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Conceptual DFT:The Linear Response Function.

Paul Geerlings

General Chemistry, Vrije Universiteit Brussel, Belgium

Quitel, 2013

Granada, June 30 - July 5, 2013

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QUITEL: 39th International Conference:Theoretical Chemists of Latin Expression

but …. Belgium is a trilingual Country

but … “Belgae” were already mentioned by Caesar in De Bello Gallico

Gallia est omnis divisa in partes tres, quarum unam incolunt Belgae, aliam

Aquitani…Horum omnium fortissimi sunt Belgae….

C.I. Caesar, De Bello Gallico, I,1

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Outline

1. Introduction: Chemical Concepts from DFT

2. The Linear response Function

3. Conclusions

4. Acknowlegements

2.1. Preliminaries2.2. Direct evaluation of the second order functional derivative2.3. A simple perturbational approach

2.3.0. Preliminaries2.3.1. Atoms revisited2.3.2. Inductive and Mesomeric effects 2.3.3. Aromaticity

2.4. Coupled perturbation KS evaluation of the Fukui function and the linear response function

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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)"

•

•

• • •

•

•

•

••

•

compatible with a single v(r)

ρ(r) for a given ground state

- nuclei - position/charge

electrons

v(r)

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• Variational Principle

FHK Universal HohenbergKohn functional

Lagrangian Multiplier normalisation

• Practical implementation : Kohn Sham equations

Computational breakthrough

ρ(r)∫ dr = N

HKF

v(r)+ =(r)

δµ

δρ

[ ] ( ) [ ]op v HKρ(r) H E=E ρ = ρ r v(r)dr+F ρ(r)→ → ∫

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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

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.

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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

identification

ρ r( )

Electronic Chemical Potential (R.G. Parr et al, J. Chem. Phys., 68, 3801 (1978))

= - χ (Iczkowski - Margrave electronegativity)

μ

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Chemical hardnessFukui function

∂E

∂N

v(r)

= µ

∂2E

∂N2

v(r)

= η∂2E

∂Nδv(r)=

δµ

δv(r)

N

=∂ρ(r)

∂N

v

E[N,v]

= f(r)

Identification of two first derivatives of E with respect to N and v in a DFT context → response functions in reactivity theory

Linear response Function

2

N

E(r, r ')

v(r) v(r ')

δ= χ

δ δ

= −

Electro-negativityElectronic

Chemical Potential

χN

E(r)

v(r)

δ= ρ

δ

• P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. , 103, 1793, 2003

• P. Geerlings, F. De Proft, PCCP, 10, 3028 (Third order derivatives)

• P. Geerlings, P.W. Ayers, A. Toro Labbé, P.K. Chattaraj, F. De Proft, Acc. Chem. Res., 55, 2012 (WH rules)

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What about the remaining second order derivative

: linear response function

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)

Softness kernel

inverted

Hardness kernel

……

( )s r

S

∫

∫

( )χ r,r'

( ) ( )( )

( )

2

' '

N N

δρ rδ E= =δv r δv r δv r

( ) ( )( ) ( )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

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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

• Early Hückel MO theory: mutual atom – atom polarizability → π-electron system

• Baekelandt, Wang: EEM

• NO direct, practical, generally applicable, nearly exact approach available or exploited

rrs

s

qΠ =

α

∂

∂

• Cioslowski: approximate softness matrices

2.1 Preliminaries

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2.2 Direct evaluation of the second order functional derivative

• 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

( )( )

( )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.

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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

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

δQ

δv

( ) ii

i

-pw r =

r-R

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An example H2CO Fukui function f+(r)

Nucleophilic attack (including correct angle) at C

SCN- Fukui function f-(r)

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

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Turning to a second functional derivative

[ ] [ ] [ ]i iE v+w -2E v +E v-w ~ ( ) ( ) ( )i iχ r,r' w r w r' drdr'∫ ∫

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∑ ∫ ∫

( ) ( ) ( )kl k l

kl

χ r,r' = q β r β r∑

( )( ) ( )

2

N

δ Eχ r,r' =

δv r δv r'

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• 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

? Visualisation /interpretation of this six dimensional kernel

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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

… but computing times prohibitively large use as standard

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2.3. A simple perturbational approach; an independent particle model

(P.W.Ayers, Faraday Discussions, 135, 161, 2007)

: occupied orbitals : unoccupied orbitals : orbital energies

( )( )

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

i a a i

i=1 a=N 2+1 i aN

δρ r φ r φ r φ r' φ r' = χ r,r' =4 δv r' 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

