Conceptual Density Functional Theory:

Redox Reactions RevisitedRedox Reactions Revisited

9-9-2008 1Herhaling titel van presentatie

Conceptual Density Functional Theory:Redox reactions Revisited

Paul Geerlings, Jan Moens, Pablo Jaque*, Goedele Roos

Department of Chemistry, Faculty of SciencesVrije Universiteit BrusselPleinlaan 2, 1050 - Brussels

Pag.

Pleinlaan 2, 1050 - BrusselsBelgium

*Laboratorio de Quimica Téorica Computacional – Pontificia Universidad Católica deChile, Stantiago, Chile

TACC 2008TACC 2008Shanghai, ChinaShanghai, China

September September 2323--27, 27, 20082008

9-9-2008 2

Outline

1. Introduction: Chemical Concepts from DFT

2. Redoxpotentials of Men+/Me(n-1)+ couples: the role of electrophilicity.

3. An EEM approach based on the Grand Canonical Ensemble Theory

Pag.

3. An EEM approach based on the Grand Canonical Ensemble Theory

4. Some disgressions about vertical quantities

5. Conclusions

6. Acknowledgements

9-9-2008 3

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)"

Pag.9-9-2008 4

•

•

• • •

•

•

•

••

•

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

( )( )HKF

v rr

δµ

δρ+ =

Pag.9-9-2008 59-9-2008 5

• 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.9-9-2008 69-9-2008 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

identification first order perturbation theory

( )

( )( )v r N

E EdE dN v r dr

N v r

∂ δδ

∂ δ

= +

∫

Pag.9-9-2008 79-9-2008 7

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)

= µδE

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

=

Pag.9-9-2008 89-9-2008 8

Chemical Chemical hardnesshardness

ChemicalChemical SoftnessSoftness Fukui functionFukui function

Local softnessLocal softness

∂ E

∂N2

v(r)

= η∂ E

∂Nδv(r)=

δµ

δv(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

Pag.9-9-2008 99-9-2008 9

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, Br∅nsted-Lowry)

- generalized acid base (Lewis) → complexation reactions

- organic chemistry

• electrophilic/nucleopilic• substitution/addition/elimination

Pag.9-9-2008 109-9-2008

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

• Pericyclic Reactions (WATOC talk)

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

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

• 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)

Pag.

V. Subramanian, S. Giri, P. Geerlings, J.Chem.Theory Comput., 4, 595 (2008)

• Redox Reactions

Today’s talk

9-9-2008 11

2. Redox potential of Men+/Me(n-1)+ complexes: the role of electrophilicity

• Redoxreactions are the Archetypes of reactions involving a change in the number of electrons perturbative approach

• Appoperate desciptor?

• Electrophilicity ω:

[ ],E E N v=

Pag.

• Electrophilicity ω: R.G. Parr, L.v.Szentpaly, S.Liu, J.Am.Chem.Soc., 121, 1922 (1999)

P.K. Chattaraj, U.Sarkar, D.B.Roy, Chem.Rev., 106, 2065 (2006)

• Local version?

9-9-2008 12

Systems: Me3+/ Me2+(aq) redox couples

First row transition metals: Sc → Zn

ω = electrophilicity

= energy lowering at maximal uptake of electrons

2

2E

µω

η∆ = − = −

Interaction with a perfect electron donor

Pag.9-9-2008 13

Nucleofugality=ω-A

• Quadratic model:

• Me3+ : highly prone to electron uptake

( ) ( )2 2

2 3

2 8( ) 8( )

I A I AA

I A I A

µω

η

+ −= = = +

− −

( )2( )ideal

I AN

I A

µ

η

+∆ = − =

−

1idealN∆ >>

Pag.9-9-2008 14

14

System readily accepts more than one electron

Use of ωScaled :2

1Scaled

idealN

ωω =

+ ∆

• System in aqueous solution

• First (6) and second (12) solvation sphere

• PCM

Pag.9-9-2008 15

First solvation sphere First +second solvation sphere

Correlation analysis E° vs. ω

Correlation

Coefficient

R2

Gas 6 H2O 6 H2O +

PCM

18H2O 18 H2O +

PCM

ω 0.76 0.88 0.76 0.87 0.79

Pag.9-9-2008 16

ω 0.76 0.88 0.76 0.87 0.79

ωScaled

0.83 0.95 0.95 0.90 0.91

∧ ∧ ∧ ∧ ∧

Pag.9-9-2008 17

F

Absolute values of E° versus ωScaled for metal ions embedded in PCM and a first solvation sphere

Local Level (e.g. Fe3+/Fe2+)

