• For p-electrons, l = 1 and s = 1/2
j = 1/2 and 3/2 with former lowest in energy
j = 1/2
j = 3/2
Slide 15/16
j-j Coupling
• For p-electrons, l = 1 and s = 1/2
j = 1/2 and 3/2 with former lowest in energy
j = 1/2
j = 3/2
• If electrostatic >> magnetic overall increase due to increasing nuclear charge decrease in ionization energy for p4 due to pairing (1st rule)
• If magnetic > electrostatic overall increase due to increasing nuclear charge decrease in ionization energy for p3 due to repulsive magnetic
interaction (3rd rule)
Summary
Spin and orbital magnetism • Electrons have intrinsic magnetism due to spin• Electrons may also have orbital magnetism Spin-orbit coupling• Usually weak magnetic coupling between spin and orbit• Characterized by levels with total angular momentum, J Hund’s 3rd Rule• Lowest J lies lowest for < 1/2 filled shells• Highest J lies lowest for > 1/2 filled shellsConsequences• Magnitude of magnetism due to J, L and S• Stabilization of p1 and p2, destabilization of p4 – p6
Task!• Work out ground levels and magnetism for fn elements