Ligand Field Theory, Density Functional Theory and Molecular Mechanics:

Adventures with d-electronsDr Rob Deeth

Inorganic Computational Chemistry Group

Overview• The Density Functional Theory revolution• Practical applications• Beyond DFT: Classical modelling for TM systems

and ‘smart’ approaches• LFT v. CFT• LFT v. DFT• Acknowledgements

DFTThe Density Functional Theorem — (Hohenberg-Kohn, 1964) the ground state total energy, E0, is a unique functional of the electron density, ρ.

E0 = E[ρ]

The theorem includes ALL electron correlation (Quantum Mechanics).

Pre-1988• Slater exchange-only Xα model• Vx ∝ ρ1/3

• Fast, accurate electronic structure• No automatic geometry optimisation• [CuCl4]2-, [VOCl2(urea)2]

d-d transitions energiesmolecular orbitalscharge distributions

Energy Gradients and Functionals

• Versluis and Ziegler: J Chem Phys, 1988• Exact DFT solution for a Uniform Electron Gas• Slater exchange plus fitted correlation – LOCAL

DENSITY APPROXIMATION• LDA overbinds• GENERALISED GRADIENT APPROXIMATION (e.g.

BP86, B3LYP)• GGA includes ∇ρ corrections

Structural Chemistry: 1994

TcO

ONNH

NHN

OHO

[TcO2Pent(AO)2]

199458 atoms297 basis functions10 hrs per geometry step on DEC 3000/700 workstation

Overlay: Obs/Calc(RMS ~0.03 Å)

Structural Chemistry: 2002

Overlay: Obs/Calc(RMS ~0.04 Å)

Zr

OBzBz

ONN

iPr

iPr2002113 atoms586 basis functions2 hrs per geometry step on 4 CPUs of Pentium III cluster

• DFT gives ~ X-ray structure accuracy for TM complexes

Aziridination

Cu

N

N

N

Ts

*

*

R'NR

R = TsR' = H, CO2Me

A Cu+-nitrene diimine is the proposed catalyst.

No experimental structure— it’s a catalyst!! — the structure must be computed.

Real Substrates: CinnimatesOnly the ester group was modelled.Quadrant 2 system spontaneously gives a Cu-O=C interaction at 2.26 Å and a structure 20 kJ mol-1more stable than the comparable styrene complex.DFT predicts much better e.e.s (Obs. e.e. >95%) Obs. and calc. absolute configurations identical

2.26 Å

Mechanism

2.00

2.08

2.3382°

Pd

‡2.01

2.1995°

1.99

‡

DFT provides direct access to Transition States

Activation barriersExchange on d8 centres: H2O, MeCN, MeNC, C2H4,

CO, CN-

Excellent correlation with ∆H‡ but GGA energies vital!

Activation Energies

-80

-60

-40

-20

0

20

40

60

80

100

120

0 20 40 60 80 100 120

Energy (kJ/mole)

Ener

gy (k

J/m

ole)

LDA

GC

Exp

4

3

1

6

52

89

Oxo-Transfer Mechanism

Products: -72

+91 [Exp(PR3) ~ 75]

+29

+35

0 kJ mol-1

‡

‡Energies in kJ mol-1

MECHANISM OK

Conclusions: DFT• DFT is the best QM model for TM systems.• Excellent structures and direct access to Transition

States• “Static” DFT modelling a good starting point for catalyst

design• Functionals not specifically designed for TM systems

(too covalent) so require validation• DFT (or any QM method) is too slow for really big

molecules, VHTS or MD

Need for Speed

• DFT is accurate but ALL QM methods, including DFT, are relatively slow

• QM poorly suited to:Conformational searchingVirtual high-throughput screeningBiomoleculesMolecular Dynamics

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

Fast (big systems, dynamics)Accurate (experimental information built in to Force Field parameters)Works well for organics and TM complexes with “regular” coordination environmentsConventional MM has fundamental problems with Transition Metal systems

Challenges• Coordination numbers > 4

• Multiple oxidation states

• Multiple spin states

• Organometallic versus classical Werner complexes

• Electronic effects (Jahn-Teller)