Bare atom 6 H2O

First solvation

sphere

12 H2O

Second solvation

sphere

Charge (NPA) 2.136 0.240 0.625

Fukui function f+ 0.524 0.324 0.152

k kfω ω += Local electrophilicity ( : electron uptake)f

+

Pag.9-9-2008 18

• Two electrophilic regions

• Central metal ion

• First solvation sphere

• Second solvation sphere: “take-and-give”, acts as a continuum

Bare atom 6 H2O

First solvation

sphere

12 H2O

Second solvation

sphere

Local ω

(ω+)

0.263 0.163 0.076

Pag.9-9-2008 19

Group Philicity

,group Metal First shell ii

ω ω ω= +∑

• Local electrophilicity (condensed)

• Additive

kω+

k

k

ω ω+ =∑

Pag.9-9-2008 20

• Completely mimics the redox potential (R2=0.90)

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).

3. An EEM Appraoch based on the Grand Canonical Ensemble Theory

Previous Approach

• Electrophilicity: interaction between a system and a perfect electron donor⇒ ideal system with zero chemical potential.

• Now: simulation of chemical environment of oxidized and reduced species by an electron bath with constant chemical potential µelectrode =electrode in experimental electrochemistry

• Open systems: exchange of electrons when µelectrode is changed

Pag.9-9-2008 21

Grand Potential

A= Helmholtz free energy functional

Equilibrium,

0T VN

∂Ω =

∂

T=0

( ) ( ) electroder A r Nρ ρ µΩ = Ω = −

,

electrode

T V

A

Nµ

∂ =

∂

,

electrode

T V

E

Nµ

∂ =

∂

e-

e-

e-

e-

e-

e-

e-

e-

e-

e-

µelectrode

Ox Red

Oxidized species Reduced species

oxN∆ redN∆

Second order model 0 0 Nµ µ η= + ∆

Pag.9-9-2008 22

µelectrode

0 0electrode ox ox oxNµ µ η= + ∆

0 0electrode red red redNµ µ η= + ∆

0 0

0 0red ox

ox red

red ox

N Nµ µ

η η

−∆ = −∆ =

+

Equilibrium point:

with

µox= µred=

Cfr. Electronegativity, Equalization

One-electron redox potential for 44 different organic and inorganic systems:

0 0electrodeNHEE E

F

µ= − −

0 0 0 0

0 0ox red red ox

electrode

ox red

µ η µ ηµ

η η

+=

+

-3 -2 -1 0 1 2 3 4

-2

-1

0

1

2

3

4

E° e

xpt (

V)

E°calc

E°expt.

= 1.174 E°calc.

+ 0.190V

Now µelectrode : change in Helmholtz Free Energy at 0K ≅Gibbs Free Energy related to one-electron half reaction

Pag.9-9-2008 23

organic and inorganic systems:– aliphatic group (acetonitril (PCM)) (12)

– aromatic group (acetonitril (PCM)) (16)

– aniline group (water(PCM)) (7)

– inorganic systems: First row transition metal ions

(18 H2O +water(PCM)) (9)

• Remarkable accuracy is achieved based on solely vertical quantities (I and A used to evaluate µ0 and η0 of oxidized and reduced species) and neglecting entropic and thermal contributions.

•No problems with scaling of highly electrophilic species as in previous method.

-3

expt. calc.

R2=0.960

4. Some disgression about vertical quantities

• More common approach in the calculation of redox potentials is a Thermodynamic cycle: linking proces in gas phase and solution

• Treatment of solvent effects during the electron transfer proces

• ?Accurate determination of solvation energy of reduced and oxidized species

• Electron transfer takes place rapidly geometric relaxation after the electron transfer

Pag.

transfer

Electron uptake / loss step

Reorganization step

• How can the vertical quantities I and A used in the first method be exploited in this approach?

9-9-2008 24

•

• Re Red d e Ox Ox−− →

Re ReOx Ox e d d−+ →

, Re ,Re Re Re ,1 ReOx Ox Ox d d d org dE E E I E E∆ = − = + + ∆ = −∆

,1 ,2 Rereorg reorg Ox dE E A I∆ + ∆ = −

Hypothesis ,1 ,2reorg reorgE E∆ = ∆

( )Re

1

2reorg Ox d

E A I∆ = −

E

Re Re Re ,2d dred E Ox Ox Ox orgE E A E=∆ − = − + ∆

Pag.9-9-2008 25

2

( )Re Re R

1

2d d Ox edE I A I∆ = − − −

( )R

1

2Ox Ox ed

A A I= − + −

( )Re R

1

2d Ox ed Ox

E A I E∆ = − + = −∆

→ Reduction/Oxidation proces can be approximated by vertical electron affinity and ionization energy taking into account a constant reorganisation term, identical for oxidation and reduction

0 0RedNHE

EE E

F

∆= − −

Slope close to unity deviating only by 5%

1

2

3

4

E° ex

pt.

