Electronic Effects andLigand Field Theory

Dr Rob DeethInorganic Computational Chemistry Group

University of WarwickUK

Overview

• Introduction• Electronic effects in TM chemistry• Classical v. Organometallic compounds• Ligand Field Stabilisation Energy• d orbitals• Spin states and Jahn-Teller effects• Generalised ligand field theory• Ligand Field Molecular Mechanics• DommiMOE

Electronic Effects

• Geometric preferences• Obvious ones:

– Jahn-Teller effect = distorted, especiallyCu(II) – four short, two long

• Less obvious ones:– Low-spin d8 = planar, especially Pd(II),

Pt(II), Rh(I)– Low-spin d6 = octahedral, Co(III)

• First row TMs particularly complicated

Plasticity

• M-L bonds weaker than C-C• Higher coordination numbers• More flexible geometry – angular variations

– [CuCl4]2-

– High spin NiL4 – tetrahedral– Low spin NiL4 – planar– Five coordination – small energy difference

between square pyramidal and trigonal bipyramidal

Classical v. Organometallic

• Werner-type:– Relatively ionic– Electronic effects focussed on d orbitals– IONS

• Organometallic– Relatively covalent– More general electronic effects – spndm

– Neutral or +-1• For classical coordination complexes, need to

consider d orbitals

d orbitals

dx2-y2

Z

Y

X X

Y

Z

d2z2-x2-y2

X

Y

Z

dxz

Y

Z

X

dxy

Z

X

Y

dyz

d Orbital splittings

• In octahedral symmetry, the five d orbitals split

• Barycentre relative to average d orbital energy

Mn+

Point charge q = ze

Free Mn+ ion

d

eg

t2g

10Dq

Mn+ in octehdral crystal field

+3/5

-2/5

∆oct

Ligand Field Stabilisation Energy

• Structural preferences and Jahn-Teller instabilities can be traced to LFSE

• LFSEd0: 0 d1: -2/5∆oct

d2: -4/5∆oct

d3: -6/5∆oct

d4: -3/5∆oct

d5: 0

∆Hhyd

Ca Mn ZnV NiSc Ti Cr Fe Co Cu

Spin States• For dn configurations with 2 ≤ n ≤ 8, multiple spin states are

possible• Spin depends on symmetry and ligands• Consider octahedral complexes

– Spin state a balance between d orbital splitting and spin pairing energy

d1 d2 d3

d4 d4 d5 d5 d5

S = 1/2 S = 1 S = 3/2

S = 2 S = 1 S = 5/2 S = 1/2 S = 3/2

high low high low intermediate

π Bonding Affects ∆oct

Metal

3d

4s

4p

Ligands

σ

t2g

eg*

a1g*

t1u*

eg

a1g

t1u

Octahedral ML6

t2g

eg*eg* eg*

Ligands

π (filled)

Ligands

empty π*

σ only

π donor10Dq decreases

π acceptor10Dq increases

t2g

t2g

t2g*

t2g*

• σ-only ligand leaves t2gorbitals degenerate

• π donors decrease ∆oct

• π acceptors increase ∆oct

Jahn-Teller Effect• The d electrons are structurally and energetically

non-innocent.• Complexes with a ground state orbital degeneracy

unstable with respect to a vibration which removes the degeneracy - Jahn-Teller theorem

eg

t2g

∆EJT

∆EJT

L

CuL L

L

L

L+2δ

-δ

dx2-y2

dz2

Molecular Mechanics• Etot = ΣEstr + ΣEbend + ΣEtor + ΣEvdw + ΣEC

Fast (big systems, dynamics)Accurate (experimental information built in to Force Field parameters)ParameterisedWorks well for organics and TM complexes with “regular” coordination environmentsProblems with “plastic” systemsProblems with electronic effects

Extending MM to the d-block• Problem: conventional MM requires

independent FF parameters for high spin d8

(octahedral) Ni-N 2.1Å versus low spin d8

(planar) Ni-N 1.9Å• Answer: add LFSE directly to MM

Ligand Field Molecular Mechanics (LFMM)

• LFMM captures d electronic effects directly• Etot = ΣEstr + ΣEbend + ΣEtor + ΣEvdw + ΣEC +

LFSE

d-orbital energies

• Crystal Field Theory is global symmetryapproach – all ligands simultaneously

• MM is bond centred• Need to express d orbital energies as

function of individual bonds• Angular Overlap Model describes each

bond´s contribution to the total ligand fieldpotential

Getting LF Parameters• Each M-L bond is described by up to three parameters — eσ, eπx, eπy.

