Conceptual Density Functional Theory

Paul Geerlings

Department of Chemistry, Faculty of Sciences

Pag.18-8-2011 1

Department of Chemistry, Faculty of SciencesVrije Universiteit Brussel

Pleinlaan 2, 1050 - BrusselsBelgium

STC 2011, STC 2011, SurseeSursee, Switzerland, SwitzerlandAugust 21August 21--25, 201125, 2011

Outline

1. Introduction: Chemical Concepts from DFT

2. The Linear response Function

2.1. Direct evaluation of the second order functional derivates2.2. An Orbital based Approach2.3. From inductive and mesomeric effects to aromaticity

Pag.18-8-2011 2

3. Higher order Derivates

4. Conclusions

5. Acknowlegements

2.3. From inductive and mesomeric effects to aromaticity

3.1. The Dual Descriptor3.2. Reconciling the dual descriptor and the initial hardness response3.3. The Chemical relevance of higher (n≥3) order derivatives

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.18-8-2011 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.18-8-2011 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.18-8-2011 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.18-8-2011 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.18-8-2011 718-8-2011 718-8-2011 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

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

Pag.18-8-2011 818-8-2011 8

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)

• P. Geerlings, F.De Proft, PCCP, 10, 3028 (2008)

•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)Maximum Hardness Principle ( Pearson)

χ

1η S=η

→

Pag.18-8-2011 9

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

Pag.18-8-2011 10

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.

Pag.18-8-2011 11

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

• some examples of the computation / use of

( )χ r,r'

Pag.18-8-2011 12

• some examples of the computation / use of third order derivatives (the dual descriptor)

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

Pag.18-8-2011 13

• 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

2.1 Direct evaluation of the second order functional derivative

In line with earlier work concerning the transformation of to expressionsN

∂

∂

• Fukui-functions ( )( )

( )v N

ρ r δµf r

N δv r

∂ = = ∂

Difficulties with derivativeN

∂

∂

( ) ( )

2

N

δ E

δv r δv r'

( )

δ

δv r

Pag.18-8-2011 14

• Dual descriptor ( ) ( )( )

( )2

v N

f r δf r

N δv r

η ∂ = = ∂

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.

T. Fievez, N. Sablon, F. De Proft, P.W. Ayers, P. Geerlings, J.Chem.Theor. Comp., 45, 1065, 2008

Methodology: earlier work (e.g. Fukui Function)

[ ] [ ][ ]( )

( ) ( )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

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[ ] [ ] ( ) ( )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.18-8-2011 16

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'∫ ∫

[ ] [ ] [ ] ( ) ( ) ( ) ( )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'

Pag.18-8-2011 17

perturbations

expansions

( )klq χ r,r'→condensation

ABχ

(in practice: direct evaluation in atom-condensed version

Examples:

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.18-8-2011 18

H2O

O H1 H2

O -3.77

H1

1.88 -1.39

H2

1.88 -0.49 -1.39

2.2 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.18-8-2011 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

∗∗∗∗

• 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

Pag.18-8-2011 20

• 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

Pag.18-8-2011 21

• 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:

Some examples:

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.18-8-2011 22

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.18-8-2011 23

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

Transmission of a perturbation through a carbon chain

2.3 From inductive and mesomeric effects to aromaticity

Pag.18-8-2011 24

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.18-8-2011 25

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)

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.18-8-2011 26

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.18-8-2011 27

( )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 above studied

N. Sablon, F. De Proft, P. Geerlings, JPCLett.,1, 1228, 2010

Some recent work: substituted benzenes vs cyclohexanes

Pag.18-8-2011 28

• 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.18-8-2011 29

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∫ ∫

Pag.18-8-2011 30

• 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.18-8-2011 31

• typical benzene-type pattern encountered before

Pag.18-8-2011 32

χ

Pag.18-8-2011 33

→ Linear response function as a descriptor of aromaticity

PDI

2.5 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.18-8-2011 34

• 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. Higher order derivatives

