Electronic Structure of Metal Atoms and Ions –Further Splitting of Levels

e-

… and spin

… orbital angularmomentum (l > 0) …

electrons can have…

magnetic moments (whichcan couple) are associatedwith these motions

this isthe spin orbitinteraction

Electronic Structure of Metal Atoms and Ions –Further Splitting of LevelsTo understand the electronic spectra of transition metal complexes, oneneeds to know the electronic states of the system.

The following effects lead to a splitting of the energy levels of transitionmetal atoms (and ions):

- Electron-electron repulsion → spectroscopic terms- Crystal field → spectroscopic terms- Spin-Orbit Interaction → multiplet- Spin-Spin Interaction → magnetic states- Zeeman Effect → magnetic states- Hyperfine Interaction

Of these, only the first two effects are of importance in UV/Vis (electron) spectroscopyas they lead to a splitting of the energy levels which are on the same order of magnitudeas the energy of the applied UV/vis light.

The other effects are of importance in magnetism and EPR spectroscopy(see the upcoming lectures)

Electronic Structure of Metal Atoms and Ions

e- dn

ATOMICTERM

weak ligandfield

Interelectronrepulsion

CRYSTAL-FIELD TERMS

dn

strong ligandfield

ElectronConfiguration

Interelectronrepulsion

Spin-OrbitCoupling

Multiplets

Magnetic Field

Magnetic States

Zeeman States

Spin-OrbitCoupling

MultipletsAtomic

Magnetic Field

Zeeman States

Spin-Orbit Coupling

Naground states = 1/2, l = 0

2S1/2

Naexited states = 1/2, l = 1

2P1/22P3/2

e-

2P3/2 3λ22P1/2

589,16 nm589,76 nm

(16973 cm-1)(16956 cm-1)

E

17 cm-1∆E =

∆E =

λ = 11 cm-1

Spin-Orbit coupling parameters, λ, for TM ions (ground state)

Ion dn Ground state λ / cm–1

Ti3+ d1 2D 154V3+ d2 3F 104V2+ d3 4F 55Cr3+ d3 4F 87Cr2+ d4 5D 57Mn3+ d4 5D 85Fe2+ d6 5D –100Co2+ d7 4F –180Ni2+ d8 3F –335Cu2+ d9 2D –852

• Spin-Orbit coupling is a small perturbation, applies well for 3d elements(but not for lanthanides, actinides)

• Russel-Saunders-Coupling LS-Coupling

Total spin J = L + S (vectors)Total spin quantum number: J (scalar) : J = |L-S| to |L+S|Terms Symbol: (2S+1)LJEnergy of the term: E(J, L, S) = 1/2λ[J(J+1)-L(L+1)-S(S+1)]

HSO = ζl·sHSO = λL·S

λ = ±ζ/(2S)

ζ = one-electron spin-orbit coupling constant(always positive), dimension= cm-1 (energy)

λ = many-electron spin-orbit coupling constant(always positive), dimension= cm-1 (energy)

Only valid forthe ground term

positive for n = 1-4

Spin-Orbit coupling parameters, ζ, for some elements

Cr0 ζ3d = 227 cm–1

Cr2+ ζ3d = 230 cm–1

Mo2+ ζ4d = 749 cm–1

Mo3+ ζ4d = 818 cm–1

B ζ2p = 11 cm–1

Tl ζ6p = 5195 cm–1

Values taken from

Spin-orbit coupling of the d2 configuration

d2

E

1S

3P

1G

1D

3F

e- - e- repulsion

spin-orbit

-4λJ = 2(5)

J = 3-λ

3λJ = 4

(5)

(7)

(9)

J = 2-2λ

J = 0(1)

J = 1-λ

λJ = 2(3)

(5)J = 4(9)

J = 0

d2, λ > 0

The spin orbitinteraction splits

the atomic terms intoj levels

Russel-Saunders Coupling(LS-coupling)

Spin-Orbit Coupling in Crystal Field TermsExample: Terms of the d2 configuration

d2

E

1S

3P

1G

1D

3F

e- - e- repulsion

crystal field

2ε0

6Dq

3T1g

3T2g

3A2g

2Dq

12Dq

3T1g (P)

In coordination compounds spin-orbit coupling can be of• First order (only for orbitally degenerate states, e.g. T terms)• Second order (for orbitally non-degenerate states, e.g. E and A terms)

Problem: Meaning of L in cystal field terms is lost

dn

ATOMICTERM

weak ligandfield

Interelectronrepulsion

CRYSTAL-FIELD TERMS

dn

strong ligandfield

ElectronConfiguration

Interelectronrepulsion

Spin-OrbitCoupling

Multiplets

Magnetic Field

Magnetic States

Zeeman States?

