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.

Pag.15-7-2011 32

• 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

−

∑

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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)

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AcknowledgementsDr. M. Torrent – Sucarrat (Barcelona)

Prof. M. Sola (Girona)

Fund for Scientific Research-Flanders (Belgium) (FWO)