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Pag.
Conceptual DFT:The Linear Response Function.
Paul Geerlings
General Chemistry, Vrije Universiteit Brussel, Belgium
Quitel, 2013
Granada, June 30 - July 5, 2013
28-1-2015 1
Pag.
QUITEL: 39th International Conference:Theoretical Chemists of Latin Expression
but …. Belgium is a trilingual Country
but … “Belgae” were already mentioned by Caesar in De Bello Gallico
Gallia est omnis divisa in partes tres, quarum unam incolunt Belgae, aliam
Aquitani…Horum omnium fortissimi sunt Belgae….
C.I. Caesar, De Bello Gallico, I,1
28-1-2015 2
Pag.
Outline
1. Introduction: Chemical Concepts from DFT
2. The Linear response Function
3. Conclusions
4. Acknowlegements
2.1. Preliminaries2.2. Direct evaluation of the second order functional derivative2.3. A simple perturbational approach
2.3.0. Preliminaries2.3.1. Atoms revisited2.3.2. Inductive and Mesomeric effects 2.3.3. Aromaticity
2.4. Coupled perturbation KS evaluation of the Fukui function and the linear response function
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Pag.
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)"
•
•
• • •
•
•
•
••
•
compatible with a single v(r)
ρ(r) for a given ground state
- nuclei - position/charge
electrons
v(r)
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Pag.
• Variational Principle
FHK Universal HohenbergKohn functional
Lagrangian Multiplier normalisation
• Practical implementation : Kohn Sham equations
Computational breakthrough
ρ(r)∫ dr = N
HKF
v(r)+ =(r)
δµ
δρ
[ ] ( ) [ ]op v HKρ(r) H E=E ρ = ρ r v(r)dr+F ρ(r)→ → ∫
28-1-2015 5
Pag.
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
µµµµ as (the opposite of) the electronegativity.
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Pag.
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
identification
ρ r( )
Electronic Chemical Potential (R.G. Parr et al, J. Chem. Phys., 68, 3801 (1978))
= - χ (Iczkowski - Margrave electronegativity)
μ
28-1-2015 7
Pag.28-1-2015 828-1-2015 8
Chemical hardnessFukui function
∂E
∂N
v(r)
= µ
∂2E
∂N2
v(r)
= η∂2E
∂Nδv(r)=
δµ
δv(r)
N
=∂ρ(r)
∂N
v
E[N,v]
= f(r)
Identification of two first derivatives of E with respect to N and v in a DFT context → response functions in reactivity theory
Linear response Function
2
N
E(r, r ')
v(r) v(r ')
δ= χ
δ δ
= −
Electro-negativityElectronic
Chemical Potential
χN
E(r)
v(r)
δ= ρ
δ
• P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. , 103, 1793, 2003
• P. Geerlings, F. De Proft, PCCP, 10, 3028 (Third order derivatives)
• P. Geerlings, P.W. Ayers, A. Toro Labbé, P.K. Chattaraj, F. De Proft, Acc. Chem. Res., 55, 2012 (WH rules)
28-1-2015 8
Pag.
What about the remaining second order derivative
: linear response function
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)
Softness kernel
inverted
Hardness kernel
……
( )s r
S
∫
∫
( )χ r,r'
( ) ( )( )
( )
2
' '
N N
δρ rδ E= =δv r δv r δv r
( ) ( )( ) ( )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
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Pag.
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-electron system
• Baekelandt, Wang: EEM
• NO direct, practical, generally applicable, nearly exact approach available or exploited
rrs
s
qΠ =
α
∂
∂
• Cioslowski: approximate softness matrices
2.1 Preliminaries
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Pag.
2.2 Direct evaluation of the second order functional derivative
• 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
( )( )
( )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.
28-1-2015 11
Pag.
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
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
δQ
δv
( ) ii
i
-pw r =
r-R
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Pag.
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
28-1-2015 13
Pag.
Turning to a second functional derivative
[ ] [ ] [ ]i iE v+w -2E v +E v-w ~ ( ) ( ) ( )i iχ r,r' w r w r' drdr'∫ ∫
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∑ ∫ ∫
( ) ( ) ( )kl k l
kl
χ r,r' = q β r β r∑
( )( ) ( )
2
N
δ Eχ r,r' =
δv r δv r'
28-1-2015 14
Pag.
• Condensation to an atom-condensed linear response matrix ABχ
• multi-center numerical integration scheme using Becke’s “fuzzy” Voronoi polyhedra
( )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
? Visualisation /interpretation of this six dimensional kernel
28-1-2015 15
Pag.
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
… but computing times prohibitively large use as standard
28-1-2015 16
Pag.
2.3. A simple perturbational approach; an independent particle model
(P.W.Ayers, Faraday Discussions, 135, 161, 2007)
: occupied orbitals : unoccupied orbitals : orbital energies
( )( )
( )( ) ( ) ( ) ( )* *N 2
i a a i
i=1 a=N 2+1 i aN
δρ r φ r φ r φ r' φ r' = χ r,r' =4 δv r' 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
2.3.0. Preliminaries
• Can we calculate in a simpler way thereby creating the possibility to explore in detail its physical/chemical content, starting from atoms going to molecules
• Closed shell N-electron system in the KS ansatz; frozen orbital approach
• 1ste order Perturbation Theory → → taking functional derivative w.r.t
( )χ r,r'
( ) ( )1ρ r ( )v r
28-1-2015 17
Pag.
