DFT Reactivity Descriptors and Catalysis - Vrije Universiteit Brussel

DFT Reactivity Descriptors
and Catalysis
30-9-2008 Herhaling titel van presentatie 1

DFT Reactivity Descriptors and Catalysis
Paul Geerlings
Department of Chemistry, Faculty of Sciences
Vrije Universiteit Brussel
Pleinlaan 2, 1050 - Brussels
Belgium
Emeritaatsviering Professor Robert Schoonheydt
Leuven, October 1, 2008
30-9-2008 Pag. 2

Outline
1. Robert Schoonheydt and the VUB: some reminiscences
2. Chemical concepts from DFT
3. Some applications of global and local softness and
hardness in Catalysis
4. Towards the inclusion of confinement effects
5. Conclusion
30-9-2008 Pag. 3

1. Robert Schoonheydt and the VUB:
some reminiscences
It all started in the late seventies as a collaboration with
Wilfried Mortier on the use of quantum chemical methods
(at that time CNDO/2) to get more insight into the
structure, chemistry and catalytic – activity of zeolites.
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and turned, in the nineties, into a collaboration with Robert
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finally leading to ± 25 joint (KULeuven- VUB) publications
on the application of quantum chemistry
in zeolite chemistry, the most recent one being
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Joint publications on Quantumchemical studies on Zeolites
• predominant role of
• Chemical Concepts from DFT
• Conceptual DFT
• gradual transition
• Electronegativity
hardness/softness
• Global
local
What are these descriptors, how do they originate,
what is the overall context in which they are to be used?
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2. 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))
ρ(r) as basic variable
ρ(r) determines N (normalization)
"The external potential v(r) is determined, within a trivial additive constant, by the
electron density ρ(r)"
electrons
• • •
• • • •
• •
•
•
ρ(r) for a given
ground state
compatible
with a
single v(r)
v(r)
- nuclei
- position/charge
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30-9-2008 11

ρ(r) → H op → E = E[ ρ]= ∫ ρ( r)
v(r)dr + F HK ρ(r)
• Variational Principle
[ ]
δF v( r)
+ HK
=
δρ( r)
µ
F HK Universal Hohenberg Kohn functional;
contains unknown E xc , exchange correlation
energy functional
Lagrangian Multiplier
normalisation
∫ ρ(r) dr = N
• Practical implementation : Kohn Sham equations
Computational breakthrough
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30-9-2008 12

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
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
µ as (the opposite of) the electronegativity.
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Starting point for DFT perturbative approach to chemical reactivity
Consider
E = E[N,v]
Atomic, molecular system, perturbed in number
of electrons and/or external potential
⎛ ∂ E ⎞ ⎛ δ E ⎞
dE = ⎜ ⎟ dN + δ v( r)
dr
∂ N
∫ ⎜ ⎟
δ v( r)
⎝ ⎠v( r ) ⎝ ⎠N
µ
identification
first order perturbation theory
ρ( r)
identification
Electronic Chemical Potential (R.G. Parr et al, J. Chem. Phys., 68, 3801 (1978))
= - χ (Iczkowski - Margrave electronegativity)
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Identification of two first derivatives of E with respect to N and v in a DFT
context → response functions in reactivity theory
E[N,v]
⎛ ∂E ⎞
⎜ ⎟
⎝ ∂N⎠
v(r)
= µ
⎛ δE ⎞
⎜ ⎟
⎝ δv(r) ⎠
N
= ρ(r)
⎛ ∂ 2 E ⎞
⎜
⎝ ∂N 2 ⎟
⎠
v(r)
= η
Chemical hardness
∂ 2 E
∂Nδv(r) = ⎛ δµ ⎞
⎜ ⎟
⎝ δv(r) ⎠
= f(r)
N
⎛
= ∂ρ(r) ⎞
⎜ ⎟
⎝ ∂N ⎠
v
2
⎛ δ E ⎞
⎜
⎟
⎝ δ v( r) δ v( r ') ⎠ N
= χ( r, r ')
Linear response
Function
S = 1 η
Chemical Softness
Fukui function
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Softness and hardness: from global to local level
2
E
η ⎛ ∂ ⎞
2
⎝ ∂N
⎠v
⎛ ∂ µ ⎞
= ⎜ ⎟ = ⎜ ⎟
⎝ ∂N
⎠
v
η S =
1
( r)
⎛ ∂N
⎞ ∂
⎛ ∂ρ
⎞
S = ⎜ ⎟ = ρ ( r)
d r = ⎜ ⎟ d r = s( r)
d r
∂µ ∂µ ∫ ∫
∂µ
∫
⎝ ⎠v
⎝ ⎠v
s( r ) = local softness
Chain rule ( )
( ) ∂ρ
( )
∂ρ
r ⎛ r ⎞ ⎛ ∂N
⎞
s r = = ⎜ ⎟ ⎜ ⎟ = Sf r
∂µ ⎝ ∂N
⎠ ∂µ
v ⎝ ⎠v
( )
⎛
δµ
⎞
• Local Hardness η ( r)
= ∫η ( r) s ( r) d r = 1 but η
⎜
!!
