Conceptual Density Functional Theory
Paul Geerlings
Department of Chemistry, Faculty of Sciences
Vrije Universiteit Brussel
Pleinlaan 2, 1050 - Brussels
Belgium
STC 2011, Sursee,
Sursee, Switzerland
August 2121-25, 2011
18-8-2011
Pag.1
Outline
1.
Introduction: Chemical Concepts from DFT
2.
The Linear response Function
2.1. Direct evaluation of the second order functional derivates
2.2. An Orbital based Approach
2.3. From inductive and mesomeric effects to aromaticity
3.
Higher order Derivates
3.1. The Dual Descriptor
3.2. Reconciling the dual descriptor and the initial hardness response
3.3. The Chemical relevance of higher (n≥3) order derivatives
4.
Conclusions
5.
Acknowlegements
18-8-2011
Pag.2
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))
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
18-8-2011
Pag.3
compatible
with a
single v(r)
v(r)
- nuclei
- position/charge
ρ (r ) → H op → E = Ev [ ρ ] = ∫ ρ ( r )v(r )d r + FHK [ ρ (r ) ]
• Variational Principle
v(r)+
δ FHK
δρ(r)
FHK Universal Hohenberg
Kohn functional
=µ
Lagrangian Multiplier
normalisation
∫ ρ(r)dr = N
• Practical implementation : Kohn Sham equations
Computational breakthrough
18-8-2011
Pag.4
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 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.
18-8-2011
Pag.5
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
 δE 
 ∂E 
dE =  
dN + ∫ 
 δv(r)d r
 ∂N  v(r)
 δv(r)  N
identification
µ
first order perturbation theory
identification
ρ(r)
Electronic Chemical Potential (R.G. Parr et al, J. Chem. Phys., 68, 3801 (1978))
=-χ
18-8-2011
Pag.6
(Iczkowski - Margrave electronegativity)
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 
=µ =− χ
 
 ∂N v(r)
Electronegativity
Electronic
Chemical
Potential
 ∂2 E 
=η
 2
 ∂N v(r)
 δµ 
 ∂ρ(r) 
∂2E
=
=



∂Nδv(r)  δv(r) N  ∂N v
Chemical hardness
S=
1
Chemical Softness
η
18-8-2011
Pag.7
 δE 

 = ρ(r)
 δv(r)  N
= f(r)
Fukui function


δ2E

 = χ(r, r ')
 δv(r)δv(r ') 

N
Linear response
Function
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
µ2
∆E = − = −ω
2η
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)
•J. L. Gazquez, J.Mex.Chem.Soc., 52, 1 (2008)
• S. Liu, Acta Physico-Chimica Sinica, 25, 590 (2009)
18-8-2011
Pag.8
Until now: focus on first and second order derivates
• µ = − χ → Electronegativity Equalization Principle
(EEM) (Sanderson, Mortier, …)
1
• η → S= η
→ Chemical Hardness and Softness
→ HSAB Principle
Maximum Hardness Principle
( Pearson, Parr)
( Pearson)
• f (r)
s ( r ) =Sf ( r )
→ Fukuifunction and local Softness.
→ Local reactivity / selectivity (Parr, Yang)
18-8-2011
Pag.9
What about the remaining second order derivative
χ ( r,r' ) : linear response function

δ2E   δρ( r) 
 =

=
'
'
 δv( r) δv( r )   δv( r ) 

N 
N
• Computationally demanding
• ? Chemical Interpretation: six dimensional kernel
• ? Condensed Version
Fundamental Importance
1. Information about propagation of an (external potential) perturbation on position r’
throughout the system
2. Berkowitz Parr relationship (JCP 88, 2554, 1988)
χ ( r, r ' ) = −s ( r, r ') +
s ( r ) s ( r' )
S
 δρ ( r ) 
Softness kernel s ( r,r' ) = - 

 δv ( r') µ
∫
s(r)
∫
S
All information
18-8-2011
Pag.10
f(r)
inverted → η ( r,r' )
Hardness kernel
……
What about third order derivatives?
•
 ∂ 3E   ∂η 
 3  =

