Electronic Structure and Angular Momentum of Transition Metal Complexes

Angular Momentum vs. Number of Electron Spins

To understand the energy of paramagnetic systems (i.e., ones with unpaired electrons), we must describe them in terms of their angular momentum.

S = 0 Singlet Total Angular Momentum: S = 1/2 Doublet S = 1 Triplet J = Ms + ML S = 3/2 Quartet S = 2 Quintet

Total Multiplicity: (2S+1)(2L+1) L = 0 “S term” singly orb. deg. Each “microstate” has the same L = 1 “P term” triply L and S, but different J. L = 2 “D term” pentuply L = 3 “F term” L = 4 “G term”

1 Spin

For the Total Spin of an atom or molecule the rules apply: 1. Doubly occupied orbitals do NOT contribute to the total Spin 2. Singly occupied orbitals can be occupied with either spin-up or spin-down e- 3. Unpaired e- can be coupled parallel or antiparallel, giving a total spin S 4. For a state with total spin S there are 2S+1 “components” with M = S,S-1,...,-S. Hence terms singlet, doublet, triplet, …

5. The MS quantum number is always the sum of all individual ms QNs.

Spin Names

“Russell-Saunders” Term Symbols

for atoms; Irr Rep Mulliken for molecules

Examples for dn configurations: Atoms Molecules doublet 2H

sextet 6A

triplet 3 l=1 or 3

2 L–S Coupling L-S Coupling = Russell-Saunders Coupling

If coupling of the spin angular momentum and orbital angular momentum is relatively weak (and it usually is), then L and S remain “good” Quantum Numbers and can be treated independently of each other.

Each electronic state has its own term symbol

spin multiplicity 2S+1 L = 0 “S term” singly orb. deg. L = 1 “P term” triply L L = 2 “D term” pentuplicately L = 3 “F term” L = 4 “G term”

(Within each term, there can be several degenerate microstates with different ML and MS.)

L–S Coupling L = 0, 1, 2…total orbital angular momentum (“term”)

ML = 0, 1, 2, L components of L (ML = ml for each e).

For example, for L = 1, there are three ML values: 1, 0, -1. (analogous to l = 1 and its three ml values: 1, 0, -1) # of ML states is 2L+1 = orbital degeneracy S = total spin angular momentum

Ms = S, S-1, ….-S components of S (MS = ms).

For example, for S = 1, there are three Ms values, 1, 0, -1. Each electronic state has its own term symbol

spin multiplicity 2S+1 L = 0 “S term” singly orb. deg. L = 1 “P term” triply L L = 2 “D term” pentuplicately L = 3 “F term” L = 4 “G term”

(Within each term, there can be several degenerate microstates with different ML and MS.)

3 Hund’s Rules

1. The ground state (GS ‘term’) has the highest spin multiplicity (S).

2. If two or more terms have the same spin multiplicity, then the GS will have the highest value of L.

3. For subshells less than half-filled (e.g., p2), lowest J is preferred; for subshells more than half-filled, highest J is preferred.

Of all the states possible from degenerate orbitals, the lowest energy one will have the highest spin multiplicity (i.e., most unpaired spins).

For states with the same spin multiplicity, the highest orbital degeneracy will be lowest in energy.

The Problem: Electron-Electron Repulsion. d2

Consider as an example, 2 d electrons, one in z2

e e z2 x2-y2 g z2 x2-y2 g

t2g t2g xy xz yz xy xz yz

z z xz + z2 xy + z2 y y x x

overlapping lobes, lobes far apart, large inter-electron repulsion small inter-electron repulsion These two electron configurations differ in energy. © K. S. Suslick, 2013

4 Microstates and Spin Orbit Coupling

For a given L, the allowed values of ML and Ms are called microstates. 2N from spin (2No)! o # of microstates = up vs. down (2No –Ne)! Ne!

Where No = degeneracy of orbitals in set of subshell and Ne = number of electrons

e.g., for free atoms/ions, No for d orbitals = 5

Microstates and Spin Orbit Coupling

2N from spin (2No)! o up vs. down (2No –Ne)! Ne!

