Conceptual DFT: The Woodward Hoffmann rules revisited

3-9-2009 1Herhaling titel van presentatie

rules revisited

Conceptual DFT: The Woodward Hoffmann rules revisited

Paul Geerlings, Frank De Proft, Paul Ayers*

Department of Chemistry, Faculty of SciencesVrije Universiteit Brussel

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Vrije Universiteit BrusselPleinlaan 2, 1050 - BrusselsBelgium

*Department of Chemistry, Mc Master UniversityHamilton, Canada

WATOC 2008WATOC 2008Sydney, AustraliaSydney, Australia

September 14September 14--19, 200819, 2008

Outline

1. Introduction: Chemical Concepts from DFT

2. Hardness and Aromaticity

3. The Woodward Hoffmann rules for pericyclic reactions

3.1. Introduction3.2. Initial Hardness response

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4. Conclusions

5. Acknowlegements

3.2. Initial Hardness response3.3. The dual descriptor

1. Introduction : Chemical Concepts from DFT

Fundamentals of DFT : the Electron Density Function as Carrier of Information

Hohenberg Kohn Theorems (P. Hohenberg, W. Kohn, Phys. Rev. B136, 864 (1964))

ρρρρ(r) as basic variable

� ρρρρ(r) determines N (normalization)

� "The external potential v(r) is determined, within a trivial additive constant, by the electron density ρρρρ(r)"

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•

•

• • •

•

•

•

••

•

compatible with a single v(r)

ρ(r) for a given ground state

- nuclei - position/charge

electrons

v(r)

ρ(r) → Hop → E = E ρ[ ]= ρ r( )∫ v(r)dr + FHK ρ(r)[ ]

• Variational Principle

FHK Universal HohenbergKohn functional

Lagrangian Multiplier normalisation

( )( )HKFv rr

δµ

δρ+ =

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• Practical implementation : Kohn Sham equations

Computational breakthrough

ρ(r)∫ dr = N

Conceptual DFT (R.G. Parr, W. Yang, Annu. Rev. Phys. Chem. 46, 701 (1995)

That branch of DFT aiming to give precision to often well known but rather

vaguely defined chemical concepts such as electronegativity,

chemical hardness, softness, …, to extend the existing descriptors and to use

them, to describe chemical bonding and reactivity, either as such or within

Pag.3-9-2009 6

the context of principles such as the Electronegativity Equalization Principle,

the HSAB principle, the Maximum Hardness Principle …

Starting with Parr's landmark paper on the identification of

µµµµ as (the opposite of) the electronegativity.

Starting point for DFT perturbative approach to chemical reactivity

E = E[N,v]Consider Atomic, molecular system, perturbed in number of electrons and/or external potential

dE =∂E

∂N

v(r)

dN +δE

δv(r)

∫

N

δv(r)dr

identification first order perturbation theory

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identification

ρ r( )

Electronic Chemical Potential (R.G. Parr et al, J. Chem. Phys., 68, 3801 (1978))

= - χ (Iczkowski - Margrave electronegativity)

µ

∂E

∂N

v(r)

= µ δE

δv(r)

N

= ρ(r)

∂2E

= η ∂2E

=δµ

=

∂ρ(r)

E[N,v]

Identification of two first derivatives of E with respect to N and v in a DFT context → response functions in reactivity theory

2

( , ')E

r r∂

χ

=

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Chemical Chemical hardnesshardness

ChemicalChemical SoftnessSoftness Fukui functionFukui function

Local softnessLocal softness

∂ E

∂N2

v(r)

= η ∂ E

∂Nδv(r)=

δµδv(r)

N

=∂ρ(r)

∂N

v

S =1

η

= f(r)

Sf(r) = s(r)

Linear response Linear response FunctionFunction

( , ')( ) ( ')

N

Er r

v r v r

∂χ

δ δ

=

Combined descriptors

electrophilicity (R.G.Parr, L.Von Szentpaly, S.Liu, J.Am.Chem.Soc.,121,1992 (1999))

energy lowering at maximal uptake of electrons

∆E = −µ2

2η= −ω

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Reviews :

• H.Chermette, J.Comput.Chem., 20, 129 (1999)

