Crystal-Field Theory Ligand-Field or Molecular Orbital Theory

How can we explain bonding between
metals and ligands?

Crystal-Field Theory
Ligand-Field or

Molecular Orbital Theory

October 4, 2010

The Periodic Table

Electronic structure of transition
metals

Filled ns, np shells; chemistry defined by unoccupied nd-orbitals
Number of d-electrons = Z – nearest noble gas e’s – charge

e.g.:

Fe(II) = 26 – 18 – 2 =6
Fe(III) = 26 – 18 – 3 = 5



d2z2-y2-x2

Valence Bond (VB) Theory

developed by Pauling in the 1930s - used in VSEPR

Useful in d-complexes sp3d, sp3d2, sp2d useful in CN = 5 (sp, tbp), 6 (octahedral), and 4 (square plana

Not much used for d- coplexes today, but some of the terminology is essential

Note limitations: M needs 3dx2-y2, 3dz2, 4s, 4px, 4py, 4pz to be unoccupied

applying VB:

3d 4s 4p
d2sp3
Cr+3
[Cr(NH3)6]+3

OK for all d3

Fe3+ d5 LS: ↑↓ ↑↓ ↑ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ OK
3d d2sp3

Now consider Fe 3+ d5 HS octahedral: d2sp3

↑↑↑↑↑ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ sp3d2, but 4d?

d8 octahedral case

3d 4s 4p 4d

Ni+2 high spin sp3d2 again 4d
[Ni(NH3)6]+2

[Ni(L)4]2+

Tetrahedral case: 4L e- pair goes to sp3, OK, paramagnetic

Square planar case: dsp2 V
B

↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓
(dx2-y2)sp2
3d(z2, xy, xz, yz)

VB can rationalize geometry and magnetic properties on a simple level
cannot say anything about electronic spectra (color)

and why some ligands lead to HS and some to LS complexes
why LS d6 complexes are kinetically inert etc..

Need better theory!

Crystal-Field Theory
(gross over-simplification)

d2z2-y2-x2

INTERACTION OF d-ORBITALS with OCTAHEDRAL CRYSTAL FIELD of LIGANDS

Chemistry of TM complexes
controlled by where the
d-electrons are
d-electron energies
controlled by ligand
electrons

charges

charges





10Dq, a.k.a. ∆

Definition: energy splitting between the t2g and eg orbitals

Measurement:

eg

∆E = 10Dq hν

t2g

[Ti(III)(H2O)6]3+ d-electrons = 22 - 18 – 3 = 1

hν = 20,300 cm-1 = green light (complex = violet)
= 58.04 kcal/mol (on the order of a bond)



∆oct = E= hν

Ti3+ d1

l

Spectrochemical series of ligands



Spectrochemical series of metal ions

Mn2+<Ni2+<Co2+<Fe2+<V2+<Fe3+,Co3+<Mn4+<Mo3+<Rh3+<Ru3+<Pd4+<Ir3+<Pt4+

Strong and weak field

Factors affecting 10 Dq

-Increased charge on metal: draws ligands in more closely; more effect
on d-orbital splitting
-Ligand type: spectrochemical series

-So named because “strong field” ligands have higher energy d-d
transitions (shorter wavelength UV/vis maxima)

I- < Br- < S2- < Cl- < NO3- < OH- ~ RCOO- < H2O ~ RS- < NH3 ~ Im
< 1,2-diaminoethane < bipy < CN-, CO

d2 Case

Electronic configuration is
same for strong or weak field case

d3 Case

Electronic configuration is
same for strong or weak field case

d4 Weak Field – High Spin (HS)
Case

∆o < Ep, pairing energy

S=2

d4 Strong Field – Low Spin
Case

∆o > Ep, pairing energy

S=1



Electron pairing

Ions with equal and greater than d4 electronic configuration in octahedral (Oh)
coordination can have low and high spin forms depending on value of ∆o (10Dq)

octahedral Low spin High spin
Fe3+ (d5) ∆o=10 Dq
eg
eg ∆o =10 Dq

t2g t2g

P = spin pairing energy spin pairing energy larger
smaller than ∆o: S = ½ than ∆o: S = 5/2

LS: ∆o > P HS: ∆o < P

Unpaired (non-integer) spin: paramagnetic
- detectable by magnetic measurement or sometimes by EPR

Crystal-field Stabilization Energy (CFSE)

CFSE = x(-4Dq) + y(+6Dq)

or since
∆o = 10 Dq

CFSE = x(-0.4∆o) +
y(+0.6∆o)

where

x = number of electrons in t2g (lower levels)
y = number of electrons in eg (upper levels)

