Electronic Structure and Angular Momentum of Transition...

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© K. S. Suslick, 2013

Electronic Structureand

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 SingletS = 1/2 DoubletS = 1 TripletS = 3/2 QuartetS = 2 Quintet

L = 0 “S term” singly orb. deg.

L = 1 “P term” triply

L = 2 “D term” pentuply

L = 3 “F term”L = 4 “G term”

Total Angular Momentum:

J = Ms + ML

Total Multiplicity: (2S+1)(2L+1)

Each “microstate” has the sameL and S, but different J.

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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


Spin Names

“Russell-Saunders” Term Symbols

Examples for dn configurations:

doublet

sextet

triplet

for atoms; Irr Rep Mulliken for molecules

Atoms Molecules

2H

6A

3l=1 or 3

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Each electronic state has its own term symbol

2S+1

Lspin multiplicity L = 0 “S term” singly orb. deg.

L = 1 “P term” triplyL = 2 “D term” pentuplicatelyL = 3 “F term”L = 4 “G term”

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

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.

L–S Coupling


Each electronic state has its own term symbol

2S+1

Lspin multiplicity L = 0 “S term” singly orb. deg.

L = 1 “P term” triplyL = 2 “D term” pentuplicatelyL = 3 “F term”L = 4 “G term”

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

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.

L–S Coupling

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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

eg

t2g

xy xz yz

z2 x2-y2 eg

t2g

xy xz yz

z2 x2-y2

xz + z2 xy + z2

overlapping lobes, large inter-electron repulsion

lobes far apart, small inter-electron repulsion

These two electron configurations differ in energy.

z

x

y

x

z

y

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

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Microstates and Spin Orbit Coupling

For a given L, the allowed values of ML and Ms

are called microstates.

# of microstates =

Where No = degeneracy of orbitals in set of subshelland Ne = number of electrons

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

(2No)! (2No – Ne)! Ne!

2No from spin up vs. down



(2No)! (2No – Ne)! Ne!

2No from spin up vs. down

etc.for 20more

Pauli X

etc.for 25

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Shorthand way to describe determinantal Wave Functions,

e.g., for 2 electrons:

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

mL = 1 0 -1 ( 1- 0+ )

mL of electron #1

mS of electron #1



To find all the terms for a specificconfiguration:

3) Repeat #2 for microstates remainingwith next greater ML etc.

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Microstates and Spin Orbit Coupling: p3

examples:

p3


Microstates and Spin Orbit Coupling: d2

MsmL

1G3F3P1D1S

So, the allowedterms states for d2 are 1G, 3F,3P, 1D, and 1S.

What’s the GS?!Hund’s Rules.

ML

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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.



G.S. is 3F.

What’s the lowest excited state?

The allowed terms states for d2 are 1G, 3F, 3P, 1D, and 1S.

What’s the GS? Hund’s Rules.

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

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Energy Separations of Free Ion States


Microstates and Spin Orbit Coupling, d2

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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

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



“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.)

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“center of gravity preserved”



microstates:


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L=3, S=1 J = 4, 3, 2Degeneracy = 2J+1

SOCmagnitudefrom centerof gravity



Spin Orbit Coupling, d2 3F ground state

L=3, S=1, J = 4, 3, 2Degeneracy = 2J+1

SOCmagnitudefrom centerof gravity

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= 9-fold degenerate

7-fold degenerate

5-fold degenerate




The inter-electron repulsion integrals can be broken into 3 radially and angular forms:

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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

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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.

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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 Coupling120 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).

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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 correctionin Ligand Field Theory)

5. Useful for optical spectramagnetismEPRMössbauer


Approximations in CFT

kinetic coulombic inter-electron repulsion

spin-orbit coupling ligand field

comparable magnitudes

= net pt. charge on ligand, l.

= distance between ith electronand ligand, l.

= ligand or crystal field

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1. Free ion with SOC:

2. Weak field

3. Strong field



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

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Spin states through microstate analysis

As an example, consider d2


Energy Correlation Diagrams: Orgel Diagram

Ab

solu

te E

ner

gy

of

Sta

tes

freeion

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The Spectrochemical Series: “Tuning the Gap”

eg

t2g

eg

t 2g

weak field ligand(π bases)

high spin complexes“strong field” ligands

(π acids)low spin complexes

Δ Δ

In the middle(σ only donors)

KNOW THIS SERIES, AT LEAST ROUGHLY.

Ligating atom: halogen < oxygen < nitrogen < carbonsmall o large o

less e- donating more e- donatingless covalent more covalent

I- < Br- < Cl-< OH- < RCO2- < F-

< H2O < NCS- < NH3 < en < bipy

<NO2- < phen < PR3 < CN- < CO <NO


The Spectrochemical Series: “Tuning the Gap”

eg

t2g

eg

t 2g

weak field ligand(π bases)

high spin complexes“strong field” ligands

(π acids)low spin complexes

Δ Δ

In the middle(σ only donors)

Ligating atom: halogen < oxygen < nitrogen < carbonsmall o large o

less e- donating more e- donatingless covalent more covalent

I- < Br- < Cl-< OH- < RCO2- < F-

< H2O < NCS- < NH3 < en < bipy

< phen < PR3 < CN- < CO <NO

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Ab

solu

te E

ner

gy

of

Sta

tes

freeion

Triplet states only

Energy Correlation Diagrams: d2


Energy Correlation Diagrams: Orgel Diagram, d2

Ab

solu

te E

ner

gy

of

Sta

tes

d2, d7 tetrahedrald3, d8 octahedral

d3, d8 tetrahedrald2 and high spin d7octahedral

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

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Energy Correlation Diagrams: d2

Ab

solu

te E

ner

gy

of

Sta

tes

free weak strong ion field field

All spin states


Energy Correlation Diagrams

d2 Octahedral

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Spin Selection Rule

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

Assumes (1) spin can be separated from orbital functions, and(2) dipole operator does notaffect spin.

(l = GS, m = XS)


Tanabe-Sugano Diagram for d2 Ions

En

erg

y R

elat

ive

to G

S

free weak strong ion field field

Ground Statetriplet

1st Excited Triplet State

1st ExcitedState

2nd Excited Triplet State

3rd Excited Triplet State

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Remember: in TS Diagrams,Ground state assignedzero energy


En

erg

y R

elat

ive

to G

S

tripletstates

quintetstates


Ground state assigned to E = 0

Energy scale in units of B, the“Racah parameter” (measure of inter-term repulsion).

Critical value of To the left - weak field (or no) ligands:

high spin.

To the right – strong field ligands:higher up in spectro-chemical serieslow spin.

E/B

o

2T2g

4A1g, 4E

4T2g

4T1g

4T2g

4T1g

2A1g

4T2g

2T2g

6A1g

2Eg

4A2g, 2T1g

2T1g

2A1g

4Eg

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


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Tanabe-Sugano format:Always make the ground state y=0

2T2g

6A1g

If we had kept4A1g at y=0, thenthe new ground state moves “below” y=0.

2T2g

6A1g

The 2T2g-6A1g gap is the same forthe two formats(of course!)


4T2g

4T1g

2T2g

4A2g, 2T1g

4T2g

4T1g

4A2g, 2T1g

The 2T2g-6A1g gap is the same forthe two formats(of course!)

Electronic Structure and Angular Momentum of Transition...

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