The Linear Response Function. - Vrije Universiteit Brussel

The Linear Response Function.Paul Geerlings, Nick Sablon, Frank De ProftEenheid Algemene Chemie, Vrije Universiteit Brussel, BelgiumSummer Talks in Santiago IIIJanuary 2012Pag.6-1-2012 1

Outline1. Introduction: Chemical Concepts from DFT2. The Linear response Function2.1. Preliminaries2.2. Direct evaluation of the second order functional derivative2.3. An Orbital based Approach2.4. Inductive and Mesomeric effects and the concept of nearsightedness2.5. Aromaticity3. Conclusions4. AcknowlegementsPag.6-1-2012 2

1. Introduction : Chemical Concepts from DFTFundamentals of DFT : the Electron Density Function as Carrier of InformationHohenberg Kohn Theorems (P. Hohenberg, W. Kohn, Phys. Rev. B136, 864 (1964))ρ(r) as basic variableρ(r) determines N (normalization)"The external potential v(r) is determined, within a trivial additive constant, by the electrondensity ρ(r)"electrons• • •• • • •• •••ρ(r) for a givenground statecompatiblewith asingle v(r)v(r)- nuclei- position/chargePag.6-1-2012 3

• Variational Principlev[ ] ( ) [ ]ρ( r) → H → E = E ρ = ρ r v( r) d r + F ρ( r)op∫HKδF v(r)+ HK= µδρ(r)F HK Universal HohenbergKohn functionalLagrangian Multipliernormalisation∫ ρ(r) dr = N• Practical implementation : Kohn Sham equationsComputational breakthroughPag.6-1-2012 4

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 rathervaguely defined chemical concepts such as electronegativity,chemical hardness, softness, …, to extend the existing descriptors andto use them either as such or within the context of principles such asthe Electronegativity Equalization Principle, the HSAB principle,the Maximum Hardness Principle …Starting with Parr's landmark paper on the identification ofµ as (the opposite of) the electronegativity.Pag.6-1-2012 5

Starting point for DFT perturbative approach to chemical reactivityConsiderE = E[N,v]Atomic, molecular system, perturbed in numberof electrons and/or external potentialdE =⎛ ∂E ⎞⎜ ⎟⎝ ∂N⎠dN +v(r)⎛ δE ⎞∫ ⎜ ⎟⎝ δv(r) ⎠δv(r)drNµidentificationfirst order perturbation theoryρ( r)identificationElectronic Chemical Potential (R.G. Parr et al, J. Chem. Phys., 68, 3801 (1978))= - χ (Iczkowski - Margrave electronegativity)Pag.6-1-2012 6

Identification of two first derivatives of E with respect to N and v in a DFTcontext → response functions in reactivity theoryE[N,v]⎛ ∂E ⎞⎜ ⎟⎝ ∂N⎠v(r)ElectronicChemicalPotential= µ= −χElectronegativity⎛ δE⎞⎜ ⎟⎝ δv(r)⎠N= ρ(r)⎛ ∂ 2 E ⎞⎜⎝ ∂N 2 ⎟⎠v(r)= η∂ 2 E∂Nδv(r) = ⎛ δµ ⎞ ⎛⎜ ⎟ = ∂ρ(r) ⎞⎜ ⎟⎝ δv(r) ⎠ ⎝ ∂N ⎠Nv⎛ 2 δE⎞= χ(r, r ')⎜ δv(r) δv(r ') ⎟⎝ ⎠NChemical hardness= f(r)Linear responseFunctionS = 1 ηChemical SoftnessFukui function6-1-2012 Pag.7

Until now: focus on first and second order derivates• µ = −χ→ Electronegativity Equalization Principle(EEM) (Sanderson, Mortier, …)• η → S= 1 η→ Chemical Hardness and Softness→ HSAB Principle ( Pearson, Parr)Maximum Hardness Principle ( Pearson)• f ( r)s( r ) =Sf ( r)→ Fukuifunction and local Softness.→ Local reactivity / selectivity (Parr, Yang)P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103, 1793 (2003)Pag.6-1-2012 8