2.3.0. Preliminaries

• Can we calculate in a simpler way thereby creating the possibility to explore in detail its physical/chemical content, starting from atoms going to molecules

• Closed shell N-electron system in the KS ansatz; frozen orbital approach

• 1ste order Perturbation Theory → → taking functional derivative w.r.t

( )χ r,r'

( ) ( )1ρ r ( )v r

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2.3.1. Atoms revisited

• Earlier work: A. Savin, F. Colonna, M. Allatena, J. Chem. Phys, 115, 6827, 2001: some light elements

• Light elements (KS ansatz; PBE). Spherical potential perturbation

. plot : radial distribution of the linear response kernel( )2 ' '2r χ r,r r

He Be

• Similar to Savin’s plots

• Positive and negative region, for He, duplicated in Be shell structure

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One dimensional version for a fixed ( )2 'r χ r,r ( )r'=0 2r' r χ r,0→

He

Be

• Positive perturbation δv(r) v(r) becomes less negative electron density depletion in the vicinity of the nucleus

• ( ) ( ) ( )ρ r dr' χ r,r' δv r'∆ = ∫( ) ( )δv r Aδ r 0 A>0= −Pointlike

perturbation

( ) ( )ρ r Aχ r,0∆ =

Alternating positive and negative regions due to conservation of number of electrons

also in 2D plot

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Extending → the noble gases

• Shell structure upon increasing Z

• Similar results Be, Mg, Ca

• Isoelectronic Series

Ne F- Na+

Ar Cl- K+

Same “structure”

Contraction/dilatation of the contours → cf. decreasing/ increasing polarizability

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χ(r,0) for Ar in the x,y plane

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A direct application → polarizability calculations

( )ij i jα = - dr dr' r χ r,r' r'∫ i,j= x,y,z

• Comparison with high level calculations

• Absolute values deviate, trend is respected

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• First systematic study of non condensed linear response function on atoms

→ closed shell atoms throughout the periodic table

• Relation to the shell structure → physical information

• How to extend its use for molecules to look for chemical information

→ condensation at stake

polarizability

Z. Boisdenghien, C. Van Alsenoy, F. De Proft, P. Geerlings, J.Chem.Theor. Comp., 9, 1007 (2013)

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An example:

Numerical methodPresent method

Correlation coefficient between the matrix elements of the two methods:

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

Y = 0.99x - 0.01 R2= 0.96 Slope ∼ 1

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

N. Sablon, P.W.Ayers, F. De Proft, P. Geerlings, J.Chem.Theor. Comp., 6, 3671, 2010

For other simple molecules (H2O, 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 switch from f(r) to s(r).

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Ethanal (PBE-6-31+G*)

• 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.

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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Transmission of a perturbation through a carbon chain

NXχ

OXχ

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 …)

2.3.2. Inductive and mesomeric effects

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Unsaturated systems

• alternating values• C1, C3, C5 of the chain: minimaC0, C2, C4, C6 : maxima

R= OH, NH2: resonance structures

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)

135

6 4 2

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

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Substituted benzenes vs cyclohexanes

• 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

1,4 effect ? Aromaticity

N. Sablon, F. De Proft, P. Geerlings, Chem.Phys.Lett., 498, 192, 2010

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2.3.3. 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

(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

( ) ( )1 2 1 2

A B XC

δ A,B = -2 r ,r dr drΓ∫ ∫

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16 non equivalent sixrings studied by Sola et al.

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• typical benzene-type pattern encountered before

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

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An application: inorganic rings

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Valence Bond Structures

X= NH, O, PH

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

J. Engelberts, R. Havenith, J. Van Lenthe, L. Jenneskens, P. Fowler, Inorg. Chem., 44, 5266, 2005

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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)

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)

N. Sablon, F. De Proft, M. Sola, P. Geerlings, PCCP, 14, 3960 (2012)

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Digging further in aromaticity →→→→ σσσσ,ππππ aromaticity

Condensed version of the linear response function

( )A B

A B

v v

χ = χ r , r' d r d r'∫ ∫

N/2 MAB A B

ia aiNi i aa= +12

1χ = 4 S S

ε -ε∑ ∑ ( ) ( )

A

A *

ia i a

v

S = r r drφ φ∫

( ) ( )B

B *

ai a i

v

S = r' r' dr 'φ φ∫

Written as sum of contributions of different occupied MO’s → σ,π separation (planar molecules)

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Benzene

• Decreasing σ contribution (cfr. Cyclohexane)

• Zig/zag for π and total

• Anti-aromatic molecules:Cyclobutadiene, COT

“inverted zig zag”

S. Fias, P. Geerlings, P. Ayers, F. De Proft, PCCP, 15, 2882 (2013)

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Planar Metallic Systems

Aromaticity of all metal systems and involving a planar tetracoordinate carbon atom2-

4CE

2-

4E E= Al, Ga

• Large interest in submit in clustersclassified as aromatic: debate:

• Systematic replacement Al→ Ge double σ and π aromaticity (M.Sola)

• Influence of insertion of C in (Merino)

2-

4Al

-

4MAl (M= Li, Na, ...)