• P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev., 103, 1793 (2003)

• P.W.Ayers, J.S.M.Anderson and J.L.Bartolotti, Int.J.Quantum Chem, 101, 520 (2005)

• P.Geerlings, F.De Proft, PCCP, 10, 3028 (2008)

Applications until now mostly on (generalized) acid base reactions

- acid/base (Arrhenius, BrØnsted-Lowry)

- generalized acid base (Lewis) → complexation reactions

- organic chemistry

• electrophilic/nucleopilic• substitution/addition/elimination

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Use of Principles

- EEP (Electronegativity Equalization Principle)- HSAB (Hard and Soft Acids and Bases Principle; global/local)- MHP (Maximum Hardness Principle)

Not or nearly not discussed with Conceptual DFT

- Redox reactions (prototype of reactions involving electron transfer)

Recent work:

• J.Moens, P.Geerlings, G.Roos, Chem.Eur.J., 13, 8147 (2007)

• J.Moens, G.Roos, P.Jaque, F.De Proft, P.Geerlings, Chem.Eur.J., 13, 9331 (2007)

• J.Moens, P.Jaque, F.De Proft, P.Geerlings, J.PhysChem.A, 112, 6023 (2008)

Pag.3-9-2009 11

• J.Moens, P.Jaque, F.De Proft, P.Geerlings, J.PhysChem.A, 112, 6023 (2008)

Electrophilicity Concept

- Pericyclic reactions

Today’s Talk

3.Woodward-Hoffmann Rules for Pericyclic Reactions

3.1 Introduction

• Pericyclic reactions : cyclic rearrangement of electrons

- systematization by Woodward and Hoffmann:

- decisive role of the symmetry of the wave function and the symmetry and phase (nodal structure) of its constituent orbitals

Conservation of Orbital Symmetry

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• DFT : fundamental descriptor is the electron density:

• Conceptual DFT : responses of E and its derivatives e.g. µ and ρ(r)

• Reactivity indices : essentially orbital free, not obscured by increasingcomplexity associated to increasing accuracy of wave function

to perturbations

( )( )Nr( E v ) )= δ δ

3.2 The Initial Hardness response

• Electron density : • strictly positive• trivial symmetry (Totally Symmetric Irreducible Representation)

How to cast WH rules in a "Density" context ??

• Third approach (non-symmetry based) to the WH rules : aromaticity

- Examination of aromaticity of Transition State

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- Allowed : aromatic TS : (4n+2)π electrons (Hückel System)

- Not allowed : anti-aromatic TS : (4n)π electrons (Hückel System)

"Conceptual DFT" approach ?

- Examination of aromaticity of Transition State (H.E.Zimmerman, Acc. Chem.Res., 4, 272 (1971)

Aromaticity in Conceptual DFT (F.De Proft, P.Geerlings, Chem.Rev., 101, 1451 (2001))

- Hardness ~ Stability

- Aromaticity ~ Stability

Relation Hardness – AromaticityZ.Zhou, R.G.Parr, J.F.Garst, Tetr.Lett., 4843 (1988)

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Correlation of REPE with Hückel hardness for benzenoid hydrocarbons

Z.Zhou, R.G.Parr, J.Am.Chem.Soc., 111, 7371 (1989)

-Focussing on TS properties

Activation Hardness Z. Zhou, R.G. Parr, J.Am.Chem.Soc., 112, 5720 (1991)

Electrophilic Aromatic Substitution

The smaller the activation hardness, ηa,the more predominant the reaction product

2NO++X

NO2X

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a Reac tan ts TSη = η − η

• Manifestation of Maximum Hardness PrincipleTS with high η→ stabilization

→ηa↓→rate constant↑

o,m,p directing effects for X= F, Cl,Br,OH,CH3, CHO, COOH

Investigation of TS hardness

Simplified procedure in the context of perturbational appraoch to chemical reactivity:

• consider evolution of η at the initial stage of the reaction

- early TS- non crossing (Klopman)

• use model reaction coordinate R

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• use model reaction coordinate R

Evaluation of

• Initial Hardness Response

• Third order energy derivative

( )N

R∂η ∂

? Initial hardness Response as a way to reformulate WH rules in a Conceptual DFT Context

