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The Linear Response Function.
Paul Geerlings, Nick Sablon, Frank De Proft
Pag.6-1-2012 1
Eenheid Algemene Chemie, Vrije Universiteit Brussel, Belgium
Summer Talks in Santiago IIISummer Talks in Santiago IIIJanuary 2012January 2012
Outline
1. Introduction: Chemical Concepts from DFT
2. The Linear response Function
2.1. Preliminaries2.2. Direct evaluation of the second order functional derivative
Pag.6-1-2012 2
3. Conclusions
4. Acknowlegements
2.2. Direct evaluation of the second order functional derivative2.3. An Orbital based Approach2.4. Inductive and Mesomeric effects and the concept of nearsightedness2.5. Aromaticity
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.6-1-2012 3
•
•
• • •
•
•
•
••
•
compatible with a single v(r)
ρ(r) for a given ground state
- nuclei - position/charge
electrons
v(r)
• Variational Principle
FHK Universal HohenbergKohn functional
Lagrangian Multiplier normalisation
HKF
v(r)+ =(r)
δµ
δρ
[ ] ( ) [ ]v( ) ( ) ( )op HKr H E E r v r d r F rρ ρ ρ ρ→ → = = +∫
Pag.6-1-2012 4
• 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 either as such or within the context of principles such as
Pag.6-1-2012 5
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
Pag.6-1-2012 6
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
∂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
2E(r, r ')
δ= χ
= −
Electro-negativity
Electronic Chemical Potential
χN
E(r)
v(r)
δ= ρ
δ
Pag.6-1-2012 76-1-2012 76-1-2012 7
Chemical Chemical hardnesshardness
ChemicalChemical SoftnessSoftness Fukui functionFukui function
∂ E
∂N2
v(r)
= η∂ E
∂Nδv(r)=
δµ
δv(r)
N
=∂ρ(r)
∂N
v
S =1
η
= f(r)Linear response Linear response FunctionFunction
N
E(r, r ')
v(r) v(r ')
δ= χ
δ δ
Until now: focus on first and second order derivates
• µ = − → Electronegativity Equalization Principle
(EEM) (Sanderson, Mortier, …)
• → Chemical Hardness and Softness
→ HSAB Principle ( Pearson, Parr)Maximum Hardness Principle ( Pearson)
χ
1η S=η
→
Pag.6-1-2012 8
Maximum Hardness Principle ( Pearson)
•
→ Fukuifunction and local Softness.→ Local reactivity / selectivity (Parr, Yang)
( )f r
( ) ( )s r =Sf r
P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103, 1793 (2003)
What about third order derivatives?
• : hyperhardness: known to be small
• : awkward …
? Importance of mixed “2+1” derivatives
•Dual descriptor
3
3
vv
E η
N N
∂ ∂ =
∂ ∂
( ) ( ) ( )( )( )
3
N N
δχ r,r'δ E
δv r δv r' δv r'' δv r''
=
( ) ( )( ) ( )23
2 2
vvN
ρ r f rE δ
Nδv r δv r N N
η ∂ ∂ ∂= = = ∂ ∂ ∂ C. Morell, A. Grand, A. Toro-Labbé,
J. Phys. Chem. A 109, 205, 2005.
Pag.6-1-2012 9
Already a large number of applications: one shot representation of nucleophilic and electrophilic regions
•
unexplored
Might play a role in describing change in polarizability upon ionization, electron attachment, …
Fukui kernelPolarization effects on Fukui function( ) ( )
( ) ( )( )
3
vv N
χ r, r' δf rδ E
Nδv r δv r' N δv r'
