Methodological Advances in Conceptual DFTwe.vub.ac.be/~algc/algc_new/Geerlings/watoc 2011... ·...
Transcript of Methodological Advances in Conceptual DFTwe.vub.ac.be/~algc/algc_new/Geerlings/watoc 2011... ·...
Methodological Advances in Conceptual DFT
Paul Geerlings1, Nick Sablon1, Frank De Proft1, Aron Cohen2, Weitao Yang3
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Aron Cohen , Weitao Yang
1Eenheid Algemene Chemie, Vrije Universiteit Brussel, Belgium2Department of Chemistry, University of Cambridge, UK3Department of Chemistry, Duke University, Durham, USA
WATOC, 2011WATOC, 2011
Outline
1. Introduction: Chemical Concepts from DFT
2. The Linear response Function
2.1. Preliminaries2.2. Direct evaluation of the second order functional derivative2.3. An Orbital based Approach2.4. Inductive and Mesomeric effects and the concept of nearsightedness
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3. Analytical Evaluation of Fukui functions
4. Conclusions
5. Acknowlegements
2.4. Inductive and Mesomeric effects and the concept of nearsightedness2.5. Aromaticity2.6. Conclusions
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)"
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•
•
• • •
•
•
•
••
•
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→ → = = +∫
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• 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
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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
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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
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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
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
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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)
• J. L. Gazquez, J.Mex.Chem.Soc., 52, 1 (2008)
• S. Liu, Acta Physico-Chimica Sinica, 25, 590 (2009)
Until now: focus on first and second order derivates
• µ= − → Electronegativity Equalization Principle
(EEM) (Sanderson, Mortier, …)
• → Chemical Hardness and Softness
→ HSAB Principle ( Pearson, Parr)
χ
1η S=η
→
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→ HSAB Principle ( Pearson, Parr)
Maximum Hardness Principle ( Pearson)
•
→ Fukuifunction and local Softness.
→ Local reactivity / selectivity (Parr, Yang)
( )f r
( ) ( )s r =Sf 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
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2. Berkowitz Parr relationship (JCP 88, 2554, 1988)
Softness kernel
inverted
Hardness kernel
……
( )s r
S
All information
∫
∫
( ) ( )( ) ( )s r s r'
χ r, r ' s r, r 'S
= − +
( )η r,r'→
( )( )( )
µ
δρ rs r,r' = -
δv r'
f(r)
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.
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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.
Today’s talk
• evaluation / interpretation of
• new, general, strategies for evaluation of
( )χ r,r'
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• new, general, strategies for evaluation of reactivity descriptors
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
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• 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
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• 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.
An example H2CO Fukui function f+(r)
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Nucleophilic attack (including correct angle) at C
T. Fievez, N. Sablon, F. De Proft, P.W. Ayers, P. Geerlings, J.Chem.Theor. Comp., 45, 1065, 2008
• for the linear response function promising result were obtained for small molecules
• computationally demanding ; not yet suitable for medium and large systems
N. Sablon, P.W. Ayers, F. De Proft, P. Geerlings, J. Chem. Theor. Comp., 6, 3671, 2010
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• Looking for an alternative: analytical evaluation (?)
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
∞
∑ ∑ ∗∗∗∗
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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
∗∗∗∗
• Condensation to an atom-condensed linear response matrix ABχ
Exact ( ) ( )( )KS Nδρ r δv r'
? Visualisation /interpretation of this six dimensional kernel
∗∗∗∗Zeroth order approximation to the linear response kernel for the interacting system
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• 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.