Electronic Effects: d-electronsThe d electrons are structurally and energetically non-innocent.The effect can be correlated with changes in the LIGAND FIELD STABILISATION ENERGY (LFSE)E.g.: d9 [CuL6]: ∆EJT electronic driving force

eg

t2g

∆EJT

∆EJT

L

CuL L

L

L

L+2δ

-δ

dx2-y2

dz2

Extending MM to the d-block• Many TM properties have a LFSE component

(‘double hump’ behaviour)• Add LFSE directly to MM

Ligand Field Molecular Mechanics (LFMM)

• Etot = ΣEstr + ΣEbend + ΣEtor + ΣEvdw + ΣEC+ LFSE

• LFMM captures d electronic effects directly

Dr Veronica Paget (nee Burton)

Getting LF Parameters• The LFSE is computed using the Angular Overlap Model

(AOM).• Each M-L bond described by LOCAL parameters — eσ, eπx, eπy

• Values from ‘d-d’ spectra: e.g. ∆oct = 3eσ- 4eπ

• Fit to general expression:eλ = a0 + a1 r + a2 r-2 + a3 r-3 + a4 r-4 + a5 r-5 + a6 r-6

L

eσ

dz2

deπx

dxz

d eπyd

L

M

L

MX

Y

Z

dyz

M

LFSE Gradients• Ε ’ = tr(NW’)• W’ = Q’VQT + QV’QT + QVQT

• = tr(NQ QT)

• LFSE includes σ and π parameters and d-s mixing effects

δV δx(λκ)

i

δΕ δx(λκ)

i

Dr Dave Foulis

Parameter FittingCannot simply take existing TM FF parametersBond length, r, is a balance of conventional MM terms (e.g. Morse function D0, α and r0) and LFSELow spin d6 has maximum LFSE: hence r < r0

-120

-80

-40

0

40

80

120

160

1.50 1.70 1.90 2.10 2.30 2.50

Bond Length

Ener

gy

MorseCLFSETotal

[Co(NH3)6]3+

r0r

r = 1.93 År0 = 2.11 Å

Spectroscopic AccuracyParameter a0 a1

eσ (Cl) 14400 -4000 eπ(Cl) 4300 -1500 eds(Cl) 9500 -3500

[CuCl4]2- Calca (no π or d-s) Calc Obs Cu-Cl /Å 2.26 2.26 2.26 dxy → dx2-y2 10050 12440 12500 dxz/yz → dx2-y2 10050 14260 14300 dz2 → dx2-y2 6700 17080 17000 [CuCl6]4- Cu-Cl(eq) /Å - 2.28 2.30 Cu-Cl(ax) /Å - 2.86 2.90 dz2 → dx2-y2 - 10720 10800 dxy → dx2-y2 - 12320 12220 dxz/yz → dx2-y2 - 14080 13300

Morse parameters: D0, r0, α: 80.0 kcal mol-1, 2.50 Å, 0.30

LFMM: d9 Cu(II)

MOE parametersAll Cu-N 1.93Å

Molecular OperatingEnvironmentDOMMIMOE

LFMM parameters(MMFF94-TM)Cu-Nax 2.29Å (2.32)Cu-Neq 2.05Å (2.06)Dr Natalie Fey

Ben Williams-Hubbard

Auto J-T Effect: d9

Cu

NH3

NH3

NH3

NH3

NH3

NH3

Cu

N

N

NN

NN

Cu

N

N

NN

N

N

CuN

NNN

N

N

Cu

NN

N

N

N

N

1 2 3 4 5

Cu-Neq (av) 2.042.15

1.992.04

2.052.06

2.062.07

2.012.07

Cu-Nax (av) 2.482.45

2.462.41

2.292.32

2.412.41

2.282.35

cis N-Cu-N (av) 9090

8081

8181

8081

8997

trans N-Cu-N (av) 180180

170169

177178

164166

180180

No other MM scheme gives J-T distortions automatically

High-spin/low-spin d8

eg

t2g

L

NiL L

L

L

L

dx2-y2

dz2

The structures of d8

Ni(II) complexes are determined by the LFSE

eg

t2g

2∆EJT

L

NiL L

L

L

L

dx2-y2

dz2

High and Low Spin States: d8

NH3H3N

H3N NH3

Ni

NH3

NH3

NN

NN

N

N

Ni Ni

N

NN

N

N NNi

N

NN

N

N N

Ni

N

NN

N

N N NN

NN

N

N

NiNN

N NNi

NN

N NNi

NN

N NNi

NN

N NNi

RMS ErrorsNi-N: 0.01ÅN-Ni-N: 0.6°

Ni-N(hs) ~2.1ÅNi-N(ls) ~1.9Å

JACS, 117, 8407, (1995)