2.3.1. Atoms revisited
• Earlier work: A. Savin, F. Colonna, M. Allatena, J. Chem. Phys, 115, 6827, 2001: some light elements
• Light elements (KS ansatz; PBE). Spherical potential perturbation
. plot : radial distribution of the linear response kernel( )2 ' '2r χ r,r r
He Be
• Similar to Savin’s plots
• Positive and negative region, for He, duplicated in Be shell structure
28-1-2015 18
Pag.
One dimensional version for a fixed ( )2 'r χ r,r ( )r'=0 2r' r χ r,0→
He
Be
• Positive perturbation δv(r) v(r) becomes less negative electron density depletion in the vicinity of the nucleus
• ( ) ( ) ( )ρ r dr' χ r,r' δv r'∆ = ∫( ) ( )δv r Aδ r 0 A>0= −Pointlike
perturbation
( ) ( )ρ r Aχ r,0∆ =
Alternating positive and negative regions due to conservation of number of electrons
also in 2D plot
28-1-2015 19
Pag.
Extending → the noble gases
• Shell structure upon increasing Z
• Similar results Be, Mg, Ca
• Isoelectronic Series
Ne F- Na+
Ar Cl- K+
Same “structure”
Contraction/dilatation of the contours → cf. decreasing/ increasing polarizability
28-1-2015 20
Pag.
χ(r,0) for Ar in the x,y plane
28-1-2015 21
Pag.
A direct application → polarizability calculations
( )ij i jα = - dr dr' r χ r,r' r'∫ i,j= x,y,z
• Comparison with high level calculations
• Absolute values deviate, trend is respected
28-1-2015 22
Pag.
• First systematic study of non condensed linear response function on atoms
→ closed shell atoms throughout the periodic table
• Relation to the shell structure → physical information
• How to extend its use for molecules to look for chemical information
→ condensation at stake
polarizability
Z. Boisdenghien, C. Van Alsenoy, F. De Proft, P. Geerlings, J.Chem.Theor. Comp., 9, 1007 (2013)
28-1-2015 23
Pag.
An example:
Numerical methodPresent method
Correlation coefficient between the matrix elements of the two methods:
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
Y = 0.99x - 0.01 R2= 0.96 Slope ∼ 1
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
N. Sablon, P.W.Ayers, F. De Proft, P. Geerlings, J.Chem.Theor. Comp., 6, 3671, 2010
For other simple molecules (H2O, 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 switch from f(r) to s(r).
28-1-2015 24
Pag.
Ethanal (PBE-6-31+G*)
• 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.
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
28-1-2015 25
Pag.
Transmission of a perturbation through a carbon chain
NXχ
OXχ
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 …)
2.3.2. Inductive and mesomeric effects
28-1-2015 26
Pag.
Unsaturated systems
• alternating values• C1, C3, C5 of the chain: minimaC0, C2, C4, C6 : maxima
R= OH, NH2: resonance structures
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)
135
6 4 2
N. Sablon, F. De Proft, P. Geerlings, JPCLett., 1, 1228, 2010
28-1-2015 27
Pag.
Substituted benzenes vs cyclohexanes
• 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
1,4 effect ? Aromaticity
N. Sablon, F. De Proft, P. Geerlings, Chem.Phys.Lett., 498, 192, 2010
28-1-2015 28
Pag.
2.3.3. 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
(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
( ) ( )1 2 1 2
A B XC
δ A,B = -2 r ,r dr drΓ∫ ∫
28-1-2015 29
Pag.
16 non equivalent sixrings studied by Sola et al.
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Pag.
• typical benzene-type pattern encountered before
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Pag.
→ Linear response function as an “electronic” descriptor of aromaticity
28-1-2015 32
Pag.
An application: inorganic rings
28-1-2015 33
Pag.
Valence Bond Structures
X= NH, O, PH
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
J. Engelberts, R. Havenith, J. Van Lenthe, L. Jenneskens, P. Fowler, Inorg. Chem., 44, 5266, 2005
28-1-2015 34
Pag.
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)
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)
N. Sablon, F. De Proft, M. Sola, P. Geerlings, PCCP, 14, 3960 (2012)
28-1-2015 35
Pag.
Digging further in aromaticity →→→→ σσσσ,ππππ aromaticity
Condensed version of the linear response function
( )A B
A B
v v
χ = χ r , r' d r d r'∫ ∫
N/2 MAB A B
ia aiNi i aa= +12
1χ = 4 S S
ε -ε∑ ∑ ( ) ( )
A
A *
ia i a
v
S = r r drφ φ∫
( ) ( )B
B *
ai a i
v
S = r' r' dr 'φ φ∫
Written as sum of contributions of different occupied MO’s → σ,π separation (planar molecules)
28-1-2015 36
Pag.