First Order Spin-Orbit Coupling, splitting of terms

In coordination compounds orbital momentum means:electron can move from one orbital to another, degenerate orbital. However, dxy, dxz, dyz, and dzz, dx2-y2 are no longerdegenerate in a complex.

In an octahedral complex, e-can only move within anopen t2g shell (first order orbital momentum => of importance in magnetochemistry)

d1, d2, (l.s.)-d4, (l.s.)-d5, etc have first order orbital momentum (T ground terms), d3, d4 have no first order orbital momentum (A, E ground terms

Terms with T symmetryexhibit orbital angular momentum

can show spin-orbit couplingThis rule is only applicable in Oh

Symmetry.

Terms with T symmetryexhibit L = 1, HSO = -AλLS

EJ = -1/2Aλ[J(J+1)-L(L+1)-S(S+1)For (t2g)n less than half occupied: λ positive

more than half occupied: λ negative

dx2-y2

dxy

(leer)

First Order Spin-Orbit Coupling, splitting of terms, d1 configuration

Ti3+, d1, has a 2T2g ground term, shows first order spin-orbit coupling

eg

t2g

∆o

2Eg

2T2g

d1 d1

e--e- -repulsion

crystal field

λ

−λ/2

Energy S L J

1/2

1/2

1

1

1/2

3/2

spin orbit coupling

First Order Spin-Orbit Coupling, splitting of terms, d1 configuration

Quenching of the orbital contribution, to the magnetic moment, due to ligand field

n ground ent2m ligand field quenchingterm term

1 2D e1 2E Yes2 3F e2 3A2 Yes3 4F e2t21 4T1 No4 5D e2t22 5T2 No5 6S e2t23 6A1 Yes6 5D e3t23 5E Yes7 4F e4t23 4A2 Yes8 3F e4t24 3T1 No9 2D e4t25 2T2 No

These ionshave L = 1

Tetrahedral symmetry

First Order Spin-Orbit Coupling, splitting of terms, d1 configuration

Quenching of the orbital contribution, to the magnetic moment, due to ligand field

n ground t2gneg

m ligand field quenchingterm term

1 2D t2g1 2T2g No

2 3F t2g2 3T1g No

3 4F t2g3 4A2g Yes

4 5D t2g3eg

1 5Eg Yest2g

4 3T1g No5 6S t2g

3eg2 6A1g Yes

t2g5 2T2g No

6 5D t2g4eg

2 5T2g Not2g

6 1A1g Yes7 4F t2g

5eg2 4T1g No

t2g6eg

1 2Eg Yes8 3F t2g

6eg2 3A2g Yes

9 2D t2g6eg

3 2Eg Yes

These ionshave L = 1

Octahedral symmetry

Second Order Spin-Orbit Coupling

TM ions with E and A ground terms show second order spin-orbit coupling

The s.o. s-o-coupling leads to a stabilization of the ground term and a destabilization of the excited state

A terms are stabilized by –8λ2/∆o

E terms are stabilized by –4λ2/∆o

The Zero field splitting D is also associated with thesecond order spin-orbit coupling (see lectures on Magnetochemistry and EPR)

∑ −

ΦΦΦΦ

EeELSLS

g

egggee λλ

Energy levels of Co2+: e-/e-repulsion, crystal field, 1st order spin-orbit coupling and Zeeman splitting

d7

4F

4P

2H

2G

2F

2D

4T1g

4T2g

4A2g

4T1g

J =1/2

J =3/2

J =5/2

-5Aλ/2

-Aλ

3Aλ/2e-/e-repulsion

crystalfield

1st orderspin-orbitcoupling

ZeemanSplitting

This will be part of the EPRlecture