First Case: Cycloadditions

- Cycloaddition of ethene to ethene :

Ground state forbidden, excited state allowed (supra/supra)

+

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Ground state forbidden, excited state allowed (supra/supra)

- Diels-Alder cycloaddition of 1,3-butadiene to ethene :

Ground state allowed, excited state forbidden (supra/supra)

+

• Model initial hardness profile of :

- [π2s+π2s] cycloaddition of ethene to ethene

- [π 4s+ π 2s] cycloaddition of 1,3-butadiene to ethene

• Choice of model reaction coordinate “R”

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• Evaluate hardness of the reacting system at early stage

of the reaction: approximate evaluation as

( ) ( )LUMO HOMO S , IE Tε − ε

• Ground state forbidden, excited state allowed [π2s+π2s] :Diels-Alder cycloaddition of ethene to ethene

C C

R

Singlet approach :

2.500

2.700

2.900

3.100

3.300

3.500

3.0 3.2 3.4 3.6 3.8 4.0

R (Å)

η↓ when R↓

0NR

η∂ > ∂

η(eV)

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C C

R (Å)

2.000

2.200

2.400

2.600

2.800

3.000

3.0 3.2 3.4 3.6 3.8 4.0

R (Å)

Triplet approach :

η↑ when R↓

0NR

η∂ < ∂

η(eV)

• Ground state allowed, excited state forbidden [π4s+π2s] : Diels-Alder cycloaddition of 1,3-butadiene to ethene

2.500

2.520

2.540

2.560

2.580

2.600

3.0 3.2 3.4 3.6 3.8 4.0

Singlet approach :

C C

C C

R

η↑ with R↓

0NR

η∂ < ∂

η(eV)

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R (Å)

2.800

2.820

2.840

2.860

2.880

2.900

3.0 3.2 3.4 3.6 3.8 4.0

R (Å)

Triplet approach :

C C

R

η↓ with R↓

0NR

η∂ > ∂

η(eV)

Summary for cycloadditions :

Ethene + Ethene

1 + Forbidden

3 − Allowed

1,3-butadiene + Ethene

Multiplicity

Sign of

NR

η∂ ∂

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1,3-butadiene + Ethene

1 − Allowed

3 + Forbidden

F. De Proft, P.W. Ayers, S. Fias, P. Geerlings, J.Chem.Phys., 125,214101 (2006).

• Sign of as an indicator for allowed/forbidden character of this reaction?

• Justification by a perturbational appraoch

( )N

Rη∂ ∂

Second Case: Electrocyclisations

WH rulesSinglet : conrotatory Singlet : disrotatory

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WH rulesSinglet : conrotatoryTriplet : disrotatory

Singlet : disrotatoryTriplet : conrotatory

Model reaction path R = R (θ, θ')

torsion angles

Model reaction path : conrotatory or disrotatorymovement mimicking the initial phase of theelectrocyclisation (θ, θ' : 0 → 30°)

Conrotatory

Disrotatory

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Disrotatory

1,3,5-hexatriene 1,3-cyclohexadiene

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Both modes lead to a decreasedecrease in hardness• Disrotatory mode: higher hardness• shows less negative less negative initial ∂η/∂R slope ∂η/∂R slope

Initial hardness response in agreement with WH rules

Both modes lead to an increaseincrease in hardness• Conrotatory mode: higher hardness shows largestlargest initial

SINGLET TRIPLET

1,3 – butadiene Cyclobutene

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SINGLET TRIPLET

Hardness in the forbidden disrotatory mode is higher than in the allowed conrotatory mode

Two profiles almost

indistinguishable

? Model reaction coordinate; hardness evaluation

IRC path at CASSCF(4,4)/6-31G*

Hardness via PBE/6-311+G** and Tozer –De Proft approach(*)(D.J.Tozer, F.De Proft, J.Phys.Chem.A 109, 8923 (2005))

• η= I-A A problematic for closed shell systems (metastable anion)

εLUMO, εHUMO: KS orbital energies with a pure density functional corresponding to

(*)

( ) ( )LUMO HOMO HOMO2 Iε − ε + ε +

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εLUMO, εHUMO: KS orbital energies with a pure density functional corresponding to

( )LUMO HOMOA I− ε + ε −�

Only calculation of neutral and cationic systems involved

Reasonable estimates for electron affinities of systems with metastable anions

(F.De Proft, N.Sablon, D.J.Tozer, P.Geerlings, Faraday Discussions, 135, 151 (2007))

1,3- butadiene → cyclobutene

Pag.3-9-2009 27

• Hardness profiles in agreement with allowed or forbidden character of the reactions: hardness along the allowed conrotatory mode is always larger than the hardness of the forbidden, disrotatory mode( also its slope).