Benzene
• Decreasing σ contribution (cfr. Cyclohexane)
• Zig/zag for π and total
• Anti-aromatic molecules:Cyclobutadiene, COT
“inverted zig zag”
S. Fias, P. Geerlings, P. Ayers, F. De Proft, PCCP, 15, 2882 (2013)
28-1-2015 37
Pag.
Planar Metallic Systems
Aromaticity of all metal systems and involving a planar tetracoordinate carbon atom2-
4CE
2-
4E E= Al, Ga
• Large interest in submit in clustersclassified as aromatic: debate:
• Systematic replacement Al→ Ge double σ and π aromaticity (M.Sola)
• Influence of insertion of C in (Merino)
2-
4Al
-
4MAl (M= Li, Na, ...)
σ, π, σ+π
2-
4E
OLR Opposite Linear Response = ( )13 24
12 23 24
1χ χ χ , χ , χ
2+ >>
Aromaticity order
2- - + 2+
4 3 2 2 3 4Al > Al Ge ³ Al Ge ALGe < Ge≥ ≤
Expected order (Feixas et al)
Mainly σ aromatic (∼ 65%)
Insertion of C: decreasing aromaticity, sequence unaltered; but increasing σ component
28-1-2015 38
Pag.
• The use of unintegrated plots
σ
• in plane• perturbation
at top nucleus
π
• 0.5 au above plane• perturbation in that
plane at top nucleus
σ
Delocalized Nature of the linear Response more pronounced in the σ electron density
π
σ Aromatic character
S. Fias, Z. Boisdenghien, T. Stuyver, M. Audiffred, G. Merino, P. Geerlings, F. De Proft, J. Phys. Chem. A, 2013, 117, 3556
Benzene 2-
4Al
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Pag.
2.4 A Coupled Perturbed KS evaluation of the Fukui function and the linear response function
• Coupled perturbed perturbation ansatz starting from KS equations
( )1ρ
( )( )
( ) ( ) ( ) ( ) ( ) ( )-1 * *
j b i aiaσ,jbiaσ jbτ
δρ r χ r,r' -2 M r ' r ' r r
δv r'τ τ τ ττ
= = φ φ φ φ∑∑
with ( )ia ,jb στ ij ab aσ bσ iaσ,jbτ iaσ,bjτM δ δ δ ε -ε K Kσ τ = + +
Contain and termsJv xcv
12
1r
( ) ( )
2
xcδ E
δρ r δρ r'
• Putting
( )( ) ( ) ( ) ( )* *
i aσ iσ aσ
iaσ aσ iσ
r r' r rχ r,r' =2
ε - ε
φ φ φ φ∑ The simpler perturbational
approach regained
W. Yang, A.J. Cohen, F. De Proft, P. Geerlings, J. Chem. Phys. , 136, 144110 (2012)
K=0
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Pag.
Some examples: Benzene (LDA/6-311+G*)
σ
π
28-1-2015 41
Pag.
Note: the analogy with Fukui function evaluation
( )( ) ( )
( ) ( ) ( )2 *fσ
fσ k c
kcτ f ,kc
δμ δεf r = = r - Q r r
δv r δv rτ τ
σ τ
= φ φ φ∑
HOMO/LUMOapproximation
relaxation term
Cfr. ( ) ( )( )
( )
2 * kσ
fσ kσ
kσ v r
rf r r - 2
N
∂φ = φ φ
∂ ∑
• W. Yang, R.G. Parr, R. Pucci, J.Chem. Phys, 81, 2862 (1984)
• A. Michalak, F. De Proft, P. Geerlings, R. Nalewajski, J. Phys. Chem. A, 103, 762, (1999)
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Pag.
H2CO (LDA/cc-pVDZ)
f −
f +
HOMO
LUMO
28-1-2015 43
Pag.
4. Conclusions
• Conceptual DFT offers a broad spectrum of reactivity descriptors.
• The “missing” second order derivative, the “linear response function”, comes within reach with various techniques of increasing complexity. It is a tool to see how ∆v perturbations are propagated through an atom or molecule. Its physicalrelevance becomes apparent, thanks to various representations of the kernel for atoms revealing atomic shell structure.
• The computational results on molecules 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.
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Pag.
Acknowledgements
Acknowledgements
Prof. Frank De Proft
Prof. Paul W. Ayers (Mc Master University, Hamilton, Canada)
Prof. Weitao Yang (Duke University, USA) and Dr. A. J. Cohen (Cambridge, UK)
Dr. Nick Sablon
Drs. Z. Boisdenghien
Dr. Stijn Fias
Dr. Tim Fievez
Dr. M. Torrent – Sucarrat (Brussels, Girona)
Prof. G. Merino (Merida, Mexico)
Prof. C. Van Alsenoy (Antwerp, Belgium)
and Prof. M. Sola (Girona)
Fund for Scientific Research-Flanders (Belgium) (FWO)
and the VUB (Strategic Research Program)
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