Pauli X

etc. etc. for 25 for 20 more

5 Microstates and Spin Orbit Coupling Shorthand way to describe determinantal Wave Functions, e.g., for 2 electrons:

mS of electron #1

mL of electron #1

2 e- in p orbitals: ml = -1, 0, +1, ms = -½ or +½  ( 1- 0+ ) mL = 1 0 -1

Microstates and Spin Orbit Coupling To find all the terms for a specific configuration:

3) Repeat #2 for microstates remaining

with next greater ML etc.

6 Microstates and Spin Orbit Coupling: p3 p3

examples:

Microstates and Spin Orbit Coupling: d2

Ms mL

M L 1G 3F 3P 1D 1S So, the allowed terms states for d2 are 1G, 3F, 3P, 1D, and 1S. What’s the GS?! Hund’s Rules. © K. S. Suslick, 2013

7 Hund’s Rules

1. The ground state (GS ‘term’) has the highest spin multiplicity (S).

2. If two or more terms have the same spin multiplicity, then the GS will have the highest value of L.

3. For subshells less than half-filled (e.g., p2), lowest J is preferred; for subshells more than half-filled, highest J is preferred.

Of all the states possible from degenerate orbitals, the lowest energy one will have the highest spin multiplicity (i.e., most unpaired spins).

For states with the same spin multiplicity, the highest orbital degeneracy will be lowest in energy.

Microstates and Spin Orbit Coupling

The allowed terms states for d2 are 1G, 3F, 3P, 1D, and 1S. What’s the GS? Hund’s Rules.

G.S. is 3F.

What’s the lowest excited state?

Hund’s Rules DO NOT TELL YOU THAT!! (the XS energies you get from Condon-Shortley or Racah Parameters).

8 Energy Separations of Free Ion States

Microstates and Spin Orbit Coupling, d2

9 Spin Orbit Coupling

Size of Spin Orbit Coupling

For a single electron, the strength of Spin-Orbit Coupling (SOC) in a particular microstate is measured by

For the whole Term State, SOC is measured by

10 Energy Separations of Free Ion States

But there is a fine structure within these states due to SOC!

Energy Separations of Free Ion States

“Lande Interval Rule”

Furthermore, with L-S approximation, Spin-Orbit Coupling must preserve the “center of gravity” of the energy of the term state. (n.b., degeneracies of the J states must be counted.)

11 Energy Separations of Free Ion States

“center of gravity preserved”

Energy Separations of Free Ion States

microstates:

12 Energy Separations of Free Ion States

SOC magnitude from center of gravity

L=3, S=1 J = 4, 3, 2 Degeneracy = 2J+1 © K. S. Suslick, 2013

Spin Orbit Coupling, d23F ground state

SOC magnitude from center of gravity L=3, S=1, J = 4, 3, 2 Degeneracy = 2J+1

13 Energy Separations of Free Ion States

= 9-fold degenerate

7-fold degenerate

5-fold degenerate

Energy Separations of Free Ion States The inter-electron repulsion integrals can be broken into 3 radially and angular forms:

14 Energy Separations of Free Ion States The inter-electron repulsion integrals can be broken into 3 radially and angular forms:

(more later on Tanabe-Sugano Diagrams)

Energy Separations of Free Ion States: d2

15 © K. S. Suslick, 2013

Hole Formalism

Eletrons vs. holes: dn and d10-n will have the same possible term states!

Their energies of interaction with the environment, however, will have opposite signs.

16 Spin Orbit Coupling: J-J Coupling

If spin-orbit coupling is very strong, then it can no longer be treated as just a perturbation.

mL and ms are no longer "good" quantum numbers.

Only j = mL + ms is valid.

For multi-electron systems,

applies only (not L and S).

Applies to Lanthanides, Actinides and 3rd row metals.

Size of Spin Orbit Coupling

120 cm–1 for Ti2+ to 830 cm–1 for Cu2+ (3d) 300 cm–1 for Y2+ to 1600 cm–1 for Pd2+(4d), 640 cm–1 for Ce3+ to 2950 cm–1 for Tb3+ (4f).