δρ ( r) ⎟
( r)
≠ 1
s ( r)
⎝ ⎠ v
∫
η
( r) ( )
f r d r
= η
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• Ambiguity in the definition
various proposals
For a recent critical account:
P.K.Chattaraj, D.R. Roy, P.Geerlings, M.Torrent-Sucarrat, Theor.Chem.Acc., 118, 923 (2007)
In the work presented here: two relatively simple approaches:
• M.Berkowitz, S.K. Ghosh, R.G. Parr, J. Am.Chem.Soc., 107, 6811 (1985)
η
1
N
( r) ∼ V ( r)
el
Cfr. Molecular Electrostatic Potential
( )
MEP r
( r ')
Z ρ dr '
A
= ∑ −
R − r
∫
r ' − r
A
A
• P. Geerlings, A.M. Vos, R.A. Schoonheydt, J.Mol. Struct. (THEOCHEM), 762, 69 (2006)
(A.Goursot Volume)
η ∼
A
N
N
A
ZA
= Cte
QA
η
A
∼
N
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A final comment
• nowadays: increasing interest in higher order response functions
( )
( ∂
n E ∂
m Nδ
m' v r )
( n = m + m' )
• recently the third order response function
3
⎛ ⎞ ∂
⎜
⎝ ∂
∂ E
δ
δη
⎟ = =
⎠
( ) ⎟ δ ( )
( )
f r
2
N v r v r N
∂
Dual Descriptor
C.Morell, A. Grand, A.Toro-Labbé,
J.Phys.Chem.A, 109, 205 (2005)
turned out to yield a firm basis to regain the Woodward Hoffmann
rules for pericyclic reactions, in a density – only context.
avoiding the phase, sign, symmetry of the wave function
(cfr. ρ(r) is strictly positive and has a trivial symmetry)
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)
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Applications until now mostly on (generalized) acid base reactions
- acid/base (Arrhenius, BrØnsted-Lowry)
- generalized acid base (Lewis) → complexation reactions
importance in Zeolite chemistry/ catalysis
- organic chemistry
• electrophilic/nucleopilic
• substitution/addition/elimination
Use of Principles
- EEP (Electronegativity Equalization Principle) (Sanderson, Mortier)
- HSAB (Hard and Soft Acids and Bases Principle; global/local)(Parr, Pearson)
- MHP (Maximum Hardness Principle)(Pearson)
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Reviews
• H. Chermette, J. Comput.Chem., 20, 129 (1999)
• F. De Proft, P. Geerlings, Chem.Rev., 101, 1451 (2001)
• 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)
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3. Some applications of global and local softness and
hardness in Catalysis
Kinetics of acid catalyzed electrophilic aromatic substitution reactions
Friedel Crafts alkylation
functionalization of aromatic molecules
Traditionally: homogeneous Lewis acid catalysis
Nowadays: more and more heterogeneous Br∅nsted acid catalysis: zeolites
? Mechanism – number of steps
- nature of the alkylating reagent
• non-empirical ab initio evaluation of reaction profiles and rate constants
• interpretation of the results in conceptual DFT context (MHP, HSAB)
Alkylation of benzene/toluene
Role of various domains of Quantum Chemistry
• methanol/ethene/propene
Br∅nsted Acid Catalysis
• CH 3 F, CH 3 Cl, CH 3 OH
Lewis Acid Catalysis
in combination with BF 3 , AlCl 3 , Al(OH) 3