∂ = = ∂ ∂
J. Phys. Chem. A 109, 205, 2005.
P. Geerlings, F. De Proft, PCCP, 10, 3028, 2008.
An Interesting application
Regaining the Woodward Hoffmann rules for pericyclic reactions on the basis of
• the dual descriptor
• the initial hardness response
( )
v
f r
N
∂
∂
N
η
R
∂
∂
Pag.6-1-2012 10
P. Geerlings, P.W.Ayers, A. Toro Labbé, P.K. Chattaraj, F. De Proft, Acc. Chem. Res., 55, xxx, 2012
related to
N
( )N
δη
δv r
What about the remaining second order derivative
: linear response function
• 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'
( ) ( )( )
( )
2
' '
N N
δρ rδ E= =δv r δv r δv r
Pag.6-1-2012 11
2. Berkowitz Parr relationship (JCP 88, 2554, 1988)
Softness kernel
inverted
Hardness kernel
……
( )s r
S
∫
∫
( ) ( )( ) ( )s r s r'
χ r, r ' s r, r 'S
= − +
( )η r,r'→
( )( )( )
µ
δρ rs r,r' = -
δv r'
f(r)
All information
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)
2.1 Preliminaries
Pag.6-1-2012 12
• How to come to numerical values and to use / interpret them
• Early Hückel MO theory: mutual atom – atom polarizability → π-electron system
• Baekelandt, Wang: EEM
• NO direct, practical, generally applicable, nearly exact approach available
rrs
s
qΠ =
α
∂
∂
• Cioslowski: approximate softness matrices
2.2 Direct evaluation of the second order functional derivative
A first step: the numerical approach
• direct evaluation of the second order functional derivative
( )( ) ( )
2
N
δ Eχ r,r'
δv r δv r'
=
• methodology in line with the first order case
Pag.6-1-2012 13
• methodology in line with the first order case
( )( )
( )v N
ρ r δµf r
N δv r
∂ = = ∂
( ) ( )( )
( )2
v N
f r δf r
N δv r
η ∂ = = ∂
fukui function
dual descriptor
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.
Methodology: evaluation of
[ ] [ ][ ]( )
( ) ( )2
i i i
N
δQ vQ v+w -Q v = w r dr+ο w
δv r
∫
perturbation wi(r) (i= 1, ….P)
• Linear response regime[ ]( )
( )K
j j
j=1N
δQ v= q β r
δv r
∑• basis set expansion
[ ] [ ] ( ) ( )K
Q v+w -Q v = q w r β r dr∑ ∫
δQ
δv
Pag.6-1-2012 14
[ ] [ ] ( ) ( )i j i j
j=1
Q v+w -Q v = q w r β r dr∑ ∫
d=Bq
Choose P>K → find q via least squares fitting
• Perturbations
• Expansion functions • s on p type GTO centered on each center• exponents doubled as compared to primitive set
( ) ii
i
-pw r =
r-R
An example H2CO Fukui function f+(r)
Nucleophilic attack (including correct angle) at C
SCN- Fukui function f-(r)
Pag.6-1-2012 15
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
Turning to a second functional derivative
[ ] [ ] [ ]i iE v+w -2E v +E v-w ~ ( ) ( ) ( )i iχ r,r' w r w r' drdr'∫ ∫
[ ] [ ] [ ] ( ) ( ) ( ) ( )E v+w -2E v +E v-w = q β r w r dr β r' w r' dr'∑ ∫ ∫
( ) ( ) ( )kl k l
kl
χ r,r' = q β r β r∑
( )( ) ( )
2
N
δ Eχ r,r' =
δv r δv r'
Pag.6-1-2012 16
perturbations
expansions
( )klq χ r,r'→
[ ] [ ] [ ] ( ) ( ) ( ) ( )i i kl k i i
kl
E v+w -2E v +E v-w = q β r w r dr β r' w r' dr'l∑ ∫ ∫
• Condensation to an atom-condensed linear response matrix ABχ
• multi-center numerical integration scheme using Becke’s “fuzzy” Voronoi polyhedra
? Visualisation /interpretation of this six dimensional kernel
Pag.6-1-2012 17
( )A B
AB
v v
χ = χ r,r' drdr'∫ ∫
M.Torrent Sucarrat, P. Salvador, P. Geerlings, M.Solá, J.Comp.Chem. 28, 574, 2007
An example: H2CO
H C O H2
H -1.21 1.19 0.42 -0.40
C 1.19 -4.82 2.45 1.19
O 0.42 2.45 -3.28 0.42
H2 -0.40 1.19 0.42 -1.21
Pag.6-1-2012 18
H2 -0.40 1.19 0.42 -1.21
2.3 An Orbital based approach