L

eσ

dz2

deπx

dxz

deπyd

L

M

L

MM X

Y

Z

dyz

Angular variations

_d

dz2

eσ(L)dxz,dyz

dxy,dx2-y2M

L

eπ(L)z

xy

X

Y

Z

L

M

L

M

L

M LM

θ = 0° θ = 25°θ = 54.7°

θ = 90°

.

Z Z Z

0.00

0.25

0.50

0.75

1.00

Frac

tion

of e

(sig

ma)

0 30 60 90 120 150 180 θ

• d orbital energies for linear ligatorM-L

• Effect of moving ligand• Fσ(dz2) = 1/4(1+3cos2θ)• E(dz2) = eσ F(dz2)

= 1/16 eσ (1 + 3cos2θ)2

Other motionsL

M

L

M

L

M LM

θ = 0° θ = 25°θ = 45°

θ = 90°χσ

dxz

Z

X

0.00

0.25

0.50

0.75

1.00

Frac

tion

of e

(sig

ma)

0 30 60 90 120 150 180 θ

Octahedral symmetryAngular Coordinates

L3

ML4 L2

L1L5

L6X

Y

Z

Ligand θ φ

1 90 0

2 90 90

3 0 0

4 90 180

5 90 270

6 180 0

M Nθ

φX

Y

Z

ψy

x

z

d orbital energies• The energy of each d function will consist of the sum of all

possible symmetry contributions (σ, πx, πy) from each ligand. For N ligands, this will in general correspond to a sum of 3N terms.

• E(dz2) = ¼eσ(L1) + ¼ eσ(L2) + eσ (L3) + ¼ eσ (L4) + ¼ eσ (L5) + eσ (L6)

= 3eσ (L)• E(dx2-y2) = ¾ eσ(L1) + ¾ eσ(L2) + 0eσ (L3) + ¾ eσ

(L4) + ¾ eσ (L5) + 0eσ (L6)= 3eσ (L)

• AOM automatically recoverscorrect symmetry

3eσ

4eπ

∆oct

dz2, dx2-y2

dxz, dyz, dxy

'mean' d

Strategy and Examples

• Only develop parameters for metal-ligandbonds

• Use existing force fields for ´spinach´• [CoF6]3-

– High spin d6

• [Co(CN)6]3-

– Low spin d6

• [CuCl4]2-

• Ammonia and amine complexes

DFT Protocol for Bond Lengths• Optimised Bond lengths for [CoL6]3-

complexesCo-F Co-CN

DFT(hs) 1.97 2.12*Exp 1.94 -

DFT(ls) 1.88* 1.88Exp - 1.89

• We can use the bond lengths for high-spin [Co(CN)6]3- and low-spin [CoF6]3- to design better LFMM parameters.

Adding Chemical Unrealism:LFSE-free

• MM uses separate energy terms so it is feasible to pose questions like “What is the M-L distance in the absence of LFSE?”• LFSE = 0 if all d orbitals equally occpied• For d6 Co(III), this corresponds to t2g

3.6eg2.4

• DFT gives approximate LFSE-free bond length

Adding Chemical Realism:π Bonding

Both F- and CN- can form π bonds.Averaged configuration DFT calculations on hypothetical CoL4 species yields ‘d’ orbital energies which can be fitted to standard AOM expressions to determine eπ to eσ ratio.Co-F: ~0.3Co-CN: ~0.1 (CN π donor!)

Parameter Fitting• In general, we want to be able to handle large M-L bond length changes: use Morse function.• Angular geometry determined by 1,3-ligand-ligand (POS, VSEPR) interaction (plus LFSE contribution).• The required bond length, r, is a balance of Morse function (D0, α and r0) with the LFSE and POS. NB: r0 > r• CAN´T USE METAL PARAMETERS FROM OTHER FFs

-120

-80

-40

0

40

80

120

160

1.50 1.70 1.90 2.10 2.30 2.50

Bond Length

Ene

rgy

MorseCLFSETotal

MOE

• Scientific Vector Language• LFSE and derivatives: code written in C• Connect LFSE code to MOE via API and SVL

communication routinefunction __LFMM_potential [x, args]

// (nf) ***********************************************************************

local function LFMM_potential;

local [f,g] = LFMM_potential [lfmm_vector, x, args, 1];// type 1 = optimisation, type2 = single point

//***************************************************************************

return [f,g];endfunction

DommiMOE

•D-orbitals in molecularmechnics in inoragnics inMOE