Crystal-Field Stabilization Energy, CFSE
or

Ligand-Field Stabilization Energy, LFSE CFSE = x(-0.4 ∆o) + y(0.6 ∆o)

d1 d2 d3

LFSE = -0.4∆o = -0.8 ∆o = -1.2 ∆o
High Spin Low Spin
High Spin Low Spin

d4 d5

LFSE = -0.6 ∆o -1.6 ∆o + P 0 -2 ∆o + 2P

Pairing Energy, P

Two terms contribute to P:

1. loss of exchange energy

p3 — — — or — — —
E1 E2 E1 > E2

total exchange energy = Σ N(N −1)K
2

N = number of e- with parallel spin
K = exchange energy (characteristic of atom or ion) e.g., ∆E = E2-E1

2. coulombic repulsion between paired e-

Crystal-Field Stabilization Energy, CFSE or LFSE
CFSE = x(-0.4∆o) + y(+0.6 ∆o)

High Spin Low Spin High Spin Low Spin

d6 d7

LFSE = -0.4 ∆o + P -2.4 ∆o + 2P -0.8 ∆o -1.8 ∆o + P
d8 d9 d10

LFSE = -1.2 ∆o -0.6 ∆o 0

Note: largest CFSE-LFSE for LS, d6



Relationship between d-orbitals of TM surrounded by
r tetrahedral coordination of ligands

Octahedral vs Tetrahedral Splitting Pattern of d-Orbitals
∆tet = 4/9; ∆tet ≅ 1/2 ∆o

tetrahedral

octahedral

∆O > ∆T thus no strong field vs. weak field cases for
tetrahedral complexes; note absence of subscript g in tetrahedral (t and e)

Color of tetrahedral complexes different



Octahedral HS
tetrahedral

S=½
1 unpaired e-

Co(II) d7 S = 3/2
3 unpaired e-

Total spin quantum #:

S ≡ Σ si
N

Σ = Summing over N electrons with
si spin quantum # on each elelctron
si = ±1/2

Jahn-Teller J-T theorem: any nonlinear molecule in a degenerate
Effect: electronic state will be unstable and will undergo a
distortion to a system of lower symmetry and lower
Cu2+ d9
energy thereby removing the degeneracy

J-T active ions:

Strong JT: HS-d4, d9 [LS d7]– eg orbitals

Weak JT: d1, d2, LS-d4, LS-d5, HS d7 – t2g

J-T does not predict if
distortion is:

elongation along z (as
shown)
or
elongation along x-y,
lowering dx2-y2

Would you expect JT in Td d4 and d9?



SQUARE PLANAR COMLEXES

elongation of M-L along z

second and third row d8 complexes are
square planar. E.g., Pd (II), Pt(II), Rh(I), Ir(I)

Square-planar Splitting Pattern of d-
Orbitals

Ni2+, d8

[Ni(CN)4]-2

Square planar Ni2+ complex is diamagnetic
S=0

Chapter 21 (20), self study on pg. 646 (564)

d8 complexes:[NiCl4]2- paramagnetic, S=1; and
Ni(CN)4]2- diamagnetic

tetrahedral Square-
planar
---
-
--
-

-
--

Thermodynamic Aspects of LFSE

Lattice Energies of MCl2 compounds

M2+: Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn

Same for MF2, MF3 and [MF6]3- complexes

Deviation from d0, d5 and d10 is due to thermochemical LFSE

Agreement between e.g.: [Ni(H2O)6]2+, LFSE (thermochemical), 120 kJ/mol and LFSE
(spectroscopic), 126 kJ/mol (1.2∆oct deter. From spectra of [Ni(H2O)6]2+, ∆oct 8500 cm-1)

Must remember that LFSE is <10% of the total in
reaction energies

M2+ (aq) + H2O (l) → [M(H2O)6]2+ (aq) Ni2+

Octahedral vs Tetrahedral coordination

Fig. 20.26 suggests that for d0, d5, and d10 there should be no preference for
octahedral or tetrahedral coordination; there is a large preference for octahedral
for d3 and HS d8

Case of spinels: MgAl2O6 mineral spinel; Mg2+ in Td while Al3+ in Oh site many analogous
“normal” spinel compounds with A2+(M3+)2O6 -LFSE can favor the formation of so-called

inverse spinels: e.g. normal: Fe(II)[(Cr(III)]2O4
LFSE [Cr(III) d3 1.2 ∆oct] > LFSE Fe(II) d6 (HS, -0.4 ∆oct)
Fe3+(Fe2+, Fe3+)O4 LFSE = 0 for Fe3+ (d5), but Fe2+ (d6) provides -0.4∆oct

What about Mn3O4, Co3O4

Review Chapter MO Theory: Chapter 2 Shriver and Atkins

Molecular Orbital Theory of Octahedral Complexes-[ML6]n+

See Oh character Table on Appendix 3

s A1g or a1g totally symmetric irred. Rep

px, py, pz T1u or t1u
dxy, dxz, dyz T2g or t2g

dx2-y2, dz2 Eg or eg