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3.1 Br∅nsted Acid Catalysis by Zeolites –
Methylation of benzene/toluene
• Two – or single step proces → reaction profile
• Rate constants
• Influence of substituent
• Interpretation within conceptual DFT context
• 4T cluster model (1Al; 3 Si)
• DFT: B3LYP
rate constants
reactivity descriptors
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The Two Step Consecutive Mechanism
• First methanol adsorbs on the Br∅nsted acid site and the acid proton of the cluster
jumps to the hydroxyl group of methanol and methylation of the cluster occurs
• In the second step benzene adsorbs on the methylated cluster
and methyl migrates to benzene with
transfer of a proton to the cluster
Energy scale in kJmol -1
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The Direct Mechanism
• Methanol and benzene coadsorb on the acid site
• Zeolite proton moves to methanol; formation of
water; methylgroup of methanol moves to benzene
transferring a proton to the zeolite
Si2
Cb
Hb
O2
Al
Cm
Om
Ha
Hm
O3
Si3
TS2
O4
acid zeolite
+ methanol (g)
+ benzene (g)
acid zeolite
+ toluene (g)
84.25
40.21
+ water (g)
R2
194.87
27.28 71.32
P2
Si2
O2
Cb
Hb
Ha
O1
Al
O4
Om
O3
Cm
Hm
Si3
Si2
Cb
Hb
O2
O4
Cm
Ha
Om
Hm
Al
O1
O3
Si1
Si3
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• Rate Constant Evaluation
• Transition State Theory
k
r
k T
h
Q
Q
‡
B
= ⋅ ⋅
∏
i
i
e
−E
/ bar
RT
• Q ‡ ,Q i …: partition functions in rigid rotor, harmonic oscillator approximation
• E bar : activation barrier
Example: one step reaction, A and B reacting simultaneously with an acidic
zeolite HZ: reaction rate per acid proton
‡
( Q Q Q
)
k T
k ( )
−
r
= N
AvV × ⋅e
h Q Q Q Q Q Q Q Q Q
2 B
v r t TS
−
E /
RT
( ) ( ) ( )
v r t A v r t B v r t HZ
bar
Arrhenius type analysis
E A , A, ∆S Act
• Tested via H/D Exchange reaction of methane: 3T cluster model
fair agreement between theoretical and experimental activation energy
A.M. Vos, F.De Proft, R.A.Schoonheydt, P. Geerlings, Chem.Comm., 1108 (2001)
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Results for the 4T cluster (1Al, 3Si) model
• rate constants 300-800K E A
• 2 mechanisms consecutive* vs direct
E A
193 127 kJmol -1
* Pre-exponential factor is higher so that at higher T the two
mechanisms become competing (400K) (effect of ∆S act )
Direct mechanism in line with experimental data ( no stable intermediates) and confirmed
in periodic boundary calculations by Van Santen (J.Am.Chem.Soc.123, 2799(2001))
• Influence of substituents: the role of activation hardness (MHP at work)
∆ η = η −η
Z. Zhou, R.G.Parr, J.Am.Chem.Soc, 112, 5720 (1991)
react
TS
obtained via HOMO –LUMO gap
E A (kJmol -1 ) k (400K) ∆η (activation hardness)
Benzene 127.1 (4) 5.6 10 -24 0.0597 (4)
Toluene o 114.2 (1) 3.6 10 -23 0.0493 (1)
m 123.9 (3) 3.1 10 -24 0.0536 (3)
p 117.8 (2) 2.2 10 -22 0.0499 (2)
• electrondonating character of CH 3 -group shows up in E A .
