1. Slide 1/16 Where Are We Going…? Week 10: Orbitals and Terms  Russell-Saunders coupling of orbital and spin angular momenta  Free-ion terms for p 2 Week 11: Terms and ionization energies  Free-ion terms for d 2  Ionization energies for 2p and 3d elements Week 12: Terms and levels  Spin-orbit coupling  Total angular momentum Week 13: Levels and ionization energies  j-j coupling  Ionization energies for 6p elements

2. 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 1s 2, C 1s 2 2s 2 2p 2 r ij e2e2  H H-like = riri Ze    ½ mv i 2 + i i Inclusion of interelectron repulsion leads to terms  for example: p 2 1 D, 3 F and 1 S  characterized by S and L quantum numbers  energy given by Hund’s 1 st and 2 nd rules  (2S+1)(2L+1) degenerate  i≠j

3. 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 M S

4. 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 M L

5. 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. rotation of a p x orbital by 90° gives a p y orbital and vice versa: generating magnetism about the z-direction

6. 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 p x orbital by 90° gives a p y orbital and vice versa: generating magnetism about the z-direction L = 1 L = 0

7. 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

8. 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  l 1 + l 2 + … = L, s 1 + s 2 + … = S r ij e2e2   i≠j H = H H-like + λL.S + Then treat spin-orbit second to give levels:  L + S = J  J is the total angular momentum configurations termslevels

9. 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, M J = J, J -1, J - 2, … -J (2J+1 values) 2S+1 L J

10. Slide 10/16 Hund’s 3 rd Rule For less than half-filled shells, smallest J lies lowest  p 2 : ground term is 3 P with S = 1 and L = 1  J = 2, 1 and 0  less than half-filled: 3P3P 3P03P0 3P13P1 3P23P2

11. Slide 11/16 Hund’s 3 rd Rule For more than half-filled shells, highest J lies lowest  p 4 : ground term is 3 P with S = 1 and L = 1  J = 2, 1 and 0  more than half-filled: 3P3P 3P23P2 3P13P1 3P03P0

12. Slide 12/16 Magnetism The magnetic moment is given by:  where g is the Landé splitting factor, p 2 : ground level is 3 P 0 with J = 0, S = 1, L = 1  μ eff = 0 (p 2 is diamagnetic, at least at low temperature) p 4 : ground level is 3 P 2 with J = 2, S = 1, L = 1  g = 3/2 and μ eff = 3.68 B.M. (B.M. = “ Bohr Magnetons ” )

13. Slide 13/16 Ionization Energies: (iii) Hund’s 3 rd Rule For 6p, there is a decrease between p 2 and p 3  No half-filled shell effect! p-block ionization energies: M  M +

14. Slide 14/16 j-j Coupling For very heavy elements, magnetic coupling becomes large Then add individual j values together to give J  j 1 + j 2 + … = 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

15. 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 p 4 due to pairing (1 st rule) If magnetic > electrostatic  overall increase due to increasing nuclear charge  decrease in ionization energy for p 3 due to repulsive magnetic interaction (3 rd rule)

16. 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 3 rd Rule Lowest J lies lowest for < 1/2 filled shells Highest J lies lowest for > 1/2 filled shells Consequences Magnitude of magnetism due to J, L and S Stabilization of p 1 and p 2, destabilization of p 4 – p 6 Task! Work out ground levels and magnetism for f n elements