The Linear Response Kernel - American Chemical Society

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The Linear Response Kernel: Inductive and Resonance Effects Quantified Nick Sablon,*,†,‡ Frank De Proft,† and Paul Geerlings† †

Eenheid Algemene Chemie, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium (Member of the QCMM Ghent-Brussels Alliance Group), and ‡Aspirant of the Research Foundation - Flanders (FWO - Vlaanderen), Egmontstraat 5, 1000 Brussels, Belgium

ABSTRACT Calculations of conceptual density functional theory (DFT) reactivity indices are mainly restricted to global quantities and local functions, whereas values for the nonlocal kernels are rarely presented. We used a molecular orbitalbased expression to calculate the atom-condensed linear response kernel. The results are the first published values of this quantity that have been obtained through a direct and generally applicable methodology. This letter focuses on the off-diagonal elements, which provide insight into the nonlocal contributions to chemical reactivity. A detailed study of a set of eight functionalized alkane and polyalkene derivatives enabled us to quantify inductive and resonance effects. SECTION Molecular Structure, Quantum Chemistry, General Theory

kernel η(r,r0 ), which is R ultimately connected to the local hardness η(r) = (1/N) η(r,r0 )F(r0 ) dr0. Senet11 derived exact functional relations between the linear and nonlinear response functions and the ground state electron density in terms of the universal Hohenberg-Kohn functional F [F] and obtained formal solutions for these kernels based on Green's functions. Theoretical expressions for and the mutual relations between the linear and nonlinear responses and the softness and hardness kernels have been elaborated within a Kohn-Sham (KS) formalism.12-14 A recent paper by Liu et al.15 summarizes the most important mathematical properties of the linear response kernel. It was emphasized that the kernel is symmetric: ð4Þ χðr, r0 Þ ¼ χðr0 , rÞ

everal authors1-4 have recently been paying attention to higher order derivatives and functional derivatives of the electronic energy E within the context of conceptual or chemical density functional theory (DFT).5-9 The linear response function or polarizability kernel χ(r,r0 ), which is defined as ! δ2 E δFðrÞ 0 χðr, r Þ ¼ ¼ ð1Þ δvðrÞδvðr0 Þ δvðr0 Þ N

S

N

with v(r) being the external potential, F(r) being the electron density, and N being the number of electrons, holds a central position. The focus has, however, mainly been on the theoretical properties of this kernel and the formulation of formal solutions, whereas its effective calculation has only rarely been done. We present here the first numerical values for this kernel obtained in a direct way for a number of functionalized alkanes and alkenes. The results yield a quantitative interpretation of the inductive and resonance (or mesomeric) effects in chemistry. Before starting a detailed discussion of our results, a succinct overview of the essential, published work is due. The central position of the linear response kernel in conceptual DFT is highlighted by the next equation, proposed by Berkowitz and Parr:10 χðr, r0 Þ ¼ -sðr, r0 Þ þ

sðrÞsðr0 Þ S

that it has real eigenvalues hi: Z χðr, r0 Þωi ðr0 Þ dr0 ¼ hi ωi ðrÞ

with ωi(r) being the corresponding eigenfunctions, and that it has a zero eigenvalue: Z χðr, r0 Þ dr0 ¼ 0 ð6Þ Furthermore, the kernel is negative semidefinite, it has an infinite number of eigenvalues arbitrarily close to zero, and it is not invertible in any real sense. Although exact expressions for computing the linear response kernel can be found in the above cited work, literature only provides a few numerical data, which are, moreover,

ð2Þ

relating the linear response kernel to the softness kernel s(r,r0 ), δFðrÞ sðr, r0 Þ ¼ ð3Þ δvðr0 Þ μ

Received Date: February 16, 2010 Accepted Date: March 18, 2010 Published on Web Date: March 26, 2010

(with μ being the electronic chemical potential), and the local softness s(r). The softness kernel's inverse is the hardness

r 2010 American Chemical Society

ð5Þ

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DOI: 10.1021/jz1002132 |J. Phys. Chem. Lett. 2010, 1, 1228–1234