One set of M-L Force Field parameters handles widely different M-L bond lengths

Relative Energies‘Conventional’ MM has different parameters for each

spin state.Cannot compare MM energies directly

MM cannot predict lowest energy spin stateLFSE term in LFMM can model spin-states

E.g.:[CoF6]3-: 5T2g [Co(CN)6]3-: 1A1g

Test case: d6 Co3+ octahedral in high and low spin

Ben Williams-Hubbard

Spin State Energies

5T2

1I

3H

3T15D

E/B

∆/B

3T1

3T2

1T1

5T2

1T2

3T1

3E

1T2,1E

5E

1A2

1A1

5T2

d6 Tanabe-Sugano

Both the LFSE and the interelectron repulsion needed.Theoretical d6 spin-crossover point:

∆E(ls-hs) = 2∆oct - (5F2 + 255F4)

F2 and F4 are ier parameters

The 5T2g-1A1g splitting from DFT is ~2400 which is consistent with d-d spectroscopy and full LFT.

Getting Other Parameters:DFT to the Rescue

Experimental data are not always available

DFT can access actual and hypothetical systems with and without constrained geometries and/or spin states

Use DFT to develop ‘smart’ LFMM parameters

DFT for high-symmetry ML6 complexes very fast

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

Co-F Co-CNLDA(hs) 1.97 2.12Exp 1.94 -

LDA(ls) 1.88 1.88Exp - 1.89

• Can rapidly tune LFMM parameters to reproduce CoL6 DFT data for BOTH spin states

Experimentalvalidation

Access tounobservables

The ResultsLFMM energies can be compared directlyParameters for homoleptic complexes applied unchanged to mixed-ligand systems - 10 for 2[Co(CN)nF6-n]3-, spin crossover at n = 1.

-1560

-1540

-1520

-1500

-1480

-1460

-1440

F6CN0

F5CN1

F4CN2c

isF4C

N2tran

sF3C

N3fac

F3CN3m

erF2C

N4cis

F2CN4m

erF1C

N5F0C

N6

Ener

gy (k

cal/m

ole)

DFT(high)DFT(low)LFMM(high)LFMM(low)

-100

-80

-60

-40

-20

0

20

Ener

gy (k

cal/m

ole)

DFTLFMM

Conclusions: LFMM• A fast scheme is needed for TM systems which

handles the important d-electron effectsLFMM

• designed for TM systems for all those cases where LFSE important

• captures the essential physics around the metal and facilitates calculations which would otherwise require full-blown QM approaches - ‘smart’ parameters

• automatic Jahn-Teller distortions• single parameter sets for multiple coordination

numbers and spin states• energies can be compared directly• efficient enough for large scale simulations

LFT v. CFT• Amines σ-bonding only• AOM: no π bonding = degenerate ‘t2g’ = two ‘d-d’ bands, • CFT: electrostatic model = four bands in rhombic D2h

• [Cu(dien)2]2+: 8800, 9900, 15400, 15900 cm-1

• AOM wrong?• DFT to the rescue!?!

eg

t2g

dx2-y2

dxy

dyz

dz2

tetragonalelongation

Oh D4h

tetragonalelongation

dx2-y2

'dπ' (dxy/dxz/dyz)

dz2

D4h

[M(NH3)6]n+: σ onlyAOM Electrostatic CFT

D2h D2h

rhombicdistortion

rhombicdistortion

dxz

[Cu(dien)2]2+

• DFT optimised structure agrees with X-ray• DFT d orbitals 1:3 pattern, EPR g-values too low• DFT too covalent (SOMO: Calc 43% d, Exp (EPR)

65-70% d)

• Tune DFT g-values by optimising Cu nuclear charge -best q = 28.2. SOMO now 69% d

• Good g-values - d orbitals still 1:3 pattern• DFT agrees with AOM, not with CFT

d lig d lig

Wrong covalent/ionicbalance

Correct covalent/ionicbalance

Mostly ligandMostly metal

New Interpretation• [Cu(dien)2]2+ has two possible elongation axes• ‘d-d’ bands arise from ~6:1 mixture of two

complexes aligned at ~90° • Exp. solid state structure obtained from DFT in

vacuo geometry• Even get rhombic equatorial geometry from

superimposing asymmetric axial elongation

Cu NNN

N

N

N

Cu NNN

NN

N

Cu NNN

N

N

N

85% 15%+ = Obs

Asymmetricaxial

elongation

2.13 2.07

LFT v. DFT• DFT seems to be replacing LFT• Lots of useful knowledge derived from

Ligand Field Theory• Would be a shame to lose it all• Does the Ligand Field description of

metal-ligand bonding map onto Density Functional Theory?

