Conceptual DFT:

the Linear response Function and Third Order Derivatives

20-8-2010 1Herhaling titel van presentatie

and Third Order Derivatives

Conceptual DFT: the Linear response Function and Third Order Derivatives

Paul Geerlings, Nick Sablon, Tim Fievez, Pablo Jaque, Paul Ayers*, Frank De Proft

Department of Chemistry, Faculty of SciencesVrije Universiteit Brussel

Pag.20-8-2010 2

Vrije Universiteit BrusselPleinlaan 2, 1050 - Brussels

Belgium

*Department of Chemistry, Mc Master University, Hamilton, Canada

Challenges for DFT Challenges for DFT –– 240240thth National ACS meetingNational ACS meeting

Boston, Boston, MassachussetsMassachussets, USA, USAAugust 22August 22--26, 201026, 2010

Outline

1. Introduction: Chemical Concepts from DFT

2. The Linear response Function

2.1. An Orbital based Approach2.2. Direct evaluation of the second order functional derivates2.3. Conclusions

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3. Higher order Derivates

4. Conclusions

5. Acknowlegements

2.3. Conclusions

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.20-8-2010 4

•

•

• • •

•

•

•

••

•

compatible with a single v(r)

ρ(r) for a given ground state

- nuclei - position/charge

electrons

v(r)

ρ(r) → Hop → E = E ρ[ ]= ρ r( )∫ v(r)dr + FHK ρ(r)[ ]

• Variational Principle

FHK Universal HohenbergKohn functional

Lagrangian Multiplier normalisation

HKF

v(r)+ =(r)

δµ

δρ

Pag.20-8-2010 5

• 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.20-8-2010 6

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.20-8-2010 7

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.20-8-2010 820-8-2010 820-8-2010 8

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.20-8-2010 920-8-2010 9

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.20-8-2010 10

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.20-8-2010 11

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'

What about third order derivatives in a pertubational approach?

• : 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

ρ r f rE δ

Nδv r δv r N N

η ∂ ∂ ∂= = = ∂ ∂ ∂

Pag.20-8-2010 12

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, …

( ) ( )vvN

Nδv r δv r N N ∂ ∂ ∂

Fukui kernelPolarization effects on Fukui function( ) ( )

( ) ( )( )

3

vv N

χ r, r' δf rδ E

Nδv r δv r' N δv r'

∂ = = ∂ ∂

Today’s talk

• evaluation / interpretation of

• some examples of the computation / use of

( )χ r,r'

Pag.20-8-2010 13

• some examples of the computation / use of 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.20-8-2010 14

• 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 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, 135, 161, 2007)

( )i

i

δv r∑

( )( ) ( ) ( ) ( )* *N 2

i a a i

i=1 a=N 2+1 i a

φ r φ r φ r' φ r'χ r,r' =4

ε -ε

∞

∑ ∑ ∗∗∗∗

Pag.20-8-2010 15

: occupied orbitals

: unoccupied orbitals

: orbital energies

Exact ( ) ( )( )KS Nδρ r δv r'

? Visualisation /interpretation of this six dimensional kernel

iφ

aφ

i aε ,ε

∗∗∗∗Zeroth order approximation to the linear response kernel for the interacting system

• 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

Pag.20-8-2010 16

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.20-8-2010 17

• 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

Pag.20-8-2010 18

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.20-8-2010 19

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

Pag.20-8-2010 20

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’) with r’ outside a 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.20-8-2010 21

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.20-8-2010 22

( )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.20-8-2010 23

• 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

Pag.20-8-2010 24

: 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

Pag.20-8-2010 25

OH

OH

CH3

CH3 NH3

+

H

• Similar behaviour for o,p and m directors

• Mesomerically active atoms: 2,4,6

• Hyperconjugation vs π-mesomeric effect

• almost identical to

N. Sablon, F. De Proft, P. Geerlings, Chem.Phys.Lett., 497, xxx, 2010

2.2 An alternative approach: 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.20-8-2010 26

• 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

Pag.20-8-2010 27

[ ] [ ] ( ) ( )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.20-8-2010 28

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.20-8-2010 29

perturbations

expansions

( )klq χ r,r'→condensation

ABχ

(in practice: direct evaluation in atom-condensed version

An example

Previous method Present 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.20-8-2010 30

Correlation coefficient between the matrix elements of the two methods:

Y = 0.99x - 0.01

R2= 0.96

Slope ∼ 1

Previous method Present 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.20-8-2010 31

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, submitted

2.3 Conclusions

• The linear response function comes within reach of a general, non-empirical computational strategy: the last method presented can serve as benchmark

• 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.20-8-2010 32

• Atom condensation is at stake, but can nowadays be done in a systematic way.

• 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

• 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.20-8-2010 33

= 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.20-8-2010 34

( )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.20-8-2010 35

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.20-8-2010 36

• 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.20-8-2010 37

Butadiene

Pag.20-8-2010 38

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.20-8-2010 39

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.20-8-2010 40

• 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.20-8-2010 41

• Disrotatory mode

• Non-linear case: mode is inverted

O O

HOMO LUMO

Linear Chelotropic Reaction: 4n system

Pag.20-8-2010 42

Side view Top view

The Initial Hardness Response

Pag.20-8-2010 43

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

Confirmed in the (4n+2) case

Conrotatory disrotatory

Disrotatory conrotatory

Pag.20-8-2010 44

What about the non-linear case?

Nonlinear Chelotropic Reaction: 4n system

Pag.20-8-2010 45

Side view Top view

Pag.20-8-2010 46

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

( )η ϑ∂ ∂ ( )0.5Z <

• The dual descriptor appraoch

Linear Chelotropic Reaction: 4n System

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

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

r

Pag.20-8-2010 47

Stabilizing interactions

Disrotatory

Nonlinear Chelotropic Reaction: 4n System

Destabilizing interactions

Pag.20-8-2010 48

Conrotatory

Destabilizing interactions

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.20-8-2010 49

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.20-8-2010 50

- 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.20-8-2010 51

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

Pag.20-8-2010 52

• 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.

• Selected third order derivatives may contain important chemical information

Acknowledgements

Acknowledgements

Prof. Frank De Proft

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

Nick Sablon

Tim Fievez

Pag.20-8-2010 53

AcknowledgementsDr. P. Jaque ( Santiago, Chile)

Prof. P.K. Chattaraj (IIT – Kharagpur – India)

Dr. C. Morell (CEA, Grenoble)

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

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