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obtained through (rough) approximations. It is interesting to note that the early H€ uckel molecular orbital (HMO) theory for π-systems already contained the notion of the mutual atomatom polarizability πrs = ∂qr/∂Rs of atoms r and s, defined as the derivative of the electron population of atom r with respect to the HMO R parameter (or Coulomb integral) of atom s.16 This theory was recently revisited and applied in a study of the substituent effects in benzoic acids.17 The explicit calculation of the response ΔF(r) of the electron density upon a point charge perturbation in the external potential v(r) has been analyzed for various atoms.18 Electron density responses of CN- have been investigated19 by implicitly calculating the linear response function through the BerkowitzParr relation (eq 2) and a local approximation to the softness kernel. The relation between the linear response kernel and the change in molecular electrostatic potential imposed by a point charge perturbation has been derived and applied in the calculation of a nucleophilicity index.20 Parr and Yang5 mentioned that the linear response function can be obtained as the zero-frequency limit of the dynamic linear response function of time-dependent DFT. Baekelandt et al.21 and Wang et al.22 calculated atom-condensed linear response matrices within the context of the electronegativity equalization method (EEM),23 yielding numerical values for water and some organic compounds. The condensed linear response kernel is also used in the studies by Yang et al.24,25 concerning reaction paths, where the coupled-perturbed KS equations are solved. This method is based on the ideas of Stone and Alderton,26,27 who analyzed dipole and multipole polarizabilities, and the algorithmic developments by Morita and Kato.28,29 The interpretation of reactivity indices within conceptual DFT can often be clarified by considering a (functional) Taylor series expansion of the electronic energy E, which takes the next form when perturbations Δv(r) in the external potential are considered at constant number of electrons N:8,15 ZZ Z 1 χðr, r0 ÞΔvðrÞΔvðr0 Þ dr dr0 þ 3 3 3 ΔE ¼ FðrÞΔvðrÞdr þ 2

reasoning stresses the importance of the diagonal elements of the linear response kernel, we will, in this study, primarily be interested in the off-diagonal elements, which provide nonlocal contributions to the interpretation of chemical interactions. The linear response kernel will be calculated by means of the next orbital-based formula, which is derived for methods involving a single Slater determinant (typically the KS DFT approach30): χðr, r0 Þ ¼ 4

¥ X

φi ðrÞφa ðrÞφa ðr0 Þφi ðr0 Þ εi - εa i ¼1 a ¼N=2 þ 1

ð9Þ

φi(r) stands for the occupied molecular orbitals, whereas φa(r) for the unoccupied ones; εi and εa are the associated orbital energies. This expression, resulting from second-order perturbation theory, should be used for spin-restricted calculations, in which the occupied molecular orbitals are doubly populated. Two approximations have been made in its derivation: the excited states were constructed by substituting an unoccupied molecular orbital for an occupied one in the Slater determinant (frozen orbital approximation), and the energy difference between the excited and ground states was replaced by an orbital energy difference. These assumptions are, however, exactly applicable for the KS noninteracting reference system so that eq 9 can be seen as the exact functional derivative of the electron density with respect to the KS potential vKS(r).13,14 The infinite sum over a will in practice be replaced by a finite one due to the use of a finite basis set, which implies a negligible approximation in view of the orbital energy difference in the denominator. Other variations of this expression have been proposed.4,14,31 Instead of analyzing the kernel as a function of the position variables r and r0, we will focus on the atom-condensed linear response matrix χAB, obtained by integration of r and r0 over the atomic volumes VA and VB of atoms A and B respectively, Z Z χðr, r0 Þ dr dr0 ð10Þ χAB ¼ VA VB

ð7Þ

Various condensation schemes have been developed.7 We opted for Becke's multicenter numerical integration procedure32 as implemented in an in-house program developed by Torrent-Sucarrat.33,34 This basis set independent condensation scheme makes use of Becke's so-called “fuzzy” Voronoi polyhedra, which take the Bragg-Slater radii to weigh the volume of the atomic regions into account. All calculations were done with the Gaussian 03 program package35 and the 6-31þG* basis set devised by Pople et al.36 The B3LYP exchange-correlation functional37-39 was used to perform the geometry optimizations, while the single point calculations yielding the KS molecular orbitals necessary in eq 9 were done with PBE.40,41 The virtual orbital energies are indeed better described by this last functional, for which the lowest unoccupied orbitals are typically bound.42,43 As a preliminary test, it is interesting to compare Wang's atom-bond electronegativity equalization method (ABEEM) values22 for the linear response matrices for ethanol and ethanal with ours, obtained through eqs 9 and 10 and