• E A and activation hardness match
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• Local level
• Softness indicators fail
Charge controlled reaction
Cfr. Hardness of electrophilic species CH 3+ (η= 7.83V)
• Dominance of electrostatic interactions
η C(arom)
|q c | (arom) derived from Electrostatic potential based
∆E
cc
∼
q
q
( ar ) c( methanol )
R cc
Charges (CHELPG) for
adsorbed reactants
η c(ar)
∆E cc
Benzene 0.0259 -9.85
Toluene o 0.1522 -67.63
m 0.0033 -1.36
p 0.0548 -20.19
Hardness related quantities are adequate reactivity descriptors
for these reactions
A.M.Vos, K.H.Nulens, F.De Proft, R.A. Schoonheydt, P.Geerlings, J.Phys.Chem.B, 106, 2026 (2002)
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From Regioselectivity (Intramolecular Reactivity Sequence)
to an Intermolecular Reactivity Sequence
Comparison between ethylation and isopropylation of benzene.
starting from ethylene and propene (one step reaction)
E A (kJmol -1 ) k (400K) η c
*
η H ∆E el
114.2 4.8 10 -23 0.174 0.331 -0.0253
106.0 2.0 10 -21 0.357 0.343 -0.0564
* C-atom of ethene or propene protonated by the acid proton
of the zeolite ( values in adsorption complex)
• sequence of k and E A in agreement with experiment (Corma et al, J.Cat., 192,
163 (2000)): isopropylation more favored
• hard – hard interaction - acidic proton of zeolite
- C atom of the alkene that is protonated more
favored in isopropylation
A.M.Vos, R.A.Schoonheydt, F.De Proft, P.Geerlings, J.Phys.Chem B, 107, 2001 (2003)
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3.2 Lewis acid Catalysis: the role of softness
• Catalytic activity of Lewis acid sites in Zeolites: much less frequently studied
than Br∅nsted acid sites
• Methylation of benzene catalyzed by Lewis acids BF 3 , AlCl 3 , Al(OH) 3 and a Lewis acid
site (L1) of a zeolite involving a three coordinated Al
A.A. Lamberov et al, J.Mol.Cat.A,
158, 481 (2000)
Methylating reagents
CH 3 F, CH 3 Cl, CH 3 OH(2x) ≡
CH 3 X
AlCl
e.g. CH 3 -Cl + 3
CH 3
+ HCl
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General, schematic energy diagram
TS
benzene +
lewis acid + CH 3
X
toluene +
lewis acid + HX
E 1 E 2 E app E act E 2 ’ E 1 ’
benzene +
(lewis acid · CH 3
X) toluene +
(lewis acid ·HX)
(lewis acid · CH 3
X · benzene)
E r
(lewis acid · HX · toluene)
• formation of a Lewis Acid – CH 3 X - Benzene Complex
• single step reaction, without formation of stable intermediates
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Transition States
C m
H 3Cb
X 1
H 2
X 3
X
M 2
tsM_Al(OH) 3
tsM_BF 3
tsM_AlCl 3
C m
CbH2 O 2
O 3
O 1
Al
tsM_L-site
• concerted, not synchronous, pathway with cyclic
transition structure
• migrating methyl group
• planar
• charge ~ +0.35 a.u.
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Activation energies and rate constants
catalytic activity AlCl 3 > BF 3 > Al(OH) 3 > L1
• AlCl 3 most efficient
catalyst in agreement with
experimental studies on
relative efficiencies of
metalhalides (Russell, Olah)
• more appropriate in H-
exchange reactions (not
shown)
• influence of surrounding
atoms?
Crystal calculations
Interpretation?
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Cyclic TS characterized by several interactions of the
electrophilic/nucleophilic type
N
Calculated TS with CH 3 F/BF 3
CH 3+ attacking
benzene
E
C m
C b
E
H 2
H + leaving
benzene
X - leaving
CH 3 X
X 1
N
M
E
X 2
N
C m
H 3Cb
X 1
H 2
X 3
X
M 2
tsM_BF 3
X 3
X 4
orbital interactions (eg vacant orbital(s) of Lewis acids) > electrostatics
softness at work?
HSAB at local level A ..... B s A
+
as close possible to s B
-
+ -
Multicenter event: looking for smallest value of ∑( s -s ) 2
A,i B,i
Cfr. regioselectivity of Diels Alder reactions showing cyclic TS.