17 Effect of Ligands on State Energies

Simplest approach: “Crystal Field Theory”

1. metal d-electron Term States

2. Perturbation by point charges (i.e., ligands)

3. No consideration of other (non-d) electrons.

4. No covalency (added as an empirical correction in Ligand Field Theory) 5. Useful for optical spectra magnetism EPR Mössbauer

Approximations in CFT

kinetic coulombic inter-electron repulsion

spin-orbit coupling ligand field

comparable magnitudes

= net pt. charge on ligand, . l

= distance between ith electron and ligand, . l = ligand or crystal field

18 Approximations in CFT

1. Free ion with SOC:

2. Weak field

3. Strong field

Approximations in CFT

So, how do we assign states in each of these conditions and how do they correlate?

19 Effect of Ligands on State Energies

Effect of Ligands on State Energies

20 Effect of Ligands on State Energies

As an example, consider d2

Spin states through microstate analysis

Energy Correlation Diagrams: Orgel Diagram Absolute Energy Absolute Energy of States

free ion

21 The Spectrochemical Series: “Tuning the Gap”

eg I- < Br- < Cl-< OH- < RCO - < F- Δ 2 Δ - t2g < H2O < NCS < NH3 < en < bipy

- -

high spin complexes In the middle t 2g (σ only donors) “strong field” ligands (π acids) low spin complexes

KNOW THIS SERIES, AT LEAST ROUGHLY.

Ligating atom: halogen < oxygen < nitrogen < carbon

The Spectrochemical Series: “Tuning the Gap”

eg I- < Br- < Cl-< OH- < RCO - < F- Δ 2 Δ - t2g < H2O < NCS < NH3 < en < bipy

- < phen < PR3 < CN < CO

high spin complexes In the middle t 2g (σ only donors) “strong field” ligands (π acids) low spin complexes

Ligating atom: halogen < oxygen < nitrogen < carbon

22 Energy Correlation Diagrams: d2 Triplet states only Absolute Energy Absolute Energy of States

free ion

Energy Correlation Diagrams: Orgel Diagram, d2

Orgel diagrams show only the spin allowed transitions (in this case all triplets) and are in absolute energy of states. Absolute Energy of States of Energy Absolute

3 8 d2, d7 tetrahedral d , d tetrahedral d3, d8 octahedral d2 and high spin d7octahedral

23 Energy Correlation Diagrams: d2

All spin states Absolute Energy Absolute Energy of States

free weak strong ion field field

Energy Correlation Diagrams d2 Octahedral

24 Spin Selection Rule Assumes (1) spin can be separated from orbital functions, and (2) dipole operator does not affect spin.

( = GS, m = XS) l

This is the strictest of electronic selection rules (at least before 2nd row TMs).

Tanabe-Sugano Diagram for d2 Ions

3rd Excited Triplet State

2nd Excited Triplet State 1st Excited Triplet State

Energy RelativeEnergy GS to 1 Excited State

25 Tanabe-Sugano Diagram for d4 Ions

Remember: in TS Diagrams, Ground state assigned zero energy

triplet states

quintet states Energy RelativeEnergy GS to

Tanabe-Sugano Diagram for d5 Ions

4 T2g Ground state assigned to E = 0 2A E/B 1g 4 Energy scale in units of B, the T1g 4 Eg 4 “Racah parameter” T2g 4 4 (measure of inter-term repulsion). A1g, E 2A 2 1g T1g

2 T2g 2E g Critical value of  4 2 A2g, T1g To the left - weak field (or no) ligands:

4 high spin. T2g

6 A1g To the right – strong field ligands: 4 T1g higher up in spectro-chemical series low spin.

2 T2g

Weak field (h.s.)o Strong field (l.s.)

26 Tanabe-Sugano Diagram for d5 Ions

6 A1g

4 2 A2g, T1g 4 T2g 2 6 The T2g- A1g gap is the same for 4 T1g the two formats 4 2 A2g, T1g (of course!) 4 T2g 6 2 2 A1g T2g T2g

If we had kept 4T Tanabe-Sugano format: 4 1g A1g at y=0, then Always make the ground state y=0 the new ground state 2 6 The T2g- A1g gap moves “below” y=0. is the same for the two formats (of course!) 2 T2g