 ∂N  v  ∂N  v
•


 δχ ( r,r' ) 
δ3 E
=




δv
r
δv
r'
δv
r''
δv
r''
(
)
(
)
(
)
(
)

N 
N
: hyperhardness: known to be small
: awkward …
? Importance of mixed “2+1” derivatives
•
 ∂ 3 E   δη 
 ∂ 2ρ ( r )   ∂f ( r ) 
 ∂ 2 Nδv ( r )  =  δv ( r )  =  ∂N 2  =  ∂N 
v
v 

 
N 
Dual descriptor
C. Morell, A. Grand, A. Toro-Labbé,
J. Phys. Chem. A 109, 205, 2005.
Already a large number of applications: one shot
representation of nucleophilic and electrophilic regions

  ∂χ ( r, r' )   δf ( r ) 
δ3 E
• 
 ∂Nδv ( r ) δv ( r')  =  ∂N  =  δv ( r') 
v 

v 
N
Fukui kernel
Polarization effects on Fukui function
unexplored
Might play a role in describing change in polarizability upon ionization,
electron attachment, …
P. Geerlings, F. De Proft, PCCP, 10, 3028, 2008.
18-8-2011
Pag.11
Today’s talk
• evaluation / interpretation of χ ( r,r' )
• some examples of the computation / use of
third order derivatives (the dual descriptor)
18-8-2011
Pag.12
2. The Linear Response function
• Some more formal discussions appeared in the literature some years ago
(Parr,Senet, Cohen et al., Ayers)
• Liu and Ayers summarized the most important mathematical properties
(J.Chem Phys, 131, 114406, 2009)
• How to come to numerical values and to use / interpret them
• Early Hückel MO theory: mutual atom – atom polarizability
Π rs =
∂q r
∂αs
→ π-electron system
• Baekelandt, Wang: EEM
• Cioslowski: approximate softness matrices
• NO direct, practical, generally applicable, nearly exact approach available
18-8-2011
Pag.13
2.1 Direct evaluation of the second order
functional derivative


δ2 E


δv
r
δv
r'
(
)
(
)

N
In line with earlier work concerning the transformation of
• Fukui-functions
 ∂ρ ( r )   δµ 
f (r) = 

 = 
∂
N

 v  δv ( r )  N
Difficulties with
• Dual descriptor
∂
δ
to
expressions
δv ( r )
∂N
f(
2)
∂
derivative
∂N
 ∂f ( r )   δη 

 = 
∂
N
δv
r
(
)

v 
N
(r) = 
P.W Ayers, F. De Proft, A. Borgoo, P. Geerlings, JCP, 126, 224107, 2007.
N. Sablon, F. De Proft, P.W. Ayers, P. Geerlings, JCP, 126, 224108, 2007.
T. Fievez, N. Sablon, F. De Proft, P.W. Ayers, P. Geerlings, J.Chem.Theor. Comp., 45, 1065, 2008
18-8-2011
Pag.14
 δQ 
Methodology: earlier work (e.g. Fukui Function) 

 δv 
 δQ [ v ] 
Q [ v+w i ] -Q [ v ] = ∫ 
 w i ( r ) dr+ο w i
 δv ( r )  N
(
2
)
perturbation wi(r) (i= 1, ….P)
K
 δQ [ v ] 
•
q jβ j ( r )
 δv ( r )  = ∑

 N j=1
• Linear response regime
basis set expansion
K
Q [ v+w i ] -Q [ v ] = ∑ q j ∫ w i ( r )β j ( r ) dr
j=1
d=Bq
Choose P>K → find q via least squares fitting
• Perturbations w i ( r ) =
-pi
r-R i
• Expansion functions • s on p type GTO centered on each center
• exponents doubled as compared to primitive set
18-8-2011
Pag.15
An example H2CO
Fukui function f+(r)
Nucleophilic attack (including correct angle) at C
SCN-
Fukui function f-(r)
Softest reaction site for an attacking electrophile: S
T. Fievez, N. Sablon, F. De Proft, P.W. Ayers, P. Geerlings, J.Chem.Theor. Comp., 45, 1065, 2008
18-8-2011
Pag.16
Turning to a second functional derivative


δ2 E
χ ( r,r') = 

δv
r
δv
r'
(
)
(
)