S. Damoun, G. Vande Woude, F. Mendez, P. Geerlings, J.Phys.Chem.A, 101, 886 (1997)
i
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Sequence AlCl 3 < L 1 < BF 3 < Al(OH) 3 ; also for H-exchange reaction
?
Note: all hardness related descriptors failed in describing these reactions
Lewis acid catalyzed reactions are much better described by softness
related descriptors than the acid zeolite catalyzed ones.
AlCl 3 always more active than BF 3 in agreement with experimental data.
Ranking of Al(OH) 3 and L 1 with comparative active centers less obvious
and incites further research on structure /acid strength of Lewis acid sites
in zeolites
A.M.Vos, R.A.Schoonheydt, F.De Proft, P.Geerlings, J.Catal. 220, 333 (2003)
P. Geerlings, A.M. Vos, R. Schoonheydt, J.Mol.Struct. (Theochem), 762, 69 (2006) (A.
Goursot Festschift)
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4. Towards the Inclusion of Confinement Effects
• Relevance cfr. PCCP Thematic Issue “Molecules in Confined Space”
“ … intricate chemistry of molecules taking place in the confined space of
e.g. zeolites …”
• Zeolites with their 3D network involving cages are archetypal structures
• Early work (C.M. Zicovich- Wilson, A. Corma, P. Viruela, J.Phys.Chem., 98, 10863 (1993)
• New concept: “electronic confinement”: density of the guest molecule orbitals suddenly
drops to nearly zero when reaching the walls of the zeolite as a consequence of the shortrange
repulsion with the delocalized electronic cloud of the lattice change is orbital
energy levels
•Hückel Theory ethylene HOMO-LUMO gap, and so the hardness, decreases if the
molecule is spatially confined.
c=c
η
LUMO
HOMO
η
LUMO
HOMO
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• Confirmed by later work by Corma and coworkers, using a supermolecule
approach, on ethylene, benzene, toluene in a zeolite cage and naphtalene in
pure silica structures “ avoiding the interference of electrostatic effects”.
( F.Marquez, C.M. Zicovich-Wilson, A. Corma, E. Palomares, H. Garcia, J.Phys.Chem.B, 105, 9973 (2001)
Are these results reflecting “pure confinement” effects or
are (still) electrostatic, polarization, … effects at work due
to the direct interaction between guest molecule and
confining atoms?
B
A
Study of pure confinement effects on global
and local DFT based reactivity descriptors
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Atoms
• previous work (1992-2005) (impenetrable spherical cavity)
by Goldman, Garza, Sen, Chattaraj
• blue shifted transition frequencies
• increasing hardness
• decreasing polarizability
• decreasing electronegativity
Our approach
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DFT based – spherical Confinement approach (collaboration with D.J.Tozer)
• Atomic Radius R A =λB A
Bragg radius
• Within the sphere: convertional v xc
• outside the sphere: v xc = constant
ensuring appropriate asymptotic density decay
λ=1.7 λ=3.0 λ=4.0
λ↑ →molecule is trapped within cavity resembling an ellipsoid
Pag.
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An atom test case: helium
Radial Density Distribution
V=9.10 3 au
ensures “atom trapped in a
spherical wall”
Atom increases its HOMO-LUMO
gap upon confinement i.e.
becomes “harder”
λ= 3
B He = 0.35 Å
λB He ~ 2a.u.
cfr earlier (1992-2005) work
Number of electrons outside this region < 10 -7
Pag.
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Molecules: ethylene
• Increasing hardness upon
confinement
•Trend opposed to literature results
by Zicovich-Wilson, Corma
Planar Confinement?
Modellisation (planar
hydrocarbons)
v xc =cte (9 .10 3 au)
R c =λB c
Conventional
v xc
v xc =cte
Pag.
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Ethylene: planar confinement
• Same trend as in spherical
confinement
Pag.
30-9-2008 41

Confirmation: benzene, toluene,naphtalene: trend again opposed to literature results
• Reason for systematic
error?