Consider

• Introduced as a new descriptor, "dual descriptor"

(C. Morell, A. Grand, A. Toro Labbé, J. Phys. Chem. A109, 205 (2005))

Third order response function

( )2

2N 1 N N 12

v

(r)f (r) (r) 2 (r) (r)

N+ −

∂ ρ= ≅ ρ − ρ + ρ ∂

( )( ) ( ) ( ) ( )

232

2 2

v

ρ r f rδ E= = = f r

δv r N N N

∂ ∂ ∂ ∂ ∂

3.1 The dual descriptor

Pag.18-8-2011 35

= ρN+1(r)− ρN(r)( )− ρN(r)− ρN−1(r)( )

≅ φLUMO(r)2

− φHOMO(r)2

Large in electrophilicregions

Large in nucleophilicregions

> 0

< 0

electrophilic regions

nucleophilic regions

vN ∂

( )(2)f r→

→ a one shot picture of both electrophilic and nucleophilic regions in space

An example: Carbene (Singlet) ( +/- 0.01 surface plot)

( )2f 0> : electrophilic region (empty 2p orbital)

Pag.18-8-2011 36

( )2f 0<

Cfr. J.V. Correa, P. Jaque, J. Olah, A. Toro-Labbé, P. Geerlings, Chem. Phys. Lett., 470, 180, 2009

: Nucleophilic region (lone pair)

Favorable chemical reactions occur when regions that are good electron acceptors

(f(2)(r) > 0) are aligned with regions that are good electron donors (f(2)(r) < 0)

Minimizing

Confirmed by perturbation theoretical ansatz (P.W. Ayers)

( ) ( )2 2

BAf (r)f (r ')drdr '

r r '−∫∫

Pag.18-8-2011 37

Confirmed by perturbation theoretical ansatz (P.W. Ayers)

A way to rationalize the celebrated Woodward Hoffmann rules is an orbital free, even wave function free context

One of our points of interest in recent years, coupled to another third order derivative:

( )0

ηR

∂∂

• F. De Proft, P.W. Ayers, S. Fias, P. Geerlings, JCP, 125, 214101, 2008

• P.W. Ayers, C. Morell, F. De Proft, P. Geerlings, Chem. Eur, J., 13, 8240, 2007

• F. De Proft, P.K. Chattaraj, P.W. Ayers, M. Torrent-Sucarrat, M. Elango, V. Subramanian, S. Giri,

P. Geerlings, J. Chem. Theor. Comp., 4, 595, 2008

• N. Sablon, F. De Proft, P. Geerlings, Croat. Chem. Acta (Z.B. Maksić Volume), 82, 157, 2009

Pag.18-8-2011 38

• P. Jaque, J.V. Correa, F. De Proft, A. Toro-Labbé, P. Geerlings, Can. J. Chem. (R.J. Boyd Volume)

88, 858, 2010

Succesful application to • electrocyclisation• cycloadditions• sigmatropic reactions• chelotropic reactions

The first case: Cycloadditions

Butadiene + ethene

HOMO LUMO

Hückel B3LYP/6-31G*

Ethene

[ ]s s4 2π π+

( ) ( ) ( ) ( )2 22

LUMO HOMOf r r r= φ − φ

Pag.18-8-2011 39

Butadiene

Pag.18-8-2011 40

Molecules align so that favorable interactions (green lines)occur between their dual descriptors

"allowed"[ ]s s4 2π π+

• The case : ethene + ethene

supra/supra : "repulsive"

not allowed

supra/antara : can occur if the molecules rotate so that

[ ]s s2 2π π+

Pag.18-8-2011 41

supra/antara : can occur if the molecules rotate so thatthe double bonds become perpendicular

allowed

(but steric demands too high to occur in practice)