This equation forms the basis of the regioselectivity description of hard reactions, for which electron transfer is not important, so that the transition state is predominantly determined by electrostatic interactions. Choosing Δv(r) = ( δ(x-r) gives 1 ð8Þ ΔE ¼ ( FðxÞ þ χðx, xÞ þ 3 3 3 2 which reveals a linear relation between the change in electronic energy ΔE and the value of the electron density at the point x of the perturbation in the external potential. This observation explains the use of the electron density as a regioselectivity descriptor for hard reactions. It is also seen that the diagonal elements of the linear response kernel should be used to correct this picture for polarization effects. As these diagonal elements are always negative, polarization exerts a stabilizing effect. Higher order terms in eqs 7 and 8 provide hyperpolarizability corrections. Although the above


N=2 X

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Table 1. Atom-Condensed Linear Response Matrix for Ethanol (Values in au) C1

H1

H2

O

H3

C2

H4

H5

C1

-3.4278

H1

0.7995

H2

0.7995

0.0572

-1.1399

O H3

0.8727 0.0700

0.1498 0.0043

0.1498 0.0043

-2.1873 0.8160

-0.9680

C2

0.6299

0.0651

0.0651

0.1158

0.0479

-3.3928

H4

0.0857

0.0079

0.0554

0.0109

0.0024

0.8193

H5

0.0857

0.0554

0.0079

0.0109

0.0024

0.8193

0.0588

-1.0931

H6

0.0846

0.0007

0.0007

0.0617

0.0207

0.8299

0.0529

0.0529

-1.1399

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

H3 0.0966

0.0024

0.1707

0.7813

0.0372 -1.1413

H4 0.0892

0.0572

0.0264

0.7950

0.0532

0.7813 -1.1413 0.0532 -1.0738

Figure 1. Numbering of the various atoms in (a) ethanol, (b) ethanal, and (c) the hexane/hexa-1,3,5-triene and heptane/hepta-2,4,6-triene derivatives. The carbonyl and nitrile carbon in (c) is indicated as C0.

summarized in Table 1 and Table 2. The numbering of the atoms is clarified in Figure 1. A linear least-squares fitting between the nonsymmetry-related linear response matrix elements yields a correlation coefficient of 0.923, indicating a good agreement between both methodologies. This agreement, providing a mutual validation, is somewhat surprising given the inherent approximations in the ABEEM.7,22 The ABEEM is a semiempirical method involving several atomic and bond-related parameters that have been fitted on ab initio data. Atom and bond-condensed conceptual DFT indices are obtained by solving a set of equations that is constructed from a molecular energy expression using the atomic and bondrelated charges, electronegativities, hardnesses, etc. and Sanderson's electronegativity equalization principle. Molecular


-1.0931 -1.1039

properties are thus obtained without an explicit ab initio or DFT calculation of the system under study. The ABEEM parameters have, however, only been determined for the elements H, C, N and O, which limits its broad applicability.7 Our approach, on the other hand, is generally applicable for all elements and does not involve any major assumptions. The link between the linear response function and the polarizability of the electron density is clearly manifested in the values of the χC1C1, χOO and χOC1 elements, which are significantly higher in ethanal (-4.2080, -3.7827, and 2.7289 au) compared with ethanol (-3.4278, -2.1873, and 0.8727 au). The carbonyl group atoms in ethanal are indeed more polarizable than the corresponding ones in the alcohol group of ethanol due to the presence of the π-electron cloud. It is remarkable to notice that all other matrix elements present quite similar values for either of the molecules. A general trend is the decrease of the linear response matrix elements with bond distance. This observation encouraged us to explore the linear response kernel's ability to determine inductive and resonance effects in chemistry. We will study four sets of organic molecules with relatively long carbon chains and differing from each other by the single functional group that is attached to their carbon ends. Each set comprises the alkane and the fully conjugated trialkene variant. The chosen systems are hexan-1-ol and hexa-1,3,5trien-1-ol, hexan-1-amine and hexa-1,3,5-trien-1-amine, heptanal and hepta-2,4,6-trienal and, finally, heptanenitrile and hepta-2,4,6-trienenitrile, the structures of which are given in Figure 1c, showing that the (1E, 3E) and (2E, 4E) isomers have been chosen for the hexatriene and heptatriene derivatives respectively. Computational specifications are as previously described: Chart 1 shows the results. The linear response matrix elements χOX (with X = C0, C1, ..., C6), representing the response of the atomic population of one of the carbon atoms upon a perturbation in the external potential at oxygen's position, or vice versa, are plotted for the alcohol and aldehyde compounds, whereas the analogous χNX elements are given for the amine and nitrile derivatives. Moreover, we have incorporated the χOH and χNH contributions of the various hydrogen atoms into the χOX and χNX values of the carbon atoms to which they are attached so as to probe the global response of the -CH2-, -CH3, dCH- and dCH2 entities, which seems more relevant for this kind of systems. The exact numerical data are given in the Supporting Information for this letter; it can be verified that the χOX and