N
E [ v+w i ] -2E [ v ] +E [ v-w i ]
∫ ∫ χ ( r,r')w ( r ) w ( r') drdr'
~
i
i
χ ( r,r' ) = ∑ q klβ k ( r ) β l ( r )
kl
E [ v+w i ] -2E [ v ] +E [ v-w i ] = ∑ q kl ∫ β k ( r ) w i ( r )dr ∫ βl ( r') w i ( r' ) dr'
kl
perturbations
expansions
q kl → χ ( r,r')
condensation
χ AB
(in practice: direct evaluation in atom-condensed version
18-8-2011
Pag.17
Examples:
H2CO
H
-1.21
1.19
0.42
-0.40
H
C
O
H
C
O
H2
-4.82
2.45
1.19
-3.28
0.42
-1.21
H2 O
O
H1
H2
18-8-2011
Pag.18
O
-3.77
1.88
1.88
H1
H2
-1.39
-0.49
-1.39
2.2 An Orbital based approach
• Single Slater determinant ( say KS-DFT)
• Second order perturbation theory
V=
∑ δv ( r )
• Closed shell, spin restricted calculations
i
i
• Frozen core
•
∆E i →a ~ ε a -ε i
N 2
χ ( r,r' ) =4∑
i=1
φ*i ( r ) φ a ( r ) φ*a ( r') φi ( r' )
∑
ε i -ε a
a= N 2+1
∞
φi : occupied orbitals
φ a: unoccupied orbitals
ε i , ε a : orbital energies
18-8-2011
Pag.19
∗
(P.W.Ayers,
Faraday Discussions,
135, 161, 2007)
Exact
( δρ ( r )
δv KS ( r') ) N
∗
Zeroth order approximation to the linear response kernel for
the interacting system
? Visualisation /interpretation of this six dimensional kernel
• Condensation to an atom-condensed linear response matrix
χ AB
• multi-center numerical integration scheme using Becke’s “fuzzy” Voronoi polyhedra
χ AB = ∫
∫ χ ( r,r') drdr'
vA vB
M.Torrent Sucarrat, P. Salvador, P. Geerlings, M.Solá, J.Comp.Chem. 28, 574, 2007.
18-8-2011
Pag.20
Ethanal (PBE-6-31+G*)
C1
C1
H1
O
C2
H2
H3
H4
-4.2080
H1
0.6900 -1.2365
O
2.7289 0.3338 -3.7827
C2
0.5067
0.1507
0.3522
-3.3676
H2
0.0966
0.0024
0.1707
0.7813 -1.1413
H3
0.0966
0.0024
0.1707
0.7813
0.0372 -1.1413
H4
0.0892
0.0572
0.0264
0.7950
0.0532 0.0532 -1.0738
• diagonal elements: largest in absolute value;
χ c1c1 ;χ oo : larger than in ethanol (-3.428, -2.187)
χ c1o
(0.873)
higher polarizability of C=O vs C-O bond
• decrease in value of off-diagonal elements upon increasing interatomic distance
• correlation coefficient with ABEEM (C.S.Wang et al., CPL, 330, 132, 2000): 0.923
→ Confidence in use for exploration of (transmission) of inductive and mesomeric effects
in organic chemistry:
18-8-2011
Pag.21
Some examples: H2CO
Present method
H
C
O
H
H
-1.22
0.68
0.34
0.19
C
-4.35
2.99
0.68
Previous method
O
-3.67
0.34
H2
-1.22
H
C
O
H
H
-1.21
1.19
0.42
-0.40
C
O
H2
-4.82
2.45
1.19
-3.28
0.42
-1.21
Correlation coefficient between the matrix elements of the two methods:
Y = 0.99x - 0.01
R2= 0.96
Slope ∼ 1
18-8-2011
Pag.22
H2O
Previous method
Present method
O
-1.76
0.88
0.88
O
H1
H2
H1
-0.93
0.05
H2
O
H1
H2
-0.93
Y= 1.98x + 0.05
R2= 0.96
O
-3.77
1.88
1.88
H1
H2
-1.39
-0.49
-1.39
Slope ∼ 2
For other simple molecules (NH3, CO, HCN, NNO) always a high correlation coefficient
is obtained with a very small intercept; the slope varies between 1 and 2.
Intramolecular comparisons are similar between two approaches; for intermolecular
sequencies the transition to s(r,r’) might be advisable comparable with the swith
from f(r) to s(r).
N. Sablon, P.W. Ayers, F. De Proft, P. Geerlings, J. Chem. Theor. Comp. 6, 3671, 2010
18-8-2011
Pag.23
2.3 From inductive and mesomeric effects to aromaticity
Transmission of a perturbation through a carbon chain
χ OX
(X= C0, C1, C2 …)
χ NX
Saturated systems
• density response of C atoms on heteroatom perturbation decreases
monotonously with distance
• exponential fit: r2= 0.982 ( vide infra)
Characterizing and quantifying the inductive effect.
18-8-2011
Pag.24
Unsaturated systems
• alternating values
• C1, C3, C5 of the chain: minima
C 0 , C2 , C4 , C6
: maxima
R= OH, NH2: resonance structures
5
6
3
4
1
2
C1, C3, C5 : mesomeric passive atoms
→ follow same trend as alkane structures (inductive effect)
C0, C2, C4, C6 : mesomeric active atoms
→ effect remains consistently large even after 6 bonds (small decrease due to
superposition of inductive and mesomeric effect)
18-8-2011
Pag.25
A Comment on “the nearsightedness of electronic matter”
W. Kohn, PRL, 76, 3168, 1996
E. Prodan, W. Kohn, PNAS, 102, 11635, 2005
• For systems of many electrons and at constant µ
∆ρ(r0) induced by ∆v(r’) no matter how large with r’ outside sphere with radius R around
r0 will always be smaller than a maximum magnitude ∆ρ
→ maximum response of the electron density will decay monotonously as a function of R.
 δρ ( r )