What about third order derivatives?3⎛ ∂ E ⎞ ⎛ ∂η⎞• ⎜ 3 ⎟ = ⎜ ⎟ : hyperhardness: known to be small⎝ ∂N⎠ ⎝ ∂N⎠vv( )( )3⎛ δ E ⎞ ⎛ δχ r,r' ⎞⎜ =δv( r) δv( r' ) δv( r'' ) ⎟ ⎜ δv r'' ⎟⎝ ⎠ ⎝ ⎠• : awkward …? Importance of mixed “2+1” derivativesNN32⎛ ∂ E ⎞ ⎛ δη⎞ ⎛ ∂ ρ r ⎞•( ) ⎛ ∂f ( r)⎞⎜ Nδv( r) ⎟= ⎜ = ⎜ ⎟ = ⎜ ⎟δv( r)⎟⎝ ⎠ ⎝ ⎠ N N ⎠2 2∂N ⎝ ∂ ⎠ ∂v ⎝vDual descriptorC. Morell, A. Grand, A. Toro-Labbé,J. Phys. Chem. A 109, 205, 2005.3⎛ δ E ⎞ ⎛ ∂χ r, r' ⎞ ⎛ δf r ⎞•Already a large number of applications: one shotrepresentation of nucleophilic and electrophilic regions( ) ( )⎟( )⎜ = ⎜ =∂Nδv ( r) δv( r' ) ⎟ ∂N ⎜ δv r' ⎟⎝ ⎠⎝ ⎠vv ⎝ ⎠NunexploredFukui kernelPolarization effects on Fukui functionMight play a role in describing change in polarizability upon ionization,electron attachment, …P. Geerlings, F. De Proft, PCCP, 10, 3028, 2008.Pag.6-1-2012 9

An Interesting applicationRegaining the Woodward Hoffmann rules for pericyclic reactionson the basis of• the dual descriptor( )⎛ ∂f r⎜⎝ ∂N• the initial hardness response⎞⎟⎠v⎛ ∂η⎜⎝ ∂R⎞⎟⎠Nrelated to⎛ δη⎜⎝ δv r⎞⎟( ) ⎟⎠ NP. Geerlings, P.W.Ayers, A. Toro Labbé, P.K. Chattaraj, F. De Proft, Acc. Chem. Res., 55, xxx, 2012Pag.6-1-2012 10

What about the remaining second order derivativeχ ( r,r' ): linear response functionN( )⎛ 2δ E ⎞ ⎛ δρ r ⎞= =''⎜δv( r) δv( r ) ⎟ ⎜δv( r ) ⎟⎝ ⎠ ⎝ ⎠N• Computationally demanding• ? Chemical Interpretation: six dimensional kernel• ? Condensed VersionFundamental Importance1. Information about propagation of an (external potential) perturbation on position r’throughout the system2. Berkowitz Parr relationship (JCP 88, 2554, 1988)( ) s( r,r ')χ r,r '= − +s( r) s( r' )SSoftness kernelS∫∫s( r)All information⎛ δρ rs( r,r' ) = -⎜⎝ δv r'f(r)( ) ⎞( ) ⎟⎠ µ( )inverted → η r,r'Hardness kernel……Pag.6-1-2012 11

2. The Linear Response function2.1 Preliminaries• 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• Baekelandt, Wang: EEM∂qΠ = rrs∂ α s→ π-electron system• Cioslowski: approximate softness matrices• NO direct, practical, generally applicable, nearly exact approach availablePag.6-1-2012 12