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
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• 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
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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
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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)
R= -CHO, -C≡N : resonance structures
024
13
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OXχ NXχ
• Similar differences in trends between conjugated and nonconjugated systems highlighting fundamentally different character of inductive and mesomeric effect
• Higher values of and as compared to previous cases0OCχ 0NCχ
→ polarizability of C=O and C≡N bonds
Cfr. also OOχ R=OH -2.784 R=CHO -4.312<
NN 2χ R=NH -3.592 R=CN -6.040<
(X= C0, C1, …)
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'
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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
( ) ( )
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( )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
Substituted benzenes vs cyclohexanes
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• 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
X Cχ ( i = 1 ,2 , . . . 6 )i
Importance of Inductive Effect
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: exponential decay: inductive effectOH
OH1
23
4
5 6
: mesomeric active carbons 2,4,6
X= O, N, F
NH2
OH
F
decreasing mesomeric activity
cfr. increasing X electronegativity
OH
O
H
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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
2.4 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∫ ∫
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• 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
16 non equivalent sixrings studied by Sola et al.
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• typical benzene-type pattern encountered before
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χ
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→ Linear response function as a descriptor of aromaticity
PDI
2.6 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.
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• 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 has ( almost) been paved.( )s r,r'
3. Analytical Evaluation of Fukui functions
As stated above: most reactivity descriptors were evaluated numerically hitherto.
Typical example: N-derivatives: finite difference approximation
( )( )
v
ρ rf r
N
∂ = =
∂ Fukui function
( ) ( )N N-1ρ r ρ r−
( ) ( )N+1 Nρ r ρ r−
• Need for analytical expressions which can be implemented in standard Quantum chemical codes.
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• Need for analytical expressions which can be implemented in standard Quantum chemical codes.
( )( )
( )v N
ρ r δµf r
N δv r
∂ = = ∂
• Recently (A.J. Cohen, P. Mori-Sánchez, W. Yang, Phys. Rev. B77, 115123, 2008)
σ
fσ eff fσ fσ
v
Eφ H φ ε
N
∂ = =
∂ •
fσφ : KS frontier orbital (spin σ)
2 σ
s
1v
2− ∇ + contains excharge correlation potential
Start for a coupled Perturbed KS ansatz
σ
fσ fσ eff fσδε φ δH φ=
σ
fσ fσ fσ J xc fσφ δv φ φ δv δv φ= + +
contains elements of K and M matrices.
Evaluation of ( )fσ
N
δε
δv r
k: occ; c: unocc;τ spinlabel
( )( ) ( ) ( ) ( )
2 *fσfσ fσ,kcτ kτ cτ
kcτ
δε= f r = φ r Q φ r φ r
δv r
−
∑
Pag.15-7-2011 34
cf ( ) ( )( )
( )
2 kσ*
fσ kσ
kσ v
φ rf r φ r - 2 φ
Nr
∂ =
∂ ∑ W. Yang, R.G. Parr, R. Pucci,
J.Chem.Phys., 81, 2862, 1984
→ Evaluation of N derivative avoided is the relaxation term.
→ Replaced by integrations and matrix multiplications
→ Complete analytical expression (previous work: approximation)
A. Michalak, F. De Proft, P. Geerlings, R. Nalewajski, J.Phys.Chem A, 103, 762, 1999
Some examples
He
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Be
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H2CO
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• A similar procedure has been worked out for the linear response function, yielding the result in the “orbital approach” discussed before as a simplified case (non interacting system)
Analytical evaluation of DFT based reactivity descriptors comes into reach which can be coupled to /introduced in standard DFT or wave function programs.
W. Yang, A. Cohen, F. De Proft, P. Geerlings, Submitted
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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
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• 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 relevance becomes apparent.
• Analytical procedures for the evaluation of reactivity descriptors are on the way.
Acknowledgements
Acknowledgements
Prof. Paul W. Ayers (Mc Master University, Hamilton, Canada)
Tim Fievez (Brussel)
Dr. M. Torrent – Sucarrat (Barcelona)
Pag.15-7-2011 41
AcknowledgementsDr. M. Torrent – Sucarrat (Barcelona)
Prof. M. Sola (Girona)
Fund for Scientific Research-Flanders (Belgium) (FWO)