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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
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