Tetragonal Distortion

• Classical (point charge) CFT model predicts t2gsplitting

• Qualitative MO model for [PtCl4]2-

concurs• What about DFT?

eg

t2g

b1g (dx2-y2)

b2g (dxy)

eg (dxz/dyz)

a1g (dz2)

tetragonal elongation

Oh D4h

Planar [MCl4]2-x

y

y

xz

b2g

eg

eg*

b2g*

eg

b2g

eg

b2g

b2g*

eg*

qualitative MO DFT

d

s

ClM ClM

• Experiment places dxy above dxz/dyz for M= Cu/Pd/Pt• DFT inverts energies of nominal dπ orbitals• Is DFT wrong?

Planar [M(NH3)4]2+

• NH3 σ-bonding only, simple MO theory and CLF model predict degenerate ‘t2g’ d orbitals

• EHMO actually gives degenerate dxy/dxz/dyz

• Ground state DFT calculations for M=Cu/Pd place dxy~4000 cm-1 lower than dxz/dyz

• DFT implies NH3 is a net π acceptor!• Point charge model always splits ‘t2g’ set but dxy is

always higher than dxz/dyz

• Is DFT wrong?

Can DFT really be wrong?• DFT gives excellent description of ground state properties

(geometries, frequencies, multipole moments etc.)• Relative energies of bonding MOs imply M-Cl π donor • DFT charge distributions reasonable• But ground state DFT d-orbital sequence still looks wrong• LFT treats both ground and excited d-d states so maybe it’s an

excited state issue• DFT excited state energies are qualitatively correct

(ADF multiplet states relative to AOCs)• DFT is OK so where is the problem?

[PdCl4]2- Exp. DFT1A1g → 1A2g 21700 164691A1g → 1Eg 23200 180211A1g → 1B1g 28910 21067

[CuCl4]2- Exp. DFT2B1g → 2B2g 12500 138412B1g → 2Eg 14300 161612B1g → 2A1g 17000 22625

DFT is OK, but...• Hoffmann and Baerends argue that Khon-Sham

orbitals provide a good basis for discussing bonding• LFT naturally focuses on the (anti-bonding) d orbitals

and assumes their energies are ‘mirrored’ by equal and opposite movements of the matching bonding functions - c.f. zero-overlap approximation -antibonding orbital increases by the same amount that the bonding MO decreases

• DFT is far more sophisticated and consequently we loose this simple relationship between antibonding d orbitals and their bonding counterparts

• Thus, ground state DFT d orbital energies cannot be interpreted in the ‘usual’ way

Implications for LFMM• Cannot use ground state DFT ‘d’ orbital energies directly for FF

development• DFT includes d-d interelectron repulsion explicitly and within the

full molecular symmetry• LFT separates d orbitals from interelectron repulsion and treats

the latter within a spherical central field approximation• If we define an approximate ‘spherical average’ configuration

where each d orbital has roughly the same population - E.g.: t2g

3.6eg2.4 for an octahedral d6 complex - then LFT and DFT

agree(N.B. Software issues. Jaguar versus ADF)

Acknowledgements• Inorganic Computational Chemistry Group

Dr Veronica Paget (LFMM) — now with AcelrysDr David Foulis (LFMM development)Dr Natalie Fey (LFMM/MOE, Diels-Alder)Ben Williams-Hubbard (LFMM/MOE,Co spin states and Cu proteins)Nicola Waite (acrylate polymers)Jack Smith (DFT and Heck reaction)Joanne Hanna (Binding energies)James Burnside (acrylate polymers)

• Peter Scott, WarwickLars Ivar Elding, LundDominic Ryan, Millenium

• ££££sEPSRCUniversity of WarwickChemical Computing Group