----- Conrotatory: allowed____ Disrotatory: forbidden

Results confirmed by electrocyclisations of

2,4- hexadiene Cyclooctatetraene

Cycloheptatriene

2,4-hexadiene conrotatory (A)disrotatory (F)

Ea ŋa ( )Q∂η ∂

43.849.0

-0.343-0.028

-0.152-0.210

( )6π

( )6π

( )4π

Higher η in TS

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cyclooctatetraene

cycloheptatriene

disrotatory (A)conrotatory (F)

disrotatory (A)conrotatory (F)

30.067.2

0.3100.008

10.961.7

-0.715-0.238

-1.212-0.098

0.095-0.158

Internal consistency ( )Q∂η ∂

Confirmation of initial hardness response as indicator of (allowed) forbidden characters

F.De Proft, P.K.Chattaraj, P.W.Ayers, M.Torrent-Sucarrat, M.Elango, V.Subramanian, S.Giri, P.Geerlings, J.Chem.Theory Comput., 4, 595 (2008)

Evolution → aromatic TS

Evolution → aromatic TS

a ,ηaE ,

Third Case: Sigmatropic reactions: H-transfer

[1,5] : allowedH

A BC D

H

DCBA

H

Suprafacial

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Antara- or suprafacial distortions

R = R (θ,θ ') mimicking the onset of the sigmatropic hydrogen shifts

C C[1,5] : forbidden

H

A BC D

H

DC BA

Antarafacial

Antarafacial

Suprafacial

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Suprafacial

Cases Considered [1,3] H shift in propene[1,4] H shift in butene cation[1,5] H shift in pentadiene[1,6] H shift in hexadiene cation

[1,5] H shift in Pentadiene

Pag.3-9-2009 31

SINGLET –suprafacial allowed TRIPLET – antarafacial allowed

Highest hardness always for allowed mode ( )R∂η ∂ Correct prediction

Smaller systems also OK (if model coordinate is replaced by IRC)

N.Sablon, F.De Proft, P.Geerlings, Croatica Chemica Acta (Z.B.Maksic Volume) Submitted

Overview and ConclusionInitial hardness response vs WH

Thermochemical Photochemical

Cycloadditions 2+2 v v

4+2 v v

Electrocyclisations 1,3-butadiene/

2,4-hexadiene v v

1,3,5-hexatriene v v

Pag.3-9-2009 32

1,3,5-hexatriene v v

cyclooctatetraene v

cycloheptatriene v

Sigmatropic Reactions

H-shift Propene [1,3] v v

Butene cation [1,4] v v

Pentadiene [1,5] v v

Hexadiene cation [1,6] v v

Initial Hardness response: reactivity indicator for pericyclic reactions

3.3. The dual descriptor

Fukui function

• Recently introduced as a new descriptor, "dual descriptor"

(C. Morell, A. Grand, A. Toro Labbé, J. Phys. Chem. A109, 205 (2005))

Third order response function( ) ( )

22

N2v

v

rf (r)f (r) f (r)

N N

∂ ρ∂ → = = ∂ ∂

( ) 22

N 1 N N 1N 2v

(r)f (r) (r) 2 (r) (r)

N+ −

∂ ρ= ≅ ρ − ρ + ρ ∂

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= ρN+1(r)− ρN(r)( )− ρN(r)− ρN−1(r)( )

≅ φLUMO(r)2 − φHOMO(r)

2

Large in electrophilicregions

Large in nucleophilicregions

→ fN(2) r( )

> 0

< 0

electrophilic regions

nucleophilic regions

v

Favorable chemical reactions occur when regions that are good electron acceptors

(f(2)(r) > 0) are aligned with regions that are good electron donors (f(2)(r) < 0)