Table 2. Atom-Condensed Linear Response Matrix for Ethanal (Values in au) C1

H6

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Chart 1. Linear Response Elements (a,c) χOX (with X = C0, C1, ..., C6) between Oxygen and the Various Carbon Atoms and (b,d) χNX between Nitrogen and the Various Carbon Atomsa

The numbering of the carbon atoms is as shown in Figure 1. The contributions χOH and χNH of the various hydrogen atoms are summed into the χOX and χNX values of the carbon atoms to which they are attached. a

χNX elements without the inclusion of the hydrogen contributions lead to identical conclusions as presented here. The pairs of unconjugated and fully conjugated molecules enable us to analyze inductive and mesomeric effects in carbon chains for four common functional groups (-OH, -NH2, -CHO, and -CN). The linear response matrix elements χOX and χNX for the alkane derivatives present a similar change in all cases: the response of a carbon atom's density upon an external potential perturbation at the heteroatom decreases monotonously with internuclear distance. An exponential regression between these linear response elements and the cumulated bond distances (e.g., the sum of the bond distances between O and C1 and between C1 and C2 results in a cumulated bond distance between O and C2) yields a correlation coefficient of 0.982 on average. The linear response function, quantifying a pure inductive effect, transferred through σ-bonds, as is the case in our series of functionalized alkanes, predicts an exponential decrease of the effect. The data in Chart 1 clearly show that the significance of the inductive effect disappears after the third bond. The linear response function for the trialkene compounds, chosen to investigate the resonance effect in conjugated


systems, exhibits a totally different pattern: the plots go alternately up and down. The χOX and χNX values can, however, be divided into two groups. The linear response elements corresponding to carbon atoms C1, C3, and C5 define the local minima in the graphs, whereas the elements associated with C0, C2, C4, and C6 are at the local maxima. This division strikingly coincides with a partitioning based on the resonance structures that can be constructed for the molecules under study, as shown in Figure 2. The atoms C1, C3, and C5 are relatively unaffected by the mesomeric effect (and will be called the mesomerically passive atoms), while the other carbon atoms actively participate in the mesomeric interaction through charge partitioning with the heteroatom (and will be called the mesomerically active atoms). It is clearly seen in Chart 1 that the linear response elements associated with the mesomerically passive carbon atoms follow the trend of the inductive effect in the corresponding alkane derivatives. The fact that the χOC1 and χNC1 values are consistently lower in the conjugated case could be ascribed to a weaker response of the C1 atom due to the increased responses of the mesomerically active atoms and a normalization effect; the sum of all elements in a row or column of the linear response matrix

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Figure 2. Resonance structures of (a) hexa-1,3,5-trien-1-ol and (b) hepta-2,4,6-trienal. Analogous structures can be drawn for hexa-1,3,5trien-1-amine and hepta-2,4,6-trienenitrile, respectively.

found in Kohn and Prodan's papers.44,46,47 These authors elaborately analyzed the principle and its implications for a range of periodic and nonperiodic metals and insulators. Numerical data indicating its applicability to molecular systems have not been given yet. It follows from eq 3 that the softness kernel s(r,r0 ), which is evaluated at constant chemical potential μ, is nearsighted; this is not necessarily the case for the linear response kernel χ(r,r0 ), as it is obtained at constant number of electrons.3,48 In view of the Berkowitz-Parr relation (eq 2), the possible farsighted contributions to the linear response kernel are accounted for by the local softnesses. If electron transfer effects are insignificant, the farsighted term in eq 2 can be neglected so that the linear response kernel essentially equals the negative of the softness kernel. As mentioned by C ardenas et al.,3,48 it follows from Prodan and Kohn's analysis that the softness kernel for any system with a band gap should decline exponentially as r and r0 get farther apart. This is exactly what we have observed empirically for the inductive effect. The mesomeric effect, on the other hand, which involves electron transfer, did not present an exponential decrease at large distance. Further research is needed to analyze the intricacies of nearsightedness and its implications for the transferability principle of atoms and functional groups in chemistry. A recent paper by Bader extensively discusses this last topic.49 A second remark concerns the Fukui response kernels f ((r,r0 ), which provide polarization corrections to the Fukui functions f ((r).11,50,51 They are defined as the following leftand right-hand-side derivatives: !( ! ∂δ2 E δf ( ðrÞ ( 0 ¼ ð11Þ f ðr, r Þ ¼ ∂NδvðrÞδvðr0 Þ δvðr0 Þ