2.2 Direct evaluation of the second orderfunctional derivativeA first step: the numerical approach• direct evaluation of the second order functional derivative( )χ r,r'2⎛ δ E ⎞= ⎜δv( r) δv( r' ) ⎟⎝⎠N• methodology in line with the first order case( )f r( )⎛ ∂ρ r ⎞ ⎛ δµ ⎞= ⎜ ⎟ = ∂N⎜ δv r ⎟( )⎝ ⎠v⎝ ⎠Nfukui functionf( 2 )( r)( )⎛ ∂f r ⎞ ⎛ δη⎞= ⎜ ⎟ = ∂N⎜ δv r ⎟( )⎝ ⎠v⎝ ⎠Ndual descriptorP.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.Pag.6-1-2012 13

Methodology: evaluation of⎛⎜⎝δQδv[ ]( )⎛ δQ v ⎞Q[ v+w ]-Q[ v ] = ∫⎜w r dr+ο wδv r ⎟⎝ ⎠⎞⎟⎠( ) (2)i i iNperturbation w i (r) (i= 1, ….P)• Linear response regime[ ]( )⎛ δQ v⎜⎝ δv r⎞⎟⎠NK∑j=1jj( )= q β r• basis set expansionK∑ ∫[ ] [ ] ( ) ( )Q v+w -Q v = q w r β r dri j i jj=1d=BqChoose P>K → find q via least squares fitting• Perturbationswi-pr = r-R( )ii• Expansion functions • s on p type GTO centered on each center• exponents doubled as compared to primitive setPag.6-1-2012 14

An example H 2 COFukui function f + (r)Nucleophilic attack (including correct angle) at CSCN -Fukui function f - (r)Softest reaction site for an attacking electrophile: ST. Fievez, N. Sablon, F. De Proft, P.W. Ayers, P. Geerlings, J.Chem.Theor. Comp., 45, 1065, 2008Pag.6-1-2012 15

Turning to a second functional derivative2⎛ δ Eχ ( r,r' ) = ⎜⎝ δv r δv r'E[ v+wi ]-2E[ v ] +E[ v-wi ] ~ ( ) ( ) ( )∑( ) ( ) ( )χ r,r' = q β r β rkl( ) ( )kl k l⎞⎟⎠N∫∫χ r,r' w r w r' drdr'ii∑ ∫ ∫[ ] [ ] [ ] ( ) ( ) ( ) ( )E v+w -2E v +E v-w = q β r w r dr β r' w r' dr'i i kl k i l iklperturbationsexpansionskl( )q → χ r,r'Pag.6-1-2012 16

? Visualisation /interpretation of this six dimensional kernel• Condensation to an atom-condensed linear response matrixχAB• multi-center numerical integration scheme using Becke’s “fuzzy” Voronoi polyhedra( )χ = χ r,r' drdr'AB∫ ∫vAvBM.Torrent Sucarrat, P. Salvador, P. Geerlings, M.Solá,J.Comp.Chem. 28, 574, 2007Pag.6-1-2012 17

An example: H 2 COH C O H2H -1.21 1.19 0.42 -0.40C 1.19 -4.82 2.45 1.19O 0.42 2.45 -3.28 0.42H2 -0.40 1.19 0.42 -1.21Pag.6-1-2012 18

2.3 An Orbital based approach• Single Slater determinant ( say KS-DFT)• Second order perturbation theory V=• Closed shell, spin restricted calculations• Frozen core∆E ~ ε -ε•i→a a i( )χ r,r' =4N 2∞∑ ∑i( )∑δv r( ) ( ) ( ) ( )φ r φ r φ r' φ r'* *i a a iε -εi=1 a=N 2+1 i ai∗(P.W.Ayers,Faraday Discussions,135, 161, 2007)φi: occupied orbitals φa: unoccupied orbitals εi,εa: orbital energiesExact( δρ( r) δvKS( r' ))N∗Zeroth order approximation to the linear response kernel forthe interacting systemPag.6-1-2012 19

An example: H 2 COPresent methodH C O H2H -1.22C 0.68 -4.35O 0.34 2.99 -3.67H 0.19 0.68 0.34 -1.22Previous methodH C O H2H -1.21C 1.19 -4.82O 0.42 2.45 -3.28H -0.40 1.19 0.42 -1.21Correlation coefficient between the matrix elements of the two methods:Y = 0.99x - 0.01R 2 = 0.96Slope ∼ 1Pag.6-1-2012 20