Minimizing

Confirmed by perturbation theoretical ansatz (P.W. Ayers)

( ) ( )2 2

BAf (r)f (r ')drdr '

r r '−∫∫

Pag.3-9-2009 34

Confirmed by perturbation theoretical ansatz (P.W. Ayers)

Application to WH

• Cycloadditions

Butadiene + ethene

HOMO LUMO

Hückel B3LYP/6-31G*

Ethene

[ ]s s4 2π π+

( ) ( ) ( )2 22LUMO HOMOf r r r= φ − φ

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Butadiene

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Molecules align so that favorable interactions (green lines)occur between their dual descriptors

"allowed"[ ]s s4 2π π+

• The case : ethene + ethene

supra/supra : "repulsive"

not allowed

supra/antara : can occur if the molecules rotate so that

[ ]s s2 2π π+

Pag.3-9-2009 37

supra/antara : can occur if the molecules rotate so thatthe double bonds become perpendicular

allowed

(but steric demands too high to occur in practice)

WH rules for cycloadditions regained

• Electrocyclizations

"Generalized Diels Alder reaction" : two fragments tethered together at their ends

Hexatriene 1,3 cyclohexadiene cycloaddition

HOMO LUMO f(2)(r)

[ ]s s4 2π π+

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Butadienyl interacting Favorable interactions

Pag.3-9-2009 39

Butadienyl interactingwith ethylgroup

Favorable interactions

Ring buckles inwards in a disrotatorymotion in agreement with WH

WH regained

Analogously: Butadiene → cyclobutene: conrotatory motion

• Sigmatropic reactions

1,5-methyl shift

Pag.3-9-2009 40

1,5-methyl shift

• generalized (2+4) cycloaddition

• suprafacial (disrotatory motion)

WH regained

1,3-methyl shift: antarafacial

What about Excited States ?

LUMO

HOMO

( ) ( ) 2rρ ρ φ− = − ≅

( ) ( ) 2

1( ) ( )

r

es N N HOMOf r r rρ ρ φ+

+= − ≅

Pag.3-9-2009 41

Reactivity preferences in the excited state are exactly reversedfrom those in the ground state as is the WH rules

P.W. Ayers, C. Morell, F. De Proft, P. Geerlings, Chem. Eur.J., 13, 8240 (2007)

( ) ( ) 2

1( ) ( )r

es N N LUMOf r r rρ ρ φ−

−= − ≅

2 22 2( ) ( ) ( ) ( )es HOMO LUMO gf r r r f rφ φ≅ − ≅ −

Reconciling the two approachesReconciling the two approaches

Dual descriptor =∂f(r)

∂N

v

=∂

∂N

δµδv(r)

N

v

=δ

δv(r)

∂µ∂N

v

N

=δη

δv(r)

N

( )2 ( )f r

Changes in provoked by changes in nuclear configuration only

( )v r( )

( )N

v rdr

R v r R

η η ∂∂ ∂ = ∂ ∂ ∂ ∫

( ) ( ) ( )2 v rf r d r

∂= ∫

Pag.3-9-2009 42

General information refined via to Initial hardness Respone

( )v r

R

∂

∂

( ) ( ) ( )2 v rf r d r

R

∂=

∂∫

4.Conclusions

• WH rules regained in a conceptual DFT context, thereby avoiding

symmetry or phase based arguments

Pag.3-9-2009 43

- initial hardness response

- dual descriptor "back of the envelope" approach

- two approaches internally consistent

Acknowledgements

Acknowledgements

Prof. Frank De Proft

Prof. Paul W. Ayers (Mc Master University, Hamilton, Canada)

Prof. P.K. Chattaraj (IIT – Kharagpur – India)

Dr. C. Morell (CEA, Grenoble)

Pag.3-9-2009 44

AcknowledgementsDr. C. Morell (CEA, Grenoble)

Nick Sablon

Dr. M. Torrent – Sucarrat ( Brussels, Girona)

Prof. D.J. Tozer (University of Durham, Durham, UK)

Fund for Scientific Research-Flanders (Belgium) (FWO)