Electronic Structure of Metal Atoms and Ions – Further Splitting of Levels magnetic moments (which can couple) are associated with these motions

this is thespinorbit interaction

electrons can have… e- … orbital angular momentum (l > 0) … … and spin Electronic Structure of Metal Atoms and Ions – Further Splitting of Levels To understand the electronic spectra of transition metal complexes, one needs to know the electronic states of the system.

The following effects lead to a splitting of the energy levels of transition metal 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) spectroscopy as they lead to a splitting of the energy levels which are on the same order of magnitude as 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

dn n e- d Interelectron strong ligand repulsion field ATOMIC Electron Spin-Orbit TERM Configuration Coupling weak ligand Interelectron field repulsion

CRYSTAL-FIELD TERMS

Spin-Orbit Atomic Coupling Multiplets Multiplets

Magnetic Field

Magnetic Field Zeeman States Magnetic States

Zeeman States Spin-Orbit Coupling

Na exited state s = 1/2, l = 1 e-

2P E 3/2 3λ Na ∆E = 2P 2 2P 2 1/2 ground state 3/2 P1/2 s = 1/2, l = 0

589,16 nm (16973 cm-1) 589,76 nm (16956 cm-1)

∆E = 17 cm-1

-1 2 λ = 11 cm S1/2 Spin-Orbit coupling parameters, λ, for TM ions (ground state)

Ion dn Ground state λ / cm–1 ζ = one-electron spin- orbit coupling constant Ti3+ d12D 154 (always positive), dimension V3+ d23F 104 = cm-1 (energy) V2+ d34F55 3+ 34 Cr d F87λ = many-electron spin- Cr2+ d45D57orbit coupling constant Mn3+ d45D85(always positive), dimension Fe2+ d65D –100 = cm-1 (energy) Co2+ d74F –180 Ni2+ d83F –335 Cu2+ d92D –852

• Spin-Orbit coupling is a small perturbation, applies well for 3d elements (but not for lanthanides, actinides) H = ζl·s • Russel-Saunders-Coupling LS-Coupling SO HSO = λL·S Total spin J = L + S (vectors) λ = ±ζ/(2S) Total spin quantum number: J (scalar) : J = |L-S| to |L+S| (2S+1) Terms Symbol: LJ Only valid for Energy of the term: E(J, L, S) = 1/2λ[J(J+1)-L(L+1)-S(S+1)] the ground term positive for n = 1-4 Spin-Orbit coupling parameters, ζ, for some elements

0 –1 Cr ζ3d = 227 cm –1 2+ –1 B ζ2p = 11 cm Cr ζ3d = 230 cm –1 Tl ζ6p = 5195 cm 2+ –1 Mo ζ4d = 749 cm 3+ –1 Mo ζ4d = 818 cm

Values taken from Spin-orbit coupling of the d2 configuration

Thespinorbit interaction splits the atomic terms into 2 d , λ > 0 j levels

1S J = 0

E 1 (9) G J = 4 (5) J = 2 λ 3P (3) J = 1 -λ (1) J = 0 1 -2λ D (5) J = 2 d2 (9) J = 4 3λ 3 (7) J = 3 F -λ (5) J = 2 -4λ e- - e- repulsion spin-orbit Russel-Saunders Coupling (LS-coupling) Spin-Orbit Coupling in Crystal Field Terms Example: Terms of the d2 configuration dn dn

Interelectron strong ligand repulsion field ATOMIC Electron TERM Configuration 1 S weak ligand Interelectron field repulsion

3 E A2g CRYSTAL-FIELD TERMS

12Dq 1G Spin-Orbit Coupling 3 3P T1g (P) Multiplets 1 3T D 2D 2g 2 q d Magnetic Field 2ε0 3 F 6Dq Magnetic States 3T ? e- - e- repulsion 1g Zeeman States

crystal field

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 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 longer degenerate in a complex.

In an octahedral complex, e-can only move within an

open 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

dxy Terms with T symmetry exhibit orbital angular momentum dx2-y2 (leer) can show spin-orbit coupling

This rule is only applicable in Oh Symmetry.

Terms with T symmetry E = -1/2Aλ[J(J+1)-L(L+1)-S(S+1) J exhibit L = 1, For (t )n less than half occupied: λ positive 2g H = -AλLS more than half occupied: λ negative SO First Order Spin-Orbit Coupling, splitting of terms, d1 configuration

3+ 1 2 Ti , d , has a T2g ground term, shows first order spin-orbit coupling

e 2 g Eg

∆o d1 d1

Energy S L J 2 T2g e--e- -repulsion λ 1/2 1 1/2

t2g −λ/2 1/2 1 3/2

crystal field 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 Tetrahedral symmetry n m n ground e t2 ligand field quenching term term

1 2De12EYes 3 23 2 Fe A2 Yes 4 2 14 3 Fet2 T1 No These ions 5 2 25 4 Det2 T2 No have L = 1 6 2 36 5 Set2 A1 Yes 5 3 35 6 Det2 EYes 4 4 34 7 Fet2 A2 Yes 3 4 43 8 Fet2 T1 No 2 4 52 9 Det2 T2 No First Order Spin-Orbit Coupling, splitting of terms, d1 configuration

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

n m n ground t2g eg ligand field quenching term term

2 12 1 Dt2g T2g No 3 23 2 Ft2g T1g No 4 34 3 Ft2g A2g Yes These ions 5 3 15 4 Dt2g eg Eg Yes have L = 1 43 t2g T1g No 6 3 26 5 St2g eg A1g Yes 52 t2g T2g No 5 4 25 6 Dt2g eg T2g No 61 t2g A1g Yes 4 5 24 7 Ft2g eg T1g No 6 12 t2g eg Eg Yes 3 6 23 8 Ft2g eg A2g Yes 2 6 32 9 Dt2g eg Eg Yes 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

Φ λ LS Φ Φ λ LS Φ ∑ e e g g g e Eg − Ee

2 A terms are stabilized by –8λ /∆o

2 E terms are stabilized by –4λ /∆o

The Zero field splitting D is also associated with the second order spin-orbit coupling (see lectures on Magnetochemistry and EPR) Energy levels of Co2+: e-/e-repulsion, crystal field, 1st order spin- orbit coupling and Zeeman splitting

2D

2F 2G

2H

4 4P T1g d7 4 A2g

4 T2g 4F J =5/2 e-/e- 4 T1g 3Aλ/2 repulsion crystal J =3/2 field -Aλ 1st order spin-orbit J =1/2 coupling -5Aλ/2 This will be part of the EPR lecture Zeeman Splitting