H 2 OPresent methodO H1 H2O -1.76H 10.88 -0.93H 20.88 0.05 -0.93Previous methodO H1 H2O -3.77H 11.88 -1.39H 21.88 -0.49 -1.39Y= 1.98x + 0.05 R 2 = 0.96 Slope ∼ 2For other simple molecules (NH 3 , CO, HCN, NNO) always a high correlation coefficientis obtained with a very small intercept; the slope varies between 1 and 2.Intramolecular comparisons are similar between two approaches; for intermolecularsequencies the transition to s(r,r’) might be advisable comparable with the swithfrom f(r) to s(r).N. Sablon, P.W. Ayers, F. De Proft, P. Geerlings, J. Chem. Theor. Comp. 6, 3671, 2010Pag.6-1-2012 21

Ethanal (PBE-6-31+G*)C 1 H 1 O C 2 H 2 H 3 H 4C 1 -4.2080H 1 0.6900 -1.2365O 2.7289 0.3338 -3.7827C 2 0.5067 0.1507 0.3522 -3.3676H 2 0.0966 0.0024 0.1707 0.7813 -1.1413H 3 0.0966 0.0024 0.1707 0.7813 0.0372 -1.1413H 4 0.0892 0.0572 0.0264 0.7950 0.0532 0.0532 -1.0738• diagonal elements: largest in absolute value;χ ;χ :c c1 1χ c 1 ooolarger 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→ Confidence in use for exploration of (transmission) of inductive and mesomeric effectsin organic chemistry.Pag.6-1-2012 22

2.4 Inductive and mesomeric effects andthe concept of nearsightednessTransmission of a perturbation througha carbon chainχ OX(X= C 0 , C 1 , C 2 …)χ NXSaturated systems• density response of C atoms on heteroatom perturbation decreasesmonotonously with distance• exponential fit: r 2 = 0.982 ( vide infra)Characterizing and quantifyingthe inductive effect.Pag.6-1-2012 23

Unsaturated systems• alternating values• C 1 , C 3 , C 5 of the chain: minimaC 0 , C 2 , C 4 , C 6 : maximaR= OH, NH 2 : resonance structures5316 4 2C 1 , C 3 , C 5 : mesomeric passive atoms→ follow same trend as alkane structures (inductive effect)C 0 , C 2 , C 4 , C 6 : mesomeric active atoms→ effect remains consistently large even after 6 bonds (small decrease due tosuperposition of inductive and mesomeric effect)N. Sablon, F. De Proft, P. Geerlings, JPCLett., 1, 1228, 2010Pag.6-1-2012 24

Substituted benzenes vs cyclohexanesχC1C= (i= 2,3,...6)i• cyclohexane:• benzene:• χ decreases exponentially (inductive effect)• influence of OH small• maxima at C 2 , C 4 , C 6 : mesomerically active atoms (mesomeric effect)• minima at C 3 , C 5 : mesomerically inactive atomsPag.6-1-2012 25

OHOHOHCH 3• Similar behaviour for o,p andm directors• Mesomerically active atoms:2,4,61,4 effect ? AromaticityN. Sablon, F. De Proft, P. Geerlings, Chem.Phys.Lett., 498, 192, 2010Pag.6-1-2012 26

2.5 The linear response function as an indicator of aromaticityRelation with the para delocalization index (PDI) derived from AIM( ) ( )∫∫ Γδ A,B = -2 r ,r dr drA B1 2 1 2XCExchange 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, …δAB= δ141423para• Does Linear response function χ 1,4 contain similar information?Pag.6-1-2012 27

16 non equivalent sixrings studied by Sola et al.Pag.6-1-2012 28

Pag.6-1-2012 29• typical benzene-type pattern encountered before

Pag.6-1-2012 30→ Linear response function as an “electronic” descriptor of aromaticity