• Single Slater determinant ( say KS-DFT)
• Second order perturbation theory V=
• Closed shell, spin restricted calculations
• Frozen core
• i a a i∆E ~ ε -ε→
(P.W.Ayers, Faraday Discussions,
( )i
i
δv r∑
( )( ) ( ) ( ) ( )* *N 2
i a a iφ r φ r φ r' φ r'χ r,r' =4
∞
∑ ∑ ∗∗∗∗
Pag.6-1-2012 19
Faraday Discussions, 135, 161, 2007)
: occupied orbitals : unoccupied orbitals : orbital energies
( )( ) ( ) ( ) ( )i a a i
i=1 a=N 2+1 i a
χ r,r' =4 ε -ε
∑ ∑
iφ aφ i aε ,ε
∗∗∗∗
Exact ( ) ( )( )KS Nδρ r δv r'
∗∗∗∗Zeroth order approximation to the linear response kernel for the interacting system
An example:
Previous methodPresent method
H C O H2
H -1.22
C 0.68 -4.35
O 0.34 2.99 -3.67
H 0.19 0.68 0.34 -1.22
H2CO
H C O H2
H -1.21
C 1.19 -4.82
O 0.42 2.45 -3.28
H -0.40 1.19 0.42 -1.21
Pag.6-1-2012 20
Correlation coefficient between the matrix elements of the two methods:
Y = 0.99x - 0.01
R2= 0.96
Slope ∼ 1
Previous methodPresent method
O H1 H2
O -1.76
H1
0.88 -0.93
H2
0.88 0.05 -0.93
H2O
O H1 H2
O -3.77
H1
1.88 -1.39
H2
1.88 -0.49 -1.39
Y= 1.98x + 0.05 R2= 0.96 Slope ∼ 2
Pag.6-1-2012 21
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
Ethanal (PBE-6-31+G*)
• diagonal elements: largest in absolute value;
C1 H1 O C2 H2 H3 H4
C1 -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
Pag.6-1-2012 22
• diagonal elements: largest in absolute value;
larger than in ethanol (-3.428, -2.187)
(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
1 1c c ooχ ;χ :
1c oχ
→ Confidence in use for exploration of (transmission) of inductive and mesomeric effects
in organic chemistry.
Transmission of a perturbation through a carbon chain
2.4 Inductive and mesomeric effects and the concept of nearsightedness
Pag.6-1-2012 23
NXχOXχ
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.
(X= C0, C1, C2 …)
Unsaturated systems
• alternating values• C1, C3, C5 of the chain: minimaC0, C2, C4, C6 : maxima
R= OH, NH2: resonance structures
135
6 4 2
Pag.6-1-2012 24
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)
N. Sablon, F. De Proft, P. Geerlings, JPCLett., 1, 1228, 2010
Substituted benzenes vs cyclohexanes
Pag.6-1-2012 25
• cyclohexane:
1 iC Cχ (i= 2,3,...6)=
• decreases exponentially (inductive effect)• influence of OH small
χ
• benzene: • maxima at C2, C4, C6: mesomerically active atoms (mesomeric effect)• minima at C3, C5 : mesomerically inactive atoms
OH
O
H
Pag.6-1-2012 26
OH
OH
CH3
H
• Similar behaviour for o,p and m directors
• Mesomerically active atoms: 2,4,6
N. Sablon, F. De Proft, P. Geerlings, Chem.Phys.Lett., 498, 192, 2010
1,4 effect ? Aromaticity
2.5 The linear response function as an indicator of aromaticity
Relation with the para delocalization index (PDI) derived from AIM
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
( ) ( )1 2 1 2
A B XC
δ A,B = -2 r ,r dr drΓ∫ ∫
Pag.6-1-2012 27
• 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, …
AB 14δ = δ
1
2
3
4
para
• Does Linear response function contain similar information?1,4χ
Borazine
• Typical resonance pattern observed in aromatic systems is recovered if one of the N-atoms is chosen as the reference atom.
• Purely inductive behaviour (exponential decay of electron delocalization with internuclear distance)
Pag.6-1-2012 32
• Purely inductive behaviour (exponential decay of electron delocalization with internuclear distance) is recovered if one of the B-atoms is chosen as the reference atom.
