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

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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.

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

16 non equivalent sixrings studied by Sola et al.

Pag.6-1-2012 28

• typical benzene-type pattern encountered before

Pag.6-1-2012 29

Pag.6-1-2012 30

→ Linear response function as an “electronic” descriptor of aromaticity

An application: inorganic rings

Pag.6-1-2012 31

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