An application: inorganic ringsPag.6-1-2012 31

Borazine• Typical resonance pattern observed in aromatic systems is recovered if one of the N-atoms is chosenas 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)Cfr. Valence bond study:J. Engelberts, R. Havenith, J. Van Lenthe, L. Jenneskens, P. Fowler,Inorg. Chem., 44, 5266, 2005Pag.6-1-2012 32

Valence Bond StructuresX= NH, O, PHMolecule E res W(I) W(II) W(III) W(IV)B 3 N 3 H 6 61.6 0.05 0.05 0.90 0.00B 3 O 3 H 3 27.6 0.02 0.02 0.96 0.00B 3 P 3 H 6 132.6 0.21 0.21 0.58 0.00C 6 H 6 161.4 0.47 0.47 0.03 0.03Resonance Energy (kJ mol -1 ) and weights W of the four Valence Bond Structures.Pag.6-1-2012 33

BorazineStructure III with lone pairs on N: prominent role• associated linear response values reflect (1,4) delocalizationresembling the aromatic situations• B atoms do not have a lone pairAssociated linear response values merely reflect an inductive effect.Pag.6-1-2012 34

Related cases• Boroxine• s-triphosphatriborinX= OX= PHBoroxine: • qualitatively similar to borazine, but even lower aromaticity• (1,4) PLR indexE res0.17 < 0.30 (borazine)27.6 < 61.6Pag.6-1-2012 35

s-triphosphatriborin (X=PH)Linear response curves: similar to benzene irrespective of the reference atom (B or P) considered• PLR0.87: important degree of aromaticity ∼ benzene• E res 132.6kJ mol -1 cfr benzene 161.4Significant contributions of structures I en II• Also in benzene I and II are the two preponderant structuress-triphosphatriborin takes an intermediate position between the aromaticbenzene and the non-aromatic borazine or boroxine cases.Qualitative trend in atom condensed linear response functions parallels thecontribution of possible Kekulé structures to the resonance energy in thecontext of VB Theory.N. Sablon, F. De Proft, P. Geerlings, PCCP, 14, xxx, 2012Pag.6-1-2012 36

4. Conclusions• Conceptual DFT offers a broad spectrum of reactivitydescriptors. Selected third order derivatives may containimportant chemical information.• The “missing” second order derivative, the “linear responsefunction”, comes within reach and is a tool to see how ∆vperturbations are propagated through a molecule its chemicalrelevance becomes apparent.• The computational results reveal that important chemicalinformation can be retrieved from the linear responsefunction: from inductive and mesomeric effects to thearomatic character of organic and inorganic rings.Pag.6-1-2012 37

AcknowledgementsProf. Frank De ProftProf. Paul W. Ayers (Mc Master University, Hamilton, Canada)Dr. Nick SablonTim FievezAcknowledgementsDr. P. Jaque ( Santiago, Chile)Dr. M. Torrent – Sucarrat (Brussels, Girona)Prof. M. Sola (Girona)Fund for Scientific Research-Flanders (Belgium) (FWO)Pag.6-1-2012 38

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(r 0 ) induced by ∆v(r’) no matter how large with r’ outside sphere with radius R aroundr 0 will always be smaller than a maximum magnitude ∆ρ→ maximum response of the electron density will decay monotonously as a function of R.⎛δρ( r)⎞Importance of softness kernel ⎜δv ( r') ⎟⎝ ⎠ µ• r’r'-r 0>> RNo numerical applications to atomic or molecular systems yetPag.6-1-2012 39

Softness kernel s(r,r’) (evaluated at constant µ!): nearsighted (Ayers)Linear response kernel χ ( r,r' ) (evaluated at constant N): not nearsightedBerkowitz - Parr relationship( ) ( )s r s r'χ ( r,r' ) =-s ( r,r' ) +Sfarsighted contribution to χ ( r,r' )? Small electrontranfer → neglect of this termχ ( 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 aboveN. Sablon, F. De Proft, P. Geerlings, JPCLett., 1, 1228, 2010Pag.6-1-2012 40