WH rules for cycloadditions regained

Similar result by considering ( )ηR

∂∂

(initial hardness response)

The fourth case: Chelotropic Reactions

LUMO

HOMO

S

O

O

Pag.18-8-2011 42

• filled and vacant frontier orbitals for bonding to other atoms onthe same atom (CO, singlet carbenes)

• σ bonds broken or formed on the same atom, in casu S

The WH Rules on chelotropic reactions in an orbital context

4n case: linear chelotropic addition

S

LUMOHOMO

Pag.18-8-2011 43

• Disrotatory mode

• Non-linear case: mode is inverted

O O

HOMO LUMO

Linear Chelotropic Reaction: 4n system

Pag.18-8-2011 44

Side view Top view

The Initial Hardness Response

Pag.18-8-2011 45

Energy and hardness profile Disrotatory mode in agreement with WH ( )η ϑ∂ ∂

Confirmed in the (4n+2) case

Conrotatory disrotatory

Disrotatory conrotatory

Pag.18-8-2011 46

What about the non-linear case?

• The dual descriptor appraoch

Linear Chelotropic Reaction: 4n System

( )2f (r) 0 (red)>

r ( )2f (r) 0 (blue)<

r

Pag.18-8-2011 47

Stabilizing interactions

Disrotatory

Dual descriptor =∂f(r)

∂N

v

=∂

∂N

δµ

δv(r)

N

v

=δ

δv(r)

∂µ

∂N

v

N

Changes in provoked by changes in nuclear configuration only

( )v r

3.2 Reconciling the dual descriptor and the initial hardness response

( )δη

δv rN

=

( ) ( )2f r

Pag.18-8-2011 48

cf.( )

( )

N N

v rdr

R v r R

∂∂η δη = ∂ δ ∂ ∫

( )( )(2) v r

f r drR

∂=

∂∫

General Information refined via to initial Hardness Response. ( )v r / R∂ ∂

WH rules regained in a conceptual DFT context for all four types of pericyclic reactions, thereby avoiding symmetry or phase based arguments

- initial hardness response

- dual descriptor "back of the envelope" approach

Pag.18-8-2011 49

- two approaches internally consistent

Importance of third order derivatives

first order: electronegativity (Pauling)

• Correction for hardness, softness, polarizability

• Incorporation of frontier MO Theory

second order: Fukui function, hardness, …

Chemistry at

3.3 The chemical relevance of higher order (n≥3) derivatives

Pag.18-8-2011 50

third order:• diagonal (N or v): small or too hard to interprete

• non diagonal: fundamentally new information,eg. dual descriptor

P.Geerlings, F.De Proft, PCCP, 10, 3028 (2008)

→ linear response function, softness kernel

4. Conclusions

• Conceptual DFT offers a broad spectrum of reactivity descriptors

• 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 significance is emerging.

• Selected third order derivatives may contain important chemical information, permitting among others to retrieve the WH rules in a density- only context.

Acknowledgements

Acknowledgements

Prof. Frank De Proft

Prof. Paul W. Ayers (Mc Master University, Hamilton, Canada)

Dr. Nick Sablon

Tim Fievez

Dr. P. Jaque ( Santiago, Chile)

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AcknowledgementsDr. P. Jaque ( Santiago, Chile)

Prof. P.K. Chattaraj (IIT – Kharagpur – India) / Prof. A. Toro Labbé (Santiago – Chile)

Dr. C. Morell (CEA, Grenoble)

Dr. M. Torrent – Sucarrat (Brussels, Girona)

Prof. M. Sola (Girona)

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

Nonlinear Chelotropic Reaction: 4n system

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Side view Top view

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Energy and hardness predict conrotatory mode Confirmed in the 4n+2 case: disrotatory mode

( )η ϑ∂ ∂ ( )0.5Z <

Nonlinear Chelotropic Reaction: 4n System

Destabilizing interactions

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Conrotatory

Destabilizing interactions