Dual picture of aromatic character of borazine (cfr. ongoing debate in the literature)
Cfr. Valence bond study: J. Engelberts, R. Havenith, J. Van Lenthe, L. Jenneskens, P. Fowler,Inorg. Chem., 44, 5266, 2005
Valence Bond Structures
X= NH, O, PH
Pag.6-1-2012 33
Resonance Energy (kJ mol-1) and weights W of the four Valence Bond Structures.
Molecule Eres W(I) W(II) W(III) W(IV)
B3N3H6 61.6 0.05 0.05 0.90 0.00
B3O3H3 27.6 0.02 0.02 0.96 0.00
B3P3H6 132.6 0.21 0.21 0.58 0.00
C6H6 161.4 0.47 0.47 0.03 0.03
Borazine
Structure III with lone pairs on N: prominent role
• associated linear response values reflect (1,4) delocalization resembling the aromatic situations
• B atoms do not have a lone pair
Pag.6-1-2012 34
Associated linear response values merely reflect an inductive effect.
Related cases • Boroxine
• s-triphosphatriborin
Pag.6-1-2012 35
X= PHX= O
Boroxine: • qualitatively similar to borazine, but even lower aromaticity
• (1,4) PLR index Eres
0.17 < 0.30 (borazine)27.6 < 61.6
s-triphosphatriborin (X=PH)
Linear response curves: similar to benzene irrespective of the reference atom (B or P) considered
• PLR 0.87: important degree of aromaticity ∼ benzene
• Eres 132.6kJ mol-1 cfr benzene 161.4
Significant contributions of structures I en II
• Also in benzene I and II are the two preponderant structures
s-triphosphatriborin takes an intermediate position between the aromatic benzene and the non-aromatic borazine or boroxine cases.
Pag.6-1-2012 36
benzene and the non-aromatic borazine or boroxine cases.
Qualitative trend in atom condensed linear response functions parallels the contribution of possible Kekulé structures to the resonance energy in the context of VB Theory.
N. Sablon, F. De Proft, P. Geerlings, PCCP, 14, xxx, 2012
4. Conclusions
• Conceptual DFT offers a broad spectrum of reactivity descriptors. Selected third order derivatives may contain important chemical information.
• The “missing” second order derivative, the “linear response function”, comes within reach and is a tool to see how ∆v
Pag.6-1-2012 37
function”, comes within reach and is a tool to see how ∆v perturbations are propagated through a molecule its chemical relevance becomes apparent.
• The computational results reveal that important chemical information can be retrieved from the linear response function: from inductive and mesomeric effects to the aromatic character of organic and inorganic rings.
Acknowledgements
Acknowledgements
Prof. Frank De Proft
Prof. Paul W. Ayers (Mc Master University, Hamilton, Canada)
Dr. Nick Sablon
Tim Fievez
Pag.6-1-2012 38
AcknowledgementsTim Fievez
Dr. P. Jaque ( Santiago, Chile)
Dr. M. Torrent – Sucarrat (Brussels, Girona)
Prof. M. Sola (Girona)
Fund for Scientific Research-Flanders (Belgium) (FWO)
A Comment on “the nearsightedness of electronic matter”
W. Kohn, PRL, 76, 3168, 1996E. 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.
Importance of softness kernel( )
( )δρ r
δv r'
Pag.6-1-2012 39
No numerical applications to atomic or molecular systems yet
• r’
Importance of softness kernel ( )µ
δv r'
0r'-r R>>
Softness kernel s(r,r’) (evaluated at constant µ!): nearsighted (Ayers)
Linear response kernel (evaluated at constant N): not nearsighted( )χ r,r'
( ) ( )( ) ( )s r s r'
χ r,r' =-s r,r' +S
Berkowitz - Parr relationship
farsighted contribution to ( )χ r,r'
? Small electrontranfer → neglect of this term
( ) ( )
Pag.6-1-2012 40
( )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 studied above
N. Sablon, F. De Proft, P. Geerlings, JPCLett., 1, 1228, 2010
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