should indeed equal zero. The linear response matrix elements corresponding to the mesomerically active carbon atoms, on the other hand, stay consistently high: even after seven bonds, values of χOC6 = 0.366 au and χNC6 = 0.361 au are found for hepta-2,4,6-trienal and hepta-2,4,6-trienenitrile. When the hydrogen contributions are not summed into the carbon ones, these values become χOC6 = 0.366 au and χNC6 = 0.332 au, respectively, which stresses their magnitude, comparing them with the data given in Tables 1 and 2. The descending trend in the linear response values of the mesomerically active atoms can partly be understood from the fact that they superimpose the inductive effects (through the σ-bonds) and the resonance effects (through the π-bonds). In addition, the resonance structures featuring high charge separation are less stable and contribute to a lesser extent to the average structure. The sharp fall in the linear response values χOX and χNX observed between the first and second mesomerically active carbon in hepta2,4,6-trienal and hepta-2,4,6-trienenitrile is not associated with a sudden drop in mesomeric activity, but rather with the high polarizability of the π-electrons in the CdO and CtN bonds, which are not present in the alcohol and amine compounds. This polarization effect can further be illustrated by the difference in χOO values for hexa-1,3,5-trien-1-ol (-2.784 au) and hepta-2,4,6-trienal (-4.312 au) and in the χNN ones for hexa1,3,5-trien-1-amine (-3.592 au) and hepta-2,4,6-trienenitrile (-6.040 au). Some final considerations should be made here. Attention has recently been paid to the nearsightedness of electronic matter, as introduced by Kohn,44 in connection with conceptual DFT reactivity indices.3,45 The nearsightedness principle quintessentially states that, for systems consisting of very many electrons and at a constant chemical potential, the change in the electron density ΔF(r0) induced by an external potential perturbation w(r0 ) outside a certain radius R around the point r0, no matter how large this perturbation is, will always be smaller than a maximum magnitude ΔF. As a result, this maximum response of the electron density will decay monotonously as a function of R. An accurate formulation of this principle can be


N

The commonly used finite difference expressions for the Fukui functions7 (f -(r) = FN(r)-FN-1(r) and (f þ(r) = FNþ1(r) -FN(r)) in combination with eq 11 show that the Fukui kernels can be obtained with our proposed methodology as the difference between two linear response kernels. It should, finally, be mentioned that the explicit calculation of the second order functional derivative of the electronic

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energy E with respect to v(r) in eq 1 offers a feasible and in principle exact alternative to obtain the linear response kernel. A methodology to evaluate functional derivatives with respect to v(r) has been proposed and extensively tested by Ayers and the present authors in studies concerning the Fukui function and dual descriptor.52-54 An extension of this method to incorporate the calculation of the linear response and Fukui response kernels is currently in progress. (unpublished results in collaboration with P. W. Ayers). In conclusion, a molecular orbital-based expression (eq 9) has been used to calculate the atom-condensed linear response matrix. The results are the first published values of this quantity that have been obtained through a direct and generally applicable methodology. Inductive and resonance effects have been analyzed for eight functionalized organic molecules. The linear response function has been shown to account for the falloff behavior of these effects and to connect them with the concept of nearsightedness.

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SUPPORTING INFORMATION AVAILABLE All numerical data and correlation graphs. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

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Corresponding Author: *To whom correspondence should be addressed. E-mail: Nick.Sablon@ vub.ac.be.

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ACKNOWLEDGMENT The authors thank M. Torrent-Sucarrat for his help with the condensation procedure and P. W. Ayers for useful discussions. N.S. acknowledges the Research Foundation - Flanders (FWO) for a position as research assistant. F.D.P. and P.G. thank the FWO and the Vrije Universiteit Brussel for continuous financial support.

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