Importance of softness kernel 
δv ( r') µ

• r’
r'-r 0 >> R
No numerical applications to atomic or molecular systems yet
18-8-2011
Pag.26
Softness kernel s(r,r’) (evaluated at constant µ!): nearsighted (Ayers)
Linear response kernel
χ ( r,r') (evaluated at constant N): not nearsighted
Berkowitz - Parr relationship
χ ( r,r') =-s ( r,r') +
s ( r ) s ( r')
S
farsighted contribution to
χ ( r,r')
? Small electrontranfer → neglect of this term
χ ( r,r')
behaves as
-s ( r,r')
declines exponentially as r and r’ get further apart
(Prodan-Kohn)
(Cardenas…Ayers JPCA, 113, 8660,2009)
Exponential decay as observed for the inductive effect in the systems above studied
N. Sablon, F. De Proft, P. Geerlings, JPCLett.,1, 1228, 2010
18-8-2011
Pag.27
Some recent work: substituted benzenes vs cyclohexanes
χ C1C i = (i= 2,3,...6)
• cyclohexane: • χ decreases exponentially (inductive effect)
• influence of OH small
• benzene:
18-8-2011
Pag.28
• maxima at C2, C4, C6: mesomerically active atoms (mesomeric effect)
• minima at C3, C5
: mesomerically inactive atoms
O
H
OH
OH
CH3
• Similar behaviour for o,p and
m directors
• Mesomerically active atoms:
2,4,6
1,4 effect
N. Sablon, F. De Proft, P. Geerlings, Chem.Phys.Lett., 498, 192, 2010
18-8-2011
Pag.29
? Aromaticity
2.4 The linear response function as an indicator of aromaticity
Relation with the para delocalization index (PDI) derived from AIM
δ ( A,B ) = -2 ∫ ∫ Γ
AB
( r1 ,r 2 ) dr1dr 2
XC
Exchange correlation density; integration over atomic basins
• quantitative idea of the number of electrons delocalized or shared between A and B
(X. Fradera, M.A. Austen, R.F.W Bader, JPCA, 103, 304, 1999.)
• investigated as a potential index of aromaticity
(J. Poater, X. Fradera, M. Duran, M. Sola, Chem.Eur. J. , 9, 400, 2003)
• six membered rings of planar PAH’s
• successful correlation of the (1,4) (para) delocalization index with NICS, HOMA, …
1
2
δ AB = δ14
para
3
4
• Does Linear response function
18-8-2011
Pag.30
χ1,4
contain similar information?
16 non equivalent sixrings studied by Sola et al.
18-8-2011
Pag.31
• typical benzene-type pattern encountered before
18-8-2011
Pag.32
χ
PDI
→ Linear response function as a descriptor of aromaticity
18-8-2011
Pag.33
2.5 Conclusions
• The linear response function comes within reach of a general, non-empirical
computational strategies.
• It is the tool by excellence to investigate how information on perturbations is propagated
through the system.
• Atom condensation is at stake, but can nowadays be done in a systematic way.
• The linear response function has been shown to account for the difference in fall off
behavior of inductive and mesomeric affects and to connect them with the concept of
nearsightedness .
• Its potential use as aromaticity descriptor has been established through comparison with
the Para Delocalization Index.
• The way to the softness kernel
18-8-2011
Pag.34
s ( r,r' )
has ( almost) been paved.
3. Higher order derivatives
3.1 The dual descriptor
 δ3 E   ∂ 2ρ ( r )   ∂f ( r ) 
( 2)
=
=