• In literature: explicit
interaction between
molecule and confining
medium in a supermolecule
approach
Naphtalene
(Corma et al,
J.Am.Chem.Soc.122, 6520 (2000)
J.Phys.ChemB, 105, 9973 (2001))
Confrontation with experiment
Upon elimination of specific interactions, exclusively focussing on
confinement effect, hardness increases effect on reactivity
remaining questions: magnitude of the effect; to which extent does it cancel
effect of specific interactions
A. Borgoo, D.J. Tozer, P. Geerlings, F.De Proft, PCCP, 10, 1406 (2008)
Naphtalene
0-0 transition 309 nm - 316 nm
∆λ= 7 nm
Pure silica
structure
Pag.
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A first try: an order of magnitude analysis
Ethlene:
Exp
η = I − A = 12.45eV = 0.458au
≅ ε LUMO - ε HOMO
Confinement: take λ=4 R B = 0.70Å → R=2.80Å
→ ± spherical confinement with radius ≅ 4.0Å
cfr. Zeolite Cavity with diameter of 8Å
∆η
≃ 0.015au
η=0.473au=12.87eV
Increasing hardness
Increasing in hardness: cfr. Particle in a Box
Cube:
1 L↓
En ∼ ⎯⎯→ E
2
n
− En−
1
↑→
L
Influence on wavelength HOMO-LUMO transition
∆λ~ -3 nm
increasing hardness
Effect opposed to “softening upon
confinement” in the literature
Pag.
30-9-2008 43

What about effects on local descriptors, eg, Fukui function?
Some introductory results on ethylene
f + (r)
f - (r)
V=9x10 3 au
Fukui function for nucleophilic attack (f + (r)) is dramatically affected by the confinement,
whereas fukui function for electrophilic attack f - (r) is hardly affected
Reason: artificial binding , due to the potential barrier, of the extra electron in the anion
(cfr. evaluation- technique for negative electron affinities)
N. Sablon, F. De Proft, P.Geerlings, D.J.Tozer, PCCP, 9, 5880 (2007)
F.De Proft, N. Sablon, D.J.Tozer, P.Geerlings, Farad.Discussions, 135, 151 (2006)
Confinement influences shape of the Fukui function and
thereby also regioselectivity reactivity
A. Borgoo, F. De Proft, P. Geerlings, D.J.Tozer, ICCMSE 2008, Am.Inst. Phys.Conf. Proceedings, in press
Pag.
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Epilogue: looking back at the 1979-2008 period from a theoretical/
computational chemistry point of view: a spectacular evolution.
• Increasing computing power (external factor)
• Moore’s law: doubling of computing power each 1.5 year
• 30 years = 20 x 1.5 years → factor 2 10 ≅ 10 6
• Theoretical/ methodological advances
• Introduction of DFT: a true computational breakthrough
• Scaling properties
DFT – B3LYP ~ N 2.5
CISD ~ N 6 ; MP2 ~ N 5
→ Performance ratio B3LYP/ other correlated techniques ~ N 2 à N 3
• Overall efficiency/quality increases by several orders of magnitude extra
as compared to 10 6
Theoretical / Computational Chemistry getting closer and closer to ‘real’
Chemistry, eg. Zeolite Chemistry
Pag.
30-9-2008 45

5. Conclusions
• Present day Quantum Chemical Computational Techniques
afford detailed studies on reaction path and rate constant of (zeolite) catalysed
reactions
• Following Parr’s dictum “to calculate a molecule is not to understand it”
interpretation of the results provides further insight. Conceptual DFT can play
a predominant role in this field.
• Global (activation hardness) and local descriptors (local softness, local
hardness) offer complementary information in combination with principles such
as HSAB an MHP.
• When discussing confinement effects care should be taken to make the
distinction between pure confinement and effects resulting from the direct
interaction with the cage atoms.
• The future of a Computational approach to (Zeolite) Catalysis is bright.
Pag.
30-9-2008 46

6. Acknowledgments
Prof. Wilfried Mortier (KULeuven)
Prof. Frank De Proft (VUB)
Dr. An Vos
Dr. Pierre Mignon
Prof. David Tozer (Durham, UK)
Alex Borgoo (VUB)
and Prof. R. Schoonheydt
… ad multos annos
30-9-2008 Pag. 47