σ, π, σ+π

2-

4E

OLR Opposite Linear Response = ( )13 24

12 23 24

1χ χ χ , χ , χ

2+ >>

Aromaticity order

2- - + 2+

4 3 2 2 3 4Al > Al Ge ³ Al Ge ALGe < Ge≥ ≤

Expected order (Feixas et al)

Mainly σ aromatic (∼ 65%)

Insertion of C: decreasing aromaticity, sequence unaltered; but increasing σ component

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• The use of unintegrated plots

σ

• in plane• perturbation

at top nucleus

π

• 0.5 au above plane• perturbation in that

plane at top nucleus

σ

Delocalized Nature of the linear Response more pronounced in the σ electron density

π

σ Aromatic character

S. Fias, Z. Boisdenghien, T. Stuyver, M. Audiffred, G. Merino, P. Geerlings, F. De Proft, J. Phys. Chem. A, 2013, 117, 3556

Benzene 2-

4Al

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2.4 A Coupled Perturbed KS evaluation of the Fukui function and the linear response function

• Coupled perturbed perturbation ansatz starting from KS equations

( )1ρ

( )( )

( ) ( ) ( ) ( ) ( ) ( )-1 * *

j b i aiaσ,jbiaσ jbτ

δρ r χ r,r' -2 M r ' r ' r r

δv r'τ τ τ ττ

= = φ φ φ φ∑∑

with ( )ia ,jb στ ij ab aσ bσ iaσ,jbτ iaσ,bjτM δ δ δ ε -ε K Kσ τ = + +

Contain and termsJv xcv

12

1r

( ) ( )

2

xcδ E

δρ r δρ r'

• Putting

( )( ) ( ) ( ) ( )* *

i aσ iσ aσ

iaσ aσ iσ

r r' r rχ r,r' =2

ε - ε

φ φ φ φ∑ The simpler perturbational

approach regained

W. Yang, A.J. Cohen, F. De Proft, P. Geerlings, J. Chem. Phys. , 136, 144110 (2012)

K=0

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Some examples: Benzene (LDA/6-311+G*)

σ

π

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Note: the analogy with Fukui function evaluation

( )( ) ( )

( ) ( ) ( )2 *fσ

fσ k c

kcτ f ,kc

δμ δεf r = = r - Q r r

δv r δv rτ τ

σ τ

= φ φ φ∑

HOMO/LUMOapproximation

relaxation term

Cfr. ( ) ( )( )

( )

2 * kσ

fσ kσ

kσ v r

rf r r - 2

N

∂φ = φ φ

∂ ∑

• W. Yang, R.G. Parr, R. Pucci, J.Chem. Phys, 81, 2862 (1984)

• A. Michalak, F. De Proft, P. Geerlings, R. Nalewajski, J. Phys. Chem. A, 103, 762, (1999)

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H2CO (LDA/cc-pVDZ)

f −

f +

HOMO

LUMO

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4. Conclusions

• Conceptual DFT offers a broad spectrum of reactivity descriptors.

• The “missing” second order derivative, the “linear response function”, comes within reach with various techniques of increasing complexity. It is a tool to see how ∆v perturbations are propagated through an atom or molecule. Its physicalrelevance becomes apparent, thanks to various representations of the kernel for atoms revealing atomic shell structure.

• The computational results on molecules 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.

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Acknowledgements

Acknowledgements

Prof. Frank De Proft

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

Prof. Weitao Yang (Duke University, USA) and Dr. A. J. Cohen (Cambridge, UK)

Dr. Nick Sablon

Drs. Z. Boisdenghien

Dr. Stijn Fias

Dr. Tim Fievez

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

Prof. G. Merino (Merida, Mexico)

Prof. C. Van Alsenoy (Antwerp, Belgium)

and Prof. M. Sola (Girona)

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

and the VUB (Strategic Research Program)

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