 = f (r)
2 
2  

 δv ( r ) ∂N   ∂N   ∂N  v
Consider
Third order response function
• Introduced as a new descriptor, "dual descriptor"
(C. Morell, A. Grand, A. Toro Labbé, J. Phys. Chem. A109, 205 (2005))
 ∂ 2ρ(r) 
2)
(
f (r) = 
 ≅ ρ N +1 (r) − 2ρ N (r) + ρ N −1 (r)
2
 ∂N


v
= (ρN+1 (r) − ρN (r)) − (ρN (r) − ρN−1(r))
2
≅ φ LUMO (r) − φ HOMO (r)
Large in electrophilic
regions
→f
18-8-2011
(2)
Pag.35
(r)
2
Large in nucleophilic
regions
>0
electrophilic regions
<0
nucleophilic regions
→ a one shot picture of both electrophilic and nucleophilic regions in space
An example: Carbene (Singlet) ( +/- 0.01 surface plot)
f ( 2) > 0
: electrophilic region (empty 2p orbital)
2
f ( ) < 0 : Nucleophilic region (lone pair)
Cfr. J.V. Correa, P. Jaque, J. Olah, A. Toro-Labbé, P. Geerlings, Chem. Phys. Lett., 470, 180, 2009
18-8-2011
Pag.36
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)
( 2)
( 2)
f (r)f B (r ')
Minimizing ∫∫ A
drdr '
r −r'
Confirmed by perturbation theoretical ansatz (P.W. Ayers)
A way to rationalize the celebrated Woodward Hoffmann rules is
an orbital free, even wave function free context
18-8-2011
Pag.37
One of our points of interest in recent years, coupled to another third order derivative:
(∂η ∂R )
0
• F. De Proft, P.W. Ayers, S. Fias, P. Geerlings, JCP, 125, 214101, 2008
• 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. Theor. Comp., 4, 595, 2008
• N. Sablon, F. De Proft, P. Geerlings, Croat. Chem. Acta (Z.B. Maksić Volume), 82, 157, 2009
• P. Jaque, J.V. Correa, F. De Proft, A. Toro-Labbé, P. Geerlings, Can. J. Chem. (R.J. Boyd Volume)
88, 858, 2010
Succesful application to •
•
•
•
18-8-2011
Pag.38
electrocyclisation
cycloadditions
sigmatropic reactions
chelotropic reactions
The first case: Cycloadditions
Butadiene + ethene
HOMO
[ π 4s + π 2s ]
LUMO
2
2
2
f ( ) ( r ) = φLUMO ( r ) − φHOMO ( r )
Hückel
Ethene
Butadiene
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Pag.39
B3LYP/6-31G*
Molecules align so that favorable interactions (green lines)
occur between their dual descriptors
[ π 4s + π 2s ]
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"allowed"
• The [ π 2s + π 2s ] case : ethene + ethene
supra/supra : "repulsive"
not allowed
supra/antara : can occur if the molecules rotate so that
the double bonds become perpendicular
allowed
(but steric demands too high to occur in practice)
Similar result by considering
(∂η ∂R )
(initial hardness response)
WH rules for cycloadditions regained
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The fourth case: Chelotropic Reactions
LUMO
O
S
HOMO
O
• filled and vacant frontier orbitals for bonding to other atoms on
the same atom (CO, singlet carbenes)
• σ bonds broken or formed on the same atom, in casu S
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The WH Rules on chelotropic reactions in an orbital context
4n case: linear chelotropic addition
LUMO
HOMO
S
HOMO
LUMO
O
• Disrotatory mode
• Non-linear case: mode is inverted
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Pag.43
O
Linear Chelotropic Reaction: 4n system
Side view
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Top view
The Initial Hardness Response
Energy and hardness profile
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Pag.45
( ∂η ∂ ϑ )
Disrotatory mode in agreement with WH
Confirmed in the (4n+2) case
Conrotatory
disrotatory
Disrotatory
conrotatory
What about the non-linear case?
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Pag.46
• The dual descriptor appraoch
Linear Chelotropic Reaction: 4n System
r
f ( 2) (r) > 0 (red)
r
f ( 2 ) (r) < 0 (blue)
Stabilizing interactions
Disrotatory
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3.2 Reconciling the dual descriptor and the initial
hardness response
Dual descriptor
2)
(
 ∂f(r)   ∂  δµ  
f (r ) = 
 = 
 