.(V)

Pag.9-9-2008 26

-3 -2 -1 0 1 2 3 4

-3

-2

-1

0

1

E°calc.

eq 19 (V)

E°expt.

= 1.050 E°calc.

+ 0.227V

R2=0.960

Avoiding the hypothesis,1 ,2reorg reorg

E E∆ = ∆

0 0

0

o o

ox red red ox

electrode o

ox red

µ η µ ηµ

η η

+=

+

( )1

2red ox redE A I∆ = − +

Split up ,1electrode red reorgI Eµ = − − ∆0

,1 0 0

2red

reorg reorg

ox red

E Eη

η η∆ = ∆

+

Pag.9-9-2008 27

,2electrode ox reorgA Eµ = − + ∆

ox red

0

,2 0 0

2ox

reorg reorg

ox red

E Eη

η η∆ = ∆

+

As → difference between two reorganization energies

Different curves for oxidized and reduced species[ ]E E N=

0 0

red oxη η≠

It turns out that the reorganization energy of the oxidized species is largely underestimated (compare with exact adiabatic calculations)

(*)

Pag.9-9-2008 28

Reorganization Energy Oxidized Species

Solution

• Refinement of the curves for oxidized and reduced species[ ]E E N=

J.L.Gazquez, A.Cedillo, A.Vela, J.Phys.Chem.A. 111, 1966(2007)

B.Gómez, N.V.Likhanova, M.A.Dominguez-Aguilar, R.Martinez-Palou, A.Vela, J.L.Gazquez, J.Phys.Chem.B. 110, 8928 (2006)

( )1

34

I Aµ + = − +

( )1

3I Aµ − = − +

Electron uptake; right derivative, more emphasis on A

Electron release; left derivative,

Pag.9-9-2008 29

( )1

34

I Aµ − = − + Electron release; left derivative, more emphasis on I

-1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

;1

( )red ox redreorganization

ox red

A IE

µ

µ µ

−

+ −

−∆ =

+

R2=0.938

∆E

reo

rgan

izat

ion

,1(i

dea

l) (

eV)

(eV)

Pag.9-9-2008 30

-2.0

-1.8

-1.6 ∆

Insertion in ( justification: µ± instead of η)reorg

E∆

,1

2red

reorg reorg

ox red

E Eµ

µ µ

−

+ −∆ = ∆

+ ,2

2ox

reorg reorg

ox red

E Eµ

µ µ

+

+ −∆ = ∆

+

Insertion

,1electrode red reorgI Eµ = − − ∆

,2ox reorgA E= − + ∆

,1

2 red reorg

reorg

ox red

EE

µ

µ µ

−

+ −

∆∆ =

+

( )2 red ox red

ox red

A Iµ

µ µ

−

+ −

−=

+

Pag.9-9-2008 31

with ( )1

34

ox ox oxI Aµ + = − +

( )1

34

red red redI Aµ − = − +

(no problem with stability of cations)

-3 -2 -1 0 1 2 3 4

-1

0

1

2

3

4

E° e

xpt.(V

)

E° (V)

• Excellent correlation• Slope still only 5% deviating from 1

Individual reorganization energy for oxidized and reduced form show fair correlation with exact adiabatic values.

Pag.9-9-2008 32

-3

-2

-1 E°calc

(V)

E°expt.

= 1.052 E°calc.

+ 0.184 V

R2=0.958

J. Moens, P. Jaque, F. De Proft & P. Geerlings , J.Phys.Chem.A, 112, 6023 (2008)

adiabatic values.

5. Conclusions

• Conceptual DFT offers a framework for the interpretation of redox

chemistry

• The electrophilicity turns out to be a key descriptor to get insight

into the redox behaviour of Men+/Me(n-1)+ couples and gives quantative

information about a system’s ability to accept electrons

• The introduction of the chemical potential of the electrode resulted

Pag.9-9-2008 33

in a quantitative estimate of the redox potential for reversible one–

electron half–reactions.

• The disgression about vertical quantities (A,I) combined with

Gazquez’s recent expressions for µ+ and µ- leads to an excellent

correlation between caculated and experimental Redox potentials with

fair account for reorganisation effects.

6. Acknowledgements

Jan Moens

Dr. Pablo Jaque (Santiago – Chile)

Dr. Goedele Roos

Pag.

Prof. F. De Proft

Prof. P. Ayers (Mc Master – Canada)

Fund for Scientific Research Flanders (FWO)

9-9-2008 34