 ∂N v
 ∂N  δv(r) N v
 δη 
 δ  ∂µ  
=

   = 
 δv(r)  ∂N v N  δv ( r )  N
()
Changes in v r provoked by changes
in nuclear configuration only

 ∂v ( r )
cf.  ∂η  =  δη 
dr
∫


 ∂R  N
 δv ( r )  N ∂R
= ∫f
(2)
(r)
∂v ( r )
∂R
dr
General Information refined via
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Pag.48
∂v ( r ) / ∂R to initial Hardness Response.
WH rules regained in a conceptual DFT context for all
four types of pericyclic reactions, thereby avoiding
symmetry or phase based arguments
- initial hardness response
- dual descriptor "back of the envelope" approach
- two approaches internally consistent
Importance of third order derivatives
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3.3 The chemical relevance of higher order
(n≥3) derivatives
Chemistry at first order: electronegativity (Pauling)
• Correction for hardness, softness, polarizability
• Incorporation of frontier MO Theory
second order: Fukui function, hardness, …
→ linear response function, softness kernel
third order: • diagonal (N or v): small or too hard to interprete
• non diagonal: fundamentally new information,
eg. dual descriptor
P.Geerlings, F.De Proft, PCCP, 10, 3028 (2008)
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Pag.50
4. Conclusions
• Conceptual DFT offers a broad spectrum of reactivity
descriptors
• The “missing” second order derivative, the “linear response
function”, comes within reach and is a tool to see how ∆v
perturbations are propagated through a molecule.
Its chemical significance is emerging.
• Selected third order derivatives may contain important
chemical information, permitting among others to retrieve the
WH rules in a density- only context.
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Pag.51
Acknowledgements
Prof. Frank De Proft
Prof. Paul W. Ayers (Mc Master University, Hamilton, Canada)
Dr. Nick Sablon
Tim Fievez
Dr. P. Jaque ( Santiago, Chile)
Acknowledgements
Prof. P.K. Chattaraj (IIT – Kharagpur – India) / Prof. A. Toro Labbé (Santiago – Chile)
Dr. C. Morell (CEA, Grenoble)
Dr. M. Torrent – Sucarrat (Brussels, Girona)
Prof. M. Sola (Girona)
Fund for Scientific Research-Flanders (Belgium) (FWO)
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Pag.52
Nonlinear Chelotropic Reaction: 4n system
Side view
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Top view
Energy and hardness ( ∂η ∂ ϑ ) predict conrotatory mode
Confirmed in the 4n+2 case: disrotatory mode
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Pag.54
(Z
< 0.5 )
Nonlinear Chelotropic Reaction: 4n System
Destabilizing